This book is printed on acid-free paper.

Copyright © 2002 by ACADEMIC PRESS

All Rights Reserved.No part of this publication may be reproduced or transmitted in any form or by any

means, electronic or mechanical, including photocopying, recording, or anyinformation storage and retrieval system, without permission in writing from the

publisher.

Academic PressA division of Harcourt, Inc.

Harcourt Place, 32 Jamestown Road, London NW1 7BY, UKhttp://www.academicpress.com

Academic PressA division of Harcourt, Inc.

525 B Street, Suite 1900, San Diego, California 92101-4495, USAhttp://www.academicpress.com

ISBN 0-12-227085-1

Library of Congress Catalog Number: 2001092782

A catalogue record for this book is available from the British Library

R Bigret M lalanne22 Rue J Varnet Laboratorie de Mecaniques des Structures93700 Draney Institut National des Sciences Appliquees de LyonFrance LMSt - Batiment 113 20

avenue Albert EinsteinP Cawley 69621 Villeurbanne Cedex, France

Imperial College of Science, Technology & MedicineDepartment of Mechanical Engineering M LinkExhibition Road Universitat Gesamthoschule KasselLondon SW7 2BX, UK Fachgebiet Leichtbau

Moenchebergstrasse 7,R Craig D34109 Kassel, GermanyUniversity of TexasAeronautical Engineering and Engineering MechanicsDepartment K McConnellAustin TX 78712 USA Iowa State University

, AEEM

B D b. Black Engineering Building

u Ulsson33 R S

·H b rt

Ames, IA 50011, USAue amt u e

60610 La Croix Saint OuenFrance D E Newland

University of CambridgeR Eshleman Department of EngineeringVibration Institute Trumpington Street6262 S. Kingery Highway Cambridge CB2 1PZ, UK

WillowbrookIL 60514, USA N Okubo

Chuo University Faculty of Science & EngineeringM Geradin Department of Precision Mechanical EngineeringUniversite de Liege 1-37-27 KasugaLTAS Dynamique des Structures Bunkyo-ku, Tokyo, JapanInstitut de Mecanique et Genie CivilChemin des Chevreuils 1 M4000 Liege Belgium Rade§, Universitatea Politehnica Bucuresti

Department of Engineering SciencesJ Hammond 313 SplaiullndependenteiUniversity of Southampton 79590 BucurestiInstitute of Sound and Vibration Research RomaniaSouthampton S09 5NH, UK

5 Hayek 5 RakhejaPennsylvania State University Concordia University112 EES Building Department of Mechanical EngineeringUniversity Park 1455 de Masonneuve Blvd WPA 16802-6812 USA Montreal, Quebec H3G 1M8

, Canada

D InmanVirginia Polytechnic Institute & State University G RosenhouseDepartment of Engineering Science and Mechanics Technion - Israel Institute of Technology310 NEB, Mail Code 0261 Faculty of Civil EngineeringBlacksburg Technion CityVA 24061-0219, USA Haifa 32000, Israel

S Shaw H S TzouMichigan State University University of KentuckyDepartment of Mechanical Engineering Department of Mechanical EngineeringA321 Engineering Building Dynamics and Systems LaboratoryEast Lansing, MI 48824-1226, USA Lexington, KY 40506, USA

M SidahmedUniversite de Technologie de CompiegneHeuristique et Diagnostic des Systemes ComplexesCentre de Recherches de RoyallieuBP 2052960205 Compiegne, France

Vibration is usually defined in dictionaries as rhythmic motion back and forth. It has attracted the curiosity ofhumans since people have had time to contemplate the natural world. In the sixth century BC, Pythagoras hadrelated harmonic intervals to ratios of lengths of a vibrating string. In 1581, Galileo had observed that theperiod of a simple pendulum is (nearly) independent of the amplitude of vibration. A century later the basicprinciples of dynamics were put on a firm basis by I. Newton. Further development of the science ofmechanics was led by L. Euler (1707-83). Most of the analytical tools now used in vibration studies wereavailable by the year 1788, when Mecanique Analytique was published by J.L. Lagrange. The first bookdevoted entirely to the theory of vibration was volume I of Lord Rayleigh's Theory of Sound (1887). Itbecame the model for the classical vibration textbooks of Timoshenko (1928), and DenHartog (1934).Subsequent texts have followed much the same pattern, with the addition of matrix notation and linearalgebra after the engineering-science revolution of the sixties.

Vibration is a fascinating physical phenomenon, well worth studying on its own merits. In manyapplications, vibration is harnessed for useful purposes, e.g., to make music, to drive vibrational transportsystems, or provide frequency standards in clocks and precision instruments. In many other applications,vibration is an undesired intruder that interferes with the normal operation of the system, creating noise, anddeveloping stresses that may cause fatigue failure.

Engineers have been dealing with vibration problems since the beginning of the industrial revolution andthe introduction of the steam engine. With each new mode of transportation and each new technology, thereoften appears an unexpected vibrational challenge. The interaction of steam-driven trains with relativelyflexible metal bridges produced unexpected problems of vibration and fatigue in the 19th century. At thebeginning of the 20th century, cities began to install central electric power stations and engineers were facedwith a variety of vibration problems associated with the rotor dynamics of turbine generator sets. By the endof World War I, the diesel engine had become a popular medium-power prime mover. A rash of fatiguefailures in diesel engine shafting spurred work on the torsional vibration of crank-shafts. A majortechnological effort of the period between World War I and II was the development of airplanes andhelicopters. The unexpected vibrational challenge for aircraft was the phenomenon of wing flutter, while thenew problem for helicopters was the phenomenon of ground resonance. In 1940 the Tacoma Narrowssuspension bridge developed large self-excited torsional vibrations in a moderate wind and faileddramatically. The challenge to explain this completely unexpected result kept vibration experts busy fordecades.

The latter half of the 20th century saw a great expansion of high technology. As new technologiesproliferated, job opportunities for vibration engineers multiplied. The new technologies also provided vastlyimproved tools for the vibration engineer. Particularly important has been the development of improvedsensors and actuators. It is interesting to compare the vast array of optical, electrical, magnetic, piezoelectric,eddy-current, and laser transducers available to the engineer today, with the situation faced by the pioneers intorsional vibration, who had no way of detecting the presence of large vibrations, superposed on the steadyrotation of the shaft, until the shaft failed.

Perhaps the most important technical advance of the last half-century has been the development of thedigital computer. The computer is now an essential part of most vibration measurement systems, and thecomputational power it has unleashed has provided important support for many theoretical advancements.

After World War II, the development of rocket propulsion and space flight introduced vibration engineersto the topic of random vibration and the requirements for large wide-band shakers and sophisticated data-processing instrumentation. Instrumentation based on analog principles was soon replaced by digitalprocessors. The fast Fourier transform (FFT) was conceived in 1965, and a decade later, commercialinstruments based on the FFT were on the market. Soon digital analyzers capable of performing experimentalmodal analysis became available. At the same time, computer software for performing finite element analysisand boundary element analysis was being applied to the dynamic analysis of complicated real structures. Thedifficulty in reconciling the experimental measurements with the analytical predictions, beyond the first fewmodes of a structure, remains a challenge for vibration engineers.

During the past half-century, important advances in the theory of nonlinear vibrations have been made,including the clarification of the concept of chaotic response from a deterministic system. Much of thesupporting work would not have been possible without high-speed digital computers.

A present trend in vibration technology is the inclusion of vibration considerations in the preliminarydesign of products. The old tradition of waiting until a prototype was available, and hoping that anyvibration problem that arose could be cured with a quick fix (which did not impact the primary design goals)is gradually giving way to early consideration of the vibrational problems of the product. In some industries,machinery is being delivered with vibration sensors already installed, so that the customer can monitor theperformance, or diagnose the cause of a malfunction. Another present trend is the miniaturization of sensorsand actuators using microelectromechanical elements (MEMS).

The field of vibration is now so broad that no single person can keep up-to-date in all its branches.Furthermore, it is impossible to squeeze all the essential information into a single book. The editors of thepresent Encyclopedia of Vibration have performed a valuable service to the field by dividing the subjectmatter into 168 chapters and selecting experts to create authoritative chapters on the individual topics. Theresult is a three-volume encyclopedia, which will be of great value to practicing vibration engineers andtheorists. The encyclopedia will also be useful as a general reference, and as a guide to students, andprofessionals in neighboring fields.

We are surrounded by vibrations, some of them generated by nature, others by human-built devices. Whilesome are harmless, many have adverse effects. These can affect human well-being or health, causemalfunctioning of devices, or result in life-threatening situations. Vibrating patterns may also carryinformation, which when deciphered correctly can convey important knowledge.

This three-volume reference work focuses on aspects of vibration which are of interest mainly to practicingand research engineers. Even within this specific audience, vibration problems span multidisciplinary aspects,and we thus believe there is a need for a reference work, geared to the specific area of vibration, covering asmany relevant examples as possible.

The more classical topics of vibration engineering are naturally covered: basic principles, sound radiationby vibrating structures, vibration isolation, damping principles, rotating systems, wave propagation,nonlinear behavior, and stability. Modern aspects developed in the last two decades also have a prominentplace. To mention just a few, these include software and computational aspects, testing and specificallymodal testing/modeling, active vibration control, instrumentation, signal processing, smart materials, andvibration standards. Specific entries address analysis/design aspects as encountered in aeronautical, civil, andmechanical engineering: vibration abatement, seismic vibrations, civil engineering structures, rotatingmachines and surveillance, aircraft flutter, and the effect of vibrations on humans.

In view of the large scope of topics addressed, the type and depth of presentation had to be adapted to thestyle of concise alphabetical entries, and some compromises were necessary in the choice of topics. More'classical' aspects suffer somewhat at the expense of more modern topics, due to our opinion that moreinformation sources are already available for them. Carefully chosen references appearing under the 'furtherreading' sections are however given for them. In addition, at the end of each entry there is a cross-referencing'see also' section where the reader is directed to other entries within the Encyclopedia which containadditional and/or related information. In view of the large scope of topics chosen, the type and depth ofpresentation had to be adapted to specific entries. Unified symbols and notation have been used to someextent, but when deemed preferable, superceded by those commonly accepted in specific areas. The table ofcontents lists entries in an intuitive alphabetical form, and any topic not found through the contents list canbe located by referring to the index which appears in each volume. Useful reference data can be found in theappendices, which also appear in each volume.

An online version of the Encyclopedia will be available to all purchasers of the print version and will haveextensive hypertext links and advanced search tools. Leading authorities have contributed to this work.Together with our advisory board they should receive the main credit for this important source of knowledge.My personal thanks are naturally extended to them, but even more so to my colleagues and co-editors, DaveEwins and Singiresu Rao, without whose constant help this task would have been impossible. Special thanksare due to Dr Carey Chapman and Lorraine Parry from the Academic Press Major Reference Work team fortheir encouragement and help extended to us at every stage.

Simon BraunEditor-in-Chief

Structure of the EncyclopediaThe material in the Encyclopedia is arranged as a series of entries in alphabetical order. Some entriescomprise a single article, whilst entries on more diverse subjects consist of several articles that deal withvarious aspects of the topic. In the latter case the articles are arranged in a logical sequence within an entry.

To help you realize the full potential of the material in the Encyclopedia we have provided three features tohelp you find the topic of your choice: contents list, cross-references, and index.

Contents listsYour first point of reference will probably be the contents list. The complete contents list appearing in eachvolume will provide you with both the volume number and the page number of the entry. On the openingpage of an entry containing more than one article, a contents list is provided so that the full details of thearticles within the entry are immediately available.

Alternatively, you may choose to browse through a volume using the alphabetical order of the entries asyour guide. To assist you in identifying your location within the Encyclopedia, a running headline indicatesthe current entry and the current article within that entry.

You will find dummy entries in the following instances:

1. Where obvious synonyms exist for entries.For example, a dummy entry appears for Structural Damping which directs you to Hysteretic Dampingwhere the material is located.

2. Where we have grouped together related topics.For example, a dummy entry appears for Magnetorheological Fluids which leads you to Electrorheologicaland Magnetorheological Fluids where the material is located.

3. Where there is debate over the given entry title and whether readers would intuitively find the topic they aretrying to locate under that title.For example, a dummy entry appears for Vibration Absorbers which directs you to Absorbers, Vibrationwhere the material is located.

4. Where there is debate over whether diverse subjects should comprise several single articles, or one entryconsisting of several articles.For example, a dummy entry appears for Smart Materials which directs you to the entriesElectrorheological and Magnetorheological Fluids, Electrostrictive Materials, Magnetostrictive Materials,Piezoelectric Materials, and Shape Memory Alloys.

Dummy entries appear in both the contents list and the body of the text.

ExampleIf you were attempting to locate material on diagnostics via the contents list the following information wouldbe provided:

DIAGNOSTICS See BEARING DIAGNOSTICS; DIAGNOSTICS AND CONDITION MONITORING, BASIC CONCEPTS;GEAR DIAGNOSTICS; NEURAL NETWORKS, DIAGNOSTIC APPLICATIONS

The page numbers of these entries are given at the appropriate location in the contents list.

If you were trying to locate the material by browsing through the text and you looked up Diagnostics thenthe following would be provided:

Cross-references

To direct the reader to other entries on related topics, a 'see also' section is provided at the end of each entry.

ExampleThe entry Wave propogation, Waves in an Unbounded Medium includes the following cross-references:

See also: Nondestructive testing, Sonic; Nondestruc-tive testing, Ultrasonic; Ultrasonics; Wave propaga-tion, Guided waves in structures; Wave propagation,Interaction of waves with boundaries.

Index

The index will provide you with the volume number and page number of where the material is located, andthe index entries differentiate between material that is a whole article, is part of an article, or is datapresented in a table. On the opening page of the index detailed notes are provided. Any topic not foundthrough the contents list can be located by referring to the index.

Color Plates

The color figures for each volume have been grouped together in a plate section. The location of this section iscited both in the contents list and before the 'See also' list of the pertinent articles.

Appendices

The appendices appear in each volume.

Contributors

A full list of contributors appears at the beginning of each volume.

Agnes, G Book, W1489 Fudge Drive Georgia Institute of TechnologyBeavercreek School of Mechanical EngineeringOH 45434 Room 472 Manufacturing Research CenterUSA 813 Ferst Drive, Atlanta, GA 30332-0405

USA

Ahmadian, M Braun SVirginia Tech .. Techni~n - Israel Institute of TechnologyAdvanced Vehicle Dynamics Laboratory Faculty of Mechanical EngineeringDepartment of Mechanical Engineering Haifa 32000Blacksburg, VA 24061-0238 IsraelUSA Cai,GQAhmed, A K W Florida Atlantic UniversityConcordia University Center for Applied StochasticsDepartment of Mechanical Engineering Boca RatonMontreal FL 33431Quebec USACanada Cardona, A

INTECBajaj, A Grupo de Technologia MecanicaPurdue University Guemes 3450Department of Mechanical Engineering RA-3000, Santa FeWest Lafayette ArgentinaIN 47907-1288 C t II" . PUSA as elm,

Universita di AnconaBanks, H T Dipartimento Di MeccanicaNorth Carolina State University Via Brecce BlancheCenter for Research _Science Comput. 1-60131 Ancona324 Harrelson Hall, Box 8205, CRSC ItalyRaleigh, NC 27695-8105 Chou, C SUSA National Taiwan University

TaipeiBauchau, 0 TaiwanGeorgia Institute of Technology Republic of ChinaAtlanta Constantinides, T~~:0332-071 0 Imperial College of Science, Technology and Medicine

Engineering Department, Room 812Exhibition Road

Baz, A London SW7 2BTUniversity of Maryland UKDepartment of Mechanical Engineering2137 Engineering Building Cooper, J ECollege Park, MD 20742 School of EngineeringUSA University of Manchester

Oxford RoadBenson, D J Manchester M13 9PL

University of California, San Diego UKDivision of Mechanical Engineering Craig, Jr R RDepartment of Applied Mechanics and University of TexasEngineering Sciences Aeronautical Engineering and Engineering Mechanics Department9500 Gilman, La Jolla, CA 92093 Austin TX 78712~A UMBert C W D'Aubrogio, WUniv~rsity of Oklahoma U~ive~sity of L'Aquila .Department of Mechanical Engineering DI~artlmen.to dl Energetlca865 Asp Avenue Room 202 ROlo PoggloNorman, OK 73019 67040 L'AquilaUM b~

Dalpiaz, GBigret, R University of Bologna22 Rue J Varnet Faculty of Mechanical Engineering93700 Drancy 2,40136 BolognaFrnooe b~

David, A Fassois, S DAuburn University University of PatrasNonlinear Systems Research Laboratory Stochastic Mechanical Systems (SMS) GroupDepartment of Mechanical Engineering Faculty of Mechanical and Aeronautical EngineeringAuburn, AL 36849 GR 265 00 PatrasUSA Greece

Devloo~ p. Feeny, B FUnlversldade Estadual ~e C~r:nPlnas Michigan State UniversityFa~uldade de Engenhana CIvil Department of Mechanical EngineeringCalxa Postal 6021 2555 Engineering Building13083-970 Campinas Sao Paulo East Lansing MI48824~~I UM '

Dimentberg, M F F Id MWorcester Polytechnic Institute e ":,an, .M h . I E" . D rt t Technion - Israel Institute of Technology

ec anlca nglneenng epa men F It f M h . IE' .Worcester a~u y 0 ec anlca nglneenngMA 01609 Haifa 32000USA Israel

Doebling, S Flatau, ALos Alamos National Laboratory National Science FoundationESA - EA, MIS P946, PO Box 1663 Dynamic Systems and Control ProgramNM 87545 4201 Wilson Blvd. Suite 545USA Arlington VA 22230

USADrew, S JThe University of Western Australia Fuller, C RDepartment of Mechanical and Materials Engineering Virginia TechNedlands 6709, Perth Department of Mechanical EngineeringWestern Australia BlacksburgAustralia VA 24061-0238

D b . B USAu Ulsson,33 rue Saint Hubert Gandhi F60610 La Croix Saint Ouen The Pe~nsYlvania State UniversityFrance Department of Aerospace EngineeringDyne, S 23~ Har:nmond BuildingUniversity of Southampton University Park, PA 16802Institute of Sound and Vibration Research USASouthampton

G F HS017 1BJ ern,UK Center for Intelligent Material Systems and Structures (CIMSS)

Virginia Polytechnic Institute and State UniversityElishakoff, I New Engineering Building, Room 303Florida Atlantic University Blacksburg, VA 24061-0261Department of Mechanical Engineering USABoca RatonFL 33431-0991 Giurgiutiu, VUSA University of South Carolina

. Department of Mechanical EngineeringElliott, S J 300 S Main Street Room A222The University of Southampton Colu~bia, SC 29208Institute of Sound and Vibration Research USASouthampton S017 1BJUK Griffin, M JEwins D J The University of Southampton

"'" .. Human Factors Research UnitImpenal College of SCI~nce, Te?hnology and Medicine Institute of Sound and Vibration ResearchDepartment of Mechanical Englneenng

S th t S017 1BJE h'b"t" R d ou amp onx I I Ion oa UK

London SW7 2BX

UK Griffin, SFarhat, C AFRL/VSSVUniversity of Colorado 3550 Aberdeen Ave SEDepartment of Aerospace Engineering Sciences Kirtland AFBCampus Box 429 NM 87117-5748Boulder, CO 80309 USAUSA

Haddow,AFarrar, C Michigan State UniversityLos Alamos National Laboratory Department of Mechanical EngineeringESA - EA, MIS P946, PO Box 1663 East LansingNM 87545 MI 48824~A UM

Hallquist, J Kobayashi, A SLivermore Software Technology Corporation (LSTC) University of Washington97 Rickenbacker Circle Department of Mechanical EngineeringLivermore Seattle, WA 98195CA 94550 USAUSA .Knshnan,RHann, F University of Maryland at College ParkUniversity of Notre Dame Department of Aerospace EngineeringDepartment of Civil Engineering 3180 Engineering Classroom BuildingNotre Dame College ParkIN 46556-0767 MD 20742-3015USA USA

Hartmann F Krousgrill, C MUniversity ~f Kassel Purdue UniversityDept. Baustatik, Fb 14 School of Mechanical EngineeringKurt-Wolters-Str 3 West LafayetteD-34109 Kassel IN 47907-1288Germany USA

leissa, A WHayek, S I ... Ohio State UniversityPennsylvania ~tate University Department of Mechanical Engineering11~ EE~ BUilding 206 W. 18th AveUniversity Park Columbus OH 43210-1154PA 16802-6812 USA 'USA

Lesieutre, G AHolmes, P J Pennsylvania State UniversityPrinceton University Department of Aerospace EngineeringDepartment of Mechanical and Aerospace Engineering 153G Hammond BuildingPrinceton University Park, PA 16802New Jersey 08544 USAUSA

Li, C JIbrahim, R Rensselaer Polytechnic InstituteWayne State University Department of Mechanical Engineering,Mechanical Engineering Aeronautical Engineering, and MechanicsDetroit, MI 48202 110 8th StreetUSA Troy, NY 12180

USAInman, DVirginia Polytechnic Institute and State University Lieven, N A JDepartment of Engineering Science and Mechanics Bristol University310 NEB, Mail Code 0261 Department of Aerospace EngineeringBlacksburg, VA 24061-0219 Queens Building, University WalkUSA Bristol, BS8 1TR

UKKapania, R .Virginia Polytechnic Institute and State University LIn,. Y K ...Department of Aerospace and Ocean Engineering Flonda Atlantic UniversityBlacksburg Center for Applied StochasticsVA 24061-0203 Boca RatonUSA FL 33431

USA

Ka~ee":1' A Link, MUniversity of Not~~ Dam~ . Universitiit Gesamthoschule KasselDepartment of CIvil Englneenng Fachgebiet LeichtbauNotre Dame M..IN 46556-0767 onchebergstrasse 7USA D34109 Kassel

Germany

Kijewski, T Lowe, M J SUniversity of Notre Dame Imperial College of Science, Technology and MedicineDepartment of Civil Engineering Department of Mechanical EngineeringNotre Dame Exhibition RoadIN 46556-0767 London SW7 2BXUSA UK '

Klapka, I Ma, FUniversite de Liege University of CaliforniaLaboratoire des Techniques Aeronautiques et Spatiales Department of Mechanical EngineeringDynamique des Structures Berkeleyrue E Solvay 21, B-4000, Liege California, CA 94720Belgium USA

Revel, G M Sidahmed, MUniversita di Ancona Universite de Technologie de CompiegneDipartimento di Meccanica Heuristique et Diagnostic des SystemesVia Brecce Bianche Complexes1-60131 Ancona Centre de Recherches de RoyallieuItaly BP 20529

60205 CompiegneRivin, E FranceWayne State UniversityMechanical Engineering Sieg, T2100W Engineering Building Paulstra Industries Inc.Detroit, Michigan 48202 Carlsbad, CAUM UM

Rixen D Silva, J M MDelft University of Technology Instituto Superior TecnicoPO Box 5 Departmento de Engenharia Mecanica2600 AA Delft Av Rovisco PaisThe Netherlands 1049-001 Lisboa

Portugal

Robert, G Sinha, SSamtech SA. Auburn UniversityRue des ~hasseurs-Ardennals, 8 Nonlinear Systems Research LaboratoryB-4031 Liege Department of Mechanical EngineeringBelgium Auburn, AL 36849

USARosenhouse,GTechnion - Israel Institute of Technology Smallwood, DFaculty of Civil Engineering Sandia National LaboratoriesTechnion City PO Box 5800Haifa 32000 AlbuquerqueIsrael NM 87185-0865

USASchmerr, Jr, L WIowa State University Snyder, RCenter for NDE, Aerospace Engineering and Engineering University of Maryland at College ParkMechanics Department of Aerospace Engineering211A Applied Science Complex 11,1915Scholl Road 3180 Engineering Classroom Building1915 Scholl Road, Ames, IA 50011 College Park, MD 20742-3015USA USA

Sciulli, D Soedel, W5725 Cedar Way #301 Purdue UniversityCenterville School of Mechanical EngineeringVA 20121 West LafayetteUSA IN 47907

USAScott, R AUniversity of Michigan Soong, ~ T .Department of Mechanical Engineering and Applied Mechanics State University of New York at Buffalo2206 G. G. Brown Building MCEERAnn Arbor MI48109 107 Red Jacket QuadrangleUSA ' Buffalo, NY 14261-0025

USA

Se~tieri,.A , ., Spencer, Jr, B FU~lve~slta Degli StUdl,dl Roma , University of Notre DameDlpartlmento Meccanlca e Aeronautlca Department of Civil EngineeringVia Eudossinia 18 Notre DameI - 00184 Roma IN 46556-0767Italy USA

Shaw, S Stanway, RMichigan State University The University of SheffieldDepartment of Mechanical Engineering Department of Mechanical EngineeringA321 Engineering Building Mappin StreetEast Lansing, MI 48824-1226 Sheffield S1 3JDUSA UK

Shteinhauz, G D Steffen, Jr, VThe Goodyear Tire Rubber Company Federal University of UberlandiaTire-Vehicle Engineering Technology Mechanical Engineering DepartmentTechnical Center D/460G Campus Santa MonicaPO Box 353, Akron, Ohio 44309-3531 PO Box 593, 38400-902 Uberlandia, MGUSA Brazil

xx CONTRIBUTORS

Steindl, A Vakakis, AVienna University of Technology University of IllinoisWiedner Hauptstrasse 8-10 Department of Mechanical and Industrial EngineeringA-1040 Vienna 140 Mechanical Engineering Building,Austria 1206 West Green Street

Stiharu, IUrbana, IL 61801USA

Concordia UniversityVaroto, P SDepartment of Mechanical Engineering

1455 de Masonneuve Blv W Escola de Engenharia de Sao Carlos, USPMontreal, Quebec H3G 1M8 Dept. Engenharia MecanicaCanada AV.Dr. Carlos Botelho, 1465, CP 359

Sao Carlos - SP - 13560-250Sun, J-Q BrasilUniversity of Delaware

Vorus, W SDepartment of Mechanical EngineeringNewark, DE 19716 University of New Orleans

USA School of Naval Architecture and Marine EngineeringCollege of Engineering

Sunar, M New Orleans, LA 70148King Fahd University of Petroleum and Minerals USADepartment of Mechanical Engineering Wang, K WPO Box 1205 The Pennsylvania State UniversityDhahran 31261 Department of Mechanical EngineeringSaudi Arabia 157 E Hammond Building, University ParkTang, J PA 16802The Pennsylvania State University USADepartment of Mechanical Engineering Wereley, N M157 E Hammond Building, University Park University of Maryland at College ParkPA 16802 Department of Aerospace EngineeringUSA 3180 Engineering Classroom Building

Tomasini, E P College Park, MD 20742-3015

Universita di Ancona USA

Dipartimento Di Meccanica White, PVia Brecce Bianche The University of Southampton1-60131 Ancona Institute of Sound and Vibration Research (ISVR)Italy Southampton, S09 5NH

UKTordon, M JUniversity of New South Wales Wickert, JSchool of Mechanical and Carnegie Mellon UniversityManufacturing Engineering Department of Mechanical EngineeringSydney 2052, New South Wales PittsburgAustralia PA 15213-3890

USATroger, H

Wright, JVienna University of TechnologyWiedner Hauptstrasse 8-10 University of Manchester

A-1040 Vienna School of Engineering

Austria Oxford RoadManchester M13 9PL

Tzou, H S UKUniversity of Kentucky Yang, BDepartment of Mechanical Engineering University of Southern CaliforniaDynamics and Systems Laboratory Department of Mechanical EngineeringLexington, KY 40506 Los AngelesUSA CA 90089-1453Uchino, K USAThe Pennsylvania State University Zacksenhouse, M134 Materials Research Laboratory Technion - Israel Institute of TechnologyUniversity Park Faculty of Mechanical EngineeringPA, 16802-4800 Haifa 32000USA Israel

Ungar, E E ZU,JWAcentech Incorporated University of Toronto33 Moulton Street Department of Mechnical and Industrial EngineeringCambridge 5 King's College RoadMA 02138-1118 Toronto, OntarioUSA Canada M5S 3G8

VOLUME 1

AABSORBERS, ACTIVE G Agnes 1

ABSORBERS, VIBRATION V Steffen, Jr 0 Rade 9

ACTIVE ABSORBERS See ABSORBERS, ACTIVE 26

ACTIVE CONTROL OF CIVIL STRUCTURES T T Soong B F Spencer, Jr. 26

ACTIVE CONTROL OF VEHICLE VIBRATION M Ahmadian 37

ACTIVE ISOLATION S Griffin 0 Sciulli 46

ACTIVE VIBRATION CONTROL See ACTIVE CONTROL OF VEHICLE VIBRATION; ACTIVE

ISOLATION; ACTIVE VIBRATION SUPPRESSION; ACTUATORS AND SMART STRUCTURES;

DAMPING, ACTIVE; FEED FORWARD CONTROL OF VIBRATION; FLUTTER, ACTIVE

CONTROL; HYBRID CONTROL. 48

ACTIVE VIBRATION SUPPRESSION 0 Inman 48

ACTUATORS AND SMART STRUCTURES V Giurgiutiu 58

ADAPTIVE FILTERS S J Elliott 81

AEROELASTIC RESPONSE J E Cooper 87

ANTI RESONANCE See RESONANCE AND ANTI RESONANCE 98

AVERAGING S Braun 98

BBALANCING R Bigret 111

BASIC PRINCIPLES G Rosenhouse 124

BEAMS R A Scott 137

BEARING DIAGNOSTICS C J Li K McKee 143

BEARING VIBRATIONS R Bigret 152

BELTS J W Zu 165

BIFURCATION See DYNAMIC STABILITY 174

BLADES AND BLADED DISKS R Bigret 174

BOUNDARY CONDITIONS G Rosenhouse 180

BOUNDARY ELEMENT METHODS F Hartmann 192

BRIDGES S S Rao 202

CCABLES N C Perkins 209

CEPSTRUM ANALYSIS R B Randall 216

CHAOS P J Holmes 227

CHATTER See MACHINE TOOLS, DIAGNOSTICS 236

COLUMNS I Elishakoff C W Bert 236

xxii CONTENTS

COMMERCIAL SOFTWARE G Robert 243COMPARISON OF VIBRATION PROPERTIES 256

Comparison of Spatial Properties M Rade~ 256Comparison of Modal Properties M Rade~ 265Comparison of Response Properties M Rade~ 272

COMPONENT MODE SYNTHESIS (CMS) See THEORY OF VIBRATION, SUBSTRUCTURING 278COMPUTATION FOR TRANSIENT AND IMPACT DYNAMICS o J Benson J Hallquist 278COMPUTATIONAL METHODS See BOUNDARY ELEMENT METHODS; COMMERCIAL

SOFTWARE; COMPUTATION FOR TRANSIENT AND IMPACT DYNAMICS; CONTINUOUS

METHODS; EIGENVALUE ANALYSIS; FINITE DIFFERENCE METHODS; FINITE ELEMENT

METHODS; KRYLOV-LANCZOS METHODS; LINEAR ALGEBRA; OBJECT ORIENTED

PROGRAMMING IN FE ANALYSIS; PARALLEL PROCESSING; TIME INTEGRATION METHODS. 286CONDITION MONITORING See DIAGNOSTICS AND CONDITION MONITORING,

BASIC CONCEPTS; ROTATING MACHINERY, MONITORING. 286CONTINUOUS METHODS C W Bert 286CORRELATION FUNCTIONS S Braun 294CRASH V H Mucino 302CRITICAL DAMPING o Inman 314

DDAMPING IN FE MODELS G A Lesieutre 321DAMPING MATERIALS E E Ungar 327DAMPING MEASUREMENT o J Ewins 332DAMPING MODELS o Inman 335DAMPING MOUNTS J-Q Sun 342DAMPING, ACTIVE ABaz 351DATA ACQUISITION R B Randall M J Tordon 364DIAGNOSTICS AND CONDITION MONITORING, BASIC CONCEPTS M Sidahmed 376DIAGNOSTICS See BEARING DIAGNOSTICS; DIAGNOSTICS AND CONDITION

MONITORING, BASIC CONCEPTS; GEAR DIAGNOSTICS; NEURAL NETWORKS,

DIAGNOSTIC APPLICATIONS 380DIGITAL FILTERS A G Constantinides 380DISCRETE ELEMENTS S S Rao 395DISKS o J Ewins 404DISPLAYS OF VIBRATION PROPERTIES M Rade~ 413DISTRIBUTED SENSORS AND ACTUATORS See SENSORS AND ACTUATORS 431DUHAMEL METHOD See THEORY OF VIBRATION: DUHAMEL'S PRINCIPLE AND CONVOLUTION 431DYNAMIC STABILITY A Steindl H Troger 431

E

EARTHQUAKE EXCITATION AND RESPONSE OF BUILDINGS F Naeim 439EIGENVALUE ANALYSIS o Bauchau 461

CONTENTS xxiii

ELECTRORHEOLOGICAL AND MAGNETORHEOLOGICAL FLUIDS R Stanway 467ELECTROSTRICTIVE MATERIALS K Uchino 475ENVIRONMENTAL TESTING, OVERVIEW o Smallwood 490ENVIRONMENTAL TESTING, IMPLEMENTATION P S Varoto 496

EQUATIONS OF MOTION See THEORY OF VIBRATION: EQUATIONS OF MOTION 504

VOLUME 2

FFATIGUE A S Kobayashi M Ramulu 505

FE MODELS See DAMPING IN FE MODELS; FINITE ELEMENT METHODS 513FEEDFORWARD CONTROL OF VIBRATION C R Fuller 513

FFT METHODS See TRANSFORM METHODS 520

FILTERS See ADAPTIVE FILTERS; DIGITAL FILTERS; OPTIMAL FILTERS 520FINITE DIFFERENCE METHODS S S Rao 520

FINITE ELEMENT METHODS S S Rao 530FLUID/STRUCTURE INTERACTION S I Hayek 544

FLUTTER J Wright 553

FLUTTER, ACTIVE CONTROL F H Gem 565

FORCED RESPONSE N A J Lieven 578

FOURIER METHODS See TRANSFORM METHODS 582

FREE VIBRATION See THEORY OF VIBRATION: FUNDAMENTALS 582

FRICTION DAMPING R Ibrahim 582

FRICTION INDUCED VIBRATIONS R Ibrahim 589

GGEAR DIAGNOSTICS CJLi 597GROUND TRANSPORTATION SYSTEMS A K Waizuddin Ahmed 603

HHAND-TRANSMITTED VIBRATION M J Griffin 621

HELICOPTER DAMPING N M Wereley R Snyder R Krishnan TSieg 629

HILBERT TRANSFORMS M Feldman 642HUMAN RESPONSE TO VIBRATION See GROUND TRANSPORTATION SYSTEMS; HAND-TRANSMITTED

VIBRATION; MOTION SICKNESS; WHOLE-BODY VIBRATION 649

HYBRID CONTROL J Tang K W Wang 649

HYSTERETIC DAMPING H T Banks G A Pinter 658

IIDENTIFICATION, FOURIER-BASED METHODS S Braun 665IDENTIFICATION, MODEL-BASED METHODS S 0 Fassois 673IDENTIFICATION, NON-LINEAR SYSTEMS See NON-LINEAR SYSTEM IDENTIFICATION 685IMPACTS, NON-LINEAR SYSTEMS See VIBRO-IMPACT SYSTEMS 686

xxiv CONTENTS

IMPULSE RESPONSE FUNCTION See THEORY OF VIBRATION: IMPULSE RESPONSE FUNCTION 686INTENSITY See VIBRATION INTENSITY 686INVERSE PROBLEMS Y M Ram 686ISOLATION, ACTIVE See ABSORBERS, ACTIVE; ACTIVE CONTROL OF VEHICLE VIBRATION; ACTIVE

ISOLATION 690ISOLATION VIBRATION - APPLICATIONS AND CRITERIA See VIBRATION ISOLATION,

APPLICATIONS AND CRITERIA 690ISOLATION VIBRATION - THEORY See VIBRATION ISOLATION THEORY 690K

KRYLOV-LANCZOS METHODS R R Craig Jr 691L

LAGRANGE METHOD See BASIC PRINCIPLES; THEORY OF VIBRATION, ENERGY METHODS 699LAPLACE TRANSFORMS See TRANSFORM METHODS 699LASER BASED MEASUREMENTS E P Tomasini G M Revel P Castellini 699LINEAR ALGEBRA C Farhat o Rixen 710LINEAR DAMPING MATRIX METHODS FMa 721LIQUID SLOSHING R A Ibrahim 726LOCALIZATION C Pierre 741M

MACHINERY, ISOLATION See VIBRATION ISOLATION, APPLICATIONS AND CRITERIA. 753MAGNETORHEOLOGICAL FLUIDS See ELECTRORHEOLOGICAL AND MAGNETORHEOLOGICAL

FLUIDS 753MAGNETOSTRICTIVE MATERIALS A Flatau 753MATERIALS, DAMPING See DAMPING MATERIALS 762MEASUREMENT See LASER BASED MEASUREMENT; SEISMIC INSTRUMENTS, ENVIRONMENTAL

FACTORS; STANDARDS FOR VIBRATIONS OF MACHINES AND MEASUREMENT PROCEDURES;TRANSDUCERS FOR ABSOLUTE MOTION; TRANSDUCERS FOR RELATIVE MOTION 762

MEMBRANES A W Leissa 762MEMS, APPLICATIONS I Stiharu 771MEMS, DYNAMIC RESPONSE I Stiharu 779MEMS, GENERAL PROPERTIES I Stiharu 794MODAL ANALYSIS, EXPERIMENTAL 805

Basic Principles o J Ewins 805Measurement Techniques J M M Silva 813Parameter Extraction Methods N M M Maia 820Construction of Models from Tests N M M Maia 824Applications o J Ewins 829

MODE OF VIBRATION o J Ewins 838MODEL UPDATING AND VALIDATING M Link 844MODELS, DAMPING See DAMPING MODELS 856

CONTENTS xxv

MODES, NON-LINEAR SYSTEMS See NON-LINEAR SYSTEMS MODES 856MODES, ROTATING MACHINERY See ROTATING MACHINERY, MODAL CHARACTERISTICS 856MONITORING See DIAGNOSTICS AND CONDITION MONITORING, BASIC CONCEPTS,

ROTATING MACHINERY, MONITORING 856MOTION SICKNESS M J Griffin 856NNEURAL NETWORKS, DIAGNOSTIC APPLICATIONS M Zacksenhouse 863NEURAL NETWORKS, GENERAL PRINCIPLES B Dubuisson 869NOISE 877

Noise Radiated from Elementary Sources M P Norton J Pan 877Noise Radiated by Baffled Plates M P Norton J Pan 887

NONDESTRUCTIVE TESTING 898Sonic S Doebling C Farrar 898Ultrasonic L W Schmerr Jr 906

NONLINEAR NORMAL MODES A F Vakakis 918NONLINEAR SYSTEM IDENTIFICATION B F Feeny 924NONLINEAR SYSTEM RESONANCE PHENOMENA A Bajaj C M Krousgrill 928NONLINEAR SYSTEMS, OVERVIEW N C Perkins 944NONLINEAR SYSTEMS ANALYSIS A Bajaj 9520OBJECT ORIENTED PROGRAMMING IN FE ANALYSIS I Klapka A Cardona P Devloo 967OPTIMAL FILTERS S J Elliott 977PPACKAGING J Marcondes 983PACKAGING, ELECTRONIC See ELECTRONIC PACKAGING 990PARALLEL PROCESSING D Rixen 990PARAMETRIC EXCITATION S C Sinha A David 1001PERTURBATION TECHNIQUES FOR NONLINEAR SYSTEMS S Shaw 1009PIEZOELECTRIC MATERIALS AND CONTINUA H S Tzou M C Natori 1011PIPES S S Rao 1019PLATES A W Leissa 1024

VOLUME 3

RRANDOM PROCESSES M F Dimentberg 1033RANDOM VIBRATION See RANDOM VIBRATION, BASIC THEORY; RANDOM PROCESSES,

STOCHASTIC SYSTEMS 1040RANDOM VIBRATION, BASIC THEORY M F Dimentberg 1040RESONANCE AND ANTI RESONANCE M Rade~ 1046

xxvi CONTENTS

RESONANCE, NON LINEAR SYSTEMS See NON LINEAR SYSTEM RESONANCE PHENOMENA;

STOCHASTIC SYSTEMS 1055

ROBOT VIBRATIONS W Book 1055

ROTATING MACHINERY See ROTATING MACHINERY, ESSENTIAL FEATURES; ROTATING

MACHINERY, MODAL CHARACTERISTICS; ROTATING MACHINERY, MONITORING;

ROTOR DYNAMICS; ROTOR STATOR INTERACTIONS; BALANCING; BLADES AND

BLADED DISKS 1064

ROTATING MACHINERY, ESSENTIAL FEATURES R Bigret 1064

ROTATING MACHINERY, MODAL CHARACTERISTICS R Bigret 1069

ROTATING MACHINERY, MONITORING R Bigret 1078

ROTOR DYNAMICS R Bigret 1085

ROTOR-STATOR INTERACTIONS R Bigret 1107

SSEISMIC INSTRUMENTS, ENVIRONMENTAL FACTORS K McConnell 1121

SENSORS AND ACTUATORS H S Tzou C SChou 1134

SHAPE MEMORY ALLOYS H S Tzou ABaz 1144

SHELLS W Soedel 1155

SHIP VIBRATIONS W S Vorus 1167

SHOCK J Marcondes P. Singh 1173

SHOCK ABSORBERS See SHOCK ISOLATION SYSTEMS 1180

SHOCK ISOLATION SYSTEMS M Rade§ 1180

SIGNAL GENERATION MODELS FOR DIAGNOSTICS M Sidahmed G Dalpiaz 1184

SIGNAL INTEGRATION AND DIFFERENTIATION S Dyne 1193

SIGNAL PROCESSING, CEPSTRUM See CEPSTRUM ANALYSIS 1199

SIGNAL PROCESSING, MODEL BASED METHODS S Braun 1199

SMART MATERIALS See ELECTRORHEOLOGICAL AND MAGNETORHELEOLOGICAL

FLUIDS; ELECTROSTRICTIVE MATERIALS; MAGNETOSTRICTIVE MATERIALS; PIEZOELECTRIC

MATERIALS; SHAPE MEMORY ALLOYS 1208

SOUND See VIBRATION GENERATED SOUND, FUNDAMENTALS; VIBRATION GENERATED SOUND,RADIATION BY FLEXURAL ELEMENTS 1208

SPECTRAL ANALYSIS, CLASSICAL METHODS S Braun 1208

SPECTRAL ANALYSIS, MODEL BASED METHODS See SIGNAL PROCESSING,

MODEL BASED METHODS 1223

SPECTRAL ANALYSIS, WINDOWS See WINDOWS 1223

STABILITY See DYNAMIC STABILITY 1224

STANDARDS FOR VIBRATIONS OF MACHINES AND MEASUREMENT PROCEDURES J Niemkiewicz 1224

STOCHASTIC ANALYSIS OF NON LINEAR SYSTEMS Y K Un GQCai 1238

STOCHASTIC SYSTEMS M F Dimentberg 1246

STRUCTURAL DAMPING See HYSTERETIC DAMPING 1252

STRUCTURAL DYNAMIC MODIFICATIONS A Sestieri W D'Ambrogio 1253

STRUCTURE-ACOUSTIC INTERACTION, HIGH FREQUENCIES A Sestieri 1265

CONTENTS xxvii

STRUCTURE-ACOUSTIC INTERACTION, LOW FREQUENCIES A Sestieri 1274SUBSTRUCTURING See THEORY OF VIBRATION, SUBSTRUCTURING 1283SUPERPOSITION See THEORY OF VIBRATION, SUBSTRUCTURING 1283SVD See LINEAR ALGEBRA 1283

TTESTING, MODAL See MODAL ANALYSIS, EXPERIMENTAL: APPLICATIONS; MODAL ANALYSIS,

EXPERIMENTAL: BASIC PRINCIPLES; MODAL ANALYSIS, EXPERIMENTAL: CONSTRUCTION OFMODELS FROM TESTS; MODAL ANALYSIS, EXPERIMENTAL: MEASUREMENT TECHNIQUES;MODAL ANALYSIS, EXPERIMENTAL: PARAMETER EXTRACTION METHODS 1285

TESTING, NONLINEAR SYSTEMS A Haddow 1285THEORY OF VIBRATION 1290

Fundamentals B Yang 1290Superposition M G Prasad 1299Duhamel's Principle and Convolution G Rosenhouse 1304Energy Methods S S Rao 1308Equations of Motion J Wickert 1324Substructuring M Sunar 1332Impulse Response Function R K Kapania 1335Variational Methods S S Rao 1344

TIME-FREQUENCY METHODS P White 1360TIRE VIBRATIONS G D Shteinhauz 1369TOOL WEAR MONITORING M Sidahmed 1379TRANSDUCERS FOR ABSOLUTE MOTION K G McConnell 1381TRANSDUCERS FOR RELATIVE MOTION G E Maddux K G McConnell 1398TRANSFORM METHODS S Braun 1406TRANSFORMS, HILBERT See HILBERT TRANSFORMS 1419TRANSFORMS, WAVELETS P White 1419TRANSMISSION See VIBRATION TRANSMISSION 1435TRANSPORTATION SYSTEMS See GROUND TRANSPORTATION SYSTEMS 1435

U

ULTRASONICS M J S Lowe 1437ULTRASONICS, NONDESTRUCTIVE TESTING See NONDESTRUCTIVE TESTING: ULTRASONIC 1441

VVARIATIONAL METHODS See THEORY OF VIBRATION: VARIATIONAL METHODS 1443VEHICLES, ACTIVE VIBRATION CONTROL See ACTIVE CONTROL OF VEHICLE VIBRATION 1443VIBRATION ABSORBERS See ABSORBERS, VIBRATION 1443VIBRATION GENERATED SOUND 1443

Fundamentals M P Norton S J Drew 1443Radiation by Flexural Elements M P Norton S J Drew 1456

VIBRATION INTENSITY S I Hayek 1480

xxviii CONTENTS

VIBRATION ISOLATION THEORY E Rivin 1487

VIBRATION ISOLATION, APPLICATIONS AND CRITERIA E Rivin 1507

VIBRATION PROPERTIES, COMPARISON See COMPARISON OF VIBRATION PROPERTIES:

COMPARISON OF MODAL PROPERTIES; COMPARISON OF VIBRATION PROPERTIES:

COMPARISON OF RESPONSE PROPERTIES; COMPARISON OF VIBRATION PROPERTIES:

COMPARISON OF SPATIAL PROPERTIES 1521VIBRATION TRANSMISSION S I Hayek 1522VIBRO-IMPACT SYSTEMS I F Peterka 1531VISCOUS DAMPING F Gandhi 1548

WWAVE PROPAGATION 1551

Guided Waves in Structures M J S Lowe 1551

Interaction of Waves with Boundaries M J S Lowe 1559

Waves in an Unbounded Medium M J S Lowe 1565WAVELETS See TRANSFORMS, WAVELETS 1570WHOLE-BODY VIBRATION M J Griffin 1570WIND-INDUCED VIBRATIONS T Kijewski F Hann A Kareem 1578WINDOWS S Braun 1587

ZZ TRANSFORMS See TRANSFORM METHODS 1597

GLOSSARY Gi-Gvi

APPENDICES Ai-Axviii

INDEX li-Ixxxii

COLOUR PLATE SECTIONS

Volume 1 292-293Volume 2 812-813Volume 3 1308-1309

Positive Position FeedbackModern control design is traditionally performedusing first-order dynamical equations. The positiveposition feedback (PPF) algorithm developed by Gohand Caughey and implemented by Fanson andCaughey uses second-order compensation, allowingphysical insight to vibration control by active modaladdition. In this algorithm, a position signal is com-pensated by a second order filter for feedback control.For linear systems, the PPF controller is stable even inthe presence of unmodeled actuator dynamics. Inaddition it is possible to transform the dynamicalequations to modal space and design independentsecond-order feedback compensators for individualmodes. Many numerical and experimental implemen-tations of the PPF control scheme may be found in thevibrations literature.

See also: Absorbers, vibration; Active control of civilstructures; Active control of vehicle vibration; Vibra-tion isolation, applications and criteria; Vibration iso-lation theory

Further ReadingAgnes GS (1995) Active/passive piezoelectric vibration

suppression. journal of Intelligent Materials, Systems,and Structures 6:482-7.

American Institute of Aeronautics and Astronautics (1987)Proceedings of the 28th AIAA/ASME/ASCE/AHS/ACSStructures, Structural Dynamics and Materials Confer-ence, April, Monterey, CA.

American Institute of Aeronautics and Astronautics (1994)Proceedings of the 35th AIAA/ASME/ASCE/AHS/ACSStructures, Structural Dynamics and Materials Confer-ence, April, Hilton Head, sc.

Baz A, Poh S, Fedor J (1992) Independent modal spacecontrol with positive position feedback. Transactions ofthe ASME 114:96-103.

Caughey TK (1995) Dynamic response of structuresconstructed from smart materials. Smart Materials andStructures 4:A101-A106.

DenHartog JP (1985) Mechanical Vibrations. New York:Dover Books.

Dusch 11 (1995) Active Vibration Suppression: Stabilityand Design in Second Order Form. PhD thesis, SUNY atBuffalo.

Fanson JL, Caughey TK (1987) Positive position feedbackcontrol for large space structures. In Proceedings of the28th AIAA/ASME/ASCE/AHS/ACS Structures, Structur-al Dynamics and Materials Conference 87-0902. April,Monterey, CA.

Fanson JL, Caughey TK (1990) Positive position feedbackcontrol for large space structures. AIAA journal 28(4).

Flotow AH Von, Beard A, BaileyD (1994) Adaptive tunedvibration absorbers: tuning laws, tracking agility, sizing,and physical implementations. Proceedings Noise-Con-ference 94.

Forward RL (1979) Electronic damping of vibrations inoptical structures. journal of Applied Optics, 18:690-7.

Goh C], Caughey TK (1985) On the stability problemcaused by finite actuator dynamics in the collocatedcontrol of large space structures. International journal ofControl 41(3):787-802.

Hagood NW, Flotow AVon (1991) Damping of structuralvibrations with piezoelectric materials and passiveelectrical networks. journal of Sound and Vibration146:243-68.

Hollkamp JJ (1994) Multimodal passive vibration suppres-sion with piezoelectric materials and resonant shunts.journal of Intelligent Materials Systems and Structures5:49-57.

Hollkamp 11, Starchville TF (1994) A self-tuning piezo-electric vibration absorber. In Proceedings of the 35thAIAA/ASME/ASCE/AHS/ACS Structures, Structural Dy-namics and Materials Conference 94-1790. April, HiltonHead, sc.

V Steffen, Jr and D Rade, Federal University of improve the comfort of users when walking on pedes-Uberlandia, Uberlandia, Brazil trian bridges, to attenuate vibrations transmitted

from the main rotor to the cockpit of helicopters,Copyright © 2001 Academic Press and to improve machine tool operation conditions, todoi:10.1006/rwvb.2001.0176 mention just a few examples. Military applications

have also been developed. The use of DVAs to reduceIntroduction t~e dynamic ~or~es transmitted to an aircr~ft due to

hIgh rates of fIre Imposed on the canon motIOn can beDynamic vibration absorbers (DVAs), also called mentioned as another example.Vibration Neutralizers or Tuned Mass Dampers, are In practical applications, DVAs can be found inmechanical appendages comprising inertia, stiffness, various configurations, intended for the attenuationand damping elements which, once connected to a of either rectilinear or angular motion. The simplestgiven structure or machine, named herein the primary setup is that formed by a single mass attached to thesystem, are capable of absorbing the vibratory energy primary system through a linear spring. This config-at the connection point. As a result, the primary uration is named the 'undamped dynamic vibrationsystem can be protected from excessively high vibra- absorber'. As will be shown later, in designing antion levels. In practice, DVAs can be included in the undamped DVA to attenuate harmonic vibrations,original system design or can be added to an existing the values of its physical parameters (stiffness andsystem, often as part of a remedial course of action. inertia) must be chosen according to the value of the

Since their invention by Frahm at the beginning of excitation frequency and it is then said that the DVAthe twentieth century, dynamic vibration absorbers is tuned. The undamped DVA may become ineffectivehave been extensively used to mitigate vibrations in when the excitation frequency deviates, even slightly,various types of mechanical systems. A very well- from the nominal tuning frequency. In order to pro-known application is the so-called Stockbridge dam- vide a mechanism for energy dissipation and toper, widely used to reduce wind-induced vibrations in enlarge the effective bandwidth of the absorber,overhead power transmission lines. In a remarkable damping can be introduced into the DVA. In mostengineering application, a 400-ton absorber has been applications, a viscous damping model is used,designed for Citicorp Center, a 274-m high office although viscoelastic and Coulomb-type dampersbuilding in New York City, for suppressing primarily can be found in certain cases. In general, a DVA isthe contribution of the first vibration mode in wind- designed to attenuate vibrations generated by ainduced oscillations. In a similar application, two purely harmonic excitation. However, in several300-ton DVAs have been installed in the John Han- situations, vibrations are produced by periodic forcescock Tower, in Boston, Massachussets. The dynamics containing various harmonic components. In thisof television towers are particularly favorable for the case, multiple DVAs can be used, each one tuned touse of pendulum-like DVAs, which have been a specific frequency component. It is also possible toapplied, for example, to the towers of Alma-Ata use distributed-parameter structural elements, such asand Riga, in the former Soviet Union. beams or plates, as dynamic absorbers. Besides the

Due to their technological relevance both in the ease of physical realization, the main interest inacademic and industrial domains, DVAs are still a using these configurations is related to the fact thatsubject of permanent interest. New applications the DVA can be tuned to various frequency valuesinclude devices used to stabilize ship roll motion, to simultaneously.

single-degree-of-freedom undamped primary systemscan be extended to these types of systems, using amodal approach, developed in the following.

The main idea is to apply an optimization criterionin the vicinity of a particular natural frequency of theprimary system. Several DVAs can be designed inde-pendently for each individual vibration mode. Forthis purpose it is assumed that the natural frequenciesof the primary system are sufficiently well separatedand that the masses of the DVAs are small enough notto modify significantly the natural frequencies of theprimary system. Figure 9 shows schematically adamped DVA attached to an undamped multi-degree-of-freedom primary system, modeled by iner-tia matrix M and stiffness matrix K. The indicatedcoordinates Xc and Xf correspond to the coordinatesto which the DVA is attached and the excitation forceis applied, respectively. In the general case of multi-dimensional systems, these coordinates may corre-

Figure 17 shows the influence of the active DVA onthe frequency system response of the primary systemas compared to the response of this system withoutDVA and with the purely passive undamped DVA.

Final Remarks and FuturePerspectives

In the previous sections, only DVAs comprising stiff-ness and damping elements exhibiting linear behaviorhave been considered. However, studies have demon-strated that nonlinear DVAs generally provide asuppression bandwidth much larger than linearabsorbers. As a result, in spite of a more involvedtheory and design procedure, nonlinear vibrationabsorbers have received much attention lately.

Although only harmonic excitations were consid-ered here, the reader should be aware of the fact thatdynamic vibration absorbers have been extensivelyused to attenuate other types of vibrations, such astransient and random. In such cases, the optimaldesign is generally carried on by using time domain-based procedures.

The study and development of techniques relatedto smart materials represent new possibilities ofvibration reduction in mechanics and mechatronics.The physical properties of such materials can bemodified by controlled modifications of some envir-onmental parameters. To mention a few examples,the viscosity (damping capacity) of electrorheologicaland magnetorheological fluids can be varied byapplying external electric and magnetic fields, respec-tively. The geometry of components made of shapememory alloys can be changed by applying tempera-ture variations. Some researchers have considered thepossibility of using such smart materials to conceiveself-tunable adaptive vibration absorbers. Further-more, the possibility of dissipating mechanical energywith piezoelectric material, such as piezoelectric cera-mics, shunted with passive electrical components hasbeen investigated by various authors in this decade.The four basic kinds of shunt circuits are: inductive,resistive, capacitive, and switched. If a piezoelectricelement is attached to a structure, it is strained as thestructure deforms and part of the vibration energy isconverted into electrical energy. The piezoelectricelement behaves electrically as a capacitor and canbe combined with a so-called shunt network in orderto perform vibration control. Shunting with a resistorand inductor forms a RLC circuit introducing anelectrical resonance which, in the optimal case, istuned to structural resonances. The scheme of suchan arrangement is depicted in Figure 18. The inductoris used to tune the shunt circuit to a given resonanceof the structure and the resistor is responsible for peakamplitude reduction of a particular mode. The induc-tive shunt or resonant circuit shunt presents a vibra-tion suppression effect that is very similar to theclassical dynamic vibration absorber. The classicalDVA stores part of the kinetic energy of the primary

Figure 18 Scheme of a resonant circuit shunt used for vibra-tion attenuation.

system, while the resonant circuit shunt is designed todissipate the electrical energy that has been convertedfrom mechanical energy by the piezoelectric. A multi-mode damper can be obtained by adding a differentshunt for each suppressed mode in such a way thatattenuation can be obtained for a given number offrequencies.

See also: Absorbers, active; Active control of civilstructures; Active control of vehicle vibration; Activeisolation; Damping, active; Flutter, active control; Shipvibrations; Shock isolation systems; Theory of vibra-tion, Fundamentals; Vibration isolation theory; Viscousdamping.

Further Reading Absorbers. Theory and Practical Applications. JohnWiley.

Den Hartog JP (1956) Mechanical Vibrations, 4th edn. Nashif AD, Jones DIG and Henderson JP (1985) VibrationMcGraw-Hill Book Company. Damping. John Wiley.

Hagood NW and von Flotow A (1991) Damping of Newland DE (1989) Mechanical Vibration Analysis andstructural vibrations with piezoelectric materials and Computation. Longman Scientific and Technical.passive electrical networks. Journal of Sound and Snowdon JC (1968) Vibration and Shock in DampedVibration 146: 243-268. Mechanical Systems. John Wiley.

Harris CM (1988) Shock and Vibration Handbook, 3rd Sun JQ, Jolly MR and Norris MA (1995) Passive, adaptiveedn, McGraw-Hill Book Company. and active tuned vibration absorbers - a survey. Trans.

Inman DJ (1989) Vibration with Control, Measurement ASME Combined Anniversary Issue Journal of Mechan-and Stability. Prentice-Hall. ical Design and Journal of Vibration and Acoustics 117:

Korenev BG and Reznikov LM (1993) Dynamic Vibration 234-242.

T T Soong, State University of New York at Buffalo, The purpose of this article is to provide an assess-Buffalo, NY, USA ment of the state-of-the-art and state-of-the-practice

B F Spencer, Jr., University of Notre Dame, Notre ?f this ex~iting, a.nd st~ll evolving, techn.ology. AlsoDame IN USA Included In the dIscussIOn are some basIc concepts,

, , the types of active control systems being used andCopyright © 2001 Academic Press deployed, and their advantages and limitations in thedoi:10.1006/rwvb.2001.0189 context of seismic design and retrofit of civil engi-

neering structures.

Introduction Active, Hybrid, and Semiactive Control

I . ·1 " 1 1·· . Systemsn CIVI engIneerIng structura app lCatlOns, actIve,semiactive, and hybrid structural control systems An active structural control system has the basicare a natural evolution of passive control technolo- configuration shown schematically in Figure lA. Itgies such as base isolation and passive energy dissipa- consists of: (1) sensors located about the structure totion. The possible use of active control systems and measure either external excitations, or structuralsome combinations of passive and active systems, so- response variables, or both; (2) devices to processcalled hybrid systems, as a means of structural pro- the measured information and to compute necessarytection against wind and seismic loads has received control force needed based on a given control algo-considerable attention in recent years. Active/hybrid rithm; and (3) actuators, usually powered by externalcontrol systems are force delivery devices integrated sources, to produce the required forces.with real-time processing evaluators/controllers and When only the structural response variables aresensors within the structure. They act simultaneously measured, the control configuration is referred to aswith the hazardous excitation to provide enhanced feedback control since the structural response is con-structural behavior for improved service and safety. tinually monitored and this information is used toRemarkable progress has been made over the last 20 make continual corrections to the applied controlyears. As will be discussed in the following sections, forces. A feedforward control results when the con-research to date has reached the stage where active trol forces are regulated only by the measured excita-systems have been installed in full-scale structures. tion, which can be achieved, for earthquake inputs,Active systems have also been used temporarily in by measuring accelerations at the structural base. Inconstruction of bridges or large-span structures (e.g., the case where the information on both the responselifelines, roofs) where no other means can provide quantities and excitation is utilized for control design,adequate protection. the term feedback-feedforward control is used.

Figure 1 Structure with various schemes. (A) Structure with active control; (8) structure with hybrid control; (C) structure withsemiactive control. PED, passive energy dissipation.

To see the effect of applying such control forces toa linear structure under ideal conditions, consider abuilding structure modeled by an n-degree-of-free-dom lumped mass-spring-dashpot system. The matrixequation of motion of the structural system can bewritten as:

are only a limited number of sensors and actuators,the dynamics of the actuators can be quite complex,the actuators are typically very large, and the systemsmust be failsafe.

It is useful to distinguish between several types ofactive control systems currently being used in practice.The term hybrid control generally refers to a com-bined passive and active control system, as depicted inFigure lB. Since a portion of the control objective isaccomplished by the passive system, less active controleffort, implying less power resource, is required.

Similar control resource savings can be achievedusing the semi active control scheme sketched inFigure lC, where the control actuators do not addmvchanical energy directly to the structure, hencebounded-input bounded-output stability is guaran-teed. Semiactive control devices are often viewed ascontrollable passive devices.

A side benefit of hybrid and semi active controlsystems is that, in the case of a power failure, the pas-sive components of the control still offer some degreeof protection, unlike a fully active control system.

Full-scale ApplicationsAs alluded to earlier, the development of active,hybrid, and semi active control systems has reachedthe stage of full-scale applications to actual struc-tures. Figure 2 shows that, up to 1999, there havebeen 43 installations in building structures andtowers, most of which are in Japan (Table 1). Inaddition, 15 bridge towers have employed activesystems during erection. Most of these full-scalesystems have been subjected to actual wind forcesand ground motions and their observed performancesprovide invaluable information in terms of: (1) vali-dating analytical and simulation procedures used topredict actual system performance; (2) verifying com-plex electronic-digital-servohydraulic systems underactual loading conditions; and (3) verifying the cap-ability of these systems to operate or shut down underprescribed conditions.

Described below are several of these systemstogether, in some cases, with their observed perfor-mances. Also addressed are several practical issues inconnection with actual structural applications ofthese systems.

Hybrid Mass Damper Systems

As seen from Table 2, the hybrid mass damper(HMD) is the most common control device employedin full-scale civil engineering applications. An HMDis a combination of a passive tuned mass damper(TMD) and an active control actuator. The ability of

this device to reduce structural responses reliesmainly on the natural motion of the TMD. The forceshom the control actuator are employed to increasethe efficiency of the HMD and to increase its robust-ness to changes in the dynamic characteristics of thestructure. The energy and forces required to operate atypical HMD are far less than those associated with afully active mass damper system of comparable per-formance.

An example of such an application is the HMDsystem installed in the Sendagaya INTES building inTokyo in 1991. As shown in Figure 3, the HMD wasinstalled atop the 11th floor and consists of twomasses to control transverse and torsional motionsof the structure, while hydraulic actuators provide theactive control capabilities. The top view of the con-trol system is shown in Figure 4, where ice thermalstorage tanks are used as mass blocks so that no extramass needs to be introduced. The masses are sup-

Active Mass Damper Systems

Design constraints, such as severe space limitations,can preclude the use of an HMD system. Such is thecase in the active mass damper or active mass driver(AMD) system designed and installed in the KyobashiSeiwa Building in Tokyo and the Nanjing Commu-nication Tower in Nanjing, China.

The Kyobashi Seiwa Building, the first full-scaleimplementation of active control technology, is an

Figure 3 Sendagaya INTES building with hybrid mass damper.AMD, active mass damper.

II-story building with a total floor area of 423 m2. As

seen in Figure 9, the control system consists of twoAMDs where the primary AMD is used for transversemotion and has a weight of 4 tons, while the second-

ary AMD has a weight of 1 ton and is employed toreduce torsional motion. The role of the active systemis to reduce building vibration under strong winds andmoderate earthquake excitations and consequently toincrease comfort of occupants in the building.

Semiactive Damper Systems

Control strategies based on semiactive devices appearto combine the best features of both passive andactive control systems. The close attention receivedin this area in recent years can be attributed to the factthat semiactive control devices offer the adaptabilityof active control devices without requiring the asso-ciated large power sources. In fact, many can operateon battery power, which is critical during seismicevents when the main power source to the structuremay fail. In addition, as stated earlier, semi activecontrol devices do not have the potential to destabi-lize (in the bounded input/bounded output sense) thestructural system. Extensive studies have indicatedthat appropriately implemented semiactive systemsperform significantly better than passive devices andhave the potential to achieve the majority of theperformance of fully active systems, thus allowingfor the possibility of effective response reductionduring a wide array of dynamic loading conditions.

One means of achieving a semi active dampingdevice is to use a controllable, electromechanical,variable-orifice valve to alter the resistance to flowof a conventional hydraulic fluid damper. A sche-matic of such a device is given in Figure 10. Such asystem was implemented, for example, in a bridge todissipate the energy induced by vehicle traffic.

More recently, a semiactive damper system wasinstalled in the Kajima Shizuoka Building in Shi-zuoka, Japan. As seen in Figure 11, semi activehydraulic dampers were installed inside the walls onboth sides of the building to enable it to be used as adisaster relief base in postearthquake situations. Each

In the case of MR fluids, they typically consist ofmicron-sized, magnetically polarizable particles dis-persed in a carrier medium such as mineral or siliconeoil. When a magnetic field is applied to the fluid,particle chains form, and the fluid becomes a semi-solid and exhibits viscoplastic behavior. Transition torheological equilibrium can be achieved in a fewmilliseconds, allowing construction of devices withhigh bandwidth. Additionally, it has been indicatedthat high yield stress of an MR fluid can be achievedand that MR fluids can operate at temperatures from

F" 8 P' . I f DUOX t AMD t' d -40°C to 150°C with only slight variations in theIgure nnclp e 0 sys em. , ac Ive mass am- , .'per; TMD. tuned mass damper, Yield stress. Moreover, MR flUids are not sensitive to

impurities such as are commonly encountered duringmanufacturing and usage, and little particle/carrier

above, an advantage of controllable fluid devices is fluid separation takes place in MR fluids underthat they contain no moving parts other than the common flow conditions. Further, a wider choicepiston, which makes them simple and potentially of additives (surfactants, dispersants, friction modi-very reliable. fiers, antiwear agents, etc.) can generally be used

The essential characteristics of controllable fluids is with MR fluids to enhance stability, seal life, bearingtheir ability to change reversibly from a free-flowing, life, and so on, since electrochemistry does not affectlinear, viscous fluid to a semisolid with a controllable the magnetopolarization mechanism. The MR fluidyield strength in milliseconds when exposed to an can be readily controlled with a low-voltage (e.g.,electric (for electrorheological (ER) fluids) or mag- 12-24 V), current-driven power supply outputtingnetic (for magnetorheological (MR) fluids) field. only 1-2 A.

While no full-scale structural applications of MRdevices have taken place to date, their future for civilengineering applications appears to be bright. Therehave been published reports on the design of afull-scale, 20-ton MR damper, showing that thistechnology is scalable to devices appropriate forcivil engineering applications. At design velocities,the dynamic range of forces produced by this deviceis over 10 (Figure 14), and the total power required bythe device is only 20-50 W.

Figure 11 Semiactive hydraulic damper in the Kajima Shizuoka building.

Concluding Remarks currently deployed in structures and towers weredesigned primarily for performance enhancement

An important observation to be made in the perfor- against wind and moderate earthquakes and, inmance observation of control systems such as those many cases, only for occupant comfort. However,described above is that efficient active control systems active control systems remain to be one of only acan be implemented with existing technology under few alternatives for structural protection againstpractical constraints such as power requirements and near-field and high-consequence earthquakes.stringent demand of reliabilit~. ~hus, signifi~ant An upgrade of current active systems to thisstrides have been made, consIdermg that senous higher level of structural protection is necessary,implementational efforts began less than 15 years since only then can the unique capability of activeago. On the other hand, there remains a significant control systems be realized.distance between the state-of-the-art of active control 2. Economy and flexibility in construction. Anothertechnology and some originally intended purposes for area in which great benefit can be potentiallydeveloping such a technology. Two of these areas are realized by the deployment of active control sys-particularly noteworthy and they are highlighted terns is added economy and flexibility to structuralbelow. design and construction. The concept of active

structures has been advanced. An active structure1. Mitigating higher-level hazards. In the context of is defined here as one consisting. ~f two ty~es of

earthquake engineering, one of the original goals load-resisting members: the tra~ItlOnal statIc (orfor active control research was the desire that, passive) members that are desIgned to supportthrough active control, conventional structures basic design loads, and dynamic (or active) mem-can be protected against infrequent, but highly bers whose function is to augmen~ the structure~sdamaging earthquakes. The active control devices capability in resisting extraordmary dynamIc

Figure 12 Maximum responses (EI Centro, Taft, and Hachinohe waves with 50cms-1 and assumed Tokai waves). (A) With semi-active hydraulic damper control; (8) without control.

loads. Their integration is done in an optimalfashion and produces a structure that is adaptiveto changing environmental loads and usage.

Note that an active structure is conceptually andphysically different from a structure that is activelycontrolled, as in the cases described above. In the caseof a structure with active control, a conventionallydesigned structure is supplemented by an active

Figure 14 Force-displacement loops at maximum and zero magnetic fields.

control device that is activated whenever necessary in development of seismic-response-controlled structure.order to enhance structural performance under extra- Proceedings 1st World Conference on Structural Controlordinary loads. Thus, the structure and the active (GW H.0user, SF M~sri and AG Chassiakos, eds),control system are individually designed and opti- InternatIonal AssocIatIOn for Structural Control, Los

. d A· h h h d· Angeles, CA, pp. 19-31.mIze. n actIve structure, on t e ot er an, IS K b . T (1999) M·' d . d f... 0 on ISSlOnan perspectIve towar sutureone whose actIve and paSSIve components are mte- I I h P d' f th 2 d... structura contro researc. rocee mgs 0 e ngrated and sImultaneously optImIzed t.o ~roduce a World Conference on Structural Control, Kyoto, Japan.new breed of structural systems. ThIs Important (T Kobori Y Inoue K Seto H lemura and A Nishitanidifference makes the concept of active structures eds), John' Wiley, Chicheste~, UK, pp. 25-34. 'exciting and potentially revolutionary. Among many Patten WN (1999) The 1-35 Walnut Creek bridge: anpossible consequences, one can envision greater flex- intelligent highway bridge via semi-active structuralibilities which may lead to longer, taller, slender, or control. In: Proceedings of 2nd World Conference onmore open structures and structural forms. Structure Control, Kyoto, Japan (T Kobori, Y Inoue, K

Seto, H lemura, A Nishitani, (eds), John Wiley,See Plates 1,2,3. Chichester, UK, pp. 427--436.

Soong IT (1990) Active Structural Control: Theory andSee also: Damping, active; Hybrid control. Practice. New York, NY: Wiley.

Soong TT and Manolis GD (1987) Active structures.Journal of Structural Engineering 113: 2290-2301.

Further Reading Soong TT, Reinhorn AM, Aizawa Sand Higashino M(1994) Recent structural applications of active control

Carlson JD and Spencer BFJr. (1996) Magneto-rheological technology. Journal of Structural Control 1: 5-21.fluid dampers for semi-active seismic control. In: Spencer BF Jr, Carlson JD, Sain MK and Yang G (1997)Proceedings of the 3rd International Conference on On the current status of magnetorheological dampers:Motion and Vibration Control, Chiba, Japan. Japan seismic protection of full-scale structures. In: Proceed-Society of Mechanical Engineers, Tokyo, Japan, Vol. III, ings of American Control Conference, American Auto-pp. 35--40. matic Control Council, pp. 458--462, Albuquerque, NM.

Carlson JD and Weiss KD (1994) A growing attraction to Spencer BF Jr, Yang G, Carlson JD and Sain MK (1999)magnetic fluids. Machine Design, pp. 61-64. Smart dampers for seismic protection of structures: a

Housner GW, Bergman LA, Caughey TK et al. (1997) full-scale study. In: Proceedings of 2nd World Con-Structural control: past, present, and future. Journal of ference on Structure Control, Kyoto, Japan (T Kobori, YEngineering Mechanics 123: 897-971. Inoue, K Seto, H lemura, A Nishitani, eds), John Wiley,

Kobori T (1994) Future direction on research and Chichester, UK, pp. 417--426.

M Ahmadian, Virginia Tech, Blacksburg, VA, USA Passive SuspensionsCopyright © 2001 Academic Press A passive suspension system is one in which thedoi:10.1006/rwvb.2001.0193 characteristics of the components (springs and dam-

pers) are fixed. These characteristics are determinedby the suspension designer, according to the design

Introduction g.oals and the intende~ ap~lication. A passive s.u.spen-SlOn, such as shown III FIgure 2, has the abilIty to

Perceived comfort level and ride stability are two of store energy via a spring and to dissipate it via athe most important factors in a vehicle's subjective damper. Figure 2 represents one-quarter of a vehicle,evaluation. There are many aspects of a vehicle that and therefore is commonly referred to as 'quarter-carinfluence these two properties, most importantly the model'. The mass of the vehicle body (sprung mass)primary suspension components, which isolate the and tire-axle assembly (unsprung mass) are definedframe of the vehicle from the axle and wheel assem- respectively by mb and ma, with their correspondingblies. In the design of a conventional primary suspen- displacements defined by Xb and Xa· The suspensionsion system, there is a tradeoff between the two spring, ks, and damper, c" are attached between thequantities of ride comfort and vehicle stability, as vehicle body and axle, and the stiffness of the tire isshown in Figure 1. If a primary suspension is designed represented by kt.to optimize the handling and stability of the vehicle, The parameters of a passive suspension are gener-the operator is often subjected to a large amount of ally fixed to achieve a certain level of compromisevibration and perceives the ride to be rough and between reducing vibrations and increasing roaduncomfortable. On the other hand, if the primary holding. Once the spring has been selected, basedsuspension is designed to be soft and 'cushy', the on the load-carrying capability of the suspension,vibrations in the vehicle are reduced, but the vehicle the damper is the only variable remaining to specify.may not be too stable during maneuvers such as Low damping yields poor resonance control at thecornering and lane change. As such, the performanceof primary suspensions is always limited by the com-promise between ride and handling. Good design of apassive suspension cannot eliminate this compromisebot can, to some extent, optimize the opposing goalsof comfort and handling.

value of damping is desired to further push the bodyupward. The on-off skyhook semiactive policy emu-lates the ideal body displacement control configura-tion of a passive damper 'hooked' between the bodymass and the 'sky' as shown in Figure 9; hence, thename 'skyhook damper'.

Continuous Skyhook Control

In continuous control, the damping force is notlimited to the minimum and maximum states alone,as was the case for the on-off skyhook control. Asillustrated in Figure 7B the damper can provide anydamping force in the range between the minimum andmaximum limits. This will enable the semiactivesuspension to achieve a performance that is closerto the ideal skyhook configuration shown in Figure 9.

In continuous skyhook control, the low stateremains defined by the minimum damping value,while the high state is set equal to a constant gainvalue multiplied by the absolute velocity of the vehi-cle body, bounded by the minimum and maximumdamping force of the damper:

Figure 2 Feedback control schematic.

Equivalence of Feedback andFeedforward ControlSince the form of the feedforward control filter isdependent on both the path between the disturbanceand the system to be controlled and the nature of thedisturbance, it is often necessary to make the feedfor-ward control filter-adaptive. Adaptation of the feed-forward controller is accomplished by feeding backthe error sensor signal to an adaptive feedforwardfilter, as shown in Figure 3, applied to the samesystem considered in Figure 1.

Since the adaptation of the feedforward controlleris dependent on the error signal, a feedback path isintroduced into the system. For the case of sinusoidaldisturbance, there is an equivalent feedback control-ler which exhibits exactly the same performancecharacteristics as the feedforward controller.

Actuation ApproachesMany different actuators are available and have foundapplication in both practical and experimental activeisolation systems. Traditional active isolation actua-tors include hydraulic and electromagnetic drives.Force and displacement limits of actuators based onhydraulic drives depend on the energy-producingmechanism and the fluidic circuit which makes upthe actuator. As such, hydraulic actuators have been

employed in systems as large as earthquake simulatorsand as small as miniature valves. Linear force anddisplacement limits of actuators based on electromag-netic voice coils of 100 lb (45 kg) and 0.5 in (12.5 mm)are readily available commercially. Greater perfor-mance is possible but may require a custom design. Anexample of a hydraulic active isolation system is thefully active suspension on high-performance Lotusrace cars. In these systems, hydraulic actuators arelocated at each wheel and force is provided by areservoir and pump system. The active suspensioncar is programmed to keep the car parallel to theroad at all times, thus minimizing roll and pitch. Anexample of an electromagnetic active isolation systemis the vibration isolation and suppression system VISSexperiment. In this system, voice coil actuators form ahexapod mount which actively isolates an infraredtelescope from the spacecraft bus.

Active isolation actuators which incorporate activematerials include piezoceramic magnetostrictive andmagnetorheological materials. Table 1 compares theactuation properties of these commercially availableactive materials. Piezoceramic properties are for PZT5H and magnetostrictive properties are for Terfenol-D. Blocked stress is the product of maximum strainand modulus.

Since piezoceramic materials tend to exhibit rela-tively high force and low stroke, they are often com-bined with a hydraulic or mechanical load-couplingmechanism to multiply motion at the expense ofapplied force.

An example of an active isolation system that usespiezoceramic actuators is the satellite ultraquietisolation technology experiment (SUITE). In this

experiment, viscoelastic damped piezoceramic actua- Conversely, a properly designed switching amplifiertors form a hexapod mount to isolate a sensitive is much more efficient at driving a capacitive load andinstrument from a spacecraft bus. An example of an would result in a lower system weight.actuator that uses a magnetostrictive material is theTerfenol-D-based reaction mass actuator (RMA) See also: Actuators and smart structures; Feedfor-made by SatCon Technologies. This actuator is ward control of vibration.being investigated for use in helicopter noise andvibration control. Finally, an example of a linearactuator that uses a magnetorheological fluid is the Further Readinglinear pneumatic motion control system made by A d E E M GI R t I (1999) S t II't.... n erson , vert , aese ea. a e I e~ORD CorporatIOn. This technology ISalso available Ultra quiet Isolation Technology Experiment (SUITE):m dampers and brakes. electromechanical subsystems. Proceedings of the 1999

In applications where maximum power draw is SPIE Smart Structures and Materials Conference, 3674:constrained, stored electrical energy is limited, or 308-328.weight is constrained, it is also necessary to compare BEl Kimco Magnetics Division. Voice Coil Actuators: Anthe relative efficiency of each of the actuation Applications Guide. BEl Sensors and Systems Company.approaches. To make a fair comparison of efficiency Cobb R, Sullivan], Das A et al. (1999), Vibration isolationbetween actuation materials, it is necessary to include and suppression system for precision payloads in space.control and power electronics and energy storage. Smart MaterIals and Structures 8: 798-812.Th' b 1· t d t· . th Fenn R, Downer ], Bushko D et al. (1996) Terfenol-DIS ecomes a comp Ica e ques lOn, smce e .....h' fl' '11 d . 11 ff dnven flaps for helicopter vibratIOn reductIOn. Smart

c olce 0 power e ect~~mcs WI . rastlC~ y a ect Materials and Structures 5: 49-57.the power or energy ~ffl~lency metnc ~hat IS select~d. Fuller C, Elliot S and Nelson P (1996) Active Control ofIf the actuator that IS mcorporated mto the active Vibration. San Diego, CA: Academic Press.isolation system is considered in terms of its closed- Maciejowski] (1989) Multivariable Feedback Design.loop electrical impedance, the real part of the impe- Addison-Wesley.dance is directly related to the mechanical work and Ross CF (1980) Active control of sound. PhD thesis,mechanical losses that are associated with the actua- University of Cambridge, England.tor. This represents the minimum amount of energy Sievers Land von Flotow A. Comparison and extensions ofnecessary to accomplish active isolation, and the ideal control metho~s for .narrowband .disturbance rejection.amplifier would supply this energy with 100% effi- IEEE TransactIOns SIgnal Processing 40: 2377-2391.. T k· th 1 f· . t Warkentin, D (1995) Power amplification for piezoelectricClency. a mg e examp eo a plezoceramlC ac ua- ..

h 1 d 1 1·

1·

d.

h· hI actuators III controlled structures. PhD thesIs, Massa-

tor, t e c ose - oop e ectnca Impe ance IS Ig Y h I· t f T h I... c usetts nstItu e 0 ec no ogy.cap~cltlve and ~~nta.ms a lar?e c~~plex com~~nent. Widrow B and Stearns SD (1985) Adaptive SignalA Imear amplIfIer IS very meffIclent at dnvmg a Processing. Englewood Cliffs, N]: Prentice-Hall.capacitive load, so the total system weight using this Williams D and Haddad W (1997) Active suspensionapproach would have to include sufficient batteries control to improve vehicle ride and handling. Vehicleand passive or active cooling to allow for inefficiency. System Dynamics 28: 1-24.

D Inman, Virginia Polytechnic Institute and State Vibration suppression is a constant problem in theUniversity, Blacksburg, VA, USA design of most machines and structures. Typically

vibration reduction is performed by redesign. Rede-Copyright © 2001 Academic Press sign consists of adjusting mass and stiffness values ordoi:10.1006/rwvb.2001.0192 adding passive damping in an attempt to reduce

vibration levels to acceptable values. Vibration isola-tion, vibration absorbers and constrained layerdamping treatments are all traditional methods ofcontrolling vibration levels by passive means. Indeed,if passive redesign or add-on techniques allow desiredvibration levels to be met, then a passive approach tovibration suppression should be used. However, ifpassive techniques cannot achieve desired vibrationlevels within design and operational constraints thenan active control method should be attempted asaddressed in this article.

Active control consists of adding an applied forceto the machine, part or structure under considerationin a known way to improve the response of a system:in this case to suppress vibrations. Control conceptsfor mechanical systems originated with radar workduring World War II and the subject has developed itsown special jargon. The object to be controlled(machine, part or structure) is often called the·plant'. Control methods for linear plants (the onlytype considered here) can be divided into two maincategories: frequency domain methods (also calledclassical control) and state space methods (also calledmodern control). Control methods are further classi-fied as open-loop or closed-loop. In an open-loopcontrol, the control force is independent of theresponse of the system, while in a closed-loop systemthe control force applied to the system (called theinput) depends directly on the response of the system.dosed-loop control can be further divided into feed-forward control or feedback control. Feedforwardcontrol is most often used in acoustic and waveapplications while feedback is most often used invibration suppression. Here we focus on closed-loopfeedback control.

Closed-loop feedback control consists of measur-ing the output or response of the system and using thismeasurement to add to the input (control force) to thesystem. In this way the control input is a function ofthe output called closed-loop control. There are anriety of methods for choosing the control force andseveral common methods are presented here at anintroductory level.

The last expression guides the designer for the choiceof the gain matrix (G) and the placement of thesensors (C) and the actuators (B). Here P and Aare known from the open-loop state matrix, and ADis given by the desired modal information. For full-state feedback, the matrices C and B may be taken asthe identity matrix and eqn [28] yields a simplesolution for the gain matrix that will provide aclosed-loop system with exactly the desired dampingratios and natural frequencies.

The calculation for the gain matrix looks straight-forward, but solving eqn [28] for G can be difficultunless Band G are nonsingular and well conditioned(requiring full-state feedback). There are a variety ofpole placement algorithms each solving specializedproblems and dealing with the difficulty of using asmall number of sensors and actuators (B and Csingular, or output feedback). Another difficulty ispartial pole placement. This refers to the case where itis desired to assign only a few of the poles and to leavethe other poles alone, as is often the case in design. Itshould also be pointed out that the larger the differ-ence between the open-loop poles, A, and the desiredpoles, AD, the more gain and hence the more powerand force required by the controller.

A more radical version of pole placement is calledeigenstructure assignment. Eigenstructure assignmentrefers to a control law that aims to produce a closed-loop system with both desired eigenvalues (poles ornatural frequencies and damping ratios) and desiredeigenvectors (mode shapes). The calculation for thegain matrix G to perform eigenstructure assignment isnot presented here. Rather it is important to note thatone cannot exactly specify any desired mode shape forthe closed-loop system, but rather the assigned modeshapes are somewhat determined by the nature of theoriginal state matrix, A. Thus, while one can comple-tely specify desired natural frequencies and dampingratios and compute a closed-loop control providingthese specified poles, one cannot do the same fordesired mode shapes. Rather, the desired mode shapesmust be modified according to the original systemdynamics. This is not surprising to structural engineerswho have worked in model updating as it is wellknown there that the analytical model and the testdata must be fairly close before updating is successful.

Optimal ControlThe question of whether or not there is a best controlsystem is somewhat answered by methods of optimalcontrol. Optimal control is a closed-loop methoddevised using variational methods to find a controllaw u(t) that minimizes a quadratic 'cost function'containing the response of ·the system. One simpleapproach to optimal control is to compute the con-trol, u(t) that minimizes the square of the response,x(t), subject to the constraint that the response satis-fies the equation of motion. This constrained optimi-zation problem produces controls that areindependent of the modal model of the structure.The optimization procedure also allows one toinclude the control force as part of the cost function.In this case the cost function becomes:

Here the matrices Q and R are weighting matricesthat are chosen by the designer to balance the effect ofminimizing the response with that of minimizing thecontrol effort. Optimal control with this sort of costfunction is referred to as the linear quadratic regula-tor (LQR) problem. Specifically the LQR problem isto minimize eqn [29] subject to eqn [10] and initialconditions. If full-state feedback is used (that is ifu = -Gx), then the solution to the LQR problemleads to:

which must be solved numerically. Steady-state solu-tions are found by assuming the derivative of S(t) tobe zero and that the system is controllable (moreprecisely: that pair A, B are controllable). The restric-tions are that the matrix R must be symmetric andpositive definite and that the matrix Q must be sym-metric and positive semidefinite. Fortunately thereare several numerical algorithms for solving thesteady-state-matrix Riccati equation.

Optimal control is the only control methodologydiscussed here that allows the designer some clearmethod of restricting the amount of control energyexpended while designing the controller. The other

methods, such as pole placement, can potentially leadto solutions that call for more control effort than ispractically available. The R term in the LQR costfunction allows the excessive use of control energy tobe penalized during the optimization. The designeradjusts Q to reduce the vibration to acceptable levelsand adjusts R to reduce the control effort to meetavailable control forces. Commercial codes are avail-able for computing the optimal control given theequations of motion, sensor locations and actuatorlocations.

CompensatorsCompensation is a technique of essentially addingdynamics to a system to compensate for its open-loop performance. The idea is very similar to that of apassive vibration absorber. In the case of a vibrationabsorber an additional spring-mass system is added tothe plant. This additional degree-of-freedom is usedto absorb a harmonic input, causing the base systemor plant to remain free of vibration. In control theory,the compensator, as it is called, adds dynamics to asystem to change its performance to obtain a moredesirable response.

A good example of the difference between statefeedback and compensation is to consider the vibra-tion isolation design. It is well known that isolatorsare designed by choosing the stiffness to be lowenough to avoid the resonance peak yet high enoughto provide low static deflection. Increased damping,which lowers the peak, unfortunately raises the mag-nitude in the region of isolation. Thus the designer isoften left with the dilemma of adding damping toimprove shock isolation and at the same time redu-cing the performance of the vibration (steady-state)isolation. If state feedback is used in an attempt toprovide an active isolator that provides good perfor-mance across the entire spectrum, the same difficultyarises because state feedback for a single-degree-of-freedom system can only change damping and stiff-ness. To solve this problem, new dynamics must beadded to the system. These new dynamics are effec-tively a compensator.

One popular example of compensation in struc-tural control is called positive position feedback(PPF). To illustrate the formulation consider thesingle-degree-of-freedom system (or alternately, asingle mode of the system):

Notice that the stability condition only depends onthe natural frequency of the structure, and not on thedamping or mode shapes. This is significant in prac-tice because when building an experiment, the fre-quencies of the structure are usually available with areasonable accuracy while mode shapes and dampingratios are much less reliable.

This stability property is also important because itcan be applied to an entire structure eliminating spil-lover by rolling off at higher frequencies. That is, thefrequency response of the PPF controller has thecharacteristics of a low-pass filter. The transfer func-tion of the controller is:

illustrating that it rolls off quickly at high frequencies.Thus the approach is well suited to controlling amode of a structure with frequencies that are wellseparated, as the controller is insensitive to the un-modeled high frequency dynamics. Thus if cast in

state space, the term b2n-k in eqn [25] is zero and nospillover results. In many ways, the PPF active feed-back control scheme is much like adding a vibrationabsorber that targets each mode of interest.

Stability and RobustnessTwo important considerations in designing a controlsystem are those of stability and robustness. Most ofthe structures and machines that mechanical engi-neers deal with are open-loop stable. However, add-ing control to such systems has the potential to makethe closed-loop system unstable. This usually occursbecause the physical parameters in the system and/orthe disturbance forces are not known exactly. Thisuncertainty leads in a natural way to the concept ofstability and performance robustness. As an exampleof robustness, consider the simple suspension designthat produces a damping ratio resulting in a smoothride and not too much static deflection. Now if thevehicle is overloaded, then the damping ratiodecreases and the static deflection increases, bothserving to ruin the performance (ride). The amountthe vehicle can be loaded (mass change) before accep-table performance is lost is a measure of the perfor-mance robustness of the design.

Stability robustness refers to how much the para-meters of the system may be changed before stability islost. For instance, suppose a control system isdesigned to reduce damping in the system in order toprovide a really fast responding machine. The effec-tive control reduces damping based on the amount ofnatural damping in the system. However, if thatamount of natural damping is greatly overestimated,it is possible to design a controller that will subtracttoo much damping and render the closed-loop systemunstable. The amount of damping in the originalsystem then becomes a measure of stability robust-ness. Stability robustness indicates the amount oferror that can be tolerated in the system's parametersbefore the system is in danger of losing stability.

As an example of one robustness condition, con-sider a control law that regulates the differencebetween the closed-loop response and the desiredresponse of a system. A good measure of performanceof this control system is then whether or not thesteady-state error is zero. The system is said to berobust if there exists a control that regulates thesystem with zero steady-state error when subjectedto variations in the state matrix A, the control inputmatrix B, or the sensor matrix C.

An example of a stability robustness conditionconsider the closed-loop system with full-state feed-back defined by:

doubtful that the actuator will be capable of control-ling transient vibrations that are over in less then asecond. The key elements in control, beyond thetheoretical methods discussed here, are force, stroke,time constant, and location of actuators and sensors.These represent the hardware issues that cannot beignored in any successful application of feedbackcontrol for vibration suppression of structures andmachines.

Another limitation of control design is that ofmodel order. Most control designs are straightfor-ward to compute for plants that are of low order (sayless than five or six degrees-of-freedom). However, inorder to obtain good models of the plant, larger-orderfinite element models are often used. Hence manycontrol designs focus on ways to control a reduced-order model without losing performance or stabilitythrough spillover. Several methods for dealing withcontrol in the presence of model reduction exist andothers are still being researched.

The question of control when the plant or actuatorshave significant nonlinear effects has also beenaddressed in the literature but is not summarizedhere. Nonlinear behavior forms a constant concernand several methods exist for controlling structureswith nonlinear deformations, rigid body motions andnonlinear damping elements. Just as nonlinear struc-tural analysis is very case dependent, so is nonlinearcontrol.

Limitations. See also: Active control of civil structures; Active con-

Numerous other control methods eXIst that may be trol of vehicle vibration; Absorbers, active; Dampingadapted to the structural control problem. However, models; Vibration isolation, applications and criteria;one item often ignored in the literature and in design Vibration isolation theory.is the need to match the actuation and sensingrequirements of the control law to physically avail- •able hardware. In particular, the force and stroke and Further Readingtime constant of each actuator must be matched with Clark RL, Saunders WR, Gibbs GP (1998) Adaptivethe force, stroke and time constant required of the Structures: Dynamics and Control. New York: Wiley.structure to be controlled. For instance if the actuator Fuller CR, Elliot 5J, Nelson PA (1996) Active Control oftakes of the order of a second to respond, then it is Vibration. London: Academic Press.

Inman DJ (1989) Vibration with Control Measurement and Skelton RE (1988) Dynamic Systems and Control. NewStability. Englewood Cliffs: Prentice Hall. York: Wiley.

Meirovitch L (1990) Dynamics and Control of Structures. Soong IT (1990) Active Structural Control: Theory andNew York: Wiley. Practice. Harlow: Longman and New York: Wiley.

V Giurgiutiu, University of South Carolina, Columbia, inspired materials by addressing the goal as creatingSC, USA material systems with intelligence and life features

C . ht © 2001 A d . P integrated in the microstructure of the material sys-opyng c ca emlc resstem to reduce mass and energy and produce adaptive

doi:1 0.1 006/rwvb.2001.0197 functionali ty.. It is important to note that the science paradigm

Introduction does not define the type of materials to be utilized. Itdoes not even state definitively that there are sensors,

The subject of smart structures and induced-strain actuators, and controls, but instead describes a phi-actuators has attracted considerable attention in losophy of design. Biological systems are the result ofrecent years. Smart structures offer the opportunity a continuous process of optimization taking placeto create engineered material systems that are over millennia. Their basic characteristics of effi-empowered with sensing, actuation, and artificial ciency, functionality, precision, self-repair, and dur-intelligence features. The induced-strain active mate- ability continue to fascinate scientists and engineersrials actuators are the enabling technology that makes alike. Smart structures have evolved by biomimesis;adaptive vibration control of smart structures realiz- they aim at creating man-made structures withable in an optimal way. Some generic concepts about embedded sensing, actuation, and control capabil-smart structures and induced-strain active materials ities. Figure 1 shows how the modern engineeractuators will be given first. Then, details will be might try to duplicate nature's functionalities withpresented about piezo-electric, electrostrictive, mag- man-made material systems: composite materials tonetostrictive, and shape memory alloy materials that replicate the biological skeleton; piezo and opticalare used in the construction of induced-strain actua- sensors to duplicate the five senses; piezo and shapetors. Guidelines for the effective design and construc- memory alloy actuators to replicate the fast twitchtion of induced-strain actuation solutions will be and slow twitch muscles; artificial intelligence net-provided. Details about sensory, actuatory, and adap- works to mimic the motor control system, etc. Suchtive smart structures will be provided, together with innovative developments have been spurred by thesome examples. Conclusions and directions for revolutionary emergence of commercially availablefurther work are given last. smart active materials and their sensing and actuation

derivatives.Smart Structures: Concepts

The discipline of adaptive materials and smart struc- Active Materials Actuatorstures, recently coined as adaptronics, is an emergingengineering field with multiple defining paradigms. The term 'smart materials' incorporates a large vari-However, two definitions are prevalent. The first ety of revolutionary material systems that exhibitdefinition is based upon a technology paradigm: sensing and actuation properties similar to that of'the integration of actuators, sensors, and controls the living world. Of these, some smart materials maywith a material or structural component'. Multifunc- have only sensing properties, others may exhibit bothtional elements form a complete regulator circuit sensing and actuation. An example of the former areresulting in a novel structure displaying reduced the optical fiber sensors, and composite materialscomplexity, low weight, high functional density, as incorporating such fibers in their fibrous structure.well as economic efficiency. This definition describes Of the latter, a most obvious example is that of piezo-the components of an adaptive material system, but electric ceramics that can both sense and createdoes not state a goal or objective of the system. The mechanical strain. We will not cover the smart sen-other definition is based upon a science paradigm, sing materials here. Concentrating our attention onand attempts to capture the essence of biologically actuating smart materials, we notice that their list is

prime mover can impart a large stroke, direct actua- a kilometer of hydraulic lines spanning its body, fromtion can be effected (e.g., a hydraulic ram). When the the engines to the most remote wing tip. Such aprime mover has small stroke, but high frequency network of vulnerable hydraulic piping can presentbandwidth, a switching principle is employed to a major safety liability, under both civilian and mili-produce continuous motion through the addition of tary operation. In ground transportation, similarswitched incremental steps (e.g., some electric considerations have spurred automobile designers tomotors). Like everywhere in engineering, the quest promote the 'brake-by-wire' concept that is scheduledfor simpler, more reliable, more powerful, easier to to enter the commercial market in the next few years.maintain, and cheaper actuators is continuously on. In some other applications, the use of conventionalIn this respect, the use of active-materials solid-state actuation is simply not an option. For example, theinduced-strain actuators has recently seen a signifi- actuation of an aerodynamic servo-tab at the tip of acant increase. Initially developed for high-frequency, rotating blade, such as in helicopter applications,low-displacement acoustic applications, these revolu- cannot be achieved through conventional hydraulictionary actuators are currently expanding their field or electric methods due to the prohibitive high-gof application into many other areas of mechanical. centrifugal force field environment generated duringand aerospace design. Compact and reliable, induced- the blade rotation.strain actuators directly transform input electrical At present, electro-mechanical actuation thatenergy into output mechanical energy. One applica- directly converts electrical energy into mechanicaltion area in which solid-state induced-strain devices energy is increasingly preferred in several industrialhave a very promising future is that of translational applications. The most widely used high-power elec-actuation for vibration and aeroelastic control. At tro-mechanical actuators are the electric motors.present, the translational actuation market is domi- However, they can deliver only rotary motion andnated by hydraulic and pneumatic pressure cylinders, need to utilize gearboxes and rotary-to-translationaland by electromagnetic solenoids and shakers. conversion mechanisms to achieve translational

Hydraulic and pneumatic cylinders offer reliable motion. This route is cumbersome, leads to addi-performance, with high force and large displacement tiona I weight, and has low-frequency bandwidth.capabilities. When equipped with servovalves, Direct conversion of electrical energy into transla-hydraulic cylinders can deliver variable stroke output. tiona I force and motion is possible, but its practicalServovalve-controlled hydraulic devices are the implementation in the form of solenoids and electro-actuator of choice for most aerospace (Figure 2), dynamic shakers is marred by typically low-forceautomotive, and robotic applications. However, a performance. The use of solenoids or electrodynamicmajor drawback in the use of conventional hydraulic shakers to perform the actuator duty-cycle of hydrau-actuators is the need for a separate hydraulic power lic cylinders does not seem conceivable.unit equipped with large electric motors and hydrau- Solid-state induced-strain actuators offer a viablelic pumps that send the high-pressure hydraulic fluid alternative (Figure 3). Though their output displace-to the actuators through hydraulic lines. These fea- ment is relatively small, they can produce remarkablytures can be a major drawback in certain applica- high force. With well-architectured displacementtions. For example, a 300-passenger airplane has over amplification, induced-strain actuators can achieve

Figure 3 Schematic representation of a solid-state induced-strain actuated flight control system using electro-active materials,

output strokes similar to those of conventional piezo-electric behavior. The distortion of the crystalhydraulic actuators, but over much wider bandwidth. domains produces the piezo-electric effect. Lead zir-Additionally, unlike conventional hydraulic actua- conate titanate, PbZr03, is commercially known astors, solid-state induced-strain actuators do not PZT. To date, many PZT formulations exist, therequire separate hydraulic power units and long main differentiation being between 'soft' (e.g., PZThydraulic lines, and use the much more efficient 5-H) and 'hard' (e.g., PZT 8). Within the linearroute of direct electric supply to the actuator site. range, piezo-electric materials produce strains that

The development of solid-state induced-strain are proportional to the applied electric field or vol-actuators has entered the production stage, and tage. Induced strains in excess of 1000 ,ustrain (0.1%)actuation devices based on these concepts are likely have become common. These features make piezo-to reach the applications market in the next few years. electric materials very attractive for a variety ofAn increasing number of vendors are producing and sensor and actuator applications.marketing solid-state actuation devices based oninduced-strain principles.

Modeling of Piezo-electric BehaviorPiezo-electric and Electrostrictive F I' , 1, '1 h " ,• or mear plezo-e ectnc matena s, t e mteractlOnMaterials between the electrical and mechanical variables can

P.

I t· .

tybe described by linear relations (ANSI/IEEE Standard

lezo-e ec rici

Piezo-electricity (discovered in 1880 by Jacques andPierre Curie) describes the phenomenon of generatingan electric field when the material is subjected to amechanical stress (direct effect), or, conversely, gen-erating a mechanical strain in response to an appliedelectric field. Piezo-elee:tricproperties occur naturallyin some crystalline materials, e.g., quartz crystals(Si02), and Rochelle salt. The latter is a naturalferroelectric material, possessing an orientabledomain structure that aligns under an external elec-tric field and thus enhances its piezo-electric response.Piezo-electric response can also be induced by elec-trical poling of certain polycrystalline materials, suchas piezo-ceramics (Figure 4).

The application of a high poling field at elevatedtemperatures results in the alignment of the crystal-line domains; this alignment is locked in place whenthe high temperatures are removed. Subsequently, thepoled ceramics response to the application of anapplied electric field or mec~anical stress with typical

Advantages and Limitations of Piezo-electric andElectrostrictive Actuation Materials

Piezo-electric ceramics, e.g., PZT, are essentiallysmall-stroke large-force solid-state actuators withvery good high-frequency performance. However,they also display certain limitations. The mostobvious limitation is that, in many engineering appli-cations, some form of mechanical amplification isrequired. Other limitations are associated with elec-trical breakdown, depoling, Curie temperature, non-linearity, and hysteresis.

1. Electrical breakdown may happen when an elec-tric field applied in the poling direction exceedsthe dielectric strength of the material, resulting inelectrical arcing through the material and short-circuit. Electrical breakdown also destroys thepiezo-electric properties of the material.

2. Depoling may happen when an electric field isapplied opposite to the poling direction, resultingin degradation of the piezo-electric properties oreven polarization in the opposite direction. Thedepoling field (or coercive field) may be as low ashalf of the electrical breakdown field.

3. Curie temperature. At temperatures close to theCurie temperature, depoling is facilitated, agingand creep are accelerated, and the maximum safemechanical stress is decreased. For typical PZTmaterials, the Curie temperature is about 3S0°C.The operating temperature should generally be atleast SOGC lower than the Curie temperature.

4. Nonlinearity and hysteresis. Actual piezo-cera-mics are nonlinear and hysteretic (Figure 6).Hysteresis is due to internal sliding events in thepolycrystalline piezo-electric material. Upon re-moval of the electric field, remnant mechanicalstrain is observed. Hysteresis of common piezo-electric may range from 1 to 10%. Under highfrequency operation, hysteresis may generate ex-cessive heat, and loss of performance may occur ifthe Curie temperature is exceeded.

The main advantage of electrostrictive materials overpiezo-electric materials is their very low hysteresis.This could be especially beneficial in high-frequencydynamic applications, which could involve consider-able hysteresis-associated heat dissipation. The maindisadvantage of electrostrictive materials is the tem-perature dependence of their properties.

energizing coil and the magnetic circuit armature. Forthis fundamental reason, the power density (eitherper unit volume or per unit mass) of magnetoactiveinduced-strain actuators will always remain belowthat of their electro active counterparts.

Advantages and Limitations of Magnetostrictive Shape Memory AlloysActuators A h I f· d d· .. Inot er c ass 0 m uce -stram actuatmg matena s,Magnetostrictive materials, like Terfenol-D, are with much larger strain response but low frequencyessentially small-stroke large-force solid-state actua- bandwidth, is represented by shape memory alloys.tors that have wide frequency bandwidth. However, Shape memory alloys (SMA) materials are thermallythey also display certain limitations. The most activated ISA materials that undergo phase transfor-obvious one is that, in actuation applications, some mation when the temperature passes certain values.form of mechanical amplification is required. The. The metallurgical phases involved in this process aremain advantage of magnetoactive actuation materials the low-temperature martensite and the high-tem-over electro active materials may be found in the fact perature austenite. When phase transformationthat it is sometimes easier to create a high intensity takes place, the SMA material modifies its shape,magnetic field than a high intensity electric field. i.e., it has 'memory'. The SMA process starts withHigh electric fields require high voltages, which the material being annealed at high-temperature inraise important insulation and electric safety issues. the austenitic phase (Figure 7). In this way, a certainAccording to eqn [4], high magnetic fields could be shape is 'locked' into the material. Upon cooling therealized with lower voltages using coils with a large material transforms into the martensitic phase, andnumber of turns per unit length, through which a adopts a twinned crystallographic structure. Whenhigh-amperage current flows. mechanical deformation is applied, the twinned crys-

An important limitation of the magneto active tallographic structure switches to a skew crystallo-materials is that they cannot be easily energized in graphic structure. Strains as high as 8% can bethe two-dimensional topology, as when applied to a achieved through this de-twinning process. This pro-structural surface. This limitation stems from the cess gives the appearance of permanent plastic defor-difficulty of creating high-density magnetic fields mati on, though no actual plastic flow took place. Inwithout a closed magnetic circuit armature. For the typical actuator applications, this process is used tosame reason, the magneto active induced-strain actua- store mechanical energy by stretching SMA wires.tors will always require additional construction ele- Upon heating, the martensitic phase changes intoments besides the magneto active materials. While a austenite and the shape initially imposed by annealingbare-bones electro active actuator need not contain is recovered. In this way, the permanent deformationanything more than just the active material, the created through de-twinning of the martensitic phasebare-bones magneto active actuator always needs the is removed, and the material returns to its initial state

Figure 7 (see Plate 5). Principles of SMA materials: (A) change in crystallographic structure during cooling and heating; (8)associated component-shape changes, using a coil spring as an example.

Advantages and Limitations of SMA Actuation

The main advantage of SMA materials is their cap-ability to produce sizable (up to 8%) actuationstrains. In addition, they have inherent simplicitysince only heating (readily available through theelectric Joule effect) is needed for actuation. Mainlimitations of the SMA actuators are the poor energyconversion efficiency, and the low bandwidth of theheating/cooling process which can only achieve afew Hz, at the very best.

Effective Implementation ofInduced-strain Actuation

Design and Construction of Piezo-electric andElectrostrictive Actuators

An electro active solid-state actuator consists of astack of many layers of electroactive material (PZTor PMN) alternatively connected to the positiveand negative terminals of a high-voltage source(Figure lOA). Such a PZT or PMN stack behaveslike an electrical capacitor. When activated, the elec-troactive material expands and produces output dis-placement. The PZT or PMN stacks are constructedby two methods. In the first method, the layers ofactive material and the electrodes are mechanicallyassembled and glued together using a structural adhe-sive. The adhesive modulus (typically, 4-5 GPa) is atleast an order of magnitude lower than the modulusof the ceramic (typically, 70-90 GPa). This aspectmay lead to the stack stiffness being significantly

Figure 10 Induced strain actuator using a PZT or PMN elec-troactive stack: (A) schematic; (8) typical commercially availablecofired stack from EDO Corporation; (C) a range of Polytec PIactuators heavy-duty actuators.

lower than the stiffness of the basic ceramic material.In the second method, the ceramic layers and theelectrodes are assembled in the 'green' state. Then,they are fired together (co-fired) under a high isostatic

pressure (HIP process) in the processing oven. This stroke larger. On the other han,d, ~he electrostrictiveprocess ensures a much stiffer final product and, actuator,has,mu~h less hysteresls',l.e., less losses andhence, a better actuator performance. However, the less heatmg m high-frequency regime.processing limitations, such as oven and press size, ...ete., do not allow the application of this process to Design and Construction of Magnetostrlctlveanything else but small-size stacks. Actuators

The PZT and PMN stacks may ~e surrou~ded by a Magnetostrictive materials can be also used to pro-protective polymeric or elastomenc wrappmg. Lead duce an effective actuator. Figure 12A shows thewires protrude from the wrapping for electrical con- typical layout of a magnetostrictive actuator. It co~-nection. Steel washers, one at each end, are also sists of a Terfenol-D bar surrounded by an electncprovided for distributing the loa~ into the ,bri~tle coil and enclosed into an annular magnetic armature.ceramic material. When mounted m the a,ppbcatl?n The magnetic circuit is closed through end caps. Instructure, these stacks m~st be handle,d with ~peCla- this arrangement, the magnetic field is strongest in thelized knowledge. Prot~ctlOn from accidental Impact cylindrical inner region filled by the Terfenol-D bar.damage must be provided. Adequate structural sup- When the coil is activated, the Terfenol-D expandsport and alignment are needed. Mechanical connec- and produces output displacement. The Terfenol-Dtion to the application structure must be such that bar the coil and the magnetic armature areneither tension stresses nor bending are induced in the ass~mbled with prestress between two steel-washersstack since the active ceramic mater~al has low ten- and put inside a protective wrapping to form the basicsion strength. Hence, th~ load appbed to the stack magnetoactive induced-strain actuator. Though themust always be compressive and perfectly centered. If Terfenol-D material has been shown to be capable oftension loading is also expected, ade~uate prestres- up to 2000 J1strain, its behavior is highly nonlinear insing must be provided through spnngs or other both magnetic field response and the effect of com-means. For applications, the stack can b~ purchased pressive prestress. Manufacturers of magnetostrictiveas such (Figure lOB), or encapsulated I,nto a steel actuators optimize the internal prestress and mag-casing which provides a prestress mechamsm and the netic bias to get a quasilinear behavior in the range ofelectrical and mechanical connections (Figure 10C). 750-1000 J1m m-1. Figure 12B shows the displace-

Figure 11 compares the resp?nse of t",:,o commer- ment-magnetic field response for a typical large-cially available actuators, one Plew-electnc, the other power magnetostrictive actuator (ETREMA AA-electrostrictive. It can be seen t~at both ha~e about 140J025, 200mm long, ~lkg weight, 0.140mmthe same maximum induced-stram va~ue, as I~ can be peak-to-peak output displacement).readily verified by dividing the maximum displace-

ment by the actuator length. Both ma,terial typ~s Principles of Induced-Strain Structural Actuationdisplay quasi linear behavior. The Plew-electncactuator allows some field reversal (up to 25%, In structural applications.' the induced ISAs mustaccording to manufacturer), which make the total work in direct relation with the actuated structure,

Figure 12 (A) Schematic construction of a magnetostrictive(Terfenol-D) solid-state actuator; (B) typical response curve (ac-tuator model AA140J025-ES1, ETREMA, Inc.).

and special attention must be given to their interac-tion. The main differentiating feature between activematerials actuators and conventional actuators lies inthe amount of available displacement. The induced-strain effect present in active materials results inoutput displacements that seldom exceed 100 pm(0.1 mm). In a conventional actuator, e.g., a hydrauliccylinder, displacement of the order of several milli-meters can be easily achieved, and if more displace-ment is needed, additional hydraulic fluid could bepumped in. In contrast, an ISA has at its disposal onlythe very limited amount of displacement generated bythe induced-strain effect. This limited displacementneeds to be carefully managed, if the desired effect isto be achieved. Under reactive service loads, theinternal compressibility of the active materials actua-tor 'eats-up' part of the induced-strain displacement,and leads to reduced output displacement(Figure BA). If the external stiffness, k., is reduced,the force in the actuator is also reduced, and moredisplacement is seen at the actuator output end. For afree actuator, i.e., under no external reaction, theoutput displacement is maximum. However, noactive work is being done in this case since the force

Figure 15 (see Plate 6). Power supplies for active material actuators: (A) principle of switching power supplies for high reactiveload; (8) schematic of the supply system incorporating the switching module, current controller, pulse width modulator, and thepiezo-actuator-externalload assembly.

and a 1% internal damping. The system was assumed Smart Structuresdriven by a 1kVA amplifier, with up to 1000 Vvoltage and 1A current. As show in Figure 16, the The concept of smart structures has largely evolvedelectromechanical system incorporating the structure by biomimesis and under the influence of Asimov'sand the embedded actuator displays an electro- three laws of robotics. In the biological world, plantsmechanical resonance at 42.42 Hz. Without adaptive and animals alike react to the environment in order toexcitation, i.e., under constant voltage supply, the protect their existence, or to acquire the much neededcurrent demands are very large, the actuator experi- nourishment. For example, the sharp heat from anences a displacement peak that may lead to its open flame would instantly make someone retract hisdestruction, and the required power is excessive. hand. It would also, most probably, make the personWhen adaptive excitation was simulated, both the shout 'ouch!'. In the engineering world, smart struc-displacement and the current could be kept within tures are viewed as adaptive systems fitted withbounds, while the power requirements became more sensors, actuators, and command-control processorsmanageable and kept within the 1kVA capability of that could take automatic actions without specificthe power amplifier. human interventions.

Figure 16 Adaptive excitation of piezo-electric actuators near embedded electromechanical resonance: (A) under constantvoltage excitation, the current demands are very large, the actuator may break, and the required power is excessive; (8) withadaptive excitation, both the displacement and the current are kept within bounds, and the power becomes manageable.

SensorySmart Structures catapulting twitch is performed, the nonlinear skele-

S ·b f d . tal kinematics effects a favorable response duringensory smart structure attn utes are oun m even h· h h d· b h h . h dth . 1 t' 1 11 .. (F' 17A) w 1C t e 1stance etween t e t rown we1g t ane Slmp es smg e-ce miCro-orgamsm 19ure ..A b 'd f'tt d 'th t t t h t' t· the elbow joint decreases while the perpendiCularn ge 1 e W1 smar s ruc ure c arac ens 1CS ,h b t I' d S h ' t' b 'd distance between the muscle tendon and the Jointas een concep ua 1ze. uc a smar n ge , , .(F'g 17B) ld b t d t' , th ' mcreases, such that the overall force reqmred m the1 ure wou e expec e 0 sense e enV1r- ,onme t t d' 1 d 't 11' h t· h muscular actuator decreases as the catapultmgn , reac accor mg y, an e w a 1S appen- '

d 1' b d' 1 . 1 h h motlOn eve ops.mg y sen mg a arm slgna stat announce t at . "t th d f t d" , h' d ' tIn fact, once the we1ght has been put m motlOn, thes reng an sa e yare 1mm1Smg, an appropna e . , .t' . d d I thO 1 t t t . biCeps muscle can reduce the contractmg slgnal andac lOn1Snee e . n 1Sexamp e, a smar s ruc ure 1S ..

th t ' bl f t t· h lth 't' even stop before the end of the catapultmg motlOn. Inseen a 1Scapa eo au oma iC ea mom onng,d d t t' d f 'I t' the same time, the antagonistic triceps muscle canamage e ec lOn, an a1ure preven lOn...

start its braking action before the end pOSitlOnhas

Ad t" A t t" S rt St t been achieved, such that a smooth transition isap Ive c ua Ion ma ruc ures " ..ach1eved. Th1s remarkable feat 1S accomplishedAnother class of smart structure is that fitted with through complex adaptive control architecture, asadaptive actuation. Nature offers the ideal example depicted in Figure 18B. An engineering buildingof adaptive actuation. A pair of antagonistic muscles incorporating adaptive structural response is pre-(musculus biceps brachii and musculus triceps brahii) sented in Figure 18C. This smart structure not onlyensures exact and precise position control during the senses and processes the external stimuli, but alsomost difficult maneuvers of our arms. The skeletal takes mechanical action. When the vibration excita-muscles of the human arm are attached at a small- tion from a gust of wind or earthquake is endangeringdisplacement high-force position, well-suited for the the structural integrity through excessive resonanceinduced-strain muscle actuation (Figure 18A). As a response, action is taken in the variable stiffness

Rgure 17 Sensory smart structures: (A) single cell micro-organism viewed as a sensory smart structure; (8) smart structureconcept applied to a bridge.

members such that the structure's natural frequency is been used as force actuators and robot controls. Theyshifted and the resonance is avoided. This natural also offer vibration control potentials based on threefrequency shift achieved through stiffening or relax- important principles: (i) the three to four timesing of active structural members resembles the bra- increase in elastic modulus in the transition fromcing of one's muscles when trying to attain steadiness martensitic to austenitic phase; (ii) the creation ofin a challenging situation. Besides resonance avoid- internal stresses; and (iii) the dissipation of energyance, smart structures can also attempt to dissipate through inelastic hysteretic damping. These effectsenergy through active and passive mechanisms, or to can be practically realized either as additional com-prevent a nonlinear vibrations or aeroelastic effect ponents to be retrofitted on existing structures, or asfrom building up through vibration cancellation. hybrid composite materials containing embedded

SMA fibers.The increase in elastic modulus is used in the active

Applications of Shape Memory Alloys to Vibration t' t . (APT) 'b t' t I th d ACo t I proper Ies ullIng VI ra IOncon ro me o. sn ro SMA wires are activated by heating with electric

Because of its biocompatibility and superior resis- current or other methods, their modulus increasesrance to corrosion, shape memory alloys such as threefold from 27 GPa to 82 GPa. Depending on theNitinol have gained wide usage in the medical field structural architecture and on how much SMAas bone plates, artificial joints, orthodontic devices, material is used, this may result in a sizable changecoronary angioplasty probes, arthroscopic instru- in the effective structural stiffness, and a considerablementation, etc. In engineering, these materials have frequency shift away from an unwanted resonance.

The activation speed, which depends on the heating energy through the electromechanical interaction ofrate, is usually sufficient to achieve acceptable struc- the active materials with the host structure can betural control. The recovery, however, depends on the achieved either passively or actively. Since the activerate of cooling, and hence takes place much more material is connected to the structure undergoingslowly. vibration, the deformation of the active material

The creation of internal stresses is used in the active follows the deformation of the structure. As thestrain energy tuning (ASET) method. The activation structure deforms during vibrations, the active mate-of stretched SMA fibers can make them shrink by 4- rial takes up the strain and transforms it into an8% and thus create considerable contractile stress in oscillatory electrical field.the support structure. If the SMA fibers are placed For passive active-material vibration suppression,inside beams or plates, active frequency control can the induced electric field is used to drive currents intobe readily achieved, since the presence of inplane an external resistance thus dissipating the energycompressive stresses can considerably change the through Ohmic heating (Figure 19). In order to dis-beams' and plates' natural frequencies. One ready sipate selected frequencies, RCL tuning principles areapplication of this effect is the avoidance of critical applied. In active vibration suppression, the activespeeds during the run-up and run-down of high-speed material is used to produce vibration input in anti-shafts. phase with the external disturbance, thus resulting in

noise and vibrations cancellation. In this case, theenergy is dissipated in the heat sink of the driver-

Applications of Electro- and Magnetoactive I"f" " "Th" "I b I dM

.I t V"

b. C I

amp I Ier cIrcUIt. e actIve matena can e a so useate ria SOl ration ontro " " "to enhance the dampmg propertIes of a conventIOnalActive materials are well suited for vibration and damping material through the 'constraint layeraeroelastic control. Electroactive (PZT and PMN) damping effect'. Conventional vibration dampingand magnetoactive (Terfenol-D) actuators can be treatments utilize the dissipation properties of viscoe-used as translational actuators to replace conven- lastic materials that mainly operate in shear. Thetional devices, or as surface-bonded actuators to shear can be enhanced if, on top of the dampinginduce axial and bending strains in the host structure. layer, an extra layer of active material is added thatIn the former case, the strain induced parallel to the deforms in antiphase with the base structure. Thus,field direction is utilized, as, for example, in piezo- the damping layer between the active material layerelectric stacks. In the latter case, the strain induced and the base structure is subjected to a much largertransverse to the direction of the applied field is used. differential shear strain than in the absence of theTranslational ISAs are ideal in retrofit situations active layer.when the replacement of a conventional actuator For illustration, two current aerospace smart struc-with a 'smart' actuator is sought. Surface-bonded ture projects are discussed. One project is aimed atactuation is a completely new engineering concept, the reduction of noise and vibrations in helicopterwhich is specific to the smart structures world. rotors (Figure 20). The other project is addressing theBonded electro active actuator wafers have been buffet vibrations alleviation in a fixed wing aircraftused successfully to control the shape of deformable (Figure 21). The smart materials actuation rotormirrors. In the stack configuration, they have been technology (SMART) rotor blade program, underused for impact dot-matrix printing. Advantages over way at Boeing (Mesa), is tasked to test the feasibilityconventional electromagnetic actuators included of using active materials actuators for rotor bladeorder-of-magnitude higher printing speeds, order-of- control to reduce noise and vibrations, improve ridemagnitude lower energy consumption; and reduced qualities, and extend the service life. The conceptualnoise emissions. A tunable ultrasonic medical probe design of the SMART rotor blade program calls forcomposed of electro active elements embedded in a the simultaneous satisfaction of two important opera-polymer matrix has also been developed. Electro- and tional requirements: (i) reduction of blade vibrationmagneto-active materials can be used to enhance through in-flight rotor track and balance adjust-structural damping and reduce vibrations. Two ments; and (ii) reduction and counteraction of aero-mechanisms are available: (i) direct approach, in dynamically induced noise and vibration through anwhich the vibration energy is dissipated directly actively controlled aerodynamic surface. The firstthrough the electromechanical interaction between objective is achieved with a slow-moving trim tabthe active material and the host structure; (ii) indirect controlled through a bidirectional SMA actuatorapproach, in which the active material is used to (Figure 20). The second objective is met with a fast-enhance the damping properties of a conventional moving control flap actuated by piezo-ceramic stacksdamping treatment. The dissipation of vibration through a stroke-amplifier.

S J Elliott, The University of Southampton, Institute of Another approach to determining the coeffici~ntsSound and Vibration (ISVR), Southampton, UK of such a filter would be to make them adaptive.

Instead of using a set of data to estimate correlationCopyright © 2001 Academic Press ., h 1 1 . 1functiOns, and then usmg t ese to ca cu ate a smg edoi:10.1006/rwvb.2001.0059 set of optimal filter coefficients, the data are used

sequentially to adjust the filter coefficients graduallyso that they evolve in a direction which minimizes the

In the article on optimal filters (see Optimal filters) mean-square error. Generally, all the filter coeffi-we saw that the optimum finite impulse response cients are adjusted in response to each new set of(FIR) filter, which minimizes the mean-square error data, and the algorithms used for this adaptation usefor the model problem shown in Figure 1 of that a considerably smaller number of calculations perarticle, can be directly calculated from a knowledge sample than the total number of calculations requiredof the autocorrelation properties of the reference to compute the true optimal coefficients. As well assignal and the cross-correlation between the reference converging towards the optimal filter for stationaryand desired signal. In a practical problem, these auto- signals, an adaptive filter will also automaticallyand cross-correlation functions would have to be readjust its coefficients if the correlation propertiesestimated from the time histories of these signals, of these signals change. The adaptive filter is thuswhich would require a considerable amount of data capable of tracking the statistics of nonstationaryin order to calculate accu~ately ..It was also a~sumed signals, provided the changes in the statistics occurthat the reference and desired signals are statiOnary, slowly compared with the convergence time of thesince otherwise their correlation properties will adaptive filter.change with time. The calculation of the optimalfilter with [ coefficients involves the inversion of theIxI au~ocorrelati.on. matrix. Al~hough this m~trix has Steepest Descent Algorithma special form (it is symmetnc and Toephtz), andefficient algorithms can be used for its inversion, the The most widely used algorithm for adapting FIRcomputational burden is still proportional to [2 and digital filters is based on the fact that the error surfaceso can be significant, particularly for long filters. The for such filters has a quadratic shape, as shown inmatrix inversion may also be numerically unstable if Figure 2 of the article on optimal filters (see Optimalthe matrix is ill-conditioned. filters). This suggests that, if a filter coefficient is

J E Cooper, University of Manchester, Manchester, UK aircraft, otherwise costly redesigns will have to be

C.

h 2001 A d.

Pmade. Both the civil and military airworthiness reg-

opyng t © ca emlc ress .....ulatIOns, that dictate how aircraft must be certified to

doi:10.1006/rwvb.2001.0125 fly, have several sections devoted to loads. If newtechnologies or ideas are being applied that are not

Introduction accounte~ for in the regulations, then ~~e e?gine~rshave to discuss the way forward for certificatIOn With

Aircraft are subjected to a wide range of static (i.e., the authorities.steady) and dynamic loads in flight and also on the This item provides a brief overview of an immenseground. Combined with the inherent flexibility of the subject area, with a wealth of literature and work instructure, these dynamic loads result in vibration. the aerospace community being devoted to each ofGusts and dynamic maneuvers can cause the limit the topics covered. The problems associated with aloads to be exceeded; however, in most cases struc- number of critical phenomena are described. Thetural failure would occur due to fatigue. Considera- classical flutter phenomenon is covered in a differentcion should also be made regarding the discomfort item (see Flutter).caused by the responses to the pilot or passengers, andalso the consequent malfunctioning of equipment and Flight Loadssystems.

Care must be taken at the design stage to ensure that Flight loads include all loads that can occur in anythe static and dynamic responses stay within accept- part of the flight envelope. Typically, the most criticalable limits of deformation and load on production design cases occur in flight.

Figure 11 Tail buffet on delta wing fighter aircraft,Active Gust Alleviation Systems

A common approach to reduce gust loads, particu- fin can experience severe loads (up to 400g has beenlarly in terms of increasing passenger comfort, is to measured) due to the buffeting. The fatigue life of themake use of an active gust alleviation system. By fin can be used up in a few hours. Buffet can also besensing the response of the aircraft as it encounters caused due to airbrakes and cavities in the structure.a gust, the control surfaces may be moved to counter- Twin-fin fighter aircraft are particularly susceptibleact the effect and reduce the loads. Any flight control to buffet and there is a significant effort underwaysystem (FCS) may well perform this function anyway. worldwide to reduce/eliminate the difficulty.It should be noted that by alleviating the wing root Approaches under investigation include:bending moment, the fatigue life of the control sur-

1 Th f MEMS h'

f I', . e use 0 on t e wmg sur aces to e lm-faces and attachments are often drastIcally reduced. ,h b I fl (MEMS Imate t e tur u ent ow see , genera prop-

erties)Buffet and Buffeting d· h f' d h I '2. SMART eVlCes on t e m to re uce t e resu tmgBuffet is defined as the aerodynamic excitation due to vibrationa separated flow. It is usually random and covers a 3. Aerodynamic devices on the fin to counteract thewide frequency range, but is dependent upon the buffetinggeometry and the flight condition. In some cases 4. Aerodynamic devices upstream to influence thethe buffet can be periodic and exist at individual vortex characteristicsfrequencies.

Buffeting is the response resulting from the excita- Buffetlbuffeting of commercial aircraft Althoughtion of the buffet. The term originated in 1930 commercial aircraft do not reach such high anglesfollowing a fatal accident that was attributed to the of attack as fighter aircraft, the buffeting problem canaircraft being subjected to gust loading. These gusts still occur. In this case, passenger comfort also needsled to a sharp increase in the effective angle of to be taken into consideration. Separated flow canincidence (see section on gusts) with the consequent occur on the main wing surfaces, particularly onformation of flow separation over the wing. The tail application of wing control surfaces, and if thiswas subjected to intense vibrations (given the name impinges on the tail, then not only the tail modesbuffeting) due to the turbulent wake and this resulted may be excited but also those of the fuselage.in structural failure. Buffeting will occur on aircraft if One major difficulty with design against buffetingthe vibration modes of the tailor wing structure are is that it is still extremely difficult and impractical toexcited by the buffet. It rarely produces an instant produce a predictive model of buffet. Consequently,catastrophic failure; however, the 19ads can be severe wind tunnel model testing must be undertaken inand consequently fatigue problems can result. order to predict the buffet behavior. Once an appre-

ciation of the frequency characteristics of the buffetFin buffetlbuffeting of delta wing fighter aircraft and the geometry of the flow has been gained, it isWhen high angles of attack maneuvers are under- possible to make response predictions using the air-taken, turbulent flow detaches itself from the wings craft mathematical model (assumed mode or finiteand meets the fin (Figure 11). Although this effect is element) and to construct any design changes thatdesirable to enable lateral control of the aircraft, the may be required.

Acoustic Excitation Birdstrike

High levels of acoustic excitation due to jet efflux and All aircraft fly with the likelihood of hitting one orturbulent fluid flow have been experienced on the more birds in flight. The airworthiness regulations layskin panels of modern jet aircraft, particularly on down strength criteria that every aircraft must meet.short take-off and vertical landing (STOVL) aircraft Certification is usually met through ground testing.such as the Harrier. This high intensity noise environ- The leading edges of the wings and tailplane, as wellment is often combined with very high temperatures, as the canopy and radome, must be constructed soe.g., at some points on the structure the sound pres- that they can survive such an incident. Ingestion ofsure could be in the region of 180 dB with panel birds into jet engines is also a problem that must betemperatures over 1500 0c. There is a high likelihood considered particularly for large diameter civil air-of fatigue damage occurring in such an environment. craft engines.

Similar problems are likely to occur with the newgeneration of stealth aircraft as power-plant and Hammershockstores will b~ internal. When the c~vities have to be Jet engines cannot operate in supersonic flow condi-opened, for mstance when a store IS to be dropped, tions, therefore the intakes and ducts of supersonicextremely high noise levels will occur. Civil aircraft aircraft need to be designed so that subsonic condi-are also susceptible to such difficulti,es o~ the flaps tions occur at the engine (Figure 13). At the limits ofand rear ,fusel~ge, etc. Much work IS bel?g under- engine performance, it is possible for the airflow into~aken to mvestlgate approach~s (e.g., passl:re damp- the engine to distort, resulting in an engine surge. Thismg tech~ology, smart matenals and devIces, and surge in turn causes a pressure wave called hammer-MEMS) m order to reduce the severe problems that shock to occur in the duct. This wave advances atwill have to be overcome. high speed (typically 400 m s-1) down the duct in the

opposite direction to the airflow. The wave causes aGunfire Loads pressure in the duct of up to three times the usual

R ' , f' bl' I h steady-state pressure that in turn leads to a dynamicepetltlve gun Ire asts Impart pu ses t at can cause'b ' I I 'II 'h " response.VI ratlOn eve s typlca y tWlCe t at occurnng m nor- Th "

k d d d'dI fl' h Th ·b' d" f e engme mta e an uct structure are eSlgnema 19 t. ese VI ratlOns occur at lstmct requen- ,

, h 'F' 12 Th 'f d to wIthstand the stresses due to maneuver loads, theCles as sown m 19ure . e alrcra t structure, an , , ,h· . h' b I d

' h d steady-state operatmg pressure, and cntlcally thet e eqUlpment WIt m must e c eare to WIt stan "h 'b ' I I d' f' hammershock loads. It IS not possIble at present tot e VI ratlOn eve s ue to gun Ire., ,

predlCt the hammershock pressures, so expenmentalmeasurements on the ground and in flight are taken.

Store Release These measurements can then be used with a struc-When stores are released from military aircraft (or tural FE model to predict the resultant stresses and toeven dropping aid parcels from transport aircraft) an make any design modifications that may be necessary.impulse is imparted onto the structure and the aircraftwill respond dynamically. The designer needs to Ground Loadsensure that this response is not significant, bearingin mind that the mass distribution and aerodynamic The landing gear of an aircraft undertakes two func-characteristics have changed. Care has also to be tions: to absorb the energy due to the vertical descenttaken in the design to guarantee that the store jettisons on landing, and to enable the aircraft to maneuver onsafely rather than rebounding back onto the aircraft. the ground. A complete analysis of the behavior of the

aircraft during both phases of operation is essential sothat the undercarriage is designed to be strong enoughand also to ensure that no other component fails. Theairworthiness regulations dictate how the undercar-riage should be cleared for each aircraft.

LandingLoads

Aircraft are subjected to significant forces duringlanding, especially in the special case of landing onan aircraft carrier. The characteristics of these loadsare dependent upon the landing gear. There are anumber of variables that must be considered:

• limit descent velocity at the design landing weight Runway Loads

(typically 3 m s-1 for transports and land based These are usually defined as the loads resulting fromfighters (not trainers), 6 m s-1 for carrier based taxiing including turning, braking applied duringaircraft), . take-off (e.g., an aborted take-off) or landing, as

• undercarriage and tyre energy absorptiOn charac- well as towing, jacking and tethering (tying downteristics (nonlinear), an aircraft in a very high wind). Of particular interest

• aircraft attitude relative to the ground, are the loads on the nose undercarriage when the• distribution of the aircraft mass, brakes have been applied suddenly, causing an• lift acting on the aircraft at the impact point, increase in the vertical load on the nose gear due to• friction coefficient between the tyre and the the pitching moment of the aircraft. Similarly, the

ground, forces due to the application of the spoilers, brakes,• rotation of the aircraft on take-off. thrust reversers, brake parachute deployment and

arrestor wire engagement need to be considered.The analysis of the aircraft response should consider .the following behavior: Shimmy

In the 1930s, the air pressure in car tyres was reduced• landing gear dynamic characteristics, with the object of producing a smoother ride. This• spin up loading due to forces that accelerate the change produced an effect known as 'shimmy' that

wheel up to the ground speed, consisted of a self-excited unstable rotational oscilla-• spring back once the wheel has reached the ground tion of the front wheels about the vertical axis. The

speed and the sliding friction has gone to zero, the problem was eventually cured through independentstrain energy store? b~ the undercarriage's rear- spring suspension of the front wheels. Shimmy is n?tward bending motiOn IS released and the u~der- restricted solely to pairs of wheels on a common aXIS,carriage returns to its original orientation wIth a but has been observed on single castored wheels suchsizeable inertia load, as on supermarket trolleys and, more importantly, on

• rigid-body response of the aircraft, aircraft nose and main undercarriages where its• dynamic response of the aircraft.

A typical commercial aircraft would make a conven-tional two-point landing (nose gear does not contactthe ground until the main undercarriage has absorbedall the energy from the descent) as shown in Figure 14.However, the cases of three-wheel (nose gear criticalcase) and one-wheel landings must also be consid-ered. In each case the loads and resulting motionsmust be analyzed to ensure that structural integrity ismaintained.

occurrence can be disastrous. The coupling of the • aerodynamics - moving shocks, other transonictorsion and bending modes, analogous to flutter, aerodynamic effects,leads to an unstable motion .• control system - time delays, nonlinear control

The analysis of the shimmy problem is made very laws.complex due to the dynamics of the tyre (viscoelasticmaterial, nonlinear friction and deformation charac- As LCOs are not immediately destructive, they can beteristics). Experimental tests are performed to verify considered as a fatigue problem. However, they arethe design calculations. extremely difficult to predict as an accurate nonlinear

model is required. At present, a large amount of•• computer simulation is needed to do this. However,

Other Aircraft Aeroelastlc Phenomena as modern aircraft are becoming more flexible andClassical linear flutter is considered in Flutter. In this nonlinear, consequently, LCOs are becoming more ofsection, a number of related phenomena that have a problem. Research is currently being directed to-nonlinear response characteristics are described. wards methods that can predict LCOs without the

need for vast amounts of testing or computer simula-Umit Cycle Oscillations tion. Such methods will speed up the flight testIf an aircraft that behaves linearly is flown fast procedure ~hi~st impro:i,ng safety by giving an im-enough, flutter will occur. In practice though, aircraft proved prediCtIve capabIlity.are nonlinear in their vibration behavior and thesenonlinearities can produce phenomena known as Aeroservoelasticitylimit cycle oscillations (LCOs). LCOs are character- , , .·

d b d d . b'l' , h 'F' 15 Modern aIrcraft contam a range of actIve controllZe as oun e msta 1 ItIes, as sown m Igure , .. ,

h'

b h h 1·d

.,. 11 technology mcludmg flIght control systems (FCSs),were It can e seen t at t e amp Itu e ImtIa y . , .·

b h . Th' ff gust allevIatIOn systems, flutter suppreSSIOnsystems,mcreases ut t en stops growmg. IS e ect can .. ,.

d 1" . ,. h etc., that can combme wIth the usual aeroelastIcanse ue to non meantIes occurnng m t e: h Th FCS d '( h' h 'p enomena. e etects a motIOn w iC It• structure - freeplay, backlash, cubic stiffening, assumes is the uncommanded aircraft rigid-body

response) and tries to correct it. In the worst case,the frequency of activation commanded coincideswith the structural mode frequency being corrected,leading to a potentially unstable condition. Thisinteraction is illustrated by the 'aeroservoelastic pyr-amid' in Figure 16. The entire system including feed-back must be included in all stability calculations andthe siting of motion sensor units is critical. It is

possible for the control system to interact with the of the shock will result in a difference in aerody-structure in such a way that unstable oscillations can namic forces with a consequent change in deflectionoccur. The control system is likely to contain non- of the aerofoil, but in particular of the control sur-linearities and these can give rise to phenomenon such face itself.as LCOs. Consider the aerofoil with flap in Figure 18. In this

case, the flap is entirely in subsonic flow as the shockPanel Flutter is on the aerofoil surface. A similar type of responseWhen there is supersonic flow past a thin panel, there occurs at slightly higher speeds when the shock is onis the possibility of self-excited oscillations of the the control surface. Oscillation of the flap causes apanel normal to its plane. Consider the panel in fore-and-aft motion of the shock waves that changesFigure 17. The pressure acting on the plate is pro- the aerodynamic moments acting on the control sur-portional to the dynamic pressure and the local slope face. The resulting change in deflection of the controlof the surface. Therefore, a symmetric slope distribu- surface results in further movement of the shocktion leads to an unsymmetrical pressure loading waves. The resulting motion is amplitude limited,which tends to deform the surface of the plate into a but can result in deflections of up to ±10 degreesmore complicated shape. As the dynamic pressure causing severe problems for the pilot. The phenom-increases, the plate loses its ability to maintain equili- enon also causes undesirable vibration and fatiguebrium and a response limited LCO occurs due to damage.nonlinear stiffening effects at higher amplitudes. Thesolution to this problem is to increase the stiffness of Stall Flutterthe panel through the use of larger or more stringers/ribs. Other solutions include increasing the tension in If the angle of incidence of an airfoil exceeds its stallthe panel. Recent work has investigated the use of angle, then stall will occur. Stall flutter is a LCO-typeshape memory alloys to do this, using the temperature phenomenon that can occur in propellers, helicopterincrease associated with supersonic flight to take the rotors, compressor/turbine blades as well as aircraftalloys beyond the activation temperature. wings. It is characteristically a torsion type motion

shown in Figure 19 and consists of the followingControl Surface Buzz stages:

This is a LCO-type phenomenon that occurs in the ....transonic flight regime. At transonic speeds there will 1. th~ aero~OlI an~le of m~ldence e mcreases dyna-be a shock wave acting on the airfoil. Any movement mlCally, mcreasmg the hft;

2. the angle of attack exceeds the stalling angle,separation occurs, and the lift reduces;

3. with the lift reduced, the angle of attack reduces,until is passes the stalling angle;

4. the flow reattaches and the cycle is repeated.

Stall flutter is not immediately catastrophic, but ma_lead to fatigue problems. For structures such as aU:-craft wings, or turbine blades, the phenomenon willoccur at the tip, where the twist will be largest.

Figure 15 TDA with triggered data acquisition.

Another example occurring in practice is that of PSD, segmentation of the data (with possible over-small variations between consecutive periods. As an lapping processing) is used in order to decreaseexample, when dealing with vibrations of rotating random errors whenever random components existmachinery, speed fluctuations may be induced by in the signal. The PSDs of the segmented sections arechanges of loads and other affecting variables. averaged (see Windows; Spectral analysis, classicalLarge fluctuations can decrease and even nullify the methods).magnitude of extracted periodic components. Recursive schemes are usually implemented, and

An example is shown in Figure 16A. A random the averaging effect can often be followed in realfluctuation of 2% ( RMS) in the period is assumed. time. Both regular and exponential averaging can beAlso shown is the average of 100 periods, compared used, obviously in the frequency domain. The expo-to the case of zero fluctuation, and the resulting nential averaging is better suited to situations invol-attenuation is evident. ving nonstationarities. Figure 17 shows the PSD of a

The effect of jitter can be recognized by averaging stationary random process with an additive compo-signal sections which correspond to a multitude of the nent - a harmonic signal with time-dependentdesired period. At least two periods should be increasing frequency. Instantaneous PSD, based onextracted to notice the effect. Figure 16B shows the one segment at a time, can follow the frequencyresult of averaging 100 sections. It can be noticed that changes, but the random error of the PSD wouldthe magnitude of the first half of the averaged signal is not be controlled. In regular averagings, the 'smear-larger than that of the second one. This effect is traced ing' due to changing frequency is evident. This isto the fact that the effect of the jitter is larger for data much less pronounced in the exponential averagingpoints more distanced from the start of the averaged scheme, the decaying memory 'forgets' some of thesection. The larger the number of periods in the initial frequencies, the corresponding peak would beaveraged section, the more pronounced the effect much stronger. Commercial analyzers sometimestowards the end of the section. Obviously the price have a 'peak' averaging mode. This is not an aver-paid for increasing this section length is that more aging operation; the global maximum values of seg-data points are needed for the same number of ment's PSD is retained.averaged sections.

Extraction of Periodic Signals via Averaging in Timeand Frequency

Averaging in the Frequency Domain The time domain averaging is aimed at extractingC t t' f PSD periodic signals from a composite signal. Averaging

ompu a Ion 0 the PSD is aimed at reducing the random error of theOften there is a need to average descriptors of data in PSD estimate. However, when the frequency domainthe frequency domain. In the computations of the description of the periodic component only is sought,

Figure 16 TDA in the presence of jitter: (A) TDA with and without jitter - sections of single period averaged; (8) TDA with jitter -sections of multiple periods averaged.

then two equivalent procedures are possible. Both squaring for PSD) are performed on each segment,assume knowledge of the frequency of the signal and then averaged. Both procedures reject compo-sought, whether via hardware triggering or external nents not synchronous with the frequency soughtknowledge. Both procedures are shown in Figure 18. (whether deterministic or random). The first proce-In the first, an FFT computation follows extraction dure is slightly more efficient (one averaging vs onevia TDA. In the second, FFT operations (without FFT operation).

C J Li and K McKee, Rensselaer Polytechnic Institute, Failure ModesTroy, NY, USA

Bearing failure modes include corrosion, wear, plasticCopyright © 2001 Academic Press deformation fatigue lubrication failures electrical" ,doi:10.1006/rwvb.2001.0200 damage, fracture, and incorrect design. Out of these,

classical localized defects result from fatigue, andR II B . certain distributed defects have fairly distinct vibra-

o er earlng tion patterns which will be detailed below. However,Figure 1A illustrates the cross-section of an angular the vibration patterns of other failure modes are, incontact ball bearing. It shows the angle of contact, general, not easy to predict, let alone detect andand the diameters of outer race, rollers, inner race and diagnose at onset. While some of them do eventuallypitch circle. Figure 1B illustrates the cross-section of a lead to localized defects, others give almost no vibra-taper roller bearing. It shows the angle o'f contact, tion warning and therefore call for means other thanand the diameters of outer race (cup), inner race vibration analysis.(cone), and rollers. The primary excitations of vibra-tion are the contact forces between the rollers and the Localized Defectsraceways, and the rollers and the cage. The frequen-cies of these contacts are directly related to the roller The most common failure mode of a properlypassing frequency on the raceways, the roller spin- installed and operated rolling element bearing isning frequency, and the cage spinning frequency localized defects such as fatigue spall, which occurwhich can be theoretically estimated with the equa- when a sizable piece of material on the contact sur-tions in Table 1. face is dislodged during operation, mostly by fatigue

In general, the success of these parameters depends limited use when there are sources of shock pulseson finding a frequency band that is dominated by other than a roller bearing. Furthermore, as a timebearing ringings. Otherwise, they indicate a change in domain method, it cannot determine the location ofsomething other than the condition of the bearing. It the defect.was reported that good results can be obtained fromhigher frequency bands such as 10-20 kHz and High-frequency resonance technique (HFRT) This20-40 kHz, where the structure resonance is usual. is probably one of the best known bearing diagnosticWhile it is easy to find some band that works to some algorithms: it is also known as amplitude demodula-extent, the optimal result is difficult to obtain because tion, demodulated resonance analysis, and envelopethere is no simple way to predict a bearing structure's analysis. The technique band-pass filters a bearingnatural frequencies, and which one will be excited. signal to remove low-frequency mechanical noise and

Although trending can be used with these statistical then estimates the envelope of the filtered signal.parameters, some of them have been toted as single- (Enveloping can be accomplished with a rectifiershot diagnostic variables that have an absolute followed by a smoothing operation, or a squaringthreshold and therefore do not need historical data. operation followed by an analog or digital low-passFor example, kurtosis is said to have a value of about filter. In addition, the Hilbert transform can also be3.0 for a relatively random vibration from an unda- used if a computer is available.)maged bearing. Any value significantly higher than 3, The periodicity of the envelope signal, like the onesay 4.5, is therefore considered as a sign of a bearing shown in Figure 3, is then estimated by spectralfault. Crest factor and probability density function analysis or autocorrelation, and compared with theare sometimes considered as single-shot variables. characteristic defect frequencies. If a match is found,Realistically, these variables are just too simple to the bearing is declared to be damaged.offer single-shot diagnosis under all circumstances. Obviously, this technique will give the best result if

the passing band is chosen to include one or moreShock pulse counting This method takes advantage excited resonances. For simple machines, good resultsof transducer resonance. When a roller runs over a can often be obtained with a fixed passing band withlocalized defect, the impact produces a pressure wave. low cut-off placed between 5 and 10 kHz and highWhen this pressure wave reaches the transducer, it cut-off between 25 and 40 kHz. For complexfrequently rings the resonance of the transducer. The machines such as helicopter transmission, selectingnumber of vibration peaks above a certain threshold a passing band to include resonances and avoid gearduring a fixed length of time is then used as a damage meshing harmonics may not be easy. Varying speed,indicator. Typically, the transducer is selected to have and therefore meshing harmonics, further com-a resonance frequency between 25 and 40 kHz to pounds the problem.avoid the usual machine noise. Due to its simplicity,it is implemented in a couple of off-the-shelf bearing Synchronized averaging This algorithm can bemonitoring systems. However, there are few theore- used as a preprocessing technique to enhance thetical guidelines about setting the threshold and it has signal-to-noise ratio. The technique consists of first

is one of the most expensive algorithms in terms ofcomputation cost and data length.

Cage fault When a cage develops a weak point suchas deformation or breakage, a modulation of thewide-band bearing vibration at the cage's rotatingfrequency is common. Traditional bearing diagnosticalgorithms such as narrow-band envelope analysisare not effective in detecting this kind of wide-bandmodulation. One way to determine if a modulation ishappening at the cage rotating frequency is to per-form a synchronized average, at the cage rotatingperiod, on the envelope of the bearing vibration.The average is then examined for once-per-cage-rota-

tion phenomena. Alternatively, bandwidth-weighteddemodulation can be used.

Summary of localized defect diagnostic algorithmsA significant number of the afore-mentioned algo-rithms look for the distinct pattern of repeated ring-ings (Figure 2) associated with the onset and the earlystage of damage. Consequently, they suffer from thelimitation that they will fall back to the undamagedlevels as localized defects spread and the distinctpattern fades away as rollers strike different localdefects almost simultaneously all the time. In otherwords, they do not trend well while damage evolvesand this is a problem for the prognosis of bearing life.In addition, since a resonance excited at the onset of adefect may no longer be excited at a later stage, usinga band-pass filter to zoom on to a resonance can haveits downside of missing another defect-excited reso-nance later.

Distributed DefectsIn addition to localized defects, distributed defectssuch as off-size rollers and waviness of componentsare another class of bearing problems. While distrib-uted defects are not generally failures per se, theyoften lead to excessive contact forces and vibrationswhich in turn lead to premature failures. Fortunately,analysis of vibration spectra is often adequate todiagnose this kind of faults.

A bearing without any distributed defect is consid-ered first. Under a thrust load, roller forces act on theraces to produce a deflection. Figure 9 shows howroller forces act on the inner race. As the rollers andraces rotate, the forces and the deflection becomeperiodic. Thus, a vibration sensor located at the innerrace would sense vibrations at (rpi and its harmonics.Similarly, a sensor at the outer race would sensevibration at (rpo and its harmonics. Table 2 gives thefrequency associated with a number of conditionswhile the inner race is stationary.

Journal BearingFigure 10 shows the three basic parts to a journalbearing: the outer housing, the journal, and thelubricant. The outer housing is a cylindrical shaft,which contains a hollow core large enough to create aclose fit between itself and the journal. In addition toconfining the path of the journal's orbit, it providesradial support to the journal through direct contact orby aiding in the creation of the oil wedge. A smallclearance space between the outer housing and thejournal is necessary: in order to assist in the assemblyof the journal and the bearing, to provide space forthe addition of the lubricant, to accommodate ther-mal expansion of the journal, and to anticipate anyjournal misalignment. Within this small clearancespace resides the lubricant which provides the basicfunction of lubricating the journal and outer housingcontact, as well as producing the load-carrying cap-ability of the journal bearing and possibly attenuatingthe vibrations of the rotors. This load-carrying cap-ability is a result of the pressure that is developed by

the viscous effects within the thin film lubricant. Amore complex journal bearing that is commonly usedis the tilt-pad journal bearing. This type of journalbearing contains the same components as the simplebearing, with the added feature of tilt pads. Tilt padjournal bearings are used for their innate ability betterto handle rotor-dynamic instability problems; how-ever, they provide less damping than the simplebearing. Consequently, if vibrations occur due toother sources aside from the bearing, then the amountof damping provided by the bearing may arise as anIssue.

Diagnostic Algorithms

Unlike roller bearings, whose vibration has simpleand distinct patterns that can mostly be predictedfrom the geometry, journal bearings, while simple-looking, are rather complicated in their dynamics.Since journal bearings are only one part of a rotatingmachine, the analysis of the bearings must take intoconsideration vibrations caused by the flexible jour-nal, fluid and rotor dynamics, and any outsidesources of vibration acting on the system. Conse-quently, this type of analysis usually becomes com-plicated, and results in no clear vibration patternsassociated with most of the failure modes. This is whythe tools for the diagnosis of a journal bearing staggerare lacking.

'Whirl' refers to a circulating path the journal takes ing from the contact of the rotor with the bearing orwithin the bearing resulting from large vibrations of due to a temperature difference across the diameter ofthe journal. Proximity sensors can be placed at per- the journal. In the latter case, the temperature differ-pendicular positions to the orbit of the shaft. These ence is a result of the differential shearing in the oilsensors would allow the monitoring of the position of film. This phenomenon, in conjunction with thethe orbit, and notify the user of the misalignment, the system running near a critical speed of the journal,amount of clearance existing, and the range of motion can generate unstable vibrations in the bearing.of the journal. The user would be warned if the Under normal, low journal velocity conditions, themotion of the journal exceeds a certain allowable journal resides at an equilibrium position that isenvelope of motion. determined by its velocity. However, as the speed

An accelerometer can be placed on the outer hous- increases and approaches a threshold speed ofing to measure the frequency and amplitude of the instability, the journal's stability becomes compro-housing vibrations which do not have a straightfor- mised. Speeds above the threshold speed cause a self-ward relationship with journal whirl. Based on the excited oscillation to occur, during which the whir-vibrations taken from the bearing's normal condi- ling motion of the journal is increased by its owntions, abnormal conditions can be detected. For rotational energy. This is dangerous if the oscillationsexample, in cases where the bearing's housing is are of large magnitude. If the journal suddenlysupported on springs rather than secured directly to becomes unstable, this is called subcritical bifurca-an immobile object, chaotic motion is found at inter- tion. On the other hand, if the journal graduallymediate speed ranges, and disappears at low and high becomes unstable, it is called supercritical bifurca-speeds. At low and high speeds, different distinct tion. Subcritical bifurcation is also possible under thesubharmonic frequency components in the X- and threshold speed when the rotor is given small pertur-¥-direction of the bearing are excited. However, bations by an outside force. Consequently, factorsduring intermediate speeds, there is a rich spectrum such as constant and imbalance loads on the journalof excited frequencies in both directions, which are considered important when preventing bifurca-results in vibrations with comparatively large ampli- tion.tudes that may induce fatigue failure. Vibrations can also occur from the lack of an oil

Misalignment occurs when the journal's centerline wedge. Oil wedges, which are responsible for thedoes not coincide with the bearing's centerline. This is load-carrying capability of the journal bearing, cancaused by combinations of rotational movements be prevented from forming if the load is too heavy,about a pivot point in the longitudinal cross-section the journal speed is too slow, or there is a lack ofand translational movements of the journal in the lubrication. In all three cases, metal-to-metal contactvertical and horizontal axis in the radial cross-sec- occurs, thus causing vibrations in the bearing.tion. Misalignment results from assembling or man- Seizure, which results in a complete halt of theufacturing errors, off-centric loads, shaft deflection journal's movement, is a serious common problem.such as elastic and thermal distortions, and externally This mode of failure can result from the lack of an oilimposed misaligned moments. A side-effect of the wedge (dry rubbing) which leads to highly localizedmisalignment is the creation of a converging wedge heating, inadequate heat release from the system, andgeometry, known as the oil wedge, between the thermal expansion of the journal. In the third case,journal and the outer housing. In addition, misalign- the journal may thermally expand faster than thement contributes to the whirl by changing the thresh- bearing housing, causing the clearance between theold speed at which instability occurs. It is able to journal and the bearing to disappear and metal-to-change the journal bearing's load-carrying capability, metal contact to occur. In the case of tilt pad bearings,increase its frictional power loss, alter the fluid film thermal expansion of the tilt pads can produce thethickness, change the dynamic characteristics such as same phenomenon.system damping and critical speeds, and modify thevibrations as well as the overall stability of the See also: Balancing; Diagnostics and condition mon-system. One of the most significant effects of mis- itoring, basic concepts.alignment is its ability to produce a substantialamount of vibration when the frequency of the rotor's •vibrations is a harmonic of the rotational speed of the Further Readingjournal.

Hot spots found on a journal, which ultimately ArumugamP, SwarnamaniSand Prabhu BS(1997) Effectsdevelop into thermal bends, are results of the New- of journal misalignment on the performance character-kirk effect. These hot spots are quite common, result- istics of three-lobe bearings. Wear 206: 122-129.

Braun Sand Datner B (1979) Analysis of roller/ball bearing Meeting of Mechanical Failures Prevention Group, 7-9vibration. Journal of Mechanical Design 101: 118-125. April, Virginia Beach, Virginia, pp. 53-62. Willow-

Chen CL and Yau HT (1998) Chaos in the imbalance brook, IL: Vibration Institute.response of a flexible rotor supported by oil film bearings Li C], Ma J, Hwang B and Nickerson GW (1991)with non-linear suspension. Nonlinear Dynamics 16: 71- Pattern recognition based bicoherence analysis of90. vibrations for bearing condition monitoring. In: Liu

de Jongh FM and Morton PG (1996) The synchronous TI, Meng CH and Chao NH (eds), Proceedings ofinstability of a compressor rotor due to bearing journal Symposium on Sensors, Controls and Quality Issuesdifferential heating. Journal of Engineering for Gas in Manufacturing, ASME Winter Annual Meeting, 1-6Turbines and Power-Transactions of the ASME 118: December, Atlanta, GA, PED - vol. 55, pp. 1-11.816-823. New York: ASME.

Deepak JC and Noah ST (1998) Experimental verification Limmer JD and Li C] (1997) A case study in cage faultof subcritical whirl bifurcation of a rotor supported on a detection using bandwidth-weighted demodulation. Influid film bearing. Journal of Tribology - Transactions of Burroughs CB (ed.) Proceedings of NOISE-CON 97,the ASME 120: 605-609. 15-17 June, State-College, PA, pp. 189-194. Pough-

Elkholy AH and Elshakweer A (1995) Experimental keepsie, New York: Noise Control Foundation.analysis of journal bearings. Journal of Engineering for Meyer LD, Ahlgren FF and Weichgrodt B (1980) AnGas Turbines and Power-Transactions of the ASME analytic model for ball bearing vibrations to predict117: 589-592. vibrations response to distributed defects. Journal of

Howard I (1994) A Review of Rolling Element Bearing Mechanical Design 102; 205-210.Vibration 'Detection, Diagnosis, and Prognosis'. DSTO- Monmousseau P, Filion M and Frene J (1998) TransientRR-0013. Melbourne, Australia: DSTO Aeronautical thermoelastohydrodynamic study of tilting-pad journaland Maritime Research Laboratory. bearings - application to bearing seizure. Journal of

Jang JY, Khonsari MM and Pascovici, MD (1998) Tribology - Transactions of the ASME 120: 319-324.Thermohydrodynamic seizure: Experimental and theore- Su YT and Lin SJ (1992) On initial fault detection of atical analysis. Journal of Tribology-Transactions of the tapered roller bearing: frequency domain analysis.ASME 120: 8-15. Journal of Sound and Vibration 155(1): 75-84.

Li C] and Ma J (1992) Bearing localized defect detection White MF and Chan SH (1992) The subsynchronousthrough wavelet decomposition of vibrations. In: Pusey dynamic behavior of tilting-pad journal bearings. ASMEH and Pusey S (eds), Proceedings of 46th General Journal of Tribology 114: 167-173.

R Bigret, Draney, France Sealing systems and elements connected to the rotor

C .. P playa part in guiding the rotor. Coupling betweenopynght © 2001 Academic ress ....

rotors falls mto the Imk category. ThIs artIcle de-doi:10.1006/rwvb.2001.0092 scribes the characteristics of links.

IntroductionR .. h· ·d d b 1· k Functionsotors m rotatmg mac mery are gUl e y m s.They are classified into three categories: Links must allow:

1. Fluid • Fixed-precision guidance to avoid rotor-stator2. Roller bearings contact3. Magnetic field • Minimum wear and tear to reduce machine una-

.. vailabilityForces are transmItted to nonrotatmg parts. They are R 1 f h

.. 1 f h1 f . d· d • emova 0 eat, m partlCu ar rom t e rotora resu t 0 assocIate Impe ance: M· ...f 1 k• ImmIzatlOn 0 ea s

• To links • Neutralization or removal of impurities• To sealing systems • Minimum loss through friction, for economy and• To elements connected to the rotor, such as wheels. low-level heating

Causes and Consequences of to be cleaned: introducing grains of rice to theBreakdowns machine helps loosen any deposits.

The ~riving forces and thus th~ ~trains are w~ak when Technologythe dIstances between the traIlmg and leadmg edgesof the stator and rotor blades are sufficient for a Blades are fixed by an integrated base radiating outtrickle of fluid to reorganize, beyond the journal from the edge of the disk, in various ways. Axialzone, as also happens when there are obstacles in rivets may be used. Where gas turbines work with gasthe flow. Large gaps make it possible to avoid siren at high temperature (800-1000°C), the play betweennoises, which are particularly annoying in ventilation the base and the disk permits transient differences incircuits. However, large gaps may reduce efficiency. expansion, and low levels of strain.In general, stator blades are subject to less dangerous Blades may be connected together at the top withvibration than rotor blades. shrouds which are in contact with each other. Wires

The blades of compressors and fans which function or shrouds attached to the body of the disk may alsoat low flow and under unstable conditions (pumping) be used. Such connectivities between blades mayat frequencies of several hertz are subject to signifi- prevent dangerous vibration resonance. This connec-cant strain which may lead to breakdown. tivity may not be acceptable for particularly long

A blade breaking may lead to the machine being blades, because of the strain involved.destroyed and this may even spark off a greater In some sets of blades, in particular in turboreac-catastrophe. Breakdown is caused by fatigue as a tors, masses - so-called 'candies' - are placed on theresult of large dynamic strains which are manifested platform at the base of the disks. These masses, whichas cracks, and transiently raised thermal gradients. come into contact through centrifugal force, createWhere there is instability (flutter), the strain may be friction which reduces vibration.greater than that seen with the forced rate of flow Natural frequencies and their associated naturalcaused by the harmonics of the rotation speed, and modes characterize the disk and blade set-up and eventhus the lifespan may be noticeably shorter. If part of the rotor. When the natural frequencies of the disksa broken blade falls free in the stator, it may destroy and rotor are much greater than those frequenciesother blades, and may well end in catastrophe. which are liable to provoke unacceptable resonance,

The unbalance created by a broken blade generates only those blades, whether connected or standinga rotating force. For example, for a blade of the last alone, fixed on the disks are taken into accountdisk of a low-pressure rotor, with 8.5 bar absolute when evaluating the vibratory rate of flow. Thisdownstream of a 15 MW turbine operating at situation is common in fans where only a few blades7600 rpm: in this case, the rotating force is equal to (:::;10) are soldered on to a massive rotor element.mrQ2 = 0.3 x 0.28(n.7600/30)2 = 53150N. On certain axial compressors the influence of the

The ratio of the rotating force to the static charge at disks and rotors may have a minimal effect on theright-angles to the bearings can be high: for a turbine vibration of thin blades.operating at 15 MW, it is 53150/3826 ~ 14. Dete- Hydraulic turbine wheels, pumps, and radial com-rioration in the bearings, in the sealants, and of the pressors may be extracted from the same metalliccontacts between the rotor and the stator may lead to blocks from which the blades are cut. The frequencieschaos. and natural modes are related to the total wheel

Misalignments and rotating forces from other assembly.causes - such as the receiving structure being Technological considerations are taken intodeformed, coaxiality problems, rotor shaft, cracked account when defining procedures to determine fre-rotor, instability, short axial distance between the quencies and natural modes by theoretical modelsrotor and stator as a result of differential expansion and experiments. As a manufacturer's control or to- can lead to contact between the blades and the define the limits of a study, approaches are generallystator, as a consequence of deterioration and chaos. made on isolated elements, such as a free fitted blade

The presence of water within steam, or cavitation or a nonbladed disk.within the pump and turbine, may cause erosion oft~e blades. These erosions affect the natural frequen- Pulsations and Natural Modes of a DiskCIes; they may cause harmful resonance and greaterunbalance, leading to a breakdown in the links. Figure 1 shows a naturally axially vibrating disk. ThisFlowing fluids can leave deposits on the blades, is characterized by two nodal diameters and onesuch as soot on fan blades; when these deposits nodal circle, on which the level of vibration is verycome loose they create unbalance. The blades need low.

Moving Boundaries Nonreflecting boundariesAs has been shown up to now, the boundary condi- Earthquakes are geophysical mechanical sources oftions are a major influence on the resulting vibrating vibrations. Such sources radiate energy from distantfield. If they are forced to move in certain ways, such locations in a domain that can be considered for theas in the case of longitudinally moving support, they purpose of structural analysis to be theoreticallycan form sources of vibration and sound. Such special infinite. However, estimates of vibrations generatedboundary conditions have specific effects. Typical in such large domains are difficult, especially if theymoving boundaries of technological interest are rotat- are done numerically. Ways of confining the domaining blades. This is a rather complicated motion that of analysis were sought as a result. A finite domain,sometimes involves a longitudinal motion as well, as where the vibration field is calculated, is cut out fromin airplanes and helicopters. The forces that act on the the infinite one, yet defining boundary conditionsblades enforce vibrations. This subject relates to the that replace the removed infinite space at the cut orstudy of aerodynamics (see Further reading). artificial boundary, without changing the vibration

Moving boundaries also appear in contact pro- field within the investigated domain (Figure 8).blems of elastic bodies. When two elastic bodies Nonreflecting boundaries mean local impedancecollide, their contact area varies as a function of the matching. However, this necessity can lead to acontact force. Compatibility, equilibrium, and con- more complicated formulation in order to securestitutive relations are to be satisfied at the contact nonreflecting transmission of all the components ofarea, and the definition of these begins with the stress and displacement along the whole artificialsimple models of Hertz and are supported today by boundary. This leads to an integral (nonlocal) for-numerical techniques. mulation of the boundary conditions.

Moving boundaries may be a consequence of non-mech~nic.al proce~ses. A~ ~~am~le is the defi~ition of Semidefinite systemsdomams m a meltmg/solidIfIcatiOn of a certam mate-rial. The definition of the moving melting front is In many problems the whole system is moving underStefan's problem. This varying boundary (interface) certain excitation forces. It might be a moving elasticproblem can combine with a vibration problem. elongated body. Ignoring air resistance, the boundary

however, generates a sizable out-of-plane motioncomponent and the closed orbit shown in Figure 7C.Increasing the excitation further yet magnifies thisnon planar motion which cannot be predicted by thelinear theory.

Note from Figures 7C and 7D that the nonplanarresponse forms a closed loop (periodic motion) in thenormal-binormal plane. For this particular loop, thecable completes two cycles of motion in the normaldirection for everyone cycle of motion in the binor-mal direction. This two-to-one relation in theresponse frequencies suggests the source of this inter-esting motion. Further experimental evidence revealsthat this motion is produced by two cable modes;namely the fundamental symmetric in-plane modeand the fundamental out-of-plane mode. As men-tioned above, these two modes have natural frequen-cies in a two-to-one ratio when the cable is at the firstcrossover point as in the experiment. The nonplanar Figure 8 Summary of experimental results showing the ampli-motion observed here results from a nonlinear cou- tudes of the in-plane (a2) and out-of-plane motion (a3) com-pIing of these two cable modes. The support excita- ponents as the excitation amplitu~e is vari~d. Solid (o~en).. , . symbols denote data collected while Increasing (decreasing)

tlOn resonantly dnves the m-plane mode and the m- the excitation amplitude. Reproduced with permission fromplane mode is strongly coupled to the out-of-plane Perkins NC (1992) Modal interactions in the non-linearresponsemode through the nonlinearities associated with non- of elastic cables under parametric/external excitation. Inter-linear (finite) stretching of the cable centerline, (see national Journal of Non-linear Mechanics 27(2): 233-250.

Nonlinear system identification). In particular, thereexists a two-to-one internal resonance of these twocable modes that leads to the resulting nonplanar K equilibrium curvaturemotion (see Nonlinear system resonance phenom- P(S) equilibrium cable tensionena). This internal resonance destabilizes the (linear) T timeplanar motion through a pitchfork bifurcation. This p cable mass/lengthfact is illustrated in the experimental results ofF· 8 h· h h h h 1· d f h· See Plates 11 12Igure w lC sows ow t e amp itU es 0 t e m- ' .plane displacement (a2) and the out-of-plane displa- S I B ·d C I M b N I·.... ee a so: rI ges; 0 umns; em ranes; on mearcem~nt (a3) vary with .the excita~lOn amplitude. system identification; Nonlinear system resonanceNotlCe that the planar (lmear) motlOn corresponds phenomena; Shells;Wavepropagation, Guided wavesto the straight line in this figure that begins at the in structures.origin. This planar motion ultimately loses stabilityand is replaced by a periodic nonplanar response thatis actually dominated by the out-of-plane motion Further Readingcomponent ..... Cheng SP, Perkins NC (1992) Closed-form vibration

Numerous studies similar to this have revealed a analysis of sagged cable/mass suspensions. ASME Jour-rich variety of nonlinear responses. These include nal of Applied Mechanics 59: 923-928.other classes of internal resonances including one- Choo Y, Casarella MJ (1973) A survey of analyticalto-one internal resonances and internal resonances methods for dynamic simulation of cable-body systems.involving multiple (more than two) cable modes. It is Journal of Hydronautics 7: 137-144.important to recognize that these motions develop Irvine HM (1981) Cable Structures. Cambridge, MA: MITprecisely because of the influence of nonlinear Press.stretching. Thus, they cannot be predicted usin a Irvi~e ~M, Caughey TK (1974) The linear theory of free

1· h f bl d· g vibratiOns of a suspended cable. Proceedmgs of the

mear t eory or ca e ynamlCs. R I S· fL d A341 299 315oya oClety a on on : - .Perkins NC (1992) Modal interactions in the non-linear

Nomenclature response of elastic cables under parametric/externalexcitation. International Journal of Non-linear Me-

EA cross-sectional stiffness chanics 27(2): 233-250.g gravity Perkins NC, Mote CD Jr. (1987) Three-dimensional

vibration of travelling elastic cables. Journal of Sound Triantafyllou MS (1984) The dynamics of taut inclinedand Vibration 114(2): 325-340. cables. Quarterly Journal of Mechanics and Applied

Simpson A (1966) Determination of the in-plane natural Mathematics 37: 421-440.frequencies of multispan transmission lines by a Triantafyllou MS (1987) Dynamics of cables and chains.transfer matrix method. lEE Proceedings 113(5): Shock and Vibration Digest 19: 3-5.870-878. Triantafyllou MS (1987) Dynamics of cables, towing

Triantafyllou MS (1984) Linear dynamics of cables and cables and mooring lines. Shock and Vibration Digestchains. Shock and Vibration Digest 16: 9-17. 23: 3-8.

R B Randall, University of New South Wales, Sydney, TerminologyAustralia

In the same way as 'cepstrum' is formed from 'spec-Copyright © 2001 Academic Press trum' by reversing the phoneme of the first syllable,doi:10.1006/rwvb.2001.0055 the original authors proposed a number of terms

which are still found in the cepstrum literature, andwhich are useful to distinguish properties or opera-

I t d t" tions associated with or carried out in the cepstrumn ro UC Ion d . Th f 1 h· h d· h·omam. e most use u, w lC are use m t ISThe cepstrum has a number of variants, definitions, section, are 'quefrency' the x-axis of the cepstrum,and realizations, but all involve a (Fourier) transform which has the units and dimensions of time, 'rahmo-of a logarithmic spectrum, and are thus a 'spectrum nics' a series of uniformly spaced components in theof a spectrum'. This is in fact the reason for the name cepstrum, and often coming from a family of harmo-'cepstrum' and a number of related terms coined, by nics in the log spectrum, and a 'lifter', the equivalentreversing the first syllable, in the original paper by of a filter, but realized by wind owing in the cepstrumBogert, Healy, and Tukey, and discussed here in the domain. Thus, a 'shortpass lifter' is analogous to asection on terminology. However, the autocorrela- lowpass filter. Other terms such as 'gamnitude' andtion function is the inverse Fourier transform of the 'saphe' are of dubious usefulness.corresponding auto spectrum and so is equally aspectrum of a. spectrum. :Vha~ really di~tinguishes Definitions and Formulaethe cepstrum IS the loganthmlc conversIOn of thespectrum before the second transform. In response The original definition of the cepstrum by Bogert,spectra, this converts the multiplicative relationship Healy, and Tukey was the 'power spectrum of thebetween the forcing function and transfer function logarithm of the power spectrum', but this has been(from force to response) into an additive one which largely superseded by the definition as the 'inverseremains in the cepstrum. This gives rise to one of the Fourier transform of the logarithm of a spectrum'. Ifmajor applications of the cepstrum. Another prop- the spectrum is a power spectrum, there are twoerty of the logarithmic conversion is that it often differences with respect to the original definition:makes families of uniformly spaced components in 1 Th d f .. h h fh h f ·1' f h . d . e secon trans orm ISmverse rat er t an or-t e spectrum, suc as ami les 0 armomcs an ..·d b d h ·d h h f· 1 ward, but smce the power spectrum ISa real, evenSl e an s, muc more eVI ent, so t at t e ma f . h· 1· d'ff . 1· I·f . bl 1 d .fy h d unctIOn, t ISon y gives a I erence m sca mg. tiStrans orm IS a e to revea an quanti t em an ..h .. Th··· f h f more loglCalto carry out an mverse transform on at elr spacmg. IS gives nse to a urt er range 0 f· f f1·· f h . ·b· 1· unctIOn 0 requency.app lCatlOns 0 t e cepstrum m VI ration ana YSlS. 2 F . h (1' d d).. ormmg t e power amp Itu e square spectrumNote, however, that the cepstrum, bemg based on f h 1 k·· ·bl d....... 0 t e resu t ma es It Irreversl e an puts moreloganthmlc conversIOns of dimensIOnless ratIOs, m . h h 1 k I 1 d 1·.... welg t on t e argest pea s. t prec u es app lCa-general gives no mformatlOn on the absolute scalmg .f· 1 All h . f ... d· h tlOns mvolvmg IIftenng m the cepstrum, followedo signa s. suc m ormatIOn IS contame m t e b f . b k h 1.. y trans ormatIOn ac to t e og spectrum.first (or zero 'quefrency') component m the cepstrum,

which is often modified, or a combination of several Moreover, the new definition can be extended to thefactors. case where the spectrum (and thus the logarithmic

sinusoid without the hyperbolic weighting, so that its Applications of the Power Cepstrumform is similar to that of the IRF. This is useful in thattechniques which have been developed to curve fit The ~ajo~ ap~lica~ion of the power c~pstru~ .inparameters to the IRF can be directly applied to the mach~ne vibratIOns ISto dete~t and quantlfy.tamlhesdifferential cepstrum. This is the second advantage of of ~m~ormly s~aced harmom~s, such as ~n~e fromthe differential cepstrum. p~nodlc added Impulses (beanng ~aults, m~ssmg t~r-

Note that for functions with minimum phase prop- bme blades, faul~y valve plate m ~ reClprocatm.gerties, which applies to FRFs for many physical compressor) and sldeban~s, such a~ anse from. amph-structures, there are no poles or zeros outside the tude ~nd phase. modulatlO.n of discrete carner fre-unit circle (the hi and di vanish) and thus there are no quenCles (faults m gears whIChmodulate the commonnegative quefrency terms in eqn [8], so that the ?earmesh ~re~u~ncy at lower fr~quencies correspond-cepstrum (and differential cepstrum) are causal. By mg to the mdlvldual gear rotatIOnal speeds).normal Hilbert transform relationships (see Hilbert F It . B .

. .. au s In earmgstransforms) this means that the real and Imagmaryparts of the corresponding Fourier transform, the log Figure 3 gives an example of the development of anamplitude and phase of the spectrum, are related by a outer race fault in a ball bearing in a high-speedHilbert transform, and only one has to be measured. gearbox driven by a gas turbine, and shows theIt also means that the complex cepstrum can be (log) spectra on the left and the cepstra on the right.obtained from the corresponding power cepstrum Even at the early stages of the fault (24 August 1981)(which is real and even) by doubling positive que- there is a dramatic change in the cepstrum, with afrency terms and setting negative quefrency terms to new series of rahmonics appearing in addition to thezero. In this case also, the phase of the spectrum does component corresponding to the shaft speed (RPM).not have to be measured or unwrapped. In addition to this detection sensitivity, the other

advantages given by the cepstrum are:

1. Since the position of the first rahmonic represents(the reciprocal of) the average harmonic spacingthroughout the whole spectrum, the value is muchmore accurate than can be obtained by measuringthe spacing between individual harmonics. Ofcourse, the same accuracy can be obtained byadjusting a finely tunable harmonic cursor on tothe spectrum pattern, but even then the cepstrummay be useful in suggesting spacings to try.

2. The fact that the shaft speed quefrency is 4.1 timesthe quefrency of the unknown component meansthat its corresponding frequency is 4.1 times theshaft speed. In this case it immediately identifiedthe source as corresponding to the outer racefrequency for a particular bearing in the gearbox(which had 10 balls and an effective ball-diameter-to-pitch-diameter ratio of 0.18).

3. Once again, because of the averaging effect acrossthe whole spectrum, the first rahmonic exhibitsmuch less variation with time than the individualharmonics in the spectrum, and thus makes a morevalid trend parameter when tracing the course ofthe fault development. Figure 4 illustrates this.The higher rahmonics are affected by a numberof artifacts and do not add much more informa-tion except to confirm the periodicity.

By way of contrast, Figure 5 compares the (log)power spectrum and its corresponding cepstrum forone of the cases in Figure 3, with the (linear) powerspectrum for the same case, and its corresponding

Figure 3 Development of a bearing outer race fault with time as manifested in the (logarithmic) spectrum and cepstrum. Note thatvariation in load affects some unrelated discrete frequency components, particularly in the range, 4 - 5 kHz.

autocorrelation function. The latter contains no in- Note that the cepstrum can only be used for bear-formation about the bearing fault; only a beat corre- ing fault diagnosis when the fault generates discretesponding to the frequency difference between the two harmonics in the spectrum. This is usually the case forhighest spectral peaks. high-speed machines, where resonances excited by

Thus the cepstrum can be useful in all three phases the fault represent a relatively low harmonic orderof condition monitoring: fault detection, diagnosis, of the ballpass frequencies involved, but is often notand prognosis. the case for slow-speed machines, where this order

Figure 4 Comparison of trend information given by two typical ball pass frequency harmonics and the corresponding first rahmo-nic in the cepstrum. BPFO, ballpass frequency, outer race.

may be in the hundreds or even thousands, and these rahmonics of 20 ms in the cepstrum. Only the firsthigh harmonics are often smeared together. It should rahmonic of the cepstrum needs to be monitored tobe noted that 'envelope analysis', where the envelope detect this pattern in the spectrum.obtained by amplitude demodulation of the band-pass-filtered signal is frequency-analyzed, can be used Faults in Gearsin either case.

Note also that the cepstra in this case have been Figure 6 illustrates a number of ways in which thescaled in terms of 'dB peak-to-peak'. Such practical power cepstrum can be useful for gear analysis. Thepoints are discussed below. degree of modulation of the gearmesh signal by each

of the meshing gears is indicated by the corresponding

F It " 'Ii b h" families of rahmonics in the cepstrum, although toau s In ur omac Ines ..separate the sidebands from low harmomcs of the

The French electrical authority (Electricite de France, shaft speeds (which may have another cause), it isEDF) has demonstrated the application of the power advisable to edit the log spectrum before calculatingcepstrum to the detection of missing blades in a steam the cepstrum, for example by only retaining that partturbine. Each missing blade gives rise to an impulse of the spectrum above half the toothmesh frequencyonce per revolution as the misdirected steam flow (but perhaps extending to several harmonics of it).interacts with the stator at the measurement point. The comparison of Figure 6B with 6A shows thatThis results in the growth of a large number of such editing considerably reduces the influence of oneharmonics of the shaft speed (50 Hz) in the mid- gear (121 Hz speed) so that the other (50 Hz speed) isfrequency range, and a corresponding growth in the dominant. However, some time later, when the

Figure 5 Effect of linear vs logarithmic amplitude scales in the power spectrum. (A) Power spectrum on linear scale (lower curve)and on logarithmic scale (upper curve). (B) Autocorrelation function (obtained from linear representation). (C) Cepstrum (obtainedfrom logarithmic representation). The circled numbers are rahmonics of 4.85 ms, which corresponds to the 206 Hz spacing of theBPFO ballpass frequency harmonics that can be seen in the logarithmic spectrum (but not the linear spectrum). This frequency is4.1 times the shaft speed.

121 Hz shaft developed some misalignment, it is seen Practical Points in Calculating the Power Cepstrumto give increased components in the cepstrum, evenwith the same editing (Figure 6C). Figure 6D shows Log amplitude spectra are normally represented on ahow liftering in the cepstrum can be used to remove dB scale, and the dB units can be retained for theone family of sidebands, allowing the other to be cepstrum (as there is no interaction with the units ofmore easily visualized. The same can be achieved by phase). As in Figure 3, the amplitude of the cepstrumsynchronous averaging, but requires a tacho signal to can be scaled in 'dB peak-to-peak' on the tacitsynchronize the averaging. assumption that the harmonic pattern is continuous

Figure 7 Obtaining cepstra from zoom spectra, by the formal definition (inverse transform of log spectrum) and as the amplitudeof the analytic signal obtained from the one-sided spectrum. (A) and (8) represent two slightly displaced zoom spectra from thesame signal. Note that the amplitude cepstra indicate the sideband spacings more clearly.

In vibration signals from gears it can be shown that number of teeth on the drive pinion of a ball mill werethe force at the mesh and the transfer function from cracked. The resonances in the transfer function arethe mesh to the measurement point, largely separate virtually unchanged, demonstrating that the entirein the cepstrum, in that the forcing function is peri- change is due to the cracked teeth affecting theodic and most of it concentrates at rahmonics corre- mesh force.sponding to the toothmesh frequency and individual Another case where the forcing function and trans-shaft speeds. Removing these with a suitable 'comb fer function are well separated in the cepstrum islifter' allows the remaining part of the log spectrum, when a structure is excited by a forcing functiondominated by the transfer function, to be reproduced with a relatively flat and smooth log spectrum suchby a forward transform. This can reveal whether as the impulse from a hammer blow. In this case theresonance peaks have changed, and thus whether force cepstrum is very short, and the higher-quefrencymeasured changes are due to changes at the source part of the response cepstrum dominated by theor in the signal transmission path. Figure 9 shows the transfer function. The poles and zeros of the FRFresults of such a manipulation in a case where a small can be extracted from this region of the cepstrum (or

Figure 9 Use of liftering in the cepstrum to separate the effects of the forcing function (gearmesh signal with and without crackedteeth) and the transfer function in the response spectra. Toothmesh rahmonics have been removed by a tailored Isinxlxl comb lifter(of which the 1Ix part is a shortpass lifter). The resulting comb-liftered spectra (displaced 5 dB for ease of comparison) indicate thatresonance frequencies are unchanged.

differential cepstrum) by curve-fitting expressions of 'phantom zeros' (as they usually are in normal modalthe form of eqn [8], using a nonlinear least-squares analysis) and are relatively insensitive to smalloptimization algorithm, or from the differential cep- changes in the pole and zero positions. This meansstrum by treating it in the same way as a free decay that, once an initial measurement has been made (orimpulse response using the Ibrahim time domain perhaps an estimate by finite element modeling),(ITD)method. Note that the latter cannot distinguish changes in the modal properties of the object can bebetween poles and zeros, as there is no absolute time tracked using response measurements only. Figure 10zero, but if measurements are made at several points, gives an example where phantom zeros determineduse can be made of the fact that the poles are global from FRF measurements on a free-free beam wereparameters while the zeros are unique to the different used in conjunction with updated poles and zeros,FRFs. It has been found that the poles and zeros extracted by curve-fitting response cepstra, to makewithin the measurement range are not sufficient in estimates of the new FRFs in a case where a milledthemselves to regenerate the FRF, as the shape is also slot in the middle of the beam had changed someaffected by unmeasured out-of-band modes. How- natural frequencies by as much as 10%. An initiallyever, these effects can be compensated for by in-band determined scaling constant was also used in this

Figure 10 Updating frequency response functions (FRFs) from response measurements obtained by impulsive excitation of afree-free beam. (A) Original measurement. (B) Measurement with a half-depth slot at midspan. Dotted lines - measured FRFs. Solidlines - FRFs reconstructed from poles and zeros extracted by curve-fitting response cepstra. The reconstructed FRF in (B) usesphantom zeros and a scaling factor obtained from the original measured FRF in (A). Note the reduction in frequency of thesymmetric (Le., odd-numbered) modes in (B).

case, as the scaling factor of the FRF cannot be Further Readingseparated in the zero quefrency value of the response

t Berther T, Davies P (1991) Condition monitoring of checkceps rum. I ... 'r 'b I 'r 34 321

Th 1 h b d b L dva ves III reClprocatlllg pumps. 1rl 0 ogy 1rans. : -

e comp ex cepstrum as een use y yon an 326o~her~ to a.id in the.invers~ filtering proce~s of recon- Boger~ BP, Healy MJR, Tukey JW (1963) The quefrencystltutmg diesel engme cylmder pressure signals from alanysis of time series for echoes: cepstrum, pseudo-external measurements, typically acceleration of the autocovariance, cross-cepstrum, and saphe cracking. Inengine block or head. Small changes in the pole/zero Proceedings of the Symposium on Time Series Analysis,positions mean that they often do not cancel each by Rosenblatt M (ed), pp. 209-243, New York: Johnother in the inverse filtering process, and the resulting Wiley.pole/zero pair disrupts the estimated pressure signal. Childers DG, Skinner DP, Kemerait RC (1977) TheShort-pass liftering in the cepstrum smooths the cepstrum: a guide to processing. Proceedings of theresult giving improved estimates. IEEE 65: 1428-1443.

, Gao Y, Randall RB (1996) Determination of frequencyresponse functions from response measurements. Part I:

Scaling the Complex Cepstrum Extraction of poles and zeros from response cepstra. PartNote from eqn [3] that the complex cepstrum has II: Regeneration of frequency response functions from

t f b th th 1 l't d d h poles and zeros. Mechanical Systems and Signal Proces-componen s rom 0 e og .amp I u e an p ase sing 10: 293-317, 319-340.of the spectrum, so the log amphtude should be scaled L RH 0 d b d· A (1982) V f t '... yon, r u a 1 se 0 ceps ra IIIill nepers (natural log of the amphtude ratIO) to agree acoustical signal analysis. ASME ]. Mech. Des. 104:with the radians of the phase function. The complex 303-306.cepstrum can then also be scaled in nepers. There are Oppenheim AV, Schafer RW (1989) Discrete Time Signal8.7 dB per neper. Processing. New Jersey: Prentice-Hall.

Randall RB (1987) Frequency Analysis, 3rd edn, Chapt. 8,Nomenclature Cepstrum Analysis. Copenhagen: Bruel and Kjaer.

Sapy G (1975) Vne application du traitement numeciqueK gain factor des signaux au diagnostic vibrato ire de panne: lan quefrency index detection des ruptures d'aubes mobiles de turbines.Q9 convolution Automatisme XX: 392-399.

Tribolet JM (1977) A new phase-unwrapping algorithm.See also: Gear diagnostics; Hilbert transforms; Signal IEEE Trans. Acoust. Speech Signal Proc. ASSP-25: 170-processing, model based methods. 177.

P J Holmes, Princeton University, Princeton, NJ, USA major tool for its analysis: symbolic dynamics. Then,using regular perturbation methods, it is shown that

Copyright © 2001 Academic Press '" .chaotic solutIOns occur m a broad class of nonlmear

doi:10.1006/rwvb.2001.0039 oscillators, including Duffing's equation, and the dif-ficulty of proving the existence of 'strange attractors' -

Introduction motions displaying sensitive dependence on initialconditions that attract almost all initial conditions -

Basic ideas and techniques from the theory of dyna- is discussed. The article ends with a brief note onmical systems are reviewed and applied to analyze and sources and types of nonlinearity likely to lead tounderstand chaotic vibrations. Single-degree-of-free- chaos, and some pointers to the (enormous) literature.dom, periodically forced, nonlinear oscillators aretreated; t~e canonical examples being the 'pendulum, A Brief Historythe Duffmg and the van der Pol equatIOns. Aftersketching some history, the key ideas of Poincare Henri Poincare's studies of celestial mechanics, inmaps and invariant manifolds are introduced, fol- particular of the three-body problem, led him to dis-lowed by a simple mathematical example (the dou- cover complex motions in deterministic Hamiltonianbling map), which illustrates deterministic chaos and a classical mechanics; he also provided the groundwork

Elishakoff I (1987) Generalized Eringen problem: influence Kerr AD (1976) On the dynamic response of a prestressedof axial force in random vibration response of simply beam. Journal of Sound and Vibration 49: 569-573.supported beam. Structural Safety 4: 255-265. Kunukkasseril VX and Arumugam M (1975) Transverse

Elishakoff I (1999) Probabilistic Theory of Structures. New vibration of constrained rods with axial force fields.York: Dover. Journal of the Acoustical Society of America 57: 89-94.

Elishakoff I, Birman V and Singer J (1984) Effect of Mujumdar PM and Suryanarayan S (1989) Nondestructiveimperfections on the vibrations of loaded structures. techniques for prediction of buckling loads - a review.Journal of Applied Mechanics 51: 191-193. Journal of the Aeronautical Society of India 41: 205-

Elishakoff I, Birman V and Singer J (1985) Influence of 223.initial imperfections on nonlinear free vibrations of Nakagiri S (1984) Structural system with uncertainties andelastic bars. Acta Mechanica 55: 191-202. stochastic FEM. JSME Transactions Series A 30: 1319-

Eringen AC (1957) Response of beams and plates to 1329.random loads. Journal of Applied Mechanics 24: 46-52. Tseng W-Y and Dugundji J (1971) Nonlinear vibrations of

Hart GC (1973) Eigenvalue uncertainty in stressed a buckled beam under harmonic excitation. Journal ofstructures. Journal of Engineering Mechanics 99: 481- Applied Mechanics 37: 467-476.494. Woinowsky-Krieger S (1950) The effect of an axial force

Hohenemser K and Prager W (1933) Dynamik der on the vibration of hinged bars. Journal of AppliedStabwerke. Berlin: Springer. Mechanics 17: 35-36.

G Robert, Samtech SA, Liege, Belgium HistoryCopyright © 2001 Academic Press Software dedicated to structural analysis firstdoi:10.1006/rwvb.2001.0010 appeared in the mid 1960s. It was initially used in

advanced technology such as aeronautical and spaceindustries. At that time, software focused on efficient

I t d t" computation rather than user-friendliness: it did notn ro UC Ion .matter that the process was tortuous provided thatA lot of engineer work is done nowadays using the result achieved were correct. Nevertheless, at thiscommercial computer software. To deal with large first stage, FEM proved its ability to predict structuralprojects, engineers construct numerical models to behavior. In some situations it appeared as a cost-simulate the real world, thanks to the performances effective alternative to testing procedure and an effi-of fast digital processors. Such models allow the cient way of reducing design cost and time. As aengineer to test choices during the design process. In result, these computational techniques spread tothe field of mechanical engineering, finite element more and more engineering sectors.analysis (FEA) is the most popular way to investigate Factors contributing to software evolution includestructural behavior without experiment. In principle, the following:finite element method (FEM) allows one to build amathematical model of a given structure and to pre-d·

t·t b h . b . 1 t t· . Analysis of More and More Complex Structures

1C I S e aVlOr y numenca compu a lOn, gIVenboundary, and initial conditions. The aim is not only to analyze structures with com-

The scope of this article is to present some FEA plex geometries, but also to analyze complex assem-commercial software. We have limited the choice of blies, made up of a large number of parts. Thesoftware because the goal is not to compare software, number of degrees of freedom may range from abut rather to give an overview of FEA software few to several millions. To solve large systems, newcapabilities. The question is not Which software algorithms have been implemented, which use paral-exists? but What does the existing software offer leI and vector processing. Also Graphical User Inter-the user? It must be noted that the software is pre- face and database systems have evolved in order tosented at a given moment in its history. It is clear that process a large amount of data. This evolution hassoftware is constantly being improved: most have a followed the improvement of graphical and digitalnew version released about once a year. processors.

Analysisof More and More ComplexSituations Transient Response

To make an accurate model of a structure, it is This is the most general dynamic response. At a givennecessary to take into account many factors influen- time, arbitrarily set to zero, initial displacements andcing its behavior. For example, to analyze the effects velocities are known at any point of the structure andof an explosion on a given structure, a model is made motion or forces are prescribed at some points. Tran-of the structure itself and the surrounding fluid as sient response results from the time integration ofwell as interaction between both. Analysis of such equations of motion. Solution techniques varysituations led software designers to develop multi- according to the range of natural modes that arephysical simulation. In addition to coupled equations excited. At low frequency, modal methods and impli-between different physical variables, FEM addresses cit time integration are generally used, whereas expli-the problem of interactions between several parts of cit time integration is better suited to high frequency.the same structure, including contact detection and A typical example of transient loading is increasingsimulation. pressure in an enclosure. At low speed, implicit

integration will give good results. At high speed, if aReduction of the Time Devoted to Design shock wave ensues, it will become necessary to use

The target of high profitability has reduced the length explicit software.of the design stage. Adaptation of FEA software to P . d· Rh' 'I' " 'b'l erlo IC esponset ISeconomIC rea Ity reqUIres Improvement m usa 1-

ity and of interoperability. That is, software must be For structures subject to periodic loading, such as aneasy to use and its integration to the design environ- unbalanced mass in rotating machines, it is of interestment must be as complete as possible. to predict the response during one of the repetitive

load cycles. Excitation may include one or moreChange of Software User Itself harmonics. In the case of single harmonics, theDuring the first years of FEA, the interaction between response is called harmonic forced response. Modalprogrammers and users was so strong that often the solution methods are used with linear systems. Tech-same people both wrote and ran the software. This niques such as equivalent linearization or finite ele-time is now passed and it must be borne in mind that ments in time are used in the presence ofthe user's background does not include a good com- nonlinearities.mand of computer science and a theoretical knowl-

R d R, an om esponseedge of numencal methods. Programmers, andprograms, must provide sufficient information both In some cases, loading is so erratic that it is betterto check input and validate output. described in a probabilistic way. A typical example is

the force profile transmitted by the suspension of aApplication Fields in D namics car.ru~ning on an uneven,road. We characterize the

Y eXCItatIOn - force or motIOn - by a power spectralFEM applies to continuous media. It allows one to density (PSD), which gives the distribution of powercompute static or dynamic equilibrium of the media in a frequency range. Solution techniques are close tosubject to initial and boundary conditions. FEM is those of the single harmonic case. Response is alsobroadly used to resolve equations related to solid and expressed in terms of PSD. From the PSD, it isfluid mechanics, heat transfer, and electromagnetism. possible to compute statistical moments of the ran-In the specific domain of vibration analysis, FEM is dom process and to derive quantities of physicalused to compute natural frequencies and correspond- meaning such as root mean square (RMS) value,ing vibration modes as well as dynamic response in standard deviation or peak value.time, possibly including nonlinearities. Differenttypes of analyses are listed below. ResponseSpectrum

M d IA I. This solution technique is widely used in earthquakeo a na YSIS , , I' b d h' fengmeenng. t IS ase on t e maxImum response 0

The prediction of natural frequencies of a structure is a Single Degree of Freedom (SDOF) oscillator. Auseful so that one can understand fundamental response spectrum is obtained by plotting frequencydynamic behavior and avoid the risk of resonance versus the maximum response of several oscillatorsduring service. Direct applications of modal analysis with different natural frequencies but the sameare, for example, computation of critical speed of damping to a given input transient signal to a givenrotating machines and noise reduction by vibration input transient signal of short duration. Responsesisolation of structures. may be expressed in terms of displacement, velocity,

or acceleration. For a real structure, the response Maintenance FEA software has a long life cycle.spectrum provides the maximum response for each Most such software has been on the market formode, according to its frequency and damping. more than 10 years. To continue to be competitiveModal responses are then accumulated using combi- over a long time, software must prove its maintain-nation rules such as Square Root of Sum of Squares ability and flexibility. Maintainability refers to the(SRSS). The result is the probable maximum response ease of bug fixing, while flexibility is the ability toof the structure. Response spectra are also known as incorporate new functionalities.shock spectra.

Training and support FEA vendors should provideinformation to support the customer. This includes an

Software Quality Assurance installation guide, user's manuals, theoretical refer-.... ences, tutorial and examples, as well as release notes.

After 1mt1ally be1~g rese~rch tools, FEA soft~are h~s Documentation must be such that information is easy~ow become an mdustn.al product. Its use IS cond1- to find. In addition, it is important to organize train-tlOned not only by techmcal aspects, but also by eco- ing sessions and to propose a hot-line service ready tonomical aspects. To play its part efficiently, software help users.must fulfill quality requirements. There are severalorganizations that have established rules and criteria The User's Sidefor software quality assurance (QA), including:

There are quality factors that must be checked by the• American National Standards Institute (ANSI). customer even if the customer has no direct influence,• Federal Aviation Administration (FAA). on them. Using the same criteria:• Institute of Electrical and Electronic Engineers

(IEEE) .... Validation The user's responsibility starts with the• InternatIOnal Standards OrgamzatlOn (ISO). choice of software. The first rule is that software must• National Agency for Finite Elements and Standards meet the user's needs. For instance, if a given problem

(NAFEMS) ... requires nonlinear analysis, the user must ensure that• Nuclear Regulatory CommIssIOn (NRC). the software includes adequate formulations. The

f h..

h.

d· user's validation is not reduced to a technical choice.

Most 0 t ese orgamzatlOns ave acqmre mterna- 1· 1 d b" h h. 1 NAFEMS b d·

h UK.. 11 It a so mc u es more su Ject1ve aspects suc as t e

tIona status. , ase m t e , IS spec1a y fd·

d.

h f dease 0 use.

de 1cate to FEA. It alms 'to promote t e sa e anreliable use of finite element technology'. Since 1983,

I.

Th k h fh· .. h d h d 1 f ntegratlOn e customer must lOa e sure t at so t-t IS orgamzatIOn as promote t e eve opment 0 .

b h k h· h d Id ·d ware wIll be correctly mtegrated m ItSworkmg enV1r-many enc mar s w 1C are use wor WI e to .l'f d FEA f onment. For example, the user should check that Itqua 1y an compare so tware ..

f 1 1 .h h f

. 1 d·

mter aces comp ete y WIt ot er so tware, mc u mgoffice tools and CAD or CAM, including possible

The Developer's Side testing programs.

This section summarizes important aspects contribut- .ing to quality that are directly under the developer's Mamtenance FEA .software may.be used through-responsibility. out a long-term project. The user IS then concerned

with the availability of new releases and compatibil-... ity. For that purpose, it is important that each soft-

ValIdatIOn Software. must provIde ~ correct ware output can be clearly identified, with releaseresponse to a correct mput .. It n:ust b~ relIable. The number, date, and type of result. Possible incompat-software vendor should mamtam a lIbrary of tests ibilities must be traced and documented. Errorrelated to all its options. Most. tests must have reports must be regularly provided. They mustaccepted references, such as analytIcal references. include possible bypass when available.

Integration The developer must ensure good inte- Training and support The fact that a software com-gration of the software with the hardware. Products putes correct results when given the correct input ismust run efficiently on various available platforms. not a guarantee of success when using it. FEA soft-Efficiency implies that algorithms should take the ware is a tool and, as such, one has to learn to use it.best advantage of computers, considering the number Moreover, it is not sufficient to learn about all theof operations to perform. possible options: the analyst has to know why and

when to use them. Appropriate training must com- Availability Network and nodelock licenses areprehensively establish the correspondence between available. There are offices and representatives inphysics and models. For advanced analysis, FEA most industrialized countries worldwide.vendors generally complete their offer by direct sup-port and by engineering services. Computer environment Cray C90/T90 (Unicos 10);

Cray ]90/YMP (Unicos 10); Cray T90 IEEE (Unicos10); Digital Alpha (Unix 4.0); Fujitsu VXNPP (uxP/

Presentation of FEA Software V); HP-PA7000 (HP-UX 10.20); HP-PA8000 (HP-As stated in the introduction, this part presents a UX 10.20); HP-PA800? (HP-UX 11.0); IBM RS6000limited number of commercial FEA software (AIX 4.1.5); Intel (Wmdows-NT 4.0); NEC SX-4packages, This information has been kindly supplied (Super UX R7.2); ?GIR4000/R5000 (Irix 6.2); S~Iby software companies which have agreed to contri- R8000/R10000 (Inx 6.2); Sun UltraSparc (Solansbute to this review. The NAFEMS maintains a more 2.6).complete list of FEA software and its internet home .page contains various links to vendors' sites and other Interfaces with other softwaresites related to FEA and numerical computation, Geometr IGES SAT (ACIS) DXF.including Computational Fluid Dynamics (CFD). Y, ,

CAD and preprocessors Catia, CADFix, Femap,ABAQUS FemGV, HyperMesh, I-DEAS, MSC/Aries, MSC/

Patran, Pro/Mesh, TrueGrid.Developer Hibbitt, Karlsson & Sorensen, Inc.,1080 Main Street, Pawtucket, RI 02860, USA. Postprocessor Catia, CADFix, Ensight, Femap,

FemGV, HyperMesh, I-DEAS, MSC/Aries, MSC/General description Advanced, general-purpose, Patran, TrueGrid.finite element system for the solution of large com- A / "Ad C M ld DADS Z k

1 bl na YSIS ams, - 0 , , encrac.p ex pro ems.

ANSYSMain features ABAQUS programs are used "h h h ld "1 h h" 1 Developer ANSYS, Inc., Southpomte, 275 Technol-t roug out t e wor to ~imu a:e t e p YSlCa ogy Drive Canonsburg PA 15317 USA.response of structures and soltd bodies to load, tem- ' , ,perature, contact, impact, and other environmental G I d .. Th ANSYS "fld" "h f h f f enera escnptlon e program is a ex-con ltiOns. T ey are state-o -t e-art so tware or " """ ,I" d I" 1 " f 1 d 1 M" ible, robust package for design analysIs and optimiza-mear an non mear ana YSlS0 arge mo e s. aJor "

k " 1 d' tiOn.mar et segments mc u e automotive, aerospace,defense, offshore, civil, power generation, consumer. " ""products and medical devices. Mam features ANSYS IScapable of makmg vanous

, types of analyses, including linear and nonlinear

F' d 1 1 structural, steady-state and transient thermal, andIrst an atest re eases 1 d f" ld 1, I " M 1 "h" ,coup e - Ie ana YSlS. nits u tip YSiCSverSiOn,

First version Version 1.0, 1978. ANSYS allows design engineers and analysts to lookat the interaction of structural, thermal, fluid flow,

Current version Version 5.8, October 1998. and electromagnetic effects on the same model.

Quality assurance Complies with requirements of First and latest releasesISO 9001 and ASME/ANSI NQA-1.

/First re ease ANSYS, 1970.

Documentation Documentation includes user's Latest release ANSYS 5.4, 1998.manual, example problems manual, theory manual,verification manual, and tutorial manual. Quality assurance Complies with requirements of

ISO 9001 and ASME/ANSI NQA-1.Service and support Unlimited technical support isprovided to customers, as well as a wide range of Documentation Documentation set includes analy-training courses, engineering services, and code cus- sis guides (one for each discipline), command andtomization. element references, theory manual, workbook, and

verification manual. User's guides on advanced topics First and latest releasesare also available.

First release COSMOS/M 1.0-1985

Service and support The customer services depart- Latest release COSMOS/M 2.0-1998ment provides service and support through multipleprograms that en~ure the customer'~ ~uccess. Services Quality assurance A strict quality assurance proce-~ange from hot,-lme support t~ tr~mmg c~urses an~ dure has been in place since 1990 (over 1800 pro-mclude consultmg and customlzatlOn services to tal- ble fall t pes sizes and options have been tested~or the ~NSYS program to meet individual engineer- an;~e: one~ ar; added continuously).mg reqUirements.

A ·1 b'I' A k f d' 'b 'd Documentation A complete set of documents isval a Iity networ 0 Istn utors ProVI es '1' d' , 'd d fI' 'd ' h USA C d avallab e, mclu mg user s gUi e, cornman re erence,Icensmg an customer support m t e , ana a, b' 1, d 1 d d 1 1B 'I W E I 1 I d' E A' d aslC ana YSlsmo u es, a vanced mo u es manua s,razl estern urope srae n la ast sla an '"A 'I' "" venflCatlOn and examples, and manuals of theory.ustra la.

C . D' , 1 Al h UNIX D" Service and support The training and consultingomputer enVlfonment Iglta p a ,lgI- , ,tal Alpha NT, HP 7000/8000 Series, HP Exemplar, dep,artment offers regular trammg classes for theIBM RS/6000 S'l' G h' S S 1 ' CRAY basIc and advanced modules of the product. The, Ilcon rap ICS, un 0 ans, , h' 1 d 'd bIntel PC (Windows NT and 95). tec mca su~port epartment provi es support y

phone, e-mail, and fax on all aspects of the program

O h d ANSYS/LS DYNA 1" to all customers with maintenance contracts.t er pro ucts -, exp lCltdynamic solver. A ·1 b'l' A 'I bl h h d' 'bval a I Ity val a e t roug Istn utors,

Int f ·th th ft retailers, and directly from SRAe.er aces WI 0 er so ware

CAD and preprocessors AutoCAD, CADDS, Computer environment PCs running Windows NT/CATIA, I-DEAS, MicroStation Modeler, Pro/Engi- 95.neer, Solid Designer, Solid Edge, Unigraphics. UNIX workstations: IBM, SGI, HP, Sun, DEe.

Postprocessor I-DEAS, PATRAN, Pro/Engineer. Other products COSMOS/Works, COSMOS/Edge,Analysis ABAQUS, ALGOR, COSMOS/M, NAS- COSMOS/M for Helix, COSMOS/M DESIGNER II,TRAN, STARDYNE, WECAN. COSMOS/M for Eureka, COSMOS/M ENGINEER,

COSMOS/Wave.COSMOS/M

Developer Structural Research & Analysis Corp., Interfaces with other software12121 Wilshire Boulevard, 7th Floor, Los Angeles, CAD d P P S l'dW k S I'd EdCA 90025 SA an re rocessors 0 I or s, 0 I ge,

,U . Helix Design System, MicroStation Modeler, lron-.. , , CAD, CADKEY, AutoCAD, Eureka Gold 97, Prof

General descnptlon A general-purpose fmlte ele- ENGINEER D 'w H h FEMAP1, k h 1 fl ' , eSlgn ave, ypermes, .ment ana YSlSpac age for structural, t erma, Uldflow, and electro magnetics analysis. Postprocessor Pro/FEM-POST (Pro/ENGINEER),

FEMAP.Main features

Analysis ANSYS, NASTRAN (MSC and UAI),Structural Linear stress, strain, displacement; fre- Patran, I-Deas.quency; buckling; nonlinear static and dynamic;dynamic response; fatigue; design optimization ana- MSC/NASTRAN

lyses. Developer MacNeal-Schwendler Corp., 815 Color-Thermal Linear and nonlinear steady-state and ado Boulevard, Los Angeles, CA 90041-1777, USA.transient thermal analysis,

General description MSC's principal product,Fluids Laminar/turbulent, incompressible, subso- MSC/NASTRAN, is the industry's leading FEA pro-nic/transonic/supersonic compressible, transient and gram and has proven its accuracy and effectivenessmuch more. over many years. It is constantly evolving to take

advantage of the newest analytical capabilities and projects range from software integration and custo-algorithms for structural analysis. mization to a complete company-wide engineering

process reVIew,Main features MSC/NASTRAN is a general-pur-pose analysis program offering a wide variety of Availability MSC/NASTRAN is available from overanalysis types. These include linear statics, normal 50 MSC offices in 17 countries plus many more fullmodes, buckling, heat transfer, dynamics, frequency service distributors and agents spread worldwide.response, transient response, random response,response spectrum analysis, and aeroelasticity. Computer environment MSC/NASTRAN is avail-Most material types can be modeled, including able for a wide range of hardware and softwarecomposites and hyperelastic materials. Advanced platforms, including PC (Windows 95, 98 and NT),features include superelements (substructuring), SGI, HP, DEC, SUN, IBM unix workstations, andcomponent mode synthesis, and DMAP (a tool kit many Supercomputer platforms.for creating custom solutions in support of proprie-taryapplications). Related products MSC/Acumen, MSC/AFEA,

In addition to analyzing structures, you can use MSC/AKUSMOD, MSC/AMS, MSC/CONSTRUCT,MSC/NASTRAN to optimize designs automatically, MSC/DropTest, MSC/DYTRAN, MSC/FATIGUE,You can optimize statics, normal modes, buckling, MSC/FlightLoads, MSC/InCheck, MSC/MVISION,transient response, frequency response, acoustics, and MSC/NVH Manager, MSC/PATRAN, MSC/Super-aeroelasticity. This can be done simultaneously, with Model, MSC/SuperForge, MSC/Ultima.both shape and sizing design variables. Weight, stress,displacement, natural frequency, and many other Interfaces with other softwareresponses can be used as either the design objective CAD d MSC/NASTRAN d f( h" h b """ d "" d) d" an preprocessors ata or-w lC can e mmlmlze or maXImIze or as eSlgn h b d d" "d" Th"" " " "" mats ave ecome a stan ar m many m ustnes. ISconstramts. In addItIOn, you can synthesIze the desIgn h I d h "I b"l" f "f CAD" , ""' " as e to t e aval a 1Ity 0 mter aces to mostobjectIve and constramts VIauser-wrItten equatIOns, "" "

k" "bl b"l" " h d" d I and preprocessors. MSC also supplIes Its own mter-ma mg pOSSI e capa 1 Itles suc as up atmg a mo e ""t t h t d MSC/NASTRAN" h I FEA faces to most major systems VIa MSC/PATRAN.o ma c est ata. ISt eon y D"" f "I bl C" P E "

th d thO " II Irect mter aces are aval a e to atla, ro ngmeer,program at oes ISautomatlCa y. E I'd CADDS 5 d U" h" k IIuc 1 , , an mgrap lCSpac ages as we

F'd I I

as all Parasolid-based and ACIS-based systems.lfst an atest re eases

P" 1 I 71 Postprocessor MSC/NASTRAN is supported byIrst re ease n 19 . " I "most commerCIa postprocessmg systems.

Latest release MSC/NASTRAN V70.5 was released A 1 "MSC/NASTRAN b b" d "th" 1998 na YSIS may e com me WIm June h I" k " h "d f. many ot er ana YSISpac ages, elt er VIa ata trans er

. , " using a standard data format, or via a number ofQualIty ~ssurance MSC/N~STRAN s mternal 9A programming options that enable direct process-to-~rogram ISfully documented m a number of publIca- process communication.tIOns.

SAMCEF

Documentation A full range of documentation is Developer SAMTECH S.A., 25 Boulevard Frereavailable in both electronic and paper versions. This Orban, 4000 Liege, Belgium.includes user manuals, theory manual, numericalmethods manual, demonstration and verification pro- General description General-purpose finite elementblem manuals, and many more, including compact package for linear and nonlinear, structural, anduser guides for most analysis topics. thermal analyses.

Service and support Primary customer support and Main features SAMCEF is composed of severaltraining is provided by local MSC offices spread analysis modules that are interconnected. For theworldwide and is backed up by a comprehensive nonlinear part, module MECANO performs static,internal system. MSC is also keen to work with cus- kinematic, or dynamic analyses on flexible mechan-tomers as part of a 'quickstart' program to make them isms modeled with nonlinear structural finiteas effective as possible, as quickly as possible. As well elements and kinematic rigid or flexible joints. Theas analysis applications, MSC's engineering service module ROTOR offers a powerful design tool

dedicated to rotating machines. All modules are Interfaces with other softwarepiloted either by menus or by a fully parameterizedcommand language. Geometry IGES.

. CAD and preprocessors CATIA, I-DEAS/MasterFirst and latest releases Series, MSC/Patran.

First release SAMCEF in 1965. Postprocessor I-DEAS/Master Series, MSC/Patran.

Latest release SAMCEF 8.0 was released in 1998. Analysis MSC/NASTRAN.

Qualityassurance Since 1990, a quality control andvalidation procedure has been integrated to the life 'Ii h" ID "t" f FEA S ftwcycle of the software. ec nlca escrlp Ion0 0 are

Documentation The complete documentation is Finite Element Modelsavailable on HTML format. This includes primer M h· Th h do °d h 0

f ' es mg e mes er IVI es t e geometry mto sev-manual, re erence manual for the commands, user s 1 11 h 11b 0 f" 1 (T bl 1)1 f . d era ce s, eac ce emg a mIte e ement a e .

manua s (one or each analysIs module), an example Th dOh 0 • d'f'problems manual. e a aptlve mes e.rpermits automatIc m? 1 ICatiOn

of the mesh accordmg to error computatiOn from a

S·

d TO .. 1 1 previous analysis. The aim is to improve the precisionervlce an support rammg seSSiOnsare regu ar y f 1 . 1 . h 0 0.

d h d d b ·do 0 resu ts (mam y stresses). Rezonmg tec lllques giveorgalllze at company ea quarters an at su Sl 1- d ·1 f h . °b' 0 0 f.

Ad d.

d. h . eta1sot e stress dlstn utiOn m some regiOns 0 an

anes. vance courses are orgalllze elt er on-sIte . t· hSAMTECH A h 1·· ·bl· h eXlSmg mes .or at . ot- me IS accessl e m eacsubsidiary. In addition, experienced engineers pro-vide consultancy and customization services. Standard elements It is considered that standard

elements are those which are required to modelizeAvailability SAMCEF is available directly from most mechanical parts. T~ey are truss~s, bea~s,SAMTECH and its subsidiaries. Networks of regio- membranes, shell~, and soltds. Many.vana?ts eXIstnal and international distributors are reselling SAM- (Table .2). Accordmg to ?eometry consIderatiOns,. theTECH products in many countries. formaltsm used to descnbe structures can be partICu-

larized. We generally distinguish between two- and

C t . t All d t UNIX three-dimensional problems. Axisymmetric struc-ompu er envlronmen pro uc s run on .. ,platforms (CRAY, DEC, HP, IBM, SGI, SUN) and on ture~ expanded in Foun.er s:ne~ and semii?filllteWindows NT (PCs). medIa are other forms of IdealtzatiOn. Dependmg on

the software, all standard elements do not necessarilyOther products exist for all formalisms.

BOSS Quattro An application manager for optimi-zation parametric statistic and correlation studies. Nonstandard elements Nonstandard elements

, " include specialized elements that are useful to expressSAMCEF Design An innovative technology pre- rigid or flexible links between degrees of freedom.and postprocessor for several FEA softwares. The list given in Table 3 is not exhaustive.

Geometric nonlinearity Geometric nonlinearity Boundary conditions Once the structure geometryconditions the amplitude of motion or deformation and materials have been described, it is necessary tothat a structure can reach during simulation. Formu- specify how the structure interacts with the externallations such as Eulerian and arbitrary Eulerian world. Prescribed conditions refer to known valuesLagrangian (ALE) are adapted to flow analysis that are imposed to a set of degrees of freedom(Table 4). (Table 11). Loads refer to any forces, whether dis-

tributed or not, that are applied to the structure

M. I Th f

" I d' , , (Table 12). Such conditions are given as functions

atena semIte e ement IscretIzatlOn process f' f d' h f, , , 0 tIme or requency accor 109 to t e type 0consIders the matenal at a macroscopIC level and I' '

h " , B' , ana YSIS.supposes t at It IScontmuous. ut mICroscopIC struc-ture of material may reveal macroscopic anisotropy(Table 5). The FEM allows one to introduce aniso-tropic properties in constitutive laws. A typical ani- Analysessotropy is the one present in composite materials. L" A I

A material is said to be nonlinear when stresses are mear naysesnot proportional to strains. Material behaviors are The linearity assumption imposes that the coefficientsdivided into three categories: material elasticity sup- appearing on both sides of the equation of motion doposes that the stresses derive from a strain potential; not depend on the motion itself. This assumption isplasticity expresses the material yielding; and visco- reasonable if the stresses are proportional to strainsplasticity concerns deformations that depend on and if the displacements remain small during thestrain rate (Tables 6-10). analysis.

In linear dynamics, an economical way of solving Reactions and stresses benefit dramatically from thatthe equations of motion is to expand the solution correction (Tables 13 and 14)(displacements) as a series of lower-frequency modes.The technique applies for relatively low-frequency N I" A 1.. Th d l· h d· on mear naysesexcitatIOns. e mo e acce eratIOn met 0 Improvesthe solution by adding to the response the static The solution of a nonlinear problem usually requirescontribution of unknown highest-frequency modes. an iterative procedure on all the degrees of freedom of

the structure. In transient static analysis, excitation Interactionsdepends on time- but rate-dependent phenomena Th .. b f h(f"

d·

d"

1 ffese concern mteractlOns etween parts 0 t e

nctlOn, ampmg, ete.) an mertIa e ects are d 1 (T bl 17)1 d K· . d· h . f b d· mo e a e .neg ecte. mematlCs stu les t emotIon 0 0 les,

and does not take into account inertial effects and Coupled Analysesother forces (Table 15).

Direct coupling analysis introduces simultaneous

L· P rt b t" A I coupling between two or more different physicalanear e ur a Ion na yses k I . f· d· 1·· 1un nowns. t ISseparate rom m Irect coup mg mvo -

Linear perturbation analysis means a linear analysis ving two sequential separate analyses. The common-at the vicinity of a nonlinear equilibrium configura- est indirect coupling case is computationaltion (Table 16). An example is the computation of thermomechanics with precomputed temperatures.vibration modes of a cable submitted to large deflec- Here, we consider only coupling with mechanicstion in a gravity field. (Table 18).

Solution Methods the program. It is why solvers are generally adapted tospecific computer architectures (Table 19).

Solver.. Eigenvalue ExtractionBasically, the solver is that part of the program

dedicated to solving a linear system of equations. It The Lanczos method is widely used to computeallows one to obtain directly the linear static response. natural modes and frequencies of large systems. TheTo solve nonlinear problems or dynamic ones, it is block Lanczos method is a variant which iterates withusually coupled to iterative procedures. Solver effi- a set of vectors instead of a single one. When extract-ciency often determines the overall performance of ing many modes, the efficiency of the technique is

improved by operating several spectral shifts in order Time Integrationto localize the frequencies in successive subintervals I to do th do 1 b 0 d(T bl 20) n ranSIent ynamICs, e ISp acements are 0 tame

a e . by integrating the equations of motions (Table 22).Schemes of the Newmark family are the most widely

Nonlinear Equilibrium used. Since the range of excited frequencies may

T °lob 0

h 1 do change during the response according to changes in

o converge to an eqm 1 num at eac oa mcrement 0 0 0 0 0

(

0

) h 0 (d 0

)

0 0 the loadmg or m the structure Itself, automatIc tImestatICs or at eac tIme step ynamIcs, IteratIve 0

do 11 d h CPU 0 fsteppmg may ramatIca y re uce t e tIme or aprocedures are used that are based on the well- 0

gIven accuracy.known Newton or tangent method (Table 21). Instatics, the loads are usually applied step by step;

S b.

h h 0 0 k f b klo

h h u structurmgw en t ere IS a ns 0 uc mg or snap-t roug ,increments must be applied according to a continua- Sub structuring is a technique used to condense thetion method (arc length, Riks, etc.). In dynamics, the degrees of freedom of a linear structure (Table 23).size of the time step conditions the size of load Stiffness, mass, and other matrices and loads areincrements. In the general case, the loads depend on reduced from their initial size to a smaller numberthe configuration reached by the structure and must of degrees of freedom. Analyses are performed usingbe computed at each iteration. condensed matrices and results are recovered a pos-

teriori for the whole set of degrees of freedom. The Further Readingmethod. is static~lly exact but gives ap~roximate Adams ML Jr (1999) Rotating Machinery Vibration.results 10 dynamICs. The condensed matnces form From Analysis to Troubleshooting. New York: Marcela superelement which is useful to export a model Dekker.from one software to another. It is possible to Craveur J-C (1996) Modelisation des Structures. Paris:impose large rotations to a superelement in a non- Masson.linear analysis. Hinton E (ed.) (1992) Introduction to Nonlinear Finite

Element Analysis. Glasgow: Nafems .. Martin J-P (1987) La qualiti! des Logiciels. Du Bricalage Ii

See also: Computation for transient and Impact l'Industrialisation. Paris: Afnor Gestion.dynamics; Eigenvalue analysis; Finite element meth- Robert G (1993) Finite element quality control. In:ods; Krylov-Lanczos methods; Nonlinear systems Proceedings of the Fourth International Conference onanalysis; Structural dynamic modifications. Quality Assurance, pp. 279-290. Glasgow: NAFEMS.

SFM (1990) Guide de Validation des Progiciels de Calcu/des Structures. Paris: Afnor Technique.

Spatial properties of structural dynamic models arepresented in the form of mass, stiffness, and dampingmatrices. For gyroscopic systems, such as rotor-bear-ing systems, one can add gyroscopic and circulatorymatrices. The static flexibility matrix, i.e., the inverseof the stiffness matrix, is also of interest because it ismeasurable.

The mass and stiffness matrices can have the fulldimension of the finite element model (FEM) or

Test Orthogonality Check

An indirect comparison of the TAM mass matrices,obtained by the four reduction methods presentedabove, can be made based on the test mode shapesand the mixed orthogonality check TOR. A typicalresult is illustrated in Figures 1-4, presenting theTOR matrices calculated from four different reducedmass matrices. The reference FEM of the structure,having about 45000 DOFs, has been reduced to a120-DOFs TAM, having 15 flexible natural modesbetween 5.2 and 34.3 Hz.

The largest off-diagonal terms occur in the TORmatrix of the modal TAM (Figure 3), especially forthe higher residual modes. This can be an indicationthat the spatial resolution given by the selectedresponse measurement points (the a set) is insufficientto make the target modes linearly independent andobservable - an important outcome of such a com-panson process.

The hybrid TAM shows a slight improvement onthe modal TAM, due to the inclusion of static modeswith the target modes (Figure 4). Surprisingly, thestatic TAM performs better than the modal TAM,showing smaller off-diagonal terms (Figure 1). TheIRS TAM yields the best reduced-mass matrix, pro-ducing the lowest off-diagonal elements in the TORmatrix (Figure 2).

While the modal TAM gives the best match infrequencies and target modes, its prediction capabil-ity is low outside the frequency range spanned by theselected target modes. The static TAM, implementedas the Guyan reduction in many computer programs,performs better in orthogonality checks, but is depen-dent on the selection of a DOFs and generally requiresmore a DOFs to give comparable accuracy. However,these types of comparisons are usually problem-dependent.

Further Reading Kammer DC (1991) A hybrid approach to test-analysismodal development for large space structures. Journal of

Ewins D] (2000) Modal Testing: Theory, Practice and Vibrations and Acoustics 113: 325-332.Application, 2nd edn. Baldock, UK: Research Studies O'Callahan]C (1989) A procedure for an improvedPress. reduced system (IRS) model. Proceedings of the 7th

Freed AM and Flanigan ChC (1991) A comparison of test- International Modal Analysis Conference, Las Vegas, pp.analysis model reduction methods. Sound and Vibration 17-21.25: 30-35. O'Callahan ]C (1990) Comparison of reduced model

Friswell MI and Mottershead ]E (1995) Finite Element concepts. Proceedings of the 8th International ModalModel Updating in Structural Dynamics. Dordrecht: Analysis Conference, Kissimmee, Florida, pp. 422-430.Kluwer. O'Callahan ]C, Avitable P, Madden R and Lieu IW

Guyan R] (1965) Reduction of stiffness and mass matrices. (1986) An efficient method of determining rotationalAIAA Journal 3: 380. degrees of freedom from analytical and experimental

Irons BM (1963) Eigenvalue economisers in vibration modal data. Proceedings of the 4th International Modalproblems. Journal of the Royal Aeronautical Society 67: Analysis Conference, Los Angeles, California, USA, pp.526-528. 50-58.

Kammer DC (1987) Test-analysis model development using O'Callahan ]C, Avitable P and Riemer R (1989) Systeman exact model reduction. International Journal of equivalent reduction expansion process (SEREP). Pro-Analytical and Experimental Modal Analysis xx: 174- ceedings of the 7th International Modal Analysis179. Conference, Las Vegas, Nevada, USA, pp. 29-37.

Modal properties that are compared usually include:natural frequencies, real mode shape vectors, modalmasses, modal kinetic and strain energies. For sys-tems with complex modes of vibration one can addmodal damping ratios and complex mode shapes.Left-hand modal vectors, modal participation fac-tors, and reciprocal modal vectors are also consid-ered in some applications. A test-analysis comparisonis meaningful only for matched modes, i.e., forcorrelated mode pairs (CMPs). These are estimatesof the same physical mode shape and their entriescorrespond one-to-one with their counterparts.Mode matching (pairing) is an essential step beforeany comparison can be undertaken.

In order to make it possible to compare experi-mental and finite element method (FEM) results, areduced test-analysis model (TAM) is often used.This is represented by the mass and stiffnessmatrices computed for the test degrees-of-freedom(DOFs) only. Comparison of modal vectors can bedone at the reduced order of the TAM or at thefull order of the FEM. Reduction of the physicalmass matrix or expansion of test modal vectorsbring inherent approximations in the comparison

criteria .... Direct Graphical ComparisonThere are three mam kmds of companson: (1)

analytical-to-analytical (FEM-to-FEM, TAM-to- A straightforward way to compare two compatibleTAM, and TAM-to-FEM); (2) experimental-to- sets of data is by making an X-¥ plot of one data setexperimental; and (3) analytical-to-experimental. against the other. The method can be used to compareThe third type will be considered in more detail the natural frequencies from two different models.below. For well correlated data, the points of the resulting

It is useful to compare: (1) measured mode shapes diagram should lie close to a straight line of slopeagainst modal vectors determined by an analytical equal to 1. If the approximating straight line has amodel; (2) estimates of the same test modal vector slope different from 1, this indicates a bias error dueobtained from different excitation locations; (3) to either calibration or erroneous material propertyestimates of the same modal vector obtained from data. Large random scatter about a 45° line indicatesdifferent modal parameter identification processes poor correlation or bad modeling.using the same test data; (4) one test mode shape The procedure can be applied to the mode shapesbefore and after a change in the physical structure of correlated mode pairs. Each element of a testcaused by a wanted modification, by damage or by mode shape is plotted against the correspondingoperation over time. element of the analytical modal vector. For consistent

where L is the number of CMPs, gives an indicationthat the modal expansion fails to provide physicallysound modal vectors.

Cross-orthogonality criteria cannot locate thesource of discrepancy in the two sets of comparedmode shapes. Large off-diagonal elements in thecross-orthogonality matrices may occur simplybecause they are basically small differences of largenumbers. Also, modes having nearly equal frequen-cies may result in (linear combinations of) analysismodes rotated with respect to the test modes, case inwhich the off-diagonal elements of XOR are skew-symmetric.

Modal Vector Correlation Coefficients

The Modal Assurance Criterion

One of the most popular tools for the quantitativecomparison of modal vectors is the modal assurancecriterion (MAC). It was originally introduced inmodal testing in connection with the MSF, as anadditional confidence factor in the evaluation of amodal vector from different excitation locations.

When an FRF matrix is expressed in the partialfraction expansion form, the numerator of each termrepresents the matrix of residues or modal constants.Each residue matrix is proportional to the outerproduct of one modal vector and the correspondingvector of the modal participation factors. Each col-umn of the residue matrix is proportional to therespective modal vector. One can obtain estimatesof the same modal vector from different columns ofthe residue matrix. MAC has been introduced as ameasure of consistency and similarity between theseestimates.

If the elements of the two vectors are used ascoordinates of points in an X - Y plot, the MACrepresents the normalized least squares deviation ofcorresponding vector entries from the best straightline fitted to the data, using the MSF. The concept canbe applied to the comparison of any pair of compa-

MAC Matrix the largest entries of the MAC matrix are no more onthe leading diagonal and it resembles a permutation

Given two sets of compatible modal vectors, a MAC matrix. The two large off-diagonal elements show thematrix can be constructed, each entry defining a indices of the switched vectors, as illustrated incertain combination of the indices of the vectors Figure lB. Figure 2 is the more often used form ofbelonging to the two sets. The ideal MAC matrix Figure lB.cannot be a unit matrix because the modal vectors are The MAC can only indicate consistency, not valid-not directly orthogonal, but mass-orthogonal ity, so it is mainly used in pretest mode pairing. The(Figure lA). However, the MAC matrix indicates MAC is incapable of distinguishing between system a-which individual modes from the two sets relate to tic errors and local discrepancies. It cannot identifyeach other. If two vectors are switched in one set, then whether the vectors are orthogonal or incomplete.

The reader must be warned that in some publica-tions the RV AC is loosely referred to as the FRACand the first variant of FDAC is similar to the RV AC.The FRAC is sometimes compared to the CO MAC,but the calculation is different. The modulus in thenumerator is taken after the vector multiplication,like in the MAC, and not inside the summation, foreach term of the scalar product, as is taken in theCOMAC.

Correlation of response properties is a relativelynew technique. Frequency response correlation coef-ficients must be applied with great care, using stiff-ness factors to adjust for frequency shifts and beingaware of the approximations introduced by the inclu-sion of an arbitrary damping model in the analysis. Aglobal frequency shift between the experimental andpredicted FRFs leads to a biased correlation coeffi-cient even if the FRFs are otherwise identical. Selec-tion of frequency points is a key factor in any FRF-based correlation.

Using magnitudes or logarithm values instead ofcomplex values can give better results, especially forlightly-damped structures whose FRFs exhibit largedifferences in the order of magnitude and the phaseangles. When the damping updating is not of inter-est, it is useful to choose the frequency points awayfrom resonances and antiresonances, though thelargest discrepancies noticed visually occur in theseregions. The FRAC coefficients are more sensitive toresonances and less sensitive to antiresonances heav-ily affected by modal truncation.

matrix is used as the left singular vectors mode indi-cator function, or the U-Mode Indicator Function(UMIF), to locate frequencies of the dominant modesand to reveal multiple modes.

PRFs are left singular vectors, scale shifted inmagnitude by multiplication with the correspondingsingular value. They can be used to eliminate redun-dant, linearly dependent information and noise, andto estimate the rank and condition of the FRF testdata.

The first six, twelve and twenty PRFs of a typicalCFRF matrix are plotted in Figures 4-6. Inspection ofsuch overlays with an increased number of PRFsreveals an upper group of six noise-free curves,more or less clearly separated from a lower groupof 'noisy' curves. The number of distinct curves in theupper group is a good estimate of the rank of theCFRF matrix. Retaining only these PRFs, a rank-limited FRF matrix can be reconstructed by multi-plying the truncated PRF matrix with the Hermitianof the matrix of corresponding right singular vectors.

Correlation coefficients similar to the FRAC andRV AC can be computed for the PRFs to characterizethe average behavior of a structure in a given fre-quency band, especially in the medium frequencyrange.

See also: Comparison of Vibration Properties, Compar-ison of Spatial Properties; Comparison of VibrationProperties, Comparison of Modal Properties

Figure 6 First twenty PRFs of the same matrix as in Figures 4 and 5.

Further ReadingAllemang RJ, Brown DL (1996) Experimental modal Heylen W, Lammens S, Sas P (1997) Modal Analysis

analysis. In: Shock and Vibration Handbook, 4th edn., Theory and Testing. Leuven: K. U. Leuven.pp. 21.1-21.74. McGraw Hill. Pascual R, Golinval JC, Razeto M (1997) A frequency

Avitable P (1999) Modal Handbook. Merrimack, NH: domain correlation technique for model correlation andDynamic Decisions Inc. updating. In: Proceedings of the 15th International

Ewins DJ (2000) Modal Testing: Theory, Practice and Appli- Modal Analysis Conference, Orlando, Florida, pp 587-cation, 2nd edn. Taunton, UK: Research Studies Press. 592.

D J Benson, University of California, San Diego, set them apart from methods intended to solve quasi-La Jolla, CA, USA static and traditional dynamic problems for struc-

J Hallquist, Livermore Software Technology tures and. solid me~hanics. Most ~f the formulat~onsCorporation (LSTC), Livermore, CA, USA for tra~slent and I~pact dynamics ~a.n be. denved

from either the fimte element or fmlte dIfferenceCopyrightu 2001 Academic Press perspective. In fact, many of the algorithms used indoi:10.1006/rwvb.2001.0006 finite element impact calculations by engineers were

originally developed by physicists at the national

I d" laboratories for the finite difference formulations

ntro uctlon used to analyze nuclear weapons.Transient and impact dynamics problems typically The global solution strategy typically uses:occur over s~o~t periods of time, ranging from n~no- 1. Explicit time integration.seconds t? mllh~econds.' and have large ~eformatIOns 2. Lumped mass matrices.and rotat~~ns, high st:al~ rates, and no~lmear bound- 3. Contact algorithms.ary conditIOns. ~p.eclahzed .computatIOnal m.etho~s 4. Algorithms for mapping solutions from distortedare used as an effICIentsolutIOn to problems with this to undistorted meshes.particular combination of characteristics. Thesemethods have evolved from the finite element and The elements have:finite differ~nce methods used for so~ving .quasistatic 1. Linear interpolation functions.and dyn~mIC structural problems. This ar:lcle focuses 2. Uniformly reduced (one-point) spatial integration.on the dlfference~ betwee~ the computa.tIOnal meth- 3. Hourglass control to eliminate zero-energy modes.ods used for transient and Impact dynamICs and those 4. A shock viscosity to resolve stress waves.used to solve more traditional types of problems instructures and solid mechanics.

Typical applications of explicit codes include auto- GI b I S I t" St t "... 0 a 0 u Ion ra eglesmotive crashworthmess and occupant protectIOn,

bird strike on jet engine fan blades and aircraft Time Integrationstructures, industrial processes such as sheetmetal ..stamping, and defense applications involving ord- ~or :rPlcal structural dynamICs proble~s, the stepnance design. To handle such a wide range of pro- size. IS governed by th: accuracy requ~red. by theblems modern-day explicit codes have many e.ngmeer and the truncatIOn error of t~e time mte?ra-capabilities, including a variety of contact algorithms, tIOn ~ethod. In con~rast, the mecham~s of transl.enta large library of constitutive models for an extensive and Impact calcu~atIOns go.verns t~e time step size.range of material behavior, equations of state for For exa~ple, the time step sl~e reqUIred to res~lve themodeling the response of materials under high pres- propagatIOn of a stress wave ISthe amount of time thesure and various forms of adaptive remeshing. wave takes to cross an element. Therefore, computa-

, tiona I formulations for impact problems minimize thecost of each time step by using explicit time integra-

Computational Methods for Transient tion methods ..and Impact Dynamics The ~econd-?rde~ accu:ate central dlf~erence

method IS the time mtegratIOn method that IS mostFormulations that are intended for transient and commonly used in codes for impact calculations.impact dynamics share several characteristics that Given the accelerations and displacements at time

Since minimizing the cost of each time step is amajor concern in developing a formulation for tran-sient and impact problems, the minimization of thestorage requirements is also important because thecost of reading and writing information from a harddisk for a traditional finite element formulation islarger than the computational cost of an explicit timestep. A lumped mass matrix therefore permits thesolution of much larger problems than a consistentmass matrix for a fixed amount of memory.

When an impact applies a sudden load on theboundary of a finite element mesh, an accelerationfield that is in the direction of the applied force isexpected. A consistent mass matrix, which inertiallycouples the nodes, often results in an oscillatoryacceleration field, with nodes accelerating in thedirection opposite of the applied force. This error,which has been analyzed theoretically, can only beeliminated from impact calculations by using alumped mass matrix.

Contact Algorithms

Impact calculations require contact algorithms toimpose the contact forces required to keep exteriorsurfaces from passing through each other. Ideally, thepenetration of surfaces through each other is held tozero by an exact enforcement of the contact con-straints, but in practice this is difficult to achieve.Most of the current methods permit the surfaces topenetrate slightly, and the small violation of thecontact constraint has no adverse effect for mostproblems. During the early years of computationalmechanics, 'gap elements' required nodes on oppositesurfaces to come in direct contact to prevent penetra-tion. Large deformation problems have surfaces thatundergo large relative slip, and therefore the nodesthat were opposite each other at the beginning of thecalculation are remote from each other by the end ofit. Furthermore, during large slip, node-on-node con-tact is impossible to maintain unless the mesh movesrelative to the material, which introduces its owncomplications.

Contact algorithms for impact calculations havetwo aspects that may be considered independently:calculating the contact locations, and calculating theforces to prevent penetration.

Calculating contact locations Contact is describedin terms of a node and its location relative to theexposed edge or face of an element. In the remainingdiscussion, the exposed element boundary will bereferred to as a 'surface segment'. When there aretwo distinct surfaces, they are called 'master' and'slave' surfaces in the literature. Buckling calculationshave a single surface which contacts itself as the

from automobile crashworthiness simulations to the Periodic rezoning The calculation is stopped peri-design of munitions. odically either by manual intervention or by the

Until recently, Lagrange multiplier methods were program itself (e.g., based on some measure of meshnever used in impact calculations because they gen- distortion), and the solution is projected from the olderate systems of coupled equations over the contact- mesh to a completely new mesh. The new mesh hasing surfaces in their standard form. Their primary nothing in common with the old mesh other than theadvantage is the contact constraints are enforced shape of the material boundaries. Originally, the newexactly. Explicit Lagrange multiplier methods have mesh was generated by the analyst, but automaticrecently been proposed which avoid this difficulty by mesh generation algorithms are advanced enoughintroducing assumptions that decouple the equations. today that most of the new meshes are generatedIn their explicit form, Lagrange multiplier methods automatically. If the new mesh is generated by theresemble a penalty method with a stiffness that is a analyst, the number of rezones in a calculation isfunction of the time step size, and the contact con- typically on the order of 10, while automatic meshstraint is no longer enforced exactly. The time depen- generation schemes may redefine the mesh up to 100dence of the surface stiffness with this method results times.in small errors in the conservation of energy. The rezoning scheme has to identify which ele-

Augmented Lagrangian methods try to combine ments of the old mesh overlap an element in thethe Lagrange multiplier method with the penalty new mesh. Search algorithms that are similar to themethod to gain the advantages of both. Although global contact search algorithms are used, and theythey have enjoyed some success in implicit formula- frequently account for a major part of the mappingtions, they have yet to be used in codes for impact cost.calculations. Once the overlapping elements are found, the map-

ping of the solution usually proceeds in one of twoways. The generality of the periodic rezoning projec-

Mapping Solutions from Distorted to Undistorted t· k h·· b th t· d dM h IOn ma es ac levmg 0 conserva IOn an secon-es es order accuracy very difficult. Either the solution is

The finite element or finite difference mesh distorts as interpolated from the old mesh, which results in a lossthe calculation progresses, which reduces the accu- of conservation, or a 'completely conservative' map-racy of the solution and the time step size. Eventually, ping scheme calculates the exact integral average forthe mesh may become too distorted to continue the the new solution values. The completely conservativecalculation. A strategy for mapping the solution from scheme is significantly more expensive and difficult tothe distorted mesh to an undistorted mesh is therefore program than interpolating the new values, but forrequired. At this time, the most popular strategies are problems where strong solution gradients are present,periodic rezoning, arbitrary Lagrangian Eulerian it gives a superior answer.(ALE) formulations, and Eulerian formulations.

While the three approaches differ considerably in ALE and Eulerian formulations ALE formulationstheir implementation, they possess many similarities. permit the mesh to move relative to the materialFirst, the qualities that are desired in the remapping continuously as the solution evolves. The most com-schemes are the same: conservation of solution vari- mon form of the ALE formulation is the simplifiedabIes (e.g., momentum); second-order accuracy; and ALE or SALE formulation, which permits only oneto avoid introducing oscillations into the solution. material in an element. This simplification means thatSecond, the simulation time is fixed during the map- the nodes on a material boundary can only moveping process, i.e., it does not proceed simultaneously tangentially to the boundary, which limits its useful-with the evolution of the solution variables, an ness since elements near the boundaries are frequentlyapproach that is referred to as 'operator splitting' in the most distorted. Eulerian formulations use a spa-the literature on ALE and Eulerian formulations. tially fixed mesh, and materials flow through it. AnThird, the functional representation of the solution element may therefore contain several materials, asvariables on the old mesh is usually different (and of may a general ALE formulation.higher order) than the one used during the evolution Since the material moves relative to the mesh eachof the solution. Fourth, the sequence of the mapping time step (or sometimes every few time steps), theprocess is first, determine if a new mesh and a map- mapping procedure is performed thousands of timesping are required; second, generate the new mesh; during a calculation. Speeq. and accuracy are there-third, project the solution from the old mesh on to the fore at a premium. While a first-order accurate map-new mesh, and fourth, restart the calculation with the ping method may be adequate for 10 periodicnew mesh. rezones, it is too diffusive to be used thousands of

times during a calculation, and a minimum of second-order accuracy is a practical necessity. The mappingalgorithms, also called transport or advection algo-rithms, are based on computational fluid dynamicsalgorithms for the Euler equations. Speed is obtainedby keeping the same topology for the new and oldmeshes and limiting their relative displacement tosome fraction of the element width.

Element TechnologyLinear Interpolation Functions

The displacements, velocities, and accelerations areinterpolated linearly and the stresses are piecewiseconstants in the elements used in transient and impactcalculations. Their advantages over higher-orderapproximations are:

1. They possess symmetries and anti symmetries thatreduce their computational cost relative to higher-order elements far more than a casual inspectionwould suggest.

2. Linear elements are very robust and are not proneto the singularities that occur in higher-order ele-ments when the nodes are not uniformly spaced.

3. Linear interpolation simplifies the geometric cal-culations in the contact and mapping algorithms.

4. The zero-energy modes due to reduced integrationare more readily suppressed in comparison tohigher-order elements.

5. For a given nodal spacing, linear elements permitlarger time steps than higher-order elements.

The primary disadvantage of linear elements rela-tive to higher-order elements is that they are too stiffwith full integration and too soft with reducedintegration. Triangular and tetrahedral elements, inparticular, are especially prone to locking withincompressible (plastic) flow for some meshes.

Uniformly reduced integration Only a single inte-gration point is used for linear elements, whichreduces the required central processing unit timeand memory for the stress by factors of four andeight in two and three dimensions respectively. Forquadratic elements, the speed gain would only be afactor of two to three.

Zero-energy mode control One byproduct of uni-formly reduced integration is the occurrence of zero-energy or hourglass modes. The shape of the zero-energy modes is a function of the element geometry.Since a zero-energy mode does not produce a strain,no stress is generated to resist it, and the modes maygrow without bound unless additional stiffness or

damping terms are introduced to resist it. Conversely,since a mode produces no strain, it does not affect theaccuracy of the stresses in the calculation. Problemsonly occur when the modes become large enough toturn the elements inside out or distort the contactgeometry.

Zero-energy modes are suppressed by calculatingtheir magnitude, and then adding a force that opposesthem. Assuming that the magnitude of the mode, h, iscalculated by:

and it differs little from the one originally introducedby von Neumann and Richtmeyer to solve shockproblems in the design of the atom bomb (they didnot include the linear term). The shock viscosity istreated like a contribution to the pressure for thecalculation of the nodal forces.

Example Transient and ImpactCalculationsThe calculations shown in this section were per-formed with LS-DYNA3D, developed and marketedby Livermore Software Technology Corporation. Itincorporates many advanced capabilities that are notavailable in the public domain version of DYNA3D,which was originally developed by John Hallquist.While the code runs on everything from PCs tomassively parallel computers, the calculationsshown here were performed on workstations.

Airbag Deployment

Transportation engineering is currently one of thelargest applications areas for explicit finite elementmethods. Since the safety of the occupants is a majorconcern, accurate detailed modeling of the impactsbetween the occupants and the vehicle is a necessity.One of the more challenging modeling aspects is thedeployment of the airbags in automobiles. In thisexample, shown in Figure 1, the airbag is initiallyfolded into the center of the steering wheel. A controlvolume model of the combustion of the propellantlocated in the steering wheel hub determines the gaspressure in the bag during its inflation. Special ele-ment technology and material models were developedto model the dynamic response of the airbag materialaccurately.

Crashworthiness Simulations

The first crash simulation of a full vehicle model,including the suspension, tires, and other runninggear, was performed with DYNA3D in 1986. It hada little over 4000 elements and required over 20 h ofcomputing on the Cray-XMP supercomputer atLawrence Livermore National Laboratory. Today,calculations with 10-100 times as many elementsare routinely performed by automobile manufac-turers to enhance the safety of modern vehicles andreduce the number of prototypes required for crash-worthiness testing.

Many of the stronger components, such as theengine, are modeled as rigid bodies. In regionsremoved from the impact, analysts use a coarsecomputational mesh or rigid bodies to minimize thecost of the calculation. The impact area requiresdetailed modeling. For example, automobiles typi-cally have thousands of spot welds which may failduring an impact, and each spot weld in the impactarea is individually modeled.

As in the previous example, the largest challenge ismodeling the contact during the crash. Most of thecontact involves interior structural components whichare not visible in Figure 2. For example, some of the

Figure 1 The simulation of the deployment of an airbag.

interior sheetmetal structure is designed to buckle inan accordion mode to absorb the impact energy. Thecontact interactions are so extensive that all the sur-faces on all the front-end components are treated aspotential contact surfaces in the calculation.

Figure 2 Crashworthiness analysis of a truck. Truck graphics courtesy of National Highway Traffic Safety Administration andNational Crash Analysis Center.

Sheetmetal Forming tions were possible, engineers designed the dies basedon their experience and the dies were altered by tool

The buckling patterns, and therefore the energy and die makers on the production floor until theyabsorbed by the structure, are very sensitive to the produced acceptable parts.manufacturing process. Current crashworthiness Explicit finite element methods are capable ofsimulations do not account for the plastic work and simulating sheetmetal-forming processes withthinning caused by the sheetmetal forming process; greater speed and accuracy than current implicithowever, they will do so in the near future. methods. Very fine meshes are required to resolve

Sheetmetal-forming simulations are used to aid in the sharp bends and the wrinkles caused by in-the design of the dies, which greatly reduces the time adequate die designs. Adaptive mesh refinementand cost to bring a product to market. Before simula- automatically adds elements to the mesh (Figures 3

and 4) during the calculation. Various criteria, e.g., In comparison to the 60 ms duration of a vehicleelement distortion, determine when a single quad- crash, the timescale of a metal-forming operation isrilateral element should be split into four elements, extremely long. Since inertial forces are not importantadding one level of local mesh refinement. The in metal forming, the density of the metal blank canmaximum number of levels (typically two to four) be scaled up to permit the explicit finite element codeis specified in the input file to prevent the adaptive to take larger time steps. Care must be taken with thisalgorithm from generating a mesh that is too large strategy, since scaling the density to too large a valuefor the available computer resources. will create spurious inertial effects.

C W Bert, The University of Oklahoma, Norman, OK, dimensional (I-D) members (strings, bars, beams,USA and columns), 2-D members (membranes, thin plates,

Co . hI 2001 A d . P and thin shells), and 3-D members (blocks, thickpyng ca emlc ress

plates, and thick shells). These methods are calleddoi:10.1006/rwvb.2001.0009 continuous to distinguish them from discrete meth-

ods, such as finite difference, collocation, finite ele-Continuous methods of vibration analysis are applic- ment, transfer matrix, boundary element, differentialable to continuous structural elements such as one- quadrature, quadrature element, etc.

Plate 13 Chaos. Chaotic systems: head-oncollision of two dipolar vortices in a stratifiedfluid environment. The original vortices, dyedorange and green, have exchanged a partnerto form two new (mixed) dipoles which aremoving at roughly right angles to the originaldirection of travel, that is, towards the top andbottom of the image. The green fluid wasinjected from the right, the orange from theleft. Dipolar vortices are relevant to turbulencein large-scale geophysical systems such asthe atmosphere or oceans. Turbulence in fluidsystems is one example of a chaotic system.(With permission from Science Photo Library).

S Braun, Technion - Israel Institute of Technology,Haifa, Israel,

Copyright © 2001 Academic Press

doi: 10.1 006/rwvb.2001.0170

IntroductionThe notion of correlation is one of the most basic onesin the description of data, and especially joint descrip-tions. Joint descriptions between data points, whetherfrom single or joint data sets, can describe patternsexisting in the data. Correlation functions andmatrices are often used to define or describe thepatterns and dynamic behavior of vibration signalsand vibrating systems.

The following first recalls basic correlation con-cepts, and their application to time functions. Sto-chastic random processes can be described by theirautocorrelation as well as the spectral density func-tion, and the relation between these presentations isdescribed next. The possibility of using correlationconcepts to define non stationary random data isalmost immediate.

Processing discrete data often involves the notionof the correlation matrix, which is briefly defined.

Classic as well as modern FFT based, computa-tional schemes are described, including some notionsof the variability of the estimated parameters.

The last part of the entry briefly describes someengineering applications, all relevant to aspects ofvibration processing: the detection of delays in dis-persionless propagation, spiking filters, and AR mod-eling. Adaptive line enhancer and adaptive noisecancellation applications conclude the entry.

V H Mucino West Virginia University Morgantown ger the development of what is known today as theWV USA ' " area of crashworthiness.

, Central to the development of the crashworthinessCopyright © 2001 Academic Press area is the use of explicit finite element (FE) techni-

ques to conduct computer simulations of collisions.doi: 10.1006/rwvb.2001.0164 Th h· h b d I d ddese tec mques ave een eve ope to a ress the

nonlinear transient solid mechanics of high-speedAutomotive collisions are responsible for many fatal- colliding structures, which involve the elastic andities every year and a public safety concern in every plastic stress wave propagation through solid conti-country for automakers, transportists and govern- nua and the characteristic nonlinear crushing beha-ments alike. According to the International Road vior of structures. Crashworthiness applications haveTraffic Accident Database (IRT AD), in 1998 alone, been extended to include human body structures andaverages of 13 fatalities and 480 injury accidents per the interactions between vehicle, occupants and100000 population were registered among the 29 restraining devices.participating countries in Europe, North America, Yet, many challenges lie ahead. New advancedAsia and Australia. Yet, this statistic is the result of materials are being used in vehicles to reduce weighta decreasing trend in the past two decades reflecting, (for fuel economy purposes) and to enhance crash-among other things, the significant technological worthiness. Many of these materials are yet to beadvances in the automotive field aimed at enhancing adequately characterized in order to take full advan-'crashworthiness' of vehicles. tage of their applicability in vehicle structures. Repre-

Crashworthiness by itself is not a characteristic or sentation of human body structures for simulationfeature that can be measured or quantified, but it purposes is another area where the challenges arerelates to the capacity of a vehicle structure and plentiful, from the mechanical characterization ofoccupant restraint system to protect the occupants human tissue to the classification and determinationin the event of a collision. The main objectives being of injury thresholds for various types impact. Ulti-to reduce the likelihood and severity of injuries to mately, the integration of crashworthiness advancesvehicle occupants and ultimately, to reduce the fatal- in vehicle systems design is a challenge, which seemsity rates in vehicle collisions. Crashworthiness con- to be within reach, given the strides in computercepts are also being applied to aircraft structures and technology.heavy-duty vehicles to enhance survivability of occu-pants in the event of a crash. I t d t-.. n ro UC IonVehicle accidents have always been a concern toauto makers. But it was not until the early 1970s, Crashworthiness has emerged as a multidisciplinarytogether with the advent of supercomputers, that the area that is now at the very heart of vehicle design andkey issues relating to the mechanics of automotive transportation systems in general. It addresses thecollisions acquired such significant relevance to trig- mechanics of colliding structures and the interactions

designs, which in turn, triggered the development of a analysis and outlined the issues to be addressed invariety of engineering tools and approaches to assess order to make it possible. In the early 1970s, acrashworthiness of vehicles and occupant response. number of studies pointed out the importance ofNew standards and revisions to existing standards are local deformations and local geometry instabilitiespublished in the Federal Register. These standards to conduct the structural response of colliding struc-represent the minimum safety performance require- tures, issues that are central to crashworthiness ana-ments for motor vehicles. The basic premise is that lysis of structures. McIvor et al., Armen et al. andthe 'public is to be protected against unreasonable Welch et al. put forward important studies for crash-risk of crashes occurring as a result of the design, worthiness simulations of structural componentsconstruction, or performance of motor vehicles and is using FEs, in which the phenomena of local deform a-also protected against unreasonable risk of death or tions and nonlinear elasto-plastic material behaviorinjury in the event crashes do occur'. was considered.

Early studies of crashworthiness relied almost com- The development of one of the most widely used FEpletely on experimental barrier impact tests that codes for crashworthiness, DYNA3D, started in theproduced data, which were used later on, to help mid 70s by Hallquist at Lawrence Livermore Labora-explain the dynamic response of colliding vehicles. In tory. This code captured the essence of high-speedsome of these tests, mannequins were used to provide impact solid mechanics phenomena, for military andsome idea of human response in vehicle frontal civilian applications, from projectile penetration tocrashes. Kamal and Lin produced one of the first vehicle collisions. In 1979, DYNA3D was repro-models for vehicle collisions based on nonlinear grammed for the CRAY-1 supercomputer withmass-spring elements, whose characteristics were more sophisticated sliding interfaces than the pre-derived experimentally. vious version. In 1981, new material formulations

Frontal collisions were first addressed by the stan- and the problem of penetration of projectiles wasdard MVSS-203 (48 km h -1 rigid barrier frontal successfully simulated by the use of sliding interfaces.crash), given the frequency of their occurrence In 1990 LS-DYNA3D is released with added capabil-(roughly 50% of collisions). In such a collision, the ities to represent fabric materials for seat belts andpassenger cabin is supposed to maintain its integrity airbags and composite glass models. Parallel to that,by not allowing large structural deformations. The pre and postprocessing codes (INGRID andkinetic energy of the vehicle is to be dissipated by TAURUS) were developed to interface with LS-large plastic deformations of the front-end of the DYNA3D and currently, there are versions ofvehicle, which in turn, acts as a cushion to the DYNA3D for PC and desktop workstations. Mean-passenger compartment. Deformations in the order while, several other codes have emerged with variousof 800 mm (32 in) can take place in a time interval of capabilities that allow for similar functions. For theapproximately 120 ms, which produces strain rates most part, these codes seem to provide consistency inbetween 1 and 100 per second. Under these the applications. Other codes commercially availableconditions, the passenger cabin experiences decelera- that are capable of simulating collisions of largetions of the order of 20g. Unrestrained occupants structures effectively are PAM-CRASH, ADINA,colliding with the vehicle interior may experience MADYMO, WHAMS, and ABAQUS. The lattereven higher deceleration rates, illustrating the impor- has useful data base descriptions of crash dummytance of restraining devices. The rather costly nature models that are widely used for vehicle-occupantof experimental tests and the limited predictability interaction assessment.of collision events based on experimental data Three special features that distinguished this codetriggered the interest in simulation. The experimental were (i) explicit time integration schemes for theprograms however, highlighted some of the most equations of motion; (ii) the application of advancedimportant issues in solid mechanics that needed contact algorithms to allow for sliding surfaces withto be addressed, namely the large plastic deforma- friction and nonpenetrating contact surfaces; and (iii)tions, strain rate dependency of materials, plastic and the use of 'economic' one-poi nt-integration elementselastic stress wave propagation, local buckling and with 'hourglassing energy control' options to monitorfracture/rupture of materials in the 120 ms of colli- convergence (hourglassing energy is associated to thesion duration. zero energy deformation modes and is briefly

described below). Additionally, this program was

O th S I"dM h " T k developed to include a wide array of nonlinear mate-n e 0 I ee ames rae . I . I d· . d d . I dna s, mc u mg stram rate epen ent matena s anMelosh and Kelly envisioned the requirements for the viscoelastic materials, rubber, honeycomb, and com-development of FE methods for crashworthiness posite materials among others.

Many other standard commercial FE codes such asANSYS, IDEAS-MS, PATRAN and CATIA amongothers, now provide import/export functions to inter-face with some of the crashworthiness codes, makingthe modeling and data preparation more readilyavailable to many engineering analysts.

Brief Description of CrashworthinessCodes Based on DYNA3DAll the FE crashworthiness codes share the same basicprinciples of computational mechanics. The use ofexplicit time integration algorithms, the use of 'eco-nomic' elements, the formulation of various consti-tutive models for engineering materials, the re-zoningof the meshes using advection algorithms and the useof similar contact-impact algorithms. Some varia-tions exist on special interface representationsbetween structural components, but in general theyall share similar or equivalent features. A briefdescription of the key features of a crashworthinesscode is provided next.

Spacial Discretization

DYNA3D uses an updated Lagrangian formulationbased on the weak form integral of the virtual workprinciple. Economic isoparametric elements with oneintegration Gauss point and diagonal mass matricesare used, for which viscous or stiffness hourglassenergy can be used to control the zero energy modesassociated to the one point of integration scheme. Thestructural elements include springs, lumped masses,discrete dampers, beams, trusses, solids (tetrahedron,wedges and brick) and shells (quadrilateral and tri-angular), the Hughes and Liu shells, also theBelytscho-Schwer beam and Belytschko- Tsy shell.Solid elements use the Flanagan-Belytschko constantstress, exact volume integral. Table 2 shows the type

of elements available in DYNA3D for structuralrepresenta tions.

Arbitrary Lagrangian Eularian (ALE) Advection

This is a re-zoning approach, which is needed to main-tain consistency between solution requirements andthe FE mesh. The rezoning consists of a Lagrangiantime step followed by an advection or remap step. Inthis process the solution is stopped and the mesh isadjusted to the deformed geometry in such a way thatelements are not highly distorted (smoothing). Thesolution is then mapped from the old mesh to the newone until the next step. While advection is differentfrom adaptive FE meshing, both techniques can beused to enhance accuracy and convergence of a solu-tion where nonlinear large displacements occur andwhere high gradients of the response variables occur.

Time Integration

A central difference explicit time integration algo-rithm is used to integrate the resulting equations ofmotion. This scheme is conditionally stable but doesnot require the use of implicit iterative techniques.The central difference approach requires that for eachtime step I1t, the current solution be expressed as:

The difference with implicit methods of integration isthat in explicit schemes, the solution to the currenttime step depends only on the solution of previoussteps, thus avoiding the costly iterations on each timestep to determine the unknown solution at the currenttime step (required in implicit schemes). The result isthat no iterations are necessary at each time step. Thedrawback with explicit methods is the stability,which can only be controled by taking rather smalltime steps, as the error is proportional to the square ofthe time step. The Courant stability criterion must beused, which requires the solution not to propagatethrough an element faster than the dilatational wavespeed of the material. This requirement gives rise tothe need for rather refined meshes to provide numer-ical stability. Monitoring the energies of the systembecomes the key to controling the stability of thesolution.

Boundary Conditions

Several loading conditions can be simulated. Directnodal loads, line and surface pressures and body loadsare available. Kinematic boundary conditions are alsoallowed through prescribed displacements, velocitiesand accelerations; fixed nodes and displacement

the passenger space envelope. Reinforced vehicle useful information about general displacements, velo-doors are examples of elements that provide that cities and accelerations of body segments during akind of protection in the case of lateral impacts. collision, but do not capture the nature of the human

4. Occupant restraint systems. It was stated above body internal anatomy and thus, they cannot be usedthat in the case of a frontal collision, the decelera- to assess detailed interactions between vehicle andtions of an unrestrained occupant could be an occupants.order of magnitude larger than that of the passen- The second approach is to model the structure ofger cabin. Thus, the occupant restraint systems dummies, which in turn are designed with the solehave as main function to reduce the magnitude of purpose of mimicking the human body response inthe decelerations of the occupant. Two types of experimental vehicle collisions for crashworthinessrestraint systems are used in vehicles. The passive assessments. The development of the 'Hybrid IIIones are the seat belts, and the active ones are air Family' is the result of many years of crashworthinessbags. Modeling seat belts requires the use of ele- technology development pursuing biofidelity betweenments that can interface with the occupant and the dummies and the human body for the 50-percen-vehicle and provide membrane stiffness only. tile male, female, and children of several ages.Crashworthiness FE codes permit the use of 'fab- MADYMO™ code provides a database for modelingric' material that provides the membrane proper- validated standard dummies.ties needed to model the seat belt; the number of The third approach is to model human body struc-points of anchorage, their location and the amount tures in such a way that human body compliance inof 'slack' in the belt being the key design factors to terms of flexibility and inertia are taken into consid-consider. eration in the analyses. The FE models that have been

Air bags provide active protection in the case of developed in this area are intended to describe thea collision. At issue is the presence of a cushion dynamics of human body structures involved in vehi-between the passenger and the hard surfaces of the cle collisions and to produce a better understandingvehicle interior immediately after a collision. The of the likely injuries that occupants may possiblyissues involve the inflation pressure level required sustain. A number of models can be found in litera-for deployment of the airbag in the first few ture for head-neck complex, for thorax and spinemilliseconds following the collision and the inter- complex and for lower extremities that reflect ana to-action with the occupant afterward. But in cases of mically correct features.proximity of occupant and passenger prior to the There are two key issues in modeling human bodycollision, the air bag deployment itself may pro- structures; one is the complex geometry of bodyduce injuries to the occupant depending on the parts, including bones and internal organs. Thisinflation pressure and the resulting head accelera- task is extremely tedious but has been advancedtions after deployment. Airbags are primarily greatly by the availability of the 'visible human pro-aimed at reducing head injuries. Thus the HIC ject'. Yet, the development of accurate FE modelscan also be applied for the assessment of airbag calls for appropriate material characterization ofadequacy. Some air bags have also been considered body tissue, from various bone structures to softto protect the occupants from lateral impacts. tissue and membrane materials. One example that

illustrates the importance of material characteriza-••••• tion can be found in modeling of the human head for

Bioengineering Considerations impact assessment. Several FE models have been

H B d St t proposed with material properties for the brain tissueuman 0 y rue ures

A major effort in crashworthiness technology devel-opment has been directed at the representation ofhuman body structures for crashworthiness study.From the kinematics point of view, a human bodycan be represented as an articulated system withkinematic degrees of freedom, certain mass distribu-tion and some degree of flexibility and relativemotion stiffness.

Three approaches have been followed in crash-worthiness studies. The first approach uses modelsthat capture human body segment mobility and gen-eral mass distribution. These models can produce

that produces a different response. At issue is how thematerial properties affect the standards used to assesshead injury, specifically, the HIe.

Application ExamplesApplication examples can be found in many fields,from the solid mechanics general examples (cylindri-cal shells under axial impact) to automotive struc-tures under various collision scenarios. Figure 1shows the crushing of a square tubing under axialimpact loading. Notice the accordion folding of thewalls of the tube in such a way that the surfaceproduces contact with itself as it folds. The use of a'single surface' contact is illustrated in this example.The same figure illustrates the effect of a mild load

developed in which the brain cavity is modeled (skulland brain mass) and then subjected to a frontalimpact. The stress waves on the brain mass can beanalyzed using visco-elastic materials for the brainmass and elastic properties for the skull. At issue arethe maximum pressures developed in the brain massand the effect of the visco-elastic vs elastic brainmaterial assumptions.

ConclusionsCrashworthiness has matured into a multidisciplinaryarea with a wide range of applications: in vehicledesign, aircraft design, military ballistics, bioengi-neering, sports gear design, metal forming processes,particle impact erosion, cold working processes. Thedisciplines involved in the development of crash-worthiness include computational mechanics, fluid-structure interactions, materials science, compositematerials, honeycomb stiffened structures, polyur-ethane foam filled tubing, glass breaking, etc.

Yet certain areas remain a challenge to the crash-worthiness technologies. Specifically, the mechan-isms of fracture mechanics and the failure ofintricate materials like composite laminates, andbraided composite materials. Fluid-structure interac-tion as well as gas-structure interactions are clearlyapplications where challenges lie ahead.

DYNA3D and DYNAP (Nonlinear Dynamic Analysisof Solids in Three Dimensions). University of California:Lawrence Livermore National Laboratory, rept. UCID-17268 (Rev. 1, 1979).

Hallquist JO, Gourdeau GL, Benson DJ (1985) Slidinginterfaces with contact-impact in large-scale Lagrangiancomputations. Computer Methods in Applied Mechanicsand Engineering 51; 107-137.

Hughes TJR, Liu WK (1981) Nonlinear finite elementanalysis of shells: part I, two dimensional shells.Computer Methods in Applied Mechanics 27: 167-181.

Hughes TJR, Liu WK (1981) Nonlinear finite elementanalysis of shells: part II, three dimensional shells.Computer Methods in Applied Mechanics 27: 331-362.

Kamal MM (1970) Analysis and simulation of vehicle tobarrier impact. SAE Transactions 79: 1498-1503.

Kamal MM, Wolf JA (1977) Finite element models forautomotive vehicle vibrations. In: ASME Finite ElementApplications in Vibration Problems, presented at theDesign Engineering Technical Conference, Chicago, IL,Sept. 26-28.

Kan S, FHWA/NHTSA National Crash Analysis Center,Public Finite Element Model Archive. http://www.ncac.gwu.edularchives/model/index.html.

Lin KH (1973) A rear-end barrier impact simulation modelfor uni-body passenger cars. SAE Transactions 82: pp.628-634.

Macmillan RH (1970) Vehicle Impact Testing. SAE paperno. 700404. Warrendale, PA: International AutomobileSafety Conference Compendium.

Martin DE, Kroel CK (1967) Vehicle crash and occupantFurther Reading behavior. Paper no. 670034. Transactions of the SAE 76:

236-258.Aida T (2000) Study of Human Head Impact: Brain Tissue McHenry RR, Naab KN (1966) Computer simulation of

Constitutive Models. PhD Dissertation, University West the crash victim: a validation study. Proceedings of theVirginia, May. Tenth Stapp Car Crash Conference. November pp. 126-

Allison D (1995) Seymour Cray Interview. http://american- 163.history.si.edulcsrlcomphist/cray.htm. McIvor IK Wineman AS,Wang HC (1975) Large Dynamic

Annen H, Pifko A, Levine H (1975) Nonlinear Finite Plastic Deformation of General Frames. Twelfth Meet-Element Techniques for Aircraft Crash Analysis. Aircraft ing of the Society of Engineering Science, University ofCrashworthiness Symposium, Cincinnati, pp. 517-548. Texas, October, pp. 1181-1190.

Belytscho T, Schwer L, Klein MJ (1977) Large displace- Melosh RJ (1972) Car-Barrier Impact Response of ament transient analysis of space frames. International Computer Simulated Mustang. DOT-NHTSA ReportJournal for Numerical and Analytical Methods in DOT-HS-091-1-125A.Engineering 11: 65-84. Melosh RJ, Kelly DM (1967) The potential for predicting

Belytscho T, Tsay CS (1981) Explicit algorithms for flexible car crash response. SEA Transactions 76: 2835-nonlinear dynamics of shells. AMD 48: 209-231. 2842.

Belytscho T, Tsay CS (1983) A stabilization procedure for OECD International Road Traffic and Accident Database,the quadrilateral plate element with one-point quad- Brief Overview. IRTAD. http://www.bast.de/irtadleng-rature. International Journal of Numerical Methods in lishlirtadlan.htm.Engineering 19: 405-419. Ruan JS, Khalil T, King AI (1991) Human head dynamic

Flanagan DP, Belytscho T (1981) A uniform strain response to side impact by finite element modeling. ASMEhexahedron and quadrilateral and orthogonal hourglass Journal of Biomechanical Engineering 113: 276-283.control. International Journal of Numerical Methods in US National Library of Medicine, The Visible HumanEngineering 17: 679-706. Project. http://www.nlm.nih.gov/researchlvisible/visible-

Hallquist JO (1976) A Procedure for the Solution of Finite human.html.Deformation Contact-Impact Problems by the Finite US-DOT, National Highway Traffic SafetyAdministrationElement Method. University of California: Lawrence The Federal Motor Vehicle Safety Standards andLivermore National Laboratory, rept. UCRL-52066. Regulations Brochure. http://www.nhtsa.dot.gov/cars/

Hallquist JO (1996) Preliminary User's Manuals for rules/import/FMVSS/SN219.

USA Office of the Federal Register. The Code of Federal ings. Report no. DOT-HS-810 847. U.S. Department ofRegulations, title, 49, subtitle B, chapter V, part 571, Transportation.subpart B, section 571.216. Wilson RA (1970) A review of vehicle impact testing: how

Van Leer B (1977) Towards the ultimate conservative it began and what is being done. Paper no. 700414. SAEdifference scheme. IV. A new approach to numerical Transactions 79:1498-1503.convention. Journal of Computational Physics 23: 276- Yang KY, Wang KH (1998) Finite Element Modeling of299. the Human Thorax. http://www.nlm.nih.gov/research!

Welch RE, Bruce RW, Belytscho T. (1976) Finite Element visible/vhpconf98/AUTHORS/YANG/YANG.HTM.Analysis of Automotive Structures Under Crash Load-

D Inman, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA, USA

Copyright © 2001 Academic Press

doi: 10.1 006/rwvb.2001.0061

Critical damping is defined for linear, single-degree-of-freedom systems with viscous damping. It isbased on the three stable solutions to a second-order, ordinary differential equation with constantcoefficients and corresponds to the case of repeated,real roots in the characteristic equation. Physically,critical damping corresponds to that value of damp-ing that separates oscillation from nonoscillation ofthe free response. Thus, critical damping is also theminimum amount of damping that a spring-mass-damper system can have and not vibrate. If such asystem has a smaller than critical amount of dampingit will oscillate.

Critical damping is also the numerical value used tonondimensionalize damping parameters to produce adamping ratio. Experimentalists and analysts alikeuse the percent critical damping as a dimensionlessparameter for describing the amount of damping in asystem. Percent critical damping is also used in per-formance specifications and in design.

Although critical damping is defined for a single-degree-of-freedom, spring-mass-damper system, it isalso routinely applied to modal equations where itappears as a modal damping ratio. The modal damp-ing ratio extends the concept of critical damping tomultiple-degree-of-freedom systems. The same is truefor systems described by distributed mass models.The extension of the concept of critical damping to

nonzero damping. In addition, the general trend ofmodal damping indicated by this model firstdecreases with increasing frequency (with a slope of-Ion a log-log scale), then increases.

Finally, note that only the simplest structures willbe made of a single material or be governed by singleelastic and strain rate coefficients. And even assumingthat the viscous damping model is reasonable oversome frequency range, different materials will exhibitdifferent ratios of strain rate to elastic coefficients.Thus, the proportional damping model must beregarded in practice as a mathematical curiosity. Ananalyst who uses a viscous damping model should beprepared to use complex modes.

Despite its drawbacks, viscous damping is a simpleway of introducing damping in a structural modeland may be adequate under some circumstances (forexample, when accuracy is only needed over a smallfrequency range). Perhaps the greatest utility of theviscous damping model is the possibility of determin-ing (identifying) a viscous damping matrix fromexperiments. Although such a damping matrixwould not be element-based, a desirable feature, itmight represent damping adequately for the purposesof continuing analysis.

Structural or Hysteretic DampingThe structural, or hysteretic, or complex stiffnessdamping model is motivated by a desire to obtainmodal damping with frequency dependence weakerthan that which results from the use of the viscousdamping model. This model may developed by defin-ing a frequency-dependent viscous damping matrix,or by using the complex modulus model of materialbehavior. Common to both approaches is a funda-mental assumption of forced harmonic response. Inpractice, damping is also assumed to be closelyrelated to stiffness (because both are associated withdeformation), and independent of mass.

Frequency-dependent Viscous Damping

Observing from eqn [9] that, for stiffness-propor-tional damping, modal damping increases monotoni-cally with modal frequency, one might be inclined tomodify a single-modulus elemental viscous dampingmatrix by dividing by a frequency:

The damped structural model described by eqns[15] and [21] has some deficiencies, but also hasconsiderable utility in practice.

First of all, this model is not generally useful forobtaining direct time response, as it essentiallydescribes frequency response. However, when thetime domain forcing function in eqn [20] is notharmonic, but nevertheless has a Laplace transform,it may be possible to find the response via inversetransformation using the elastic-viscoelastic corre-spondence principle.

In practice, an eigenvalue problem based on eqn[20] may be posed. Assuming the time responseindicated in eqn [5], a complex natural vibrationfrequency results. If a single global loss factor canbe isolated, as in eqn [20], such proportional hystere-tic damping results in real eigenvectors or modeshapes. And for light damping, the modal dampingratio is approximately half of the loss factor. In thegeneral case, complex eigenvectors result. Frequency-dependent stiffness and damping can be accommo-dated via iteration.

One difficulty with this eigenvalue problem is thatthe decaying time response postulated in eqn [5]differs from the forced harmonic response assump-tion underlying this damping model [13]. In practice,the accuracy of natural frequencies and mode shapesdetermined using this method may decrease withincreasing loss factor(s).

This hysteretic damping model is ideal, however,for frequency response analysis [21]. Furthermore,loss factors and stiffnesses can, in principle, be func-tions of frequency:

use, a main problem becomes determining appropri-ate modal damping ratios.

Experience One approach is based on personal ororganizational experience with similar structures.Although there is no assurance that this approachwill yield correct values for any modes, it can providea valuable starting point for analysis. For example,values of modal damping in the range from 0.003 to0.03 might be appropriate for lightly damped, built-up aerospace structures.

Complex stiffness Alternate approaches to estimat-ing modal damping may proceed by establishing alower bound based on material loss factor contribu-tions. Such approaches are especially effective whenhigh-damping materials are used to augment nominallightly damped structures. An example of this kind ofapproach is the use of modal analysis of a complexstiffness-based model to yield a complex naturalfrequency and associated modal damping.

Modal strain energy Another materials-basedapproach is the modal strain energy method. In thisapproach, a modal loss factor is a weighted sum ofmaterial loss factors:

1 1 '" Uri [](r';::!21Jr=2 L.. lJi-U-r

27,materials

The weighting terms are the fraction of modal strainenergy stored in each material (or component),usually estimated from analysis of an undampedmodel. The modal damping ratio is approximatelyhalf of the modal loss factor.

Other Damping ModelsFor some applications that require high model fide-lity, the damping models described in the precedingmay be inadequate. An example of such an applica-tion might be a structural dynamic model that is to beused as the basis for the design of a high-performancefeedback controller. Some of the main deficiencies ofmodels considered to this point include the following.

Viscous damping yields modal damping that tendsto increase with frequency, in a manner inconsistentwith observations. Proportional damping, eitherrelated to the mass matrix, or for multicomponentstructures, should be regarded as a mathematicalcuriosity.

Hysteretic (complex stiffness) damping, while cap-able of accommodating frequency-dependent damp-

ing and stiffness, and of yielding better estimates ofmodal damping, cannot be used directly to determinestructural response to arbitrary dynamic loading.However, the combination of either the complexstiffness or modal strain energy method, along witha viscous modal damping model, often yields accep-table results. Limitations of this approach usuallystem from neglecting phase differences in response(using real modes), from neglecting frequency-depen-dent properties, from sacrificing mode orthogonalityby including frequency-dependent properties, or fromunusually high, perhaps localized, damping.

Several damping models suitable for use with finiteelement analysis have been developed to address suchshortcomings. The emphasis here is on models thatare compatible with linear analysis, thus neglectingfriction and other models.

Fractional Derivative

Fractional derivative models provide a compactmeans of representing relatively weak frequency-dependent properties in the frequency domain. Forexample, a single material complex modulus might berepresented as:

Internal Variable Viscoelastic Models

Another approach to capturing viscoelastic (fre-quency- and temperature-dependent) material beha-vior in a time domain model involves the introductionof internal dynamic coordinates. Several such modelsare available and, although they differ in somerespects, they share many common features. A one-dimensional mechanical analogy of material behavior(Figure 2) aids understanding of the general structureof such models.

The deformation of this system is described pri-marily by the stress, (J, and the total strain, c, but itsapparent stiffness is affected by the dynamics of aninternal strain, CA. If this system is subjected toharmonic forcing, its apparent stiffness and damping

vary with frequency. At very high frequencies, theinternal dashpot is essentially locked, the materialmodulus is Eu, and the damping is low. At very lowfrequencies, the internal dashpot slides freely, themodulus is lower, and the damping remains low. Atsome intermediate frequency, the damping reaches apeak value.

The peak loss factor and change in modulus bothdepend on the strength of the coupling between thetotal strain and the internal strain, 1/ (c - 1). Thefrequency at which peak damping is observed, verynearly n, is related to the inverse of the relaxationtime for the internal strain.

With a single internal variable, the loss factor isproportional to frequency at low frequencies, andinversely proportional at high frequencies. Weakerfrequency dependence can be introduced by usingmultiple internal fields, each having its own relaxa-tion dynamics. Temperature effects can be includedby using a shift function that essentially increases therelaxation rate of the internal fields with increasingtemperature.

Such internal variable models are quite compatiblewith finite element structural analysis methods. Inone approach, additional nodal displacement coordi-nates, identical to those of an elastic element, areintroduced to model the internal system. The internalfield is then interpolated in the same way as the totaldisplacement field. Additional first-order equationsof motion are then developed to describe the relaxa-tion (creep) behavior of the internal system and itscoupling to the total displacements. The boundaryconditions for the internal coordinates are implemen-ted just as those for the corresponding total displace-ments are, with the additional elimination of strain-free motion.

Eqn [29] shows the general structure of the finiteelement equations with a single set of internal coor-dinates. In this form, K is calculated using the high-frequency material stiffness. Evidently, higher accu-racy comes at a cost of additional coordinates andmaterial properties.

Further ReadingBagleyRL and Torvik PJ (1986) On the fractional calculus

model of viscoelastic behavior. Journal of Rheology 30:133-155.

Enelund M and Josefson, BL (1997) Time-domain finiteelement analysis of viscoelastic structures with fractionalderivative constitutive relations. AIAA Journal 35:1630-1637.

Johnson AR (1999) Modeling viscoelastic materials usinginternal variables. Shock and Vibration Digest 31: 91-100.

Johnson CD (1995) Design of passive damping systems.Journal of Mechanical Design 117B: 171-176.

Johnson CD and Kienholz DA (1982) Finite elementprediction of damping in structures with constrainedviscoelastic layers. AIAA Journal 20: 1284-1290.

Lesieutre GA and Bianchini E (1995) Time domainmodeling of linear viscoelasticity using anelastic displa-cement fields. Journal of Vibration and Acoustics 117:424-430.

Lesieutre GA and Govindswamy K (1996) Finite elementmodeling of frequency-dependent and temperature-dependent dynamic behavior of viscoelastic materials in

simple shear. International Journal of Solids and Nashif AD, Jones DIG and Henderson JP (1985) VibrationStructures 33: 419-432. Damping. New York, NY: John Wiley.

McTavish DJ and Hughes, PC (1993) Modeling of linear Pilkey Wand Pilkey B (eds) (1995) Shock and Vibrationviscoelastic space structure. Journal of Vibration, Acous- Computer Programs: Reviews and Summaries. Arling-tics, Stress, and Reliability in Design, 115: 103-110. ton, VA: Shock and Vibration Information Analysis

Mead DJ (1999) Passive Vibration Control. Chichester: Center, Booz Allen & Hamilton.John Wiley. Soovere J and Drake ML (1984) Aerospace Structures

Meirovitch L (1980) Computational Methods in Structural Technology Damping Design Guide. AFWAL-TR-84-Dynamics. Alphen aan den Rijn, The Netherlands: 3089. Dayton, OH: U.S. Air Force.Sijthoff & Noordhoff.

E E Ungar, Acentech Incorporated, Cambridge, MA,USA

Copyright © 2001 Academic Press

doi:1 0.1 006/rwvb.2001.0014

What is a Damping Material?A damping material is a solid material that dissipates(that is, converts into heat) a significant amount ofmechanical energy as it is subjected to cyclic strain.Most damping materials are not useful structuralmaterials themselves, but typically are combinedwith structural elements so that the resulting combi-nation is structurally viable and relatively highlydamped. A damping material configuration that isapplied to a structural component usually is called a'damping treatment'.

Although granular materials and viscous liquidscan provide considerable energy dissipation in someapplications, these usually are not regarded as damp-ing materials. Even though so-called high-dampingmetal alloys are more highly damped than commonmetals, their damping generally is not high enough forthese to be considered damping materials in thepresent sense. Almost all practical damping materialsare polymeric plastics or elastomers - but some othermaterials can also dissipate considerable energy insome temperature and frequency ranges.

Characterization of MaterialPropertiesDamping materials are often called 'viscoelastic',because they combine energy dissipation (viscous)with energy storage (elastic) behavior. Characteriza-tion of the properties of such a material requires twoparameters; one associated with energy storage, andone, with energy dissipation.

Application of this relation to the complex compres-sion and shear moduli implies that Poisson's ratio iscomplex, unless the loss factors associated with thetwo moduli are equal. For most materials the two lossfactors indeed are equal for all practical purposes;thus, their Poisson's ratio can be taken as real.

Measurement of Material PropertiesThe most direct approaches to determination of thecomplex compression or shear modulus of a materialinvolve measurements on samples whose shapes orsizes are such that application of a force in an appro-priate direction results in an essentially uniform or inanother reliably predictable strain distribution in thesample. For example, one may evaluate the complexshear modulus of a material at a given frequency byapplying a known sinusoidal shear force at the givenfrequency to one face of a thin flat sample, whoseother face is restrained from moving, and observingthe magnitude of the resulting shear displacement,together with its phase angle relative to the appliedforce. Such measurement approaches are simple inconcept, but generally involve considerable practicaldifficulties and relatively complex apparatus. Never-theless, such apparatus is necessary for the measure-ment of the amplitude-dependences of the complexmoduli.

The moduli of most viscoelastic materials, however,are practically independent of the strain amplitude upto stains of perhaps 5% or more. Thus, one can deter-mine the damping of such materials by approaches inwhich the strain amplitude is permitted to vary duringa measurement. These approaches include measure-ment of the rate of decay of free vibrations or of theresonance bandwidth of a test system that includes asample of the damping material.

The simplest and most widely-used measurementapproaches employ metal reeds (plate strips, orbeams with thin rectangular cross-sections) whichhave thin layers of damping material bonded to oneor both of their faces. A test reed is clamped to a rigidsupport at one end, and the other end is excitedmagnetically at one of its resonances at a time. Theloss factor of the coated reed is evaluated either fromthe observed rate of decay of the reed's vibration after

the excitation has been turned off, or from measure-ment of the bandwidth of the resonance. The flexuralstiffness of the coated reed is determined from itsmass-per-unit length and the resonance frequency,using the classical relations applicable to an elasticcantilever beam. The stiffness and loss factor of thebare metal reed are determined similarly. The proper-ties of the damping material then are calculated bymeans of well-established equations that indicatehow the flexural stiffness and loss factor or the reedwith attached damping materiallayer(s) depend onthe dimensions of the reed and of the attached layer(s)and on the moduli and loss factors of the bare reedand the damping material. These measurementapproaches are the subject of recent standards.

Typical Behavior of Damping MaterialsThe dynamic properties (moduli and loss factors) of adamping material generally vary markedly with fre-quency and temperature. They usually vary only alittle with strain amplitude up to quite large strainsand generally depend only to a minor extent on staticpreload and on exposure to long-duration cyclicloading.

Figures lA and IB shows how the real (storage)shear modulus and the loss factor in shear of a repre-sentative damping material vary with frequency andtemperature. Figure lA shows graphs of these twoquantities as functions of frequency at several con-stant temperatures. Figure IB shows the same dataplotted upon a temperature-frequency plane, in orderto permit one to visualize the trends more easily.

At a given frequency, the modulus varies drasticallywith temperature, starting from large values at lowtemperatures and progressing to small values at hightemperatures via a range of intermediate tempera-tures in which the rate of change is greatest. At lowtemperatures the material is said to be in its 'glassy'state and at high temperatures, in its 'rubber-like'state. At high temperatures - beyond those covered bythe plots - the material becomes very soft and tendstoward the behavior of a viscous liquid. The region inwhich rapid changes occur is called the 'glass transi-tion region'; the temperature at which the moduluschanges most rapidly is termed the 'glass transitiontemperature'. The highest loss factor values occur ator near this temperature. t

Figure 1 Dependences of shear modulus and loss factor of a damping material on frequency and temperature: (A) shown asfunctions of frequency at constant temperatures; (B) shown as plots upon temperature-frequency plane. Adapted from Ungar EE(1992) Structural damping. In: Beranek LL, Ver IL (eds) Noise and Vibration Control Engineering, ch. 12. New York: John Wiley.

The variation with frequency of the material prop- as from a suitably selected decrease (or increase) inerties at constant temperature is similar to their the frequency.variation with temperature at constant frequency. The general behavior of many materials is similarAt a constant temperature, the modulus progresses to that illustrated by Figures lA and lB, but thefrom small values at low frequencies to high values at actual values of the properties depend on the specifichigh frequencies via an intermediate region in which materials. Storage moduli as great as 108 kPa (orthe change is relatively rapid. Again, the highest loss 107 psi) may occur in the glassy region, and modulifactor values occur in the area of the most rapid as small as 10 kPa (1 psi) may occur in the rubberymodulus changes. This similarity has led to the con- region. For a given material, the modulus values incept of 'temperature-frequency equivalence'. Accord- the glassy region may be three or four orders ofing to this concept, if one starts with the material at a magnitude greater than those in the rubbery region.given temperature and frequency, one observes the The loss factor values in the glassy region usuallysame change in the material properties due to a are small, typically between 10-3 and 10-2

, whereastemperature increase (or increase) by a given amount in the rubbery region they tend to be of the order of 0.1

for many materials. In the transition region, the loss mits one to show the modulus and loss factor data forfactors of good damping materials generally approach a single-transition material for all frequencies andI and for some materials may reach 2. The transition temperatures as functions of a single 'reduced fre-region may extend over only 20°C for some materials quency' parameter, as illustrated in Figure 2.or over more than 200°C for others. Commercial The reduced frequency is defined as the product ofmaterials that are intended for use in given tempera- the actual frequency and of a 'shift factor' IY.T thatture ranges typically have their glass transition tem- depends on the temperature as determined from theperatures in the middle of these ranges. At constant shifting required to make all the curves coalesce intotemperature, the transition region may extend over continuous ones. The dependence of the shift factorone to four decades of frequency. The loss factor on temperature may be given by an equation, by acurves for materials with wider transition ranges separate plot, or - more conveniently - by a nomo-typically exhibit flatter peaks and lower maximum gram superposed on the data plots as in Figure 2.values than similar curves for materials with narrower Presentation of data as in Figure 2 is the subject of antransition ranges. international standard.

Figures IA and IB correspond to a material con- Analytical models have been developed that usesisting of a single viscoelastic component - that is, of empirical data obtained in limited frequency anda single polymeric material (with or without non- temperature ranges to characterize a material'spolymeric admixtures). Such a material has a single behavior outside of these ranges. These models gen-transition region and, correspondingly, its loss factor erally have been confined to single-transition mate-curve exhibits a single peak, as illustrated by the rials and have been of limited practical utility. Suchfigure. The behavior of materials consisting of two models, as well as plots like Figure 2, can lead toor more viscoelastic components with different tran- significant errors if they are used for extrapolationssition regions is more complex. The loss factor curve outside of the regions for which measured data arefor such a material may have two or more peaks, and available.the slope of the modulus curve for such a materialmay not change monotonically with temperature andfrequency. Practical Considerations

•• The dynamic properties of a sample of a polymericPresentation of Material Data material depend not only on its basic chemical com-Material property data may be presented in the form positi?n (that is, on the monomers that make up theof a series of curves giving values of the modulus a'nd ~atenal), but al~o, on several ot~er factors. Theseloss factor at various temperatures as functions of I~cl~de t?e matenal s molecu~ar weIght spectrum (thefrequency, as in Figure IA. Data may also be pre- dIstnb~ti~n of molecular cham lengths), the ~egree ofsented in terms of curves representing the values of cross.-l~nkmg, and.the amount~ and types of mcludedthese parameters at various frequencies as functions plastiCIzers and fillers. Expenence has shown thatof temperature. Instead of curves of the real modulus, even ~amples ta~en from t~e .same ~roductio~ batchor in addition to these curves, one may also show of a gIvenmatenal may exhIbit consIderably dIfferentcurves of the imaginary (loss) modulus or of the modulus. and loss factor values.magnitude of the complex modulus. Because the The differ~nces may be. even .grea~er for samplesvarious moduli are simply related, so that one can taken fr.omdiffere?t,. nommally Identical,.batches ~freadily be calculated from the other, the following a matena.I..1t thus I~ I~portant that. dampmg mater~-discussion focuses on the same parameter set as that als for cntic.al apphcatiOns be obtamed from expen-used in Figures lA and IB. enced .supphe~s who can guarantee the 'performan~e

By shifting the various curves that show the mod- ~f their matenals, and that these.matena~s be Sp~CI-ulus variations with frequency at different constant fled an~ ~ccepted not on the b~sls of the.ir chemIc~1temperatures along the frequency axis, these curves compos.ltIO?S,but on the basIs of theIr dynamIccan be arranged to form a single continuous smooth propertIes m the frequency and temperature rangescurve. The same is true also for the loss factor curves, of concern.if they are shifted by the same amounts as the mod-ul~s . curves. (Anal?gous s~atements also apply to NomenclatureshIftmgcurves obtamed at dIfferent constant frequen-cies along the temperature axis.) This shifting, the D energy dissipated per cycle in unit volumepossibility of which is a consequence of the afore- E' real part of E(O)); storage modulusmentioned temperature-frequency equivalence, per- E" imaginary part of E(O)); loss modulus

D Inman, VirginiaPolytechnic Institute and State dissipation in a variety of structures and machines.University,Blacksburg, VA,USA The other fact that makes damping a difficult phe-

nomenon to model is that it is inherently nonlinear,Copyright © 2001 Academic Press frequency dependent and temperature dependent.

There are numerous sources of damping in structuresdoi:10.1006/rwvb.2001.0060 and machines. Sliding friction and the energy dissipa-

tion of material moving in air provide the dominantEnergy dissipation in vibrating systems is an extre- sources of external damping. Sliding friction alsomely significant physical phenomenon. Yet damping exists as an internal cause of damping in jointedmodels remain an illusive research topic. Part of the structures. Other common sources of internal damp-reason for the lack of definitive damping models is ing are grouped together and called material damp-that measurements of damping properties must be ing. The physical sources of various material-dynamic. On the other hand, measurements of stiff- damping mechanisms are not presented here; ratherness and inertia can be made from static experiments. some general models are presented based on phenom-The need for dynamic experiments in order to verify enological data. Here some basic models that char-analytical damping models has resulted in a great deal acterize the most common methods of modeling areof difficulty in determining the nature of energy reviewed.

J-Q Sun, University of Delaware, Newark, DE, USA between the inclusion of proper damping and the

C. isolation performance of the mounting system. This

opynght © 2001 Academic Press ...artIcle presents the common desIgns of mounts wIth

doi:10.1006/rwvb.2001.0019 various built-in damping mechanisms.Special mounting systems are also discussed in this

• Encyclopedia (see Active Mounts in article ActiveIntroduction control of vehicle vibration' and Isolation of Machin-,Mounts are commonly used to reduce the energy ery in Buildings in article Active control of civil struc-transmission from one mechanical system to another. tures.They are also referred to as vibration isolators.

Mounts ar.e ?ften designe~ to .pro~ide ~amping in Basic Concepts of Mounting Systemthe transmISSIOn path for vIbratIOn IsolatIOn. Damp-ing is therefore incorporated in the mounting system There are two types of applications with a mountingfor energy dissipation. There is a compromise system, as illustrated in Figure 1. The first is to isolate

frequency response can be created to eliminate effec-tively a tonal disturbance by properly tuning thevibration absorber. The transmissibility of a typicalcommercial fluid-elastic mount with tuned fluid iner-tia track is shown in Figure 9. For more discussions onvibration absorber, see Absorbers, vibration.

Multidirectional Mounting SystemsThe mounts discussed so far are unidirectional forone-degree-of-freedom systems. Practical applica-tions often involve multiple degrees of freedom andmultidirectional motion. Schematics of two multidir-ectional and multiple-degree-of-freedom mountingsystems are shown in Figure 10.

The design and analysis of such mounts are gen-erally more difficult and involved. A useful conceptfor evaluating multidirectional mounting systems isthe mount effectiveness. This concept is particularlywell suited to mount design for flexible structuralsystems.

Consider a mounting system for isolating a vibrat-ing source, such as aircraft engines, from the base,such as the aircraft fuselage. Let Vh be the vectorconsisting of all the generalized velocity componentsof the base at the mounting points when the vibratingsource is rigidly connected to the structure. Vi is thevector consisting of the same velocity components ofthe base at the mounting points when the vibratingsource is connected to the structure via the mount.Then, a matrix Ev can be found as a function of thesystem impedance matrices, such that:

Control in the article Active control of civil structures.Active systems can provide better vibration isolation,and are more expensive and less reliable than passivesystems. Semi-active damping represents a com-promise between passive and active systems. Mountswith ER and MR fluids can provide a range ofdamping. Figures 14 and 15 show an ER fluid

Figure 14 Vibration isolator system using an antagonized bel-lows ER damper and its control system. From Jolly MR andNakano M (1998) Properties and Applications of CommercialControllable Fluids. Actuator 98. Bremen, Germany, with per-mission.

vibration isolation system and MR fluid vibrationdampers. These devices are commercially available.

Figure 16 shows typical damping forces of a MRdamper as a function of the current input to the coil.When this damper is used in a vibration isolationsystem, the transmissibility of the system can beimproved substantially.

Figure 17 shows the magnitude of transmissibilityof a system with a MR damper. The vibration isola-tion performance is comparable with the active sys-tem in the lower-frequency range. At higherfrequencies, however, the high-level passive dampingof MR fluids becomes less advantageous, and cannotbe reduced by the control system. Nevertheless, the

Further Reading Ruzicka JE and Derby TF (1971) Influence of damping invibration isolation. In: The Shock and Vibration Mono-

Avallone EA and Baumeister T III (eds) (1978) Marks' graph Series, SVM-7. Washington, DC: Shock andStandard Handbook for Mechanical Engineers. New Vibration Information Center, United States DepartmentYork: McGraw-Hill. of Defense.

Beards CF (1996) Engineering Vibration Analysis with Snowdon JC (1968) Vibration and Shock in DampedApplication to Control Systems. New York: Halsted Press. Mechanical Systems. New York: John Wiley.

Dimarogona A (1996) Vibration for Engineers. Upper Sun JQ, Norris MA and Jolly MR (1995) Passive, adaptiveSaddle River, New Jersey: Prentice Hall. and active tuned vibration absorbers - a survey. 50th

Harris CM and Crede CE (eds) (1996) Shock and Anniversary Issue of ASME Journal of MechanicalVibration Handbook. New York: McGraw-Hill. Design and Journal of Vibration and Acoustics 117:

Jolly MR and Nakano M (1998) Properties and Applica- 234-242.tions of Commercial Control/able Fluids. Actuator 98. Swanson DA and Miller LR (1993) Design and mountBremen, Germany. effectiveness evaluation of an active vibration isolation

Karnopp D (1995) Active and semi-active vibration system for a commercial jet aircraft. Proceedings ofisolation. 50th Anniversary Issue of ASME Journal of AIAA/AHS/ASEE Aerospace Design Conference paperMechanical Design and Journal of Vibration and no. AIAA-93-1145.Acoustics 117: 177-185. Swanson DA, Miller LR and Norris MA (1994) Multi-

Lord Corporation (1997) Rheonetic Linear Damper RD- dimensional mount effectiveness for vibration isolation.1001lRD-l004 Product Information Sheet. Cary, North Journal of Aircraft 31: 188-196.Carolina. Thomson WT and Dahleh MD (1998) Theory of Vibration

Rao SS(1995) Mechanical Vibrations. New York: Addison- with Applications. Upper Saddle River, New Jersey:Wesley. Prentice Hall.

A Baz, University of Maryland, College Park, MD, USA Types of Passive, Active, and HybridCopyright © 2001 Academic Press Damping Treatmentsdoi:10.1006/rwvb.2001.0195 Various passive, active, and hybrid damping control

approaches have been considered over the yearsPassive, active, and/or hybrid damping treatments are employing a variety of structural designs, dampingrecognized as essential means for attenuating exces- materials, active control laws, actuators, and sensors.sive amplitudes of oscillations, suppressing undesir- Distinct among these approaches are the passive,able resonances, and avoiding premature fatigue activ~, ~nd hybrid damping methods ...failure of critical structures and structural compo- It is important ?er~ to not~ that pas.sive dampmgnents. The use of one form of damping treatments or can be very effective m dampmg out high-frequencyanother in most of the newly designed structures is excitations whereas active damping can be utilized tobecoming very common in order to meet the pressing c?ntrol low-frequ.ency vibrations, as shown inneeds for large and lightweight structures. With such Figure 1. For effective control over a broad frequencydamping treatments, the strict constraints imposed band, hybrid damping methods are essential.on present structures can be met to insure their P . D .ff' .. d bl I f f asslve ampmge ectlVe operatIOn as qmet an sta e p at orms or

manufacturing, communication, observation, and Passive damping treatments have been successfullytransportation. used, for many years, to damp out the vibration of a

This article presents the different types of passive, wide variety of structures ranging from simple beamsactive, and hybrid damping treatments. Emphasis is to complex space structures. Examples of such pas-placed on presenting the fundamentals of active con- sive damping treatments are given below.strained layer damping (ACLD), one of the mostcommonly used class of hybrid treatments, which Free and constrained damping layers Both types ofcombines the attractive attributes of both the passive damping treatments rely in their operation on the useand active treatments. of a viscoelastic material (VEM) to extract energy

In the compression MCLD configuration of, , , . Figure SA, the magnetic strips (1 and 2) are magne-

from the vlbratmg stru~ture, as sh~wn m Figure 2. In tized across their thickness. Hence, the interactionthe free (or unconstramed) dampmg treatment, the between the strips generates magnetic forces that arev,ibrational ener?y is dissipated by virtue o~ the exten- perpendicular to the longitudinal axis of the beam.slOnal deformatIOn of the VEM, whereas m the con- These forces subject the viscoelastic layer to cross thestrained damp,ing treatment more energy is dissipated thickness loading, which makes the treatment act as athrough sheanng the VEM. Den Hartog dynamic damper. In the shear MCLD

configuration of Figure SB, the magnetic strips (3 andShunted piezoelectric treatments These treatments 4) are magnetized along their length. Accordingly, theutilize piezoelectric films, bonded to the vibrating developed magnetic forces, which are parallel to thestructure, to convert the vibrational energy into elec- beam longitudinal axis, tend to shear the viscoelastictrical energy. The generated energy is then dissipated layer. In this configuration, the MCLD acts as ain a shunted electric network, as shown in Figure 3, conventional constrained layer damping treatmentwhich is tuned in order to maximize the energy whose shear deformation is enhanced by virtue ofdissipation characteristics of the treatment. The elec- the interaction between the neighboring magnetictric networks are usually resistive, inductive, and/or strips.capacitive.

Damping with shape memory fibers This dampingDamping layers with shunted piezoelectric mechanism relies on embedding superelastic shapetreatments In these treatments, as shown in memory fibers in the composite fabric of the vibratingFigure 4, a piezoelectric film is used to constrain structures, as shown in Figure 6A. The inherent hys-passively the deformation of a viscoelastic layer teretic characteristics of the shape memory alloywhich is bonded to a vibrating structure. The film is (SMA), in its superelastic form, are utilized to dissipatealso used as a part of a shunting circuit tuned to the vibration energy. The amount of energy dissipatedimprove the damping characteristics of the treatment is equal to the area enclosed inside the stress-strainover a wide operating range. characteristics (Figure 6B). This passive mechanism

has been successfully used for damping out the vibra-Magnetic constrained layer damping (MCLD) These tion of a wid~ variety, of ,struc~ure,s, including largetreatments rely on arrays of specially arranged per- structures subject to seismICexcitatIOn.manent magnetic strips that are bonded to viscoelas-tic damping layers. The interaction between the Active Dampingmagnetic strips can improve the damping character- Al h h h 'd' h d d 'b d' , , , t oug t e passive ampmg met 0 s escn eIStlCSof the treatments by enhancmg either the com- b '1 d I' bl h' ff t' ,, ," a ove are simp e an re la e, t elr e ec Iveness ISpressIOn or the shear of the viscoelastIC dampmg I' , d 'b f1 h' F· S Imlte to a narrow operatmg range ecause 0ayers as sown m Igure . h "f' "f h d' , 1' t e slgm ICant vanatlOn 0 t e ampmg matena

Figure 5 (SeePlate 20). Configurations of the MClD treatment. (A)Compression MClD; (8) shear MClD.

properties with temperature and frequency. It is ments aim at using various active control mechanismstherefore difficult to achieve optimum performance to augment the passive damping in a way that com-with passive methods alone, particularly over wide pensates for its performance degradation with tem-operating conditions. Hence various active damping perature and/or frequency. These treatments alsomethods have been considered. All of these methods combine the simplicity of passive damping with theutilize control actuators and sensors of one form or effectiveness of active damping in order to insure ananother. The most common types are made of piezo- optimal blend of the favorable attributes of bothelectric films bonded to the vibrating structure, as damping mechanisms.shown in Figure 7.

This active control approach has been successfully Active constrained layer damping This class ofused for damping out the vibration of a wide variety treatments is a blend between a passive constrainedof structures ranging from simple beams to more layer damping and active piezoelectric damping, ascomplex space structures. shown in Figure 8. The piezofilm is actively strained

so as to enhance the shear deformation of the viscoe-Hybrid Damping lastic damping layer in response to the vibration ofBecause of the limited control authority of the cur- the base structure.rently available active control actuators, and becauseof the limited effective operating range of passive Active Piezoelectric damping composites (APDC) Incontrol methods, treatments which are a hybrid com- this class of treatments, an array of piezoceramicbination of active damping and passive damping rods embedded across the thickness of a viscoelastictreatments have been considered. Such hybrid treat-

polymeric matrix is electrically activated to controlthe damping characteristics of the matrix that isdirectly bonded to the vibrating structure, as shownin Figure 9. The figure displays two arrangements ofthe APDC. In the first arrangement, the piezorods areembedded perpendicular to the electrodes to controlthe compressional damping and in the secondarrangement, the rods are obliquely embedded tocontrol both the compressional and shear dampingof the matrix.

Electromagnetic damping composites (EMDC) Inthis class of composites, a layer of viscoelastic damp-ing treatment is sandwiched between a permanentmagnetic layer and an electromagnetic layer, asshown in Figure 10. The entire assembly is bondedto the vibrating surface to act as a smart dampingtreatment. The interaction between the magneticlayers, in response to the structural vibration, subjectsthe viscoelastic layer to compressional forces ofproper magnitude and phase shift. These forces coun-terbalance the transverse vibration of the base struc-ture and enhance the damping characteristics of theviscoelastic material. Accordingly, the EMDC acts ineffect as a tunable Den Hartog damper with the basestructure serving as the primary system, the electro-magnetic layer acting as the secondary mass, themagnetic forces generating the adjustable stiffnesscharacteristics, and the viscoelastic layer providingthe necessary damping effect.

Active shunted piezoelectric networks In this classof treatments, shown in Figure 11, the passiveshunted electric network is actively switched on andoff in response to the response of the structure/net-

work system in order to maximize the instantaneousenergy dissipation characteristics and minimize thefrequency-dependent performance degradation.

Basics and Characteristics of a TypicalHybrid TreatmentsEmphasis is placed here on presenting the theory andthe performance characteristics of one of the mostwidely used class of active/passive hybrid dampingtreatments, the ACLD treatment.

The ACLD treatment is a new class of hybrid damp-ing treatments which has a high energy dissipation-to-weight ratio compared to conventional constrained orunconstrained damping layer configurations.

The ACLD consists of a conventional passive con-strained layer damping which is augmented withefficient active control means to control the strainof the constrained layer, in response to the structuralvibrations, as shown in Figure 12. The viscoelasticdamping layer is sandwiched between two piezoelec-tric layers. The three-layer composite ACLD whenbonded to a vibrating surface acts as a smart con-straining layer damping treatment with built-in sen-sing and actuation capabilities. The sensing isprovided by the piezoelectric layer which is directlybonded to the vibrating surface. The actuation isgenerated by the other piezoelectric layer which actsas an active constraining layer. With appropriatestrain control, through proper manipulation of thesensor output, the shear deformation of the viscoe-lastic damping layer can be increased, the energydissipation mechanism can be enhanced, and struc-tural vibration can be damped out.

In this manner, the ACLD provides a viable meansfor damping out the vibration as it combines theattractive attributes of passive and active controls.This makes the ACLD particularly suitable for criticalapplications where the damping-to-weight ratio isimportant, e.g., aircraft and automobiles.

Concept of active constrained layer damping Theeffect of interaction between the sensor and theactuator on the operation of the ACLD can best beunderstood by considering the motion experienced by

R B Randall and M J Tordon, University of New South which in its simplest form involves reading valuesWales, Sydney, Australia from instruments and recording the observations on a

Copyright Ij,) 2001 Academic Press data sheet. Even in this simple form we are represent-ing a continuous signal by values recorded at discrete

doi:10.1006/rwvb.2001.0142 times with a limited number of discrete digits.Advances in digital computers have provided efficient

Background and fast means of coll.ecti?~ and proc~ssing signalsrepresented and stored 10 dIgItal form. FIgure 1 shows

In order to process analog signals digitally it is a schematic diagram of a general data acquisitionnecessary to convert them into digital form. The (DAQ) system.object of a data acquisition system is to collect and A transducer changes a physical phenomenon intorecord data from physical phenomena in the real an electrical signal. The aim is to produce electricalworld, which by nature are continuous in both ampli- signals which represent the physical phenomenatude and time. This can be a labour-intensive process, investigated, while at the same time minimizing the

provide a better representation of the original signal.To avoid aliasing, the signal must be sampled at leastat twice the frequency of the maximum frequencycomponent of the input signal, as discussed above.However, for the applications where no anti aliasingfilters are used, the sampling rate is typically muchhigher, say 10-20 times the highest frequency to beviewed, so as not to distort the appearance of thesignal.

Sampling methods Due to the cost of the ADC, theDAQ system often contains only one ADC and uses amultiplexer to switch between channels. The datachannels may then be sampled in sequence so thatonly one conversion is taking place at anyone time.This method of sampling is suitable for applicationswhere the exact time relationship between sampledsignals is not important. If the time (and phase)relationships between input signals are importantthe inputs must be sampled simultaneously. Toachieve simultaneous sampling we need sample-and-hold circuitry for each input channel, which will allowsimultaneous sampling and subsequent sequentialconversion of instantaneous values of input signals.

Multiplexing Multiplexing is a technique for mea-AID Converter Specifications suring several signals with a single ADC. The multi-

.... plexer selects one input channel at a time and routesTh: speClf1eatiOns of the NO converter determme :h.e the signal to the ADC for digitizing. The effective~aJor performance p.arameters of the data acqUlsI- sampling rate per channel is reduced by a factor equaltiOn system. The basIc parameters of the ND con- to the number of channels sampled.verter (ADC) include the resolution, maximumsampling rate, accuracy, dynamic range, nonlinearity Resolution Resolution is the smallest signal incre-(and man~ other.s~ ... ment that can be detected by a measurement system .. The ba.sIc specIfIcatiOns o! ~~e DAQ system provIde Resolution can be expressed in bits, as a proportion,mformatiOn o~ both capabIlItIes and accuracy of ~he or in percent of full scale. For example, if a system hassystem. The! mclude. number of channels, samplmg 8-bit resolution, this corresponds to one part in 256,rate, resolutiOn, and mput range. and 0.39% of full scale. The higher the resolution, the

smaller the detectable voltage change. Figure 6 showsAnalog inputs In practice we collect more than one a sine wave obtained by an ideal 3-bit ADC, whichvariable so the system should include provisions for divides the analog range into eight divisions. Eachcollecting data from multiple channels. The input division is represented by a binary code between 000channels can be single-ended or differential. Single- and 111. It can be seen that quantization of the inputended inputs are referenced to a common ground and signal introduces irreversible loss of information.due to a higher level of noise they are typically usedfor high-level (> 1 V) input signals. Differential Range Range refers to the minimum and maximuminputs on the other hand respond to a potential voltage levels that the ADC can span. The range of thedifference between two terminals. Noise or other ADC can be changed by selecting a different amplifiersignals present in both terminals, referred to as com- gain. The range, resolution, and gain of the DAQmon-mode voltage, are cancelled out. The term com- system determine the smallest detectable change inmon-mode voltage range describes the ability of the voltage. This change in voltage corresponds to thesystem to reject common-mode voltage signals. least significant bit (LSB) of the digital number and is

often called the code width. The ideal code width isSampling rate Sampling rate specifies how often the found by dividing the voltage range by the expressionconversion takes place. A faster sampling rate will (gain times number of codes). For example, DAQ

Critical Parameters of Analog Inputs

The parameters mentioned above, such as samplingrate, resolution, and range, describe the overalllim-itation of the DAQ system. Additional parameters,such as nonlinearity, relative accuracy, settling time,and noise specifications, are needed to describe theactual performance of the DAQ system.

Nonlinearity The plot of the voltage versus theoutput code for an ideal ADC should be a straightline. Deviation from the straight line is specified asnonlinearity. Several terms are used to express thisproperty. Differential nonlinearity (DNL) is a mea-sure in LSB of the worst-case deviation of the analogcode widths from their ideal value of 1 LSB.A perfectDAQ system has a DNL of 0 LSB. A good DAQsystem will have a DNL within ± 0.5 LSB. Nonli-nearity (or integral nonlinearity) is a measure of theworst-case deviation from the ideal transfer function(a straight line) of the system expressed in percent offull-scale range (FSR).

Relative accuracy The relative accuracy of an ADCis a measure in LSBsof the worst-case deviation fromthe ideal transfer function of a straight line. It can beobtained by sweeping an applied voltage through therange and digitizing it. Plotting the digitized pointsresults in an apparent straight line. If we subtract anactual straight line from the apparent straight line, asshown in Figure 7, we can see the deviations fromzero across the range. The maximum deviation fromzero as a proportion of the FSR is the relative accu-racy of the DAQ system. The relative accuracyincludes all nonlinearity and quantization errors. It

Figure 7 Deviation of encoded values from the ideal straightline. Courtesy of National Instruments 1996.

does not include offset and gain errors of the circuitryfeeding the ADC.

Settling time In systems with a multiplexer we haveto take into account the time required for the signalsat the input of the ADC to settle after changingchannels. The duration required by the system tosettle to a specified accuracy is called the settling time.

Discretization Errors, Dynamic RangeThe process of converting a discrete-time signal into adigital signal by expressing each sample value as afinite number of digits is called quantization. Theprocess of quantization is a many-to-one mappingand is both nonlinear and noninvertible and as suchresults in loss of information. Loss of information isdirectly linked to the number of discrete levelsavailable for the process. The resulting digital infor-mation is stored in words of finite length expressed inbits. A wordlength of B bits can represent N = 2B

different discrete levels. The signal-to-noise ratio(SNR) is used to quantify the effect of quantizationnoise (errors) resulting from the finite word length ofthe conversion.

Figure 8 shows an example of a quantized sinewave using a wordlength of 3 bits. The normalized(from 0 to 1) continuous signal is represented by eightcodes representing equally spaced steps. The step size

M Sidahmed, Universite de Technologie deCompiegne, Compiegne, France

Copyright © 2001 Academic Press

doi:1 0.1 006/rwvb.2001.0147

IntroductionDiagnostics and condition monitoring is a multidis-ciplinary subject combining various techniques andareas of knowledge: mechanical engineering, reliabil-ity techniques, measurement procedures, signal pro-cessing, data analysis, and expert or knowledge-basedsystems. The objectives of diagnostics and conditionmonitoring of machinery include:

• Control of the machinery, safety is vital whendealing with high power or dangerous machines.

• Optimizing the availability of machines by avoid-ing unexpected shutdowns. This is particularly truefor critical machines in a continuous productionprocess.

• Implementation of condition-based maintenance,or more specifically predictive maintenance, forwhich the operations are planned according tovarious constraints (cost, production, failure con-dition, etc.).

This explains the importance of such techniqueswhich are extensively used in industry.

Diagnostic and condition monitoring is based onthe measurements of physical parameters on themachines in order, after specific processing, to assessthe state of the machines and to identify any existingfailures that may give rise to catastrophic damage.For any physical parameters, it obeys some generalprinciples and methodology which we will describe.Vibration diagnostics and condition monitoring isthen considered.

378 DIAGNOSTICS AND CONDITION MONITORING, BASIC CONCEPTS

appears. If the parameter has a known (estimated) exist which allow us to establish the symptomsprobability distribution, then a statistical test may be related to various faults. The capability of currentderived. Note that in this case, the detection and analysis to diagnose mechanical faults has been estab-diagnostic processes are the same. Prognosis is neces- lished (the motor sensing the mechanical load varia-sary to predict the 'safety limit'. tions) but is still under investigation.

Methods Used for Machinery Condition Monitoring Vibration analysis This is the most widely usedtechnique for diagnosing faults in rotating machines,

Various methods are used for condition monitoring, this is due to three reasons:they depend on the physical parameters to be mea- • Signal generation models associated with the majorsured and processed to deliver the diagnostic. Each faults generating dynamic efforts and then vibra-method has advantages and drawbacks, thus it is tions, have been developed. This gives a goodadvantageous to combine the different methods toobtain more reliable results. knowledge of the symptoms of the faults, and

reliable reference signatures.Oil analysis Most of the machinery requires lubri- • The technique is not intrusive and does not requirecants to minimize wear, for example diesel engines, the machine and then the production to be stoppedturbines, or components such as gear boxes or rolling (thus we may consider vibration monitoring as abearings. Oil analysis may give information on the kind of nondestructive testing)lubricant quality and on the wear metal contamina- • The sensors are low cost and, in general, reliable,tion. This last point is used for diagnosis. In complex the associated data analysis systems are increas-machinery, this technique may complement vibration ingly powerful.analysis in order to provide a more reliable diagnosis. In recent years emphasis has been placed on theThis method suffers, however, from difficulties in development of advanced signal processing techni-collecting 'good' oil samples. ques for incipient fault detection and diagnosis. These

Nondestructive techniques 'Active' techniques are detailed in the next section. We may also note,

(ultrasonics, radiography, etc.) are used to determine that standards have been developed to classify

the state of various materials: homogeneity, stress, machine vibrations into acceptable or nonacceptable

cracks, and quality of welding. The control of specific vibration levels. These levels are derived from very

parts of the rotating machine (blades, shafts, etc.) simple processing such as power in the band (10-

may be carried out. Passive techniques such as acous- 1000Hz). These deterministic levels are used asdetection limits (Figure 3). The diagnostic ability oftic emission (AE) take advantage of the sound waves these limits is very limited. It should be noted thatemitted by materials when growing cracks or loca- vibration frequency analysis standards are currentlylized constraints are present. Sound waves have very under development.high frequency components and are amplified by The various techniques for condition monitoringresonant sensors (from about 100 kHz to a few

Mhz). The complexity of sound propagation in struc- and diagnostics may be used for quality control of

tures makes the interpretation difficult. AE is used for components that are to be tested during production.They are usually used for low-cost components suchincipient detection but not for diagnosis. This techni- as small DC motors, automotive gearboxes, etc.que is used extensively for tool wear monitoring. The detection and diagnostic procedures may beSome applications have been developed for incipient

detection of faults in rolling bearings. developed in real time when we deal with controlledmachinery (for machine tools it is necessary to detect a

Infrared or thermographic analysis Infrared cameras tool breakage in real time). In general, when real timeare used to detect differences in surface temperatures is required, detection is addressed rather than diag-which may be due to specific faults. This technique nosis. In most cases, for rotating machines, signatureappears to be more suitable for the diagnosis of faults analysis techniques are carried out periodically.in electrical or electromechanical equipment (e.g.,transformers) . Vibration Condition Monitoring andCurrent analysis This is used for the detection and Diagnosticdiagnosis of electrical problems in motor driven Vibration Signaturesrotating equipment. This technique is of interestbecause it is not intrusive, the information being Vibration monitoring is suitable to detect faults gen-available on the motor power cables. Various models erating dynamics giving rise to vibrations that may be

This display may be used to study the possible mode shape plots reveal important information, suchresonances of bladed disks. In a perfectly uniform as that described below.system, each point on the diagram represents a pair ofmodes with orthogonally oriented identical modal 1. The relative amplitude of displacement through-patterns. Due to imperfections or to blade variability, out the structure and the location of nodal pointsin actual bladed disks the two modes separate and of real modes. Figure 4 illustrates the lowest threetake on slightly shifted natural frequencies, com- planar mode shapes of a cantilever beam. Forplicating the picture. An interference diagram (see straight beams, there is a direct correlation be-Blades and bladed disks) helps to locate modes with tween the mode index and the number of nodalclose natural frequencies being excited at the same points, a fact which helps in measurements.engine speed. If stationary nodes are not visible on pseudoa-

M d Sh PI nimated displays, then the modes are not real.o e ape ots C I d hObo " d"omp ex mo es ex 1 It nonstatlOnary zero- IS-Probably the displays most often used are those of placement points, at locations that change inmode shapes. While animation offers 'live' displays, space periodically, at the rate of vibration fre-showing how a structure actually moves at a natural quency. These complex modes are sometimes con-frequency, hard-copy print-outs can contain only verted into equivalent real mode shapes. Thefrozen images, i.e., pseudo-animated displays. Static complexity of measured modes is observed using(nonanimated) displays, containing zero deflection so-called compass plots (Figure 5), i.e., vectorand the maximum distortion frame, are often suffi- diagrams in which each modal vector element iscient to serve the desired goal. Three-dimensional represented by a line of corresponding length andstructures may be represented in wire frame format inclination, emanating from the origin. The com-or in solid hidden-line format. plex mode shape is then rotated through an angle

From the wide range of applications in vibrations, equal to the mean phase angle of its entries.several typical examples can be chosen, where the Finally, for lightly damped structures, the real

Figure 23 Multivariate mode indicator function plot.

data as the previously presented CMIF and MIF is phase portraits in the phase plane (velocity versusshown in Figure 25. Double modes are indicated by displacement) and three-dimensional orbits in thetwo overlaid curves. phase space (see Nonlinear systems analysis).

There are some other displays used as potential Time domain analysis displays, such as auto-mode identifiers, like the stabilization diagram, the and cross-correlation function plots, supplementplot of the sum of all available FRFs, and plots of spectrum plots in spectral analysis (see Correlationother composite functions displaying the magnitude functions).squared or the imaginary part squared, to enhance thepeaks.

See also: Blades and bladed disks; Correlation func-Other Displays tions; Modal analysis, experimental, Parameter extrac-

tion methods; Nonlinear systems analysis; RotatingSpecial displays are used to describe nonlinear vibra- machinery, essential features; Rotor dynamics; The-tions and chaotic motions, like phase trajectories and ory of vibration, Fundamentals.

A Steindl and H Trager, Vienna Universityof which experience shows that an oscillatory motionTechnology,Vienna,Austria sets in. In the language of stability theory the origin-

C . hI "" 2001 A d . P ally stable state has lost stability. Now three questionsopyng ~ ca emlc ressare of interest:

doi: 10.1 006/rwvb.2001 .0047

1. What is the value of Ac at which the loss of stabilityIntroduction occurs?

2. What type of motion is setting in after loss ofOne of the key problems in stability theory is the loss stability of the originally stable state and how isof stability of a given stable state, for example an the motion changing by further increasing theequilibrium or a periodic motion, of a nonlinear parameter A?dynamical system under variation of a system para- 3. How do small changes of the system (small varia-meter, which will be called A. Let us consider, as a tions of other system parameters) affect the sys-typical example, a straight vertically downhanging tern's behavior?tube conveying fluid, clamped at the upper end andfree at the lower end under a quasistatically increas- Under mild requirements on the smoothness of theing flow rate. (In general one understands by a system description, it is possible to give completelyquasistatic parameter variation that the stability satisfying answers to these questions.behavior of the state is studied at gradually increasing The answer to the first question is supplied frombut fixed values of the parameter. This is in contrast the solution of a linear eigenvalue problem and hasto the case of (slowly) continuously varying para- been known to engineers for a long time. The answersmeter values, on which we comment later in this to the second and third questions, however, can onlyarticle). It is well known from everyday experience, recently be given, in a systematic and complete way,for example from hosing the garden, that for small due to strong progress achieved in the 1970s andflow rate the vertically downhanging equilibrium 1980s in the field of nonlinear stability theory makingposition is stable. However, increasing the flow rate use of the methods of local bifurcation theory.A it finally reaches a critical value (denoted by Ac) for Here, as pioneers among many others, the names VI

F Naeim, John A Martin& Associates, Inc., which is usually summarized in the form of designLos Angeles, CA,USA spectra.

Copyright © 2001 Academic Press While a properly established design ground motioncriteria is expected to provide a consistent expression

doi:10.1006/rwvb.2001.0067 of demand, regardless of its form, overzealous

B Old" d" d h ,". emphasis on one form over the others, without properUI mg structures are eSIgne to ave a seIsmIC .. " "" , h d h .. d'"" understandmg of the strength and lImItatIOns of eachcapacIty t at excee s t e antICIpate seIsmIc .".""d d' C "" 1 f . f h form, can result m unrealIstIc desIgn ground motIOneman . apacIty ISa comp ex unctIon 0 strengt , "

Off d d f bOlo " d b h reqUIrements.Stl ness an e orma 1 Ity conjecture y t e system E I" f h " . h d " ." .. " va uatIOn 0 t e seIsmIC azar at a gIven sIteconfIguratIOn and matenal propertIes of the struc- " " f l"k 1 h k d" " ." "" "reqUIres an estImate ole y eart qua e grounture. GIven a partIcular bUIldmg, seIsmICdemand IS " h· f h b ·ld" Th·" b11 d b h " 1 d h 'd" motIOns at t e sIte 0 t e UI mg. IS IS ecause:contro e y w at IScommon y terme t e eSIgn "" "d . ""' h· h b d f" d" (1) sItes for whICh a recorded earthquake groundgroun motIon cntena , w IC may e e me m one "" " .

f h f 11 . h d"· f motIOn IS readIly avaIlable are extremely rare, andor more 0 t e 0 owmg tree Istmct orms: ("") f h" h h d"11 even or t e sItes were suc recor mgs are

1 S "b h d 1 1 f d" ·b" available, there is no guarantee that future ground· tatIC ase s ear an atera orce Istn utIOn " " " .f 1 motIOns wIll have the same exact charactenstIcs oformu as; . I b d . P ·bl d2 A f 'd "' prevIOUSy 0 serve motIOns. OSSI e groun· set 0 eSIgn spectra; . f" "d b f"3 A " f h k 1 " '" h" "' motIOns or a sIte are estImate y use 0 vanous· sUIte0 eart qua e acce eratIOn tIme Istones. " 1" h" 1 d b fregressIOn ana YSIStec mques on a se ecte su set 0

The design ground motion may be defined in its available earthquake recordings deemed proper formost simple form by application of simple design base such estimation. The resulting mathematical formu-shear equations and static lateral force distribution las which provide estimates of maximum responseformulas such as those embodied in a typical building parameters, such as peak ground acceleration orcode. These simple formulas are in essence, simplified response spectral ordinates for a site are called 'pre-interpretations of a design spectrum of certain shape dictive relations' or 'attenuation relations'. The termand amplitude at the fundamental vibration period of attenuation is used because these empirical relationsthe building. in fact represent formulas for attenuation of seismic

For more complex analyses, the design ground waves originating from a given source, at a givenmotion criteria may be defined by a series of either distance, through a given medium (i.e., site soil con-code-specified or site-specific design spectra and rules ditions). Dozens of attenuation relations have beenon how to apply these spectra and how to interpretthe developed and are in use today (Table 1).results. If seismic design of a project requires applica-tion of dynamic time-history analysis, then an appro-priate set of earthquake records have to be selected Characteristics of Earthquake Groundand rules have to be established on how these records Motionsare to be applied in analysis and design. The earth-quake records, in this case, are needed in addition to a The number of earthquake records available hassite-specific design spectra, and rules have to be set on grown rapidly during the past decade. While obtain-how application of these records produces a demand ing earthquake accelerograms was not simple up tothat is consistent with the site-specific seismic hazard, about the mid-1980s, hundreds of earthquake records

D Smallwood, SandiaNationalLaboratories, 1. Development testing is intended to explore theAlbuquerque,NM, USA response of the test hardware to a variety of vibra-

Copyright © 2001 Academic Press tion stimulations with the goal of identifying vi-bration characteristics of the hardware or to

doi:10.1006/rwvb.2001.0106 uncover design weaknesses of the hardware.Typically, the hardware tested is not production

Environmental testing can cover a large variety of hardware but development hardware in variousnatural and artificial environments. In the context of stages of development.this article, the environments are limited to mechan- 2. Qualification testing is used to determine if theical vibration. Sometimes it is important to consider hardware will perform satisfactorily when ex-combinations of environments, but this is beyond the posed to a representative or worst-case use envir-scope of this article. Environmental testing can onment. The purpose is to determine if the designinclude the exposure of test hardware to actual use meets a set of vibration requirements. Typically,environments. This is also beyond the scope of this the hardware tested is representative of produc-article. Environmental testing in this article is the tion hardware, but is not hardware that will bereproduction in the laboratory of a vibration envir- used in the use environment. Qualification testingonment that is intended to simulate a real or potential is typically performed with methods and levelsnatural or use vibration environment. Some typical defined in specifications.environments that produce significant vibration 3. Acceptance testing is used to determine if a par-levels include: surface transportation in cars and ticular set of production hardware is satisfactorytrucks with vibration caused by irregular surfaces, for release and subsequent use by a customer. Allair turbulence, and mechanical noise from engines, production hardware is not necessarily subjectedetc.; aircraft environments with vibration caused by to acceptance testing. Some programs demandairflow and engines; rocket environments with vibra- 100% acceptance testing, and some programstion caused by airflow, shock waves, engine noise, accept lot testing. The test methods and levelsand stage separation; ocean environments with vibra- are typically defined in specifications. Sometimestion induced by wind and waves; building vibration the acceptance test levels are related to thewith motion excited by internal and external machin- qualification levels. The levels are typically lowerery, wind, and earthquakes. Structural response to than the qualification levels because the risk ofmechanical shock can be considered a transient vibra- damage to the hardware must be kept to ation and is discussed. minimum.

Environmental testing generally falls into one of 4. Stress screening is a tool used to expose hardwarefour classifications: development testing, qualifica- at various stages of manufacture to vibration en-tion testing, acceptance testing, or stress screening. vironment that will precipitate infant failures and

identify manufacturing defects early in the manu- the statistical measures are not dependent on thefacturing cycle. The methods and levels are de- location in time. As for periodic data, random datafined by the manufacturer and tailored to a theoretically extend over all time. In practice muchspecific product. The levels should not induce data can be treated as stationary if the duration isdamage to well-manufactured hardware. much longer than the period of the lowest natural

Th d 1 f' h d d frequency of the object under study. The most com-e eve opment 0 enVironment test met 0 s an ,1 1 b d' 'd d ' F' , h mon procedures used to analyze statiOnary randomeve scan e iVi e mto two stages. irst is t e , ,·d 'f" d h " f h ' data are the auto spectral densities (commonly calledi enti icatiOn an c aractenzatiOn 0 t e enViron- ,

S d' h d 1 f h d the power spectra density) and the cross-spectralments. econ is t e eve opment 0 test met 0 s , . , , , .th 'II ' f '1' 1 h' ,densities. The statistiCal moments and probabilityat Wi satis acton y simu ate t e environment m " ,th 1 b density functions are also important parameters.

e a oratory. These are described elsewhere in this volume. By farIdentification of the Use Environment the most common assu~p~ion about t,he data is t?at

they are normally distnbuted. This assumptiOnBy far the most common field measurements are with should always be checked because much data is notaccelerometers. The use of accelerometers is covered normal.elsewhere. Much less frequently used are measure- If the duration of random data is too short to bements of velocity, displacement, and force. The data treated as stationary, it must be treated as nonsta-from the field measurements are collected and stored tionary random data. Many techniques are availablefor later analysis. The most versatile method is to to treat nonstationary data. All the techniques addstore the raw unprocessed time histories. If the origi- complications to the analysis, and the results will benal data are stored, data-processing decisions can be dependent on the method and parameters used. Sev-changed at any time. Recently most data are sampled era1of these techniques are discussed elsewhere in thisand stored as digital records. All the precautions volume but are beyond the scope of this discussion.associated with this process should be carefully The most common error made in nonstationary ran-observed (see Further Reading). dom analysis is to ignore the uncertainty theorem.

The next step is to classify the data for reduction. Loosely, the uncertainty states that you cannotData will seldom fall neatly into a classification. resolve time and frequency independently. If youJudgment must be used. Sometimes data can be desire good time resolution, you will have pooranalyzed making several assumptions of the class, frequency resolution and vice versa.and the most appropriate reduction can be chosen One is tempted simply to reproduce the field envir-after the data reduction. onment in the laboratory. If this is done, the field

The data must first be classified as random or environment is being treated as deterministic. If adeterministic. Loosely deterministic data are data random component is present, the reproduction willthat, if the experiment is repeated, the data will be not necessarily be adequate.the same. Deterministic data can be a transient, Data can also be a mixture of all the types. Theseperiodic, or complex nonperiodic. A transient is problems are difficult to handle. Usually somedata that starts and ends within a few periods of attempt is made to separate the data into the compo-the lowest natural frequency of the object being nent parts and each part is analyzed separately.observed. Periodic data theoretically extend over all Periodic environments are seldom encountered intime, repeating at a regular period. In practice a signal the field. Exceptions are rotating machinery meas-that repeats itself and extends over many cycles of the urements that often contain the fundamental andlowest natural frequency can be treated as periodic. A harmonics of the rotational frequency. Randomperiodic waveform can often be treated as a Fourier environments usually treated as stationary randomseries expansion. Complex nonperiodic data is typi- include: excitation from turbulent fluid flow, rocketcally composed of the sum of periodic waveforms that and turbojet excited response, wind excitation, andare not harmonically related; thus the waveform does long-term surface transportation environments.not have a period. As for periodic waveforms, the Deterministic transient environments can includewaveform must extend over a period of time that is some shocks and chirps (a short-duration sine sweep).much longer than the lowest natural period of the Examples of nonstationary random environmentsobject being observed. include: seismic excitation (earthquakes); pyroshock

Random data are data that can be described only in (transients in structures caused by explosive hard-statistical terms. The future time history cannot be ware, like bolt-cutters); response of structures topredicted from past values except in statistical terms. non penetrating impacts; and response to bumps andThe data can be stationary, which loosely means that potholes in surface transportation. To the extent that

494 ENVIRONMENTAL TESTING, OVERVIEW

The test environment will always be larger than the is determined for a 'zone' of the structure. Care mustfield environment for this special case. If forces are be taken to insure the measurement is typical and notapplied to the test object at any location other than at greatly influenced by local phenomena. For example,the control point, we are not guaranteed that the test if an accelerometer is mounted on a thin plate, theenvironment will generate responses greater than or motion can be dominated by the local plate reso-equal to the field environment. Unfortunately, this is nances and not be representative of the structuralfrequently the case. The forces in the field are often response in the neighborhood.applied in a distributed manner over the test object, In some cases it may be desirable to measure multi-and motion is measured at one or a few points. The pIe inputs to a component or system. If this is donemotion is enveloped, and the envelope is reproduced the phase relationships between the inputs must beat the control points (usually a single point) in the preserved, as this will be of critical importance if alaboratory with the sometimes unreasonable expec- multiple-input test is designed.tation that the laboratory test is a conservative test.

Motion in the field is seldom in one direction. Even Test Methodsif the motion of the control point is in a singledirection, the motion of the test object in the labora- Many test machines are used to generate vibration intory is seldom in a single direction. A typical assump- the laboratory. Several of these are discussed intion is that the motion observed in the field can be greater detail in other articles. The most commonadequately simulated with three tests in the labora- machines are electrodynamic and electrohydraulictory, one in each of three orthogonal directions. At shakers.best this is a practical compromise and has little An electrodynamic shaker is built on the samerigorous development to justify the assumption. principle as a speaker in a radio or home entertain-

For those cases where eqn [17] is valid, the motions ment system. A magnet (either permanent or anat the response points can be very much larger than electromagnet) provides a magnetic field. A coil ofthe field response. The very high input impedance of wire is placed in the field. When a current is passedshakers, coupled with large power amplifiers and through the coil, a force is generated. This forcemodern control equipment, typically causes the overt- moves the coil and the attached structure thatest. The control system will attempt to maintain the includes the test item.motion at the control point regardless of the force An electrohydraulic shaker is essentially a double-required. In the field the driving point impedance of acting hydraulic cylinder. Most of the energy isthe interface looking back into the structure on which supplied by hydraulic fluid under pressure. A servothe test object is mounted limits the force. The overt- valve (often driven by a small electrodynamic shaker)est caused by a near-infinite impedance of the shaker controls the fluid flow in the hydraulic cylinder.can be alleviated through the use of force or response Vibration can also be generated by a variety oflimiting. mechanical shakers. These devices have limited use

As can be seen from the above discussion, practical today because the versatility and control of thevibration tests require many compromises. Even for a devices are usually not as good as for electrodynamiclinear system, vibration testing IS an imperfect and electro hydraulic shakers.science. Engineering judgment and past experience The testing of aerospace structures is often accom-are valuable guides in picking an appropriate test. plished by placing the test item in a reverberant

chamber and exposing the item to high-intensity

Identification of Measurement acoustic noise. A reverberant chamber is a large

Locations chamber with hard walls and many modes of acous-tic vibration. The chamber provides a diffuse acous-

An important consideration in gathering field infor- tic field that simulates the high-intensity noise ofmation that will later be used to characterize a vibra- many acoustic environments found in the aerospacetion environment IS the determination of the industry.measurement locations. Usually an attempt is made Many machines generate mechanical shock, whichto place the measurement location in the load path of can be considered a transient vibration. The mostthe input to the system. This is not always possible, as common of these machines are drop tables. The dropmultiple load paths may exist or the interface between table essentially generates a specified velocity. Thethe item and its foundation is not accessible. In many shock is generated when the test item impacts acases the number of input points desired is beyond the stationary structure. The characteristics of the shockcapabilities of the data-gathering system. In this case are determined by the design of the impacting struc-'typical' locations are chosen and the resulting motion tures, including the interface between items. The

ENVIRONMENTAL TESTING, OVERVIEW 495

material placed at the interface is sometimes called a Fixture Designshock programmer. Many other devices are used,essentially to generate a velocity. The kinetic energy As discussed earlier, fixtures are usually designed asis used as the energy source for the shock test. The rigid as possible for several practical reasons. First isshock produced is determined by the details of the the previously mentioned assumption of a singleimpact between the moving structure and the station- control point. If the fixture is not rigid, this assump-ary structure. The test item is usually mounted on the tion is obviously flawed. Another major practicalmoving structure. In some cases the test item is reason is control of the test. If the fixture, andstationary and a moving target is impacted into the hence a control accelerometer mounted on or neartest item. This is called a turn-around test. Machines the fixture, is involved in resonant behavior withused to generate the initial velocity include: rocket more than one participating mode, there will almostsleds, drop towers, cable pull-down facilities, actua- certainly be one or more frequencies at which thetors, and guns. modal response destructi vely interferes and the

Pyrotechnic shock is a special category of shock motion will cancel. This is called an antiresonance.that requires special care in the measurements and Very large motion of the system will result withsimulation. A discussion of pyroshock is beyond the essentially zero motion at the control point. Notscope of this article. only is this potentially destructive, but the control

The location of control accelerometers in vibration system and shaker will have great difficulty maintain-testing requires care since vibration at high frequen- ing the required motion. Care must be taken to avoidcies ('high' being defined here as frequencies above placing the control accelerometer at such a location.the first structural resonances of the test apparatus This can usually be accomplished if the fixtures arebeing used) is a local phenomenon. The vibration rigid. For the same reasons the control accelerometerlevels can vary significantly with location. The com- is often placed at the extreme end of a fixture, like amon method is to mount the control accelerometer cube or slip table. The free end of a beam is free ofnear the location where field data were measured that antiresonances. If a rigid fixture is not possible, aver-defined the environment. If this is not possible, the aging or extremal (usually means control on theaccelerometer is usually mounted on the fixture near largest) control of several accelerometers is used tothe test item-fixture interface. limit the motion caused by antiresonances. Locations

As explained earlier, the boundary conditions for can usually be found where the several control accel-the field environment and the test greatly influence erometers do not all have antiresonances at the samethe equivalence of the simulation. It is rare that the frequencies. Force limiting can also be used to limitboundary conditions of the field environment are the effect of antiresonances. Fixtures can also be builtsimulated in the laboratory. Limiting is often used with a significant amount of damping. The dampingto reduce the conservatism of a test caused by impro- limits the depth of the antiresonances. The mass ofper boundary conditions and overly conservative test the fixtures is also important. All the shaker movingspecifications. Common methods include: limiting parts, including the fixtures, must be moved, whichthe response at other locations than the control increases the force required. Fixtures with small masspoint, limiting the input force, limiting the current require less force from the shaker. This is contrary tointo an electrodynamic shaker, and averaging the the requirement for rigid fixtures. For this reasonresponse at several points. most fixtures are constructed from aluminum or

magnesium that have high stiffness-to-weight ratios.

Accelerated TestingTest compression is often required to permit reason- System Level Tests that Attempt toable test times. There are several pitfalls with these Match Field Conditionsmethods. The methods usually use some form of At the system level tests are often designed to matchMinors rule for fatigue damage. This assumes failuresare related to fatigue, and the fatigue mechanism is closely the field boundary conditions and environ-

known. Generally it is wise to keep the vibration level ment. Examples include:

at or below the highest expected field environment. 1. Automobile road simulators. These devices canThis will prevent unwarranted failures from peak employ as many as 18 electro hydraulic actuatorsloads. Lower levels of vibration can be accelerated to simulate road conditions.with reasonable confidence by raising the level to the 2. Automobile crash testing. Automobiles arehighest expected field environment and reducing c~ashed at various velocities into a variety of bar-the time. ners.

absorber, vibration: bandwidth: strictly, the frequency span encom-damped vibration absorber: an added device passed by a resonance curve (Bode diagram of

which introduces damping to a structure at a the modulus of a frequency response function) atpoint by providing an inertial sprung mass the level which is 0.707 times the peak value.against which an added damper can react, Alternatively, the frequency range of excitationthereby introducing damping to a specifi- and measurement during a vibration testcally targeted mode of vibration Bode diagram/plot: see diagram

undamped vibration absorber: an added device burst random: see signalwhich provokes an antiresonance at the burst sine: see signalpoint of attachment at a preselected fre- coherence: a measure of the degree of correlationquency of forced vibration between two signals presented in the frequency

accelerance: one of the family of different fre- domain (i.e., a form of spectrum). Has a realquency response functions: the harmonic accel- value and is normalized between 0 and 1.eration response in one degree-of-freedom to a coordinate (or co-ordinate): strictly, the location ofsingle harmonic excitation force applied in the a point on a structure in space. Often used tosame or another degree-of-freedom. Also known mean a degree-of-freedomas inertance (although this name is not approved principal (or modal) coordinate: a time-varyingby ISO). See also mobility and receptance quantity which describes the extent

acceleration: see displacement (amount) of motion in one of the systemsaliasing: the phenomenon in digital signal process- modes of vibration. Sometimes referred to as

ing, due to inadequate sampling frequency, modal coordinatewhereby a signal with a frequency fN which is correlation: the systematic quantitative comparisonhigher than half the digitization sampling fre- of two sets of like data, such as time histories, orquency, fs, appears in the resulting spectrum as if modal properties, resulting in a numericalit were a signal with a frequency (fs-N), which is measure of the degree of similarity or dissim-lower than its actual value ilarity. See coherence

analysis, modal: a procedure which extracts the critical speed: speed of a rotating rotor thatnormal mode properties of a system, either by corresponds to a resonanceanalyzing measured response functions (experi- damping: any of several physical mechanismsmental modal analysis - EMA) or by perform- whereby mechanical energy (i.e., kinetic oring an eigensolution on the system's mass and strain) is converted into heat and therebystiffness matrices (analytical modal analysis) removed from a vibrating system. In the absence

antialiasing filter: signal conditioning or processing of a steady input of energy, such as is achieveddevice which provides a low-pass filtering effect through continuous excitation of the system, theto reduce in a signal to be analyzed those vibration will decay to rest as a direct conse-frequency components which are higher than quenee of the damping mechanismsone-half the sampling frequency (the Nyquist hysteretic (also structural): a type of dampingfrequency) for which the mathematical model used to

antiresonance: a condition in forced vibration describe it is based on observation of thewhereby a specific point or OOF has zero characteristics of damping due to internalamplitude at a specific frequency of vibration hysteresis of materials

Argand diagram: see diagram proportional: refers to the distribution (ratherautocorrelation function: (see power spectral den- than the type) of damping in a structure

sity) the signal average of the product of the whereby the damping elements are assumedsignal with a delayed version of itself. The to be distributed in a way which reflects theinverse Fourier transform of the power spectral distribution of stiffness, and/or mass, of thedensity function structure. Has the convenient property that

balancing: the procedure of adjusting the mass the mode shapes (q.v.) of a proportionallydistribution of a rotor so that vibrations or damped structure are identical to the modeforces reactions during rotation are minimized shapes of the undamped version of the same

GLOSSARY Giii

applied at another degree of. freedom of the line spectrum: see spectrumstructure (such as k) MAC or modal assurance criterion: a simplepoint: a particular FRF in which the parameter used to measure the degree of

excitation and response degrees of free- correlation between two vectors of the samedom are the same (i.e., j = k). length: usually eigenvectors, or mode shapes. A

transfer: an FRF in which the excitation MAC value of 100% indicates that the twoand response degrees of freedom are not vectors being compared are parallel, or arethe same (i.e., j -I- k). perfectly correlated. MAC values of greater than

impulse response function (IRF): The response 80% generally indicate 'similar' mode shapestime history of a system, measured in one mass, modal: a quantity which relates to a givenparticular degree of freedom to a unit mode shape and which is computed from thatimpulse excitation applied at another degree mode shape vector and the mass matrix of theof freedom system model. The modal mass is automatically

harmonic: (n.) an integer multiple of a fundamental unity for mass-normalized mode shapessinusoidal signal to which it is related mobility: one of the family of frequency response(adj.) equivalent to 'sinusoidal' in describing functions: the harmonic velocity response in onesignal type degree of freedom to a single harmonic excita-

Hilbert transform: see transform tion force applied in the same or another degreeimpedance, mechanical: one of the family of of freedom

inverse frequency response functions: the ratio modal coordinate: see coordinatebetween the single harmonic excitation force modal mass: see mass, modalapplied in one degree of freedom to the harmonic modal stiffness: see stiffness, modalvelocity response in the same or another degree mode, mode shape (see eigenvector): strictly, aof freedom. pattern of vibration exhibited by a structure or

impulse response function (IRF): see function system. Generally, described as a vector ofinertance: see accelerance values, defining the relative displacement ampli-isolation, vibration: a feature of forced vibration in tudes and phases of each degree of freedom,

which vibration levels in certain regions of a which describes the motion of the systemstructure are constrained to be much lower than complex: a mode of vibration in which theat other regions by suitable distribution of mass relative displacements exhibit phase differ-and stiffness ences between one degree of freedom and

jerk: derivative of acceleration, used in conjunction the next which are not always 0 or 1800

with responses to shock excitations (situation which applies in a real mode).leakage: a phenomenon which arises during digital Complex modes do not exhibit modal lines

spectral analysis of continuous signals as a result mass-normalized (see eigenvector)of the finite duration of the sampled signal and normal, damped: the vector of relative ampli-its relationship with the frequencies present in tudes for the vibration of a damped systemthe original signal. Often, leakage occurs be- in one of its natural modes of free vibration.cause the signal being analyzed is not periodic These modes are the modes which thewith respect to the sample length (duration, structure assumes in the absence of anyperiod) used for the.analysis. The consequence of external excitation. In this case, the structureleakage is that a computed spectrum may is assumed to possess arbitrary dampingindicate the presence of signal components and, as a result, the normal modes arewhich are not in the original signal and it may complexnot accurately indicate those which are. Win- normal, undamped: the vector of relative am-dows are used to reduce the effects of leakage plitudes for the vibration of the undamped

linearity: a characteristic often assumed to be system in one of its natural modes of freepresent in structures subjected to vibration vibration. These modes are the modes whichanalysis or testing techniques: the primary the structure assumes in the absence of anyassumption is essentially that the response of external excitation. In this case, the structurethe structure to two forces applied simulta- is assumed to possess no damping and, as aneously will be identical to that derived by result, the normal modes are realadding the responses of the structure to each real: a mode of vibration in which the relativeforce applied individually. Other definitions can displacements exhibit phase differences be-also be applied tween one degree of freedom and the next

Giv Glossary

which are always 0 or 180°. Real modes eigenvalue or to (square of) natural frequency.exhibit modal lines For systems describable by a ratio of rational

mode indicator function: a function which is used polynomials in a transform domain, the value ofto indicate the existence of global system modes the transform variable for which the denomi-(those which are observed throughout the nator polynomial equals zerostructure) as opposed to local modes (which power spectral density (PSD): (see autocorrelationare only visible in a small region of the structure) function) the distribution of the squared signal

model (mathematical): a mathematical description (acceleration, velocity, etc.) with frequency (inof a system or structure in a form which can be squared units per Hz, i.e., g2/Hz). The Fourierdescribed by equations of motion which can then transform of the autocorrelation functionbe solved analytically or numerically to describe principal coordinate: see coordinatethat system's dynamic behavior receptance: one of the family of frequency responsecontinuous: a mathematical model which de- functions: the harmonic displacement response

scribes the system or structure as a con tin- in one degree of freedom to a single harmonicuous medium, using partial differential excitation force applied in the same or anotherequations with, typically, transcendental degree of freedom. See also mobility and accel-expressions for the solutions erance

discrete, lumped parameter: an alternative type resonance (see frequency): a phenomenon asso-of mathematical model (to the continuous ciated with forced vibration which relates to atype) in which the mass, stiffness, and narrow range of frequencies in which thedamping properties of the structure are response amplitude of a system attains levelsdiscretized so that the behavior can be which are much higher than elsewheredescribed by a finite series of ordinary sampling (see frequency)differential equations leading to finite dis- series, Fourier: a method of representing a periodiccrete solutions signal as the sum of separate sinusoids whose

modal: a model which is defined in terms of frequencies are integer multiples of the funda-eigenvalues and eigenvectors, or natural mental period of the original signal. See alsofrequencies and mode shapes. Usually pre- transform, Fouriersented as eigenvalue and eigenvector ma- signal (time history): a time-varying quantity whichtrices describes the excitation or response of a vibrat-

response: a model which is defined in terms of ing system. In vibration testing, a quantity whichsome characteristic response functions, such is used either to drive the excitation of a systemas FRFs, or IRFs, or equivalent into vibration or to represent the measurement

spatial: a model which is defined in terms of of one of the vibration forces or responses whichthe system's distribution of mass, stiffness, are used to describe the system's movement.and damping in space. Usually described as Types of signal commonly encountered include:mass, stiffness, and damping matrices burst:

multiple-degrees-of-freedom (MDOF) system (see random: a signal which consists of twosingle-degree-of -freedom (SDOF) system): a sections, the first being a finite durationsystem for which two or more coordinates or of a stationary random signal followeddegrees of freedom are needed to define com- by a second section which is zeropletely the position of the system at any instant sme: a signal which consists of two sec-

natural frequency: see frequency tions, the first being a finite duration ofnode: 1. a DOF which has zero amplitude of a steady sinusoid followed by a second

vibration; used when describing a mode shape section which is zeroor an operating deflection shape chirp: a signal comprising a single rapidly2. a specific grid point in a discrete (lumped swept sine wave between a minimum andparameter) model of a structure maximum frequency

nonlinearity: a form of structural behavior which harmonic: a sinusoidal signal at a singledoes not conform to the definition of linearity frequency(see linearity) periodic: A signal which repeats itself indefi-

normal mode: see mode nitely after a period, T. In signal analysis orNyquist diagram: see diagram processing, the period of the repeating signaloperating deflection shape: see mode shape is not necessarily the same as the period overpole: mathematical term which is equivalent to which the signal is measured or analyzed

GLOSSARY Gv

and in such a situation, analysis of the signal transform algorithm. Sometimes synon-may not yield a true indication of its form ymous with line spectrum(see also leakage) line: a discrete spectrum, with nonzero values

periodic random: a signal which consists of a only at discrete frequencies. Typical ofsuccession of records (sometimes called vibrations generated by rotating machinery'samples'), each of finite length, T, which is stiffness, modal: a quantity which relates to a givenusually selected to suit the sampling period mode shape and which is computed from thatof a means of analyzing the signal. Each of mode shape vector and the stiffness matrix of thethe first few records is identical to each other system model. The modal stiffness is equal to theand thus represent a periodic signal, even eigenvalue for a mass-normalized mode shapethough an individual period appears to be time history (see signal): description of a time-very complex, almost random. Then the next varying quantity, such as displacement or force,few records consist of several repetitions of as a direct function of timeanother 'complex' record. This pattern is transducer: a device which transforms one form ofrepeated for as long as the signal is required energy into another, practically used synony-

pseudorandom: a periodic signal for which the mously with the term sensorbasis record is very complex, and appears to transformbe random. However, this record is repeated discrete (DFT): a discrete transform whichindefinitely and so forms a periodic signal converts a a discrete (usually time domain)

random: a nondeterministic signal. No analy- sequence into another discrete (usuallytical expression predicting future values can frequency domain) sequence. With correctexist. It is defined only in terms of statistical sampling, the DFT approximates samples offunctions and parameters, for example an the (continuous) Fourier transform In theamplitude probability function DFT, the continuous time-varying signal is

sine, sinusoid, discrete sine: a signal which approximated by a discrete number ofconsists of a pure, single, sinusoidal or sampled values taken from the original data.harmonic component; also stepped sine The resulting spectrum has exactly the same

sme sweep: a signal which comprises a sinusoid number of components as the number ofwhose frequency is varied slowly so that a sampled data points, although not all ofrange of frequencies is covered throughout these may be displayedthe duration of the signal fast Fourier transform (FFT): a computer

stepped sine: a signal which comprise a series algorithm for the efficient computation ofof discrete sinusoids, one after the other the DFT of a sampled signal. The algorithm

transient: a signal which lasts a finite length of takes advantage of the symmetry of thetime, being zero before and after that period, transform and is very efficient when appliedand which is fully defined within the record to a data set containing 2N valueslength Fourier: a continuous transform between two

white noise: a signal with a constant power domains, usually time and frequency. One ofspectral density, independent of frequency. the most basic mathematical tools forIn practice a signal can have white noise representing the relevant features of a timecharacteristics only in a finite frequency history signal in terms of its constituentrange frequency components, and of reconstruct-

single-degree-of-freedom (SDOF) system (see also: ing a time record from a knowledge of themul ti p le-degrees-of -freedom (MDOF) sys- frequency (or spectral) components. Fortem): a system for which only one coordinate periodic signals, the transform exists onlyis needed to define completely the position of the at discrete frequencies (see line spectrum),system at any instant and is more conveniently represented via the

spectrum: description of the essential elements of a Fourier seriessignal (or time history) in terms of constituent Hilbert: a type of transform for converting asinusoidal components function from one form to another which,continuous: nonzero in a continuous range of unlike the Fourier transform, derives a

frequencies. Typical for random signals transformed function in the same domaindiscrete: a spectrum computed only at discrete as the original (i.e., a time signal transforms

frequencies. These frequencies are usually to another time signal, a frequency functionequi-spaced, a result of using the fast Fourier to another frequency function)

Gvi Glossary

validation: the process of demonstrating the ade- vibration normal modes of the systemquacy of the performance of a given prediction free: the type of vibration which ensues when acapability, usually that of a finite element model system or structure is set into motion by anconstructed to predict the vibration properties of initial disturbance and then left to vibratethe structure under the influence only of the internal

velocity (see displacement) forces due to its own stiffness, inertia, andverification: the process(es) used to demonstrate damping. This motion can be described

that an algorithm or other procedure has been completely in terms of the free vibrationcorrectly implemented in a computer program or normal modes of the systemtest sequence white noise: see signal

vibration window: an approach used in signal processing toforced: the vibration structure which is the minimize the approximations introduced when

direct result of an applied excitation force, using a discrete Fourier transformation on aor forces. Although there will be an initial continuous signal. Usually applied in the timetransient response which is governed by the domain, as a multiplying function applied to thefree vibration characteristics of the system, time signal. Various shapes of window functionsthis will die away, leaving only the steady have been developed to produce optimumresponse due entirely to the externally frequency information for particular types ofapplied excitation. This motion can be signaldescribed completely in terms of the free

ADC analog-to-digital converter MAC modal assurance criteria (also FMAC,ARMA autoregressive moving average AUTOMAC, FDAC, FRAC ... )CAD computer-aided design MDOF multi-degree-of-freedomCFD computational fluid dynamics MEMS microelectromechanical systemsCG center of gravity MIFs mode indicator functionsCWT continuous wavelet transform MIMO multi-input multi-outputDAC digital-to-analog converter MISO multi-input single-outputDOF degree of freedom NDT nondestructive testingDFT discrete Fourier transform NVH noise, vibration, harshnessDSP digital signal processing ODS operating deflection shapeDWT discrete wavelet transform PCB printed circuit boardEOM equation of motion PDF probability density functionEU engineering units; European Union PSD power spectral densityFEA finite element analysis QA quality assuranceFEM finite element method RMS root mean squareFFT fast Fourier transform SDOF single-degree-of-freedomFIR finite impulse response SEA statistical energy analysisFPE final prediction error S/H sample and holdFRF frequency response function SIMO single-input multi-outputIC integrated circuit SISO single-input single-outputIIR infinite impulse response SLDV scanning laser Doppler vibrometerIRF impulse response function SNR signal-to-noise ratioISO International Standards Organization STFT short-time (term) Fourier transformLDV laser Doppler vibrometer SVD singular value decompositionLQ linear quadratic TDA time domain (synchronous) averagingLSB least significant bit WVD Wigner-Ville distributionLVDT linear voltage differential transformer YW Yule Walker

ISO 10055:1996 Mechanical vibration - vibration tion of machine vibration by measurementstesting requirements for shipboard equip- on non-rotating parts - part 5: machine setsment and machinery components in hydraulic power generating and pumping

ISO 10068:1998 Mechanical vibration and shock - plantsfree, mechanical impedance of the human ISO 10816-6:1995 Mechanical vibration - evalua-hand-arm system at the driving point tion of machine vibration by measurements

ISO 10137:1992 Bases for design of structures; on non-rotating parts - part 6: reciprocat-serviceability of buildings against vibration ing machines with power ratings above 100

ISO 10142:1996 Carbonaceous materials for use in kWthe production of aluminium - calcined ISO 10817-1:1998 Rotating shaft vibration measur-coke - determination of grain stability ing systems - part 1: relative and absoluteusing a laboratory vibration mill sensing of radial vibration

ISO 10326-1:1992 Mechanical vibration; laboratory ISO 10819:1996 Mechanical vibration and shock -method for evaluating vehicle seat vibra- hand-arm vibration - method for thetion; part 1: basic requirements measurement and evaluation of the vibra-

ISOITS 10811-1:2000 Mechanical vibration and tion transmissibility of gloves at the palm ofshock - vibration and shock in buildings the handwith sensitive equipment - part 1: measure- ISO 10846-1:1997 Acoustics and vibration -ment and evaluation laboratory measurement of vibro-acoustic

ISOITS 10811-2:2000 Mechanical vibration and transfer properties of resilient elements -shock - vibration and shock in buildings part 1: principles and guidelineswith sensitive equipment - part 2: classifi- ISO 10846-2:1997 Acoustics and vibration -cation laboratory measurement of vibro-acoustic

ISO 10814:1996 Mechanical vibration - suscept- transfer properties of resilient elements -ibility and sensitivity of machines to un- part 2: dynamic stiffness of elastic supportsbalance for translatory motion - direct method

ISO 10815:1996 Mechanical vibration - measure- ISO 11342:1998 Mechanical vibration - methodsment of vibration generated internally in and criteria for the mechanical balancing ofrailway tunnels by the passage of trains flexible rotors

ISO 10816-1:1995 Mechanical vibration - evalua- ISO 11342:1998/Cor 1:2000 Mechanical vibration-tion of machine vibration by measurements methods and criteria for the mechanicalon non-rotating parts - part 1: general balancing of flexible rotorsguidelines ISO 13090-1:1998 Mechanical vibration and shock-

ISO 10816-2: 1996 Mechanical vibration - evalua- guidance on safety aspects of tests andtion of machine vibration by measurements experiments with people - part 1: exposureon non-rotating parts - part 2: large land- to whole-body mechanical vibration andbased steam turbine generator sets in excess repeated shockof 50 MW ISO 13753:1998 Mechanical vibration and shock -

ISO 10816-3: 1998 Mechanical vibration - evalua- hand-arm vibration - method for measur-tion of machine vibration by measurements ing the vibration transmissibility of resilienton non-rotating parts - part 3: industrial materials when loaded by the hand-armmachines with nominal power above 15 systemkW and nominal speeds between 120 r/min ISO 14964:2000 Mechanical vibration and shock -and 15000 r/min when measured in situ vibration of stationary structures - specific

ISO 10816-4:1998 Mechanical vibration - evalua- requirements for quality management intion of machine vibration by measurements measurement and evaluation of vibrationon non-rotating parts - part 4: gas turbine ISO 16063-1: 1998 Methods for the calibration ofdriven sets excluding aircraft derivatives vibration and shock transducers - part 1:

ISO 10816-5:2000 Mechanical vibration - evalua- basic concepts

Axiv APPENDIX 6: STANDARDS AND GUIDELINES

ISO 16063-11: 1999 Methods for the calibration of ISO 4866:1990/Amd.1:1994 Mechanical vibrationvibration and shock transducers - part 11: and shock - vibration of buildings - guide-primary vibration calibration by laser inter- lines for the measurement of vibrations andferometry evaluation of their effects on buildings;

ISO 1925: 2001 Mechanical vibration - balancing - amendment 1vocabulary ISO 4866:1990/Amd.2:1996 Mechanical vibration

ISO 1940-1: 1986 Mechanical vibration; balance and shock - vibration of buildings - guide-quality requirements of rigid rotors; part 1: lines for the measurement of vibrations anddetermination of permissible residual un- evaluation of their effects on buildings;balance amendment 2

ISO 1940-2: 1997 Mechanical vibration - balance ISO 5007:1990 Agricultural wheeled tractors; oper-quality requirements of rigid rotors - part ator's seat; laboratory measurement of2: balance errors transmitted vibration

ISO 2017: 1982 Vibration and shock; isolators; ISO 5008:1979 Agricultural wheeled tractors andprocedure for specifyingcharacteristics field machinery; measurement of whole-

ISO 2247:2000 Packaging - complete, filled trans- body vibration of the operatorport packages and unit loads - vibration ISO 5344:1980 Electrodynamic test equipment fortests at fixed low frequency generating vibration; Methods of describing

ISO 2247:1985 Packaging; complete, filled transport equipment characteristicspackages; vibration test at fixed low fre- ISO 5347-10:1993 Methods for the calibration ofquency vibration and shock pick-ups; part 10:

ISO 2631-1:1997 Mechanical vibration and shock- prImary calibration by high Impactevaluation of human exposure to whole- shocksbody vibration - part 1: general require- ISO 5347-11:1993 Methods for the calibration ofments vibration and shock pick-ups; part 11:

ISO 2631-2:1989 Evaluation of human exposure to testing of transverse vibration sensitivitywhole-body vibration; part 2: continuous ISO 5347-12:1993 Methods for the calibration ofand shock-induced vibration in buildings (1 vibration and shock pick-ups; part 12:to 80 Hz) testing of transverse shock sensitivity

ISO 2631-4:2001 Mechanical vibration and shock- ISO 5347-13:1993 Methods for the calibration ofevaluation of human exposure to whole- vibration and shock pick-ups; part 13:body vibration - part 4: guidelines for the testing of base strain sensitivityevaluation of the effects of vibration and ISO 5347-14:1993 Methods for the calibration ofrotational motion on passenger and crew vibration and shock pick-ups; part 14:comfort in fixed-guideway transport sys- resonance frequency testing of undampedterns accelerometers on a steel block

ISO 2671:1982 Environmental tests for aircraft ISO 5347-15:1993 Methods for the calibration ofequipment; part 3.4: acoustic vibration vibration and shock pick-ups; part 15:

ISO 2953:1999 Mechanical vibration - balancing testing of acoustic sensitivitymachines - description and evaluation ISO 5347-16:1993 Methods for the calibration of

ISO 2954:1975 Mechanical vibration of rotating vibration and shock pick-ups; part 16:and reciprocating machinery; requirements testing of mounting torque sensitivityfor instruments for measuring vibration ISO 5347-17:1993 Methods for the calibration ofseverity vibration and shock pick-ups; part 17:

ISO 3046-5:1978 Reciprocating internal combustion testing of fixed temperature sensitivityengines - performance - part 5: torsional ISO 5347-18:1993 Methods for the calibration ofvibrations vibration and shock pick-ups; part 18:

ISO 4548-7:1990 Methods of test for full-flow testing of transient temperature sensitivitylubricating oil filters for internal com- ISO 5347-19:1993 Methods for the calibration ofbustion engines; part 7: vibration fatigue vibration and shock pick-ups; part 19:test testing of magnetic field sensitivity

ISO 4866:1990 Mechanical vibration and shock; ISO 5347-20:1997 Methods for the calibration ofvibration of buildings; guidelines for the vibration and shock pick-ups - part 20:measurement of vibrations and evaluation primary vibration calibration by the reci-of their effects on buildings procity method

APPENDIX 6: STANDARDS AND GUIDELINES Axv

ISO 5347-22:1997 Methods for the calibration of torsional vibration - non-resonance methodvibration and shock pick-ups - part 22: ISO 6721-8:1997 Plastics - determination of dy-accelerometer resonance testing - general namic mechanical properties - part 8:methods longitudinal and shear vibration - wave-

ISO 5347-3:1993 Methods for the calibration of propagation methodvibration and shock pick-ups; part 3: ISO 6721-9:1997 Plastics - determination of dy-secondary vibration calibration namic mechanical properties - part 9:

ISO 5347-4:1993 Methods for the calibration of tensile vibration - sonic-pulse propagationvibration and shock pick-ups; part 4: methodsecondary shock calibration ISO 6954:2000 Mechanical vibration - guidelines

ISO 5347-5:1993 Methods for the calibration of for the measurement, reporting and evalua-vibration and shock pick-ups; part 5: tion of vibration with regard to habitabilitycalibration by earth's gravitation on passenger and merchant ships

ISO 5347-6:1993 Methods for the calibration of ISO 7096:2000 Earth-moving machinery - labora-vibration and shock pick-ups; part 6: tory evaluation of operator seat vibrationprimary vibration calibration at low fre- ISO 7096:1994 Earth-moving machinery - labora-quenCles tory evaluation of operator seat vibration

ISO 5347-7:1993 Methods for the calibration of ISO 7505:1986 Forestry machinery; chain saws;vibration and shock pick-ups; part 7: measurement of hand-transmitted vibrationprimary calibration by centrifuge ISO 7626-1:1986 Vibration and shock; experimental

ISO 5347-8:1993 Methods for the calibration of determination of mechanical mobility; partvibration and shock pick-ups; part 8: 1: basic definitions and transducersprimary calibration by dual centrifuge ISO 7626-5:1994 Vibration and shock - experi-

ISO 5348:1998 Mechanical vibration and shock - mental determination of mechanical mobi-mechanical mounting of accelerometers lity - part 5: measurements using impact

ISO 5349:1986 Mechanical vibration; guidelines for excitation with an exciter which is notthe measurement and the assessment of attached to the structurehuman exposure to hand-transmitted vibra- ISO/TR 7849:1987 Acoustics; estimation of air-tion borne noise emitted by machinery using

ISO 5982:1981 Vibration and shock; mechanical vibration measurementdriving point impedance of the human body ISO 7916:1989 Forestry machinery; portable brush-

ISO 6070:1981 Auxiliary tables for vibration gen- saws; measurement of hand-transmittederators; methods of describing equipment vibrationcharacteristics ISO 7919-1:1996 Mechanical vibration of non-

ISO 6267:1980 Alpine skis; measurement of bending reciprocating machines - measurements onvibrations rotating shafts and evaluation criteria - part

ISO 6721-3:1994 Plastics - determination of 1: general guidelinesdynamic mechanical properties - part 3: ISO 7919-2:1996 Mechanical vibration of non-flexural vibration - resonance-curve reciprocating machines - measurements onmethod rotating shafts and evaluation criteria - part

ISO 6721-3:1994/Cor 1:1995 Plastics - determina- 2: large land-based steam turbine generatortion of dynamic mechanical properties - setspart 3: flexural vibration - resonance-curve ISO 7919-3:1996 Mechanical vibration of non-method reciprocating machines - measurements on

ISO 6721-4:1994 Plastics - determination of dy- rotating shafts and evaluation criteria - partnamic mechanical properties - part 4: 3: coupled industrial machinestensile vibration - non-resonance method ISO 7919-4:1996 Mechanical vibration of non-

ISO 6721-5:1996 Plastics - determination of dy- reciprocating machines - measurements onnamic mechanical properties - part 5: rotating shafts and evaluation criteria - partflexural vibration - non-resonance method 4: gas turbine sets

ISO 6721-6:1996 Plastics - determination of dy- ISO 7919-5:1997 Mechanical vibration of non-namic mechanical properties - part 6: shear reciprocating machines - measurements onvibration - non-resonance method rotating shafts and evaluation criteria - part

ISO 6721-7:1996 Plastics - determination of dy- 5: machine sets in hydraulic power generat-namic mechanical properties - part 7: ing and pumping plants

Axvi APPENDIX 6: STANDARDS AND GUIDELINES

ISO 7962:1987 Mechanical vibration and shock; ISO 8662-13:1997 Hand-held portable power toolsmechanical transmissibility of the human - measurement of vibrations at the handle -body in the z direction part 13: die grinders

ISO 8002:1986 Mechanical vibrations; land vehicles; ISO 8662-13:1997/Cor.1:1998 Hand-held portablemethod for reporting measured data power tools - measurement of vibrations at

ISO 8041:1990 Human response to vibration; the handle - part 13: die grinders; technicalmeasuring instrumentation corrigendum 1

ISO 8041:1990/Amd 1:1999 Human response to ISO 8662-14:1996 Hand-held portable power tools-vibration - measuring instrumentation - measurement of vibrations at the handle -amendment 1 part 14: stone-working tools and needle

ISO 8041:1990/Cor 1:1993 Human response to scalersvibration; measuring instrumentation; tech- ISO 8662-2:1992 Hand-held portable power tools;nical corrigendum 1 measurement of vibrations at the handle;

ISO 8042:1988 Shock and vibration measurements; part 2: chipping hammers and rivetingcharacteristics to be specified for seismic hammerspick-ups ISO 8662-3:1992 Hand-held portable power tools;

ISO 8318:2000 Packaging - complete, filled measurement of vibrations at the handle;transport packages and unit loads - sinu- part 3: rock drills and rotary hammerssoidal vibration tests using a variable ISO 8662-4:1994 Hand-held portable power tools-frequency measurement of vibrations at the handle -

ISO 8318:1986 Packaging; complete, filled transport part 4: grinderspackages; vibration tests using a sinusoidal ISO 8662-5:1992 Hand-held portable power tools;variable frequency measurement of vibrations at the handle;

ISO 8528-9:1995 Reciprocating internal combustion part 5: pavement breakers and hammers forengine driven alternating current generating construction worksets - part 9: measurement and evaluation ISO 8662-6:1994 Hand-held portable power tools-of mechanical vibration measurement of vibrations at the handle -

ISO 8569:1996 Mechanical vibration and shock - part 6: impact drillsmeasurement and evaluation-of shock and ISO 8662-7:1997 Hand-held portable power tools -vibration effects on sensitive equipment in measurement of vibrations at the handle -buildings part 7: wrenches, srewdrivers and nut

ISO 8579-2:1993 Acceptance code for gears; runners with impact, impulse or ratchedpart 2: determination of mechanical vibra- actiontions of gear units during acceptance ISO 8662-8:1997 Hand-held portable power tools-testing measurement of vibrations at the handle -

ISO 8608:1995 Mechanical vibration - road surface Part 8: Polishers and rotary, orbital andprofiles - reporting of measured data random orbital sanders

ISO 8626:1989 Servo-hydraulic test equipment for ISO 8662-9:1996 Hand-held portable power tools-generating vibration; method of describing measurement of vibrations at the handle -characteristics part 9: rammers

ISO 8662-1:1988 Hand-held portable power tools; ISO 8727:1997 Mechanical vibration and shock -measurement of vibrations at the handle; human exposure - biodynamic coordinatepart 1: general systems

ISO 8662-10:1998 Hand-held portable power tools ISO 8821:1989 Mechanical vibration; balancing;- measurement of vibrations at the handle - shaft and fitment key conventionpart 10: nibblers and shears ISO 9022-10:1998 Optics and optical instruments -

ISO 8662-11:1999 Hand-held portable power tools environmental test methods - part 10- measurement of vibrations at the handle - combined sinusoidal vibration and dry heatpart 11: fastener driving tools - ISO 8662- or cold11:1999 ISO 9022-15:1998 Optics and optical instruments-

ISO 8662-12:1997 Hand-held portable power tools environmental test methods - part 15:- measurement of vibrations at the handle - combined digitally controlled broad-bandpart 12: saws and files with reciprocating random vibration and dry heat or coldaction and saws with oscillating or rotating ISO 9022-19:1994 Optics and optical instruments -action environmental test methods - part 19:

NOTEPage numbers in bold refer to major discussions. Page numbers suffixed by T refer to Tables; page numberssuffixed by F refer to Figures. vs denotes comparisons

This index is in letter-by-letter order, whereby hyphens and spaces within index headings are ignored in thealphabetization. Terms in parentheses are excluded from the initial alphabetization.

Cross-reference terms in italics are general cross-references, or refer to subentry terms within the same mainentry (the main entry is not repeated to save space).

Readers are also advised to refer to the end of each article for additional cross-references - not all of thesecross-references have been included in the index cross-references.

A acoustics active control (continued)ABA

QUS246-247 305 boundary integral formulation and skyhook.38-41

, 1278-1279 suppressIOn48-58absorbers ~-26 ... excitation and 897 vehicles37-45, 37F

attenuatIOncapablhtles 22-23, 23F, external problem and 1279 ActiveControl of Civil Structures26-25F FEM and 1277-1278 36

AVA4F, 5-6, 7F, 8F fluid/structural interaction and ActiveControl of VehicleVibrationdynamic 9-26 545-550 37-45f~ture pers?~ctives 24, 25F internal problem 1279-1282 ActiveIsolation 46-48piezoelectriCItyand 1-3, IF, 3F MEMS and 774 775F actuation 47-48 47Tpositive position feedback 3-5, 4F, noise and 887-898 feedback 46-47,' 47F

~F, 6F .. parallel processing and 1000 feedforward 46-47, 46Fspecial conftguratlons of 18-21, radiation impedance and 891 activelycontrolled response of buffet

20T, 20F, 21F, 22F SPL1268, 1268F affected tails (ACROBAT)78undamped 10-17, 10F, 11F, 12F, structural interactions and 1265- active-passivedevices653-658

14F, 15F, 16T, 16F, 18T, 18F, 1283 active piezoelectricdamping19F subsonic waves and 897 composites (APDC)354

Absorbers, Active1-9 tire vibration and 1375-1376 active states 925-926Absorbers,Vibration 9-26 See also sound ActiveVibration Suppression48-57acausal filters 1196 active constrained layer (ACL)361F, compensators and 55-56accelerance1129 362F, 363F, 656-658 control and 53, 55acceleration 604 active constrained layer damping degree of freedom and 49-51, 49F,accelerograms439-441, 440T, 443- (ACLD)353-360 355F 356F 50F

446 359T,359F, , , , limitations of 57accelerometers active control modal control and 51-52

absolute motion and 1383, 1390- active suspensions 38, 39F pole placement 541392, 1395-1396 adjustable suspensions 38, 38F spillover 53-54

ADXL50 772, 773F applications 28-33, 29T, 29F, 30F, stability and 56-57bearing diagnostics and 151 31F 32F 33F 34F 35F 36F actuated joints 1059calibration of 1130-1132 civil str~ctur~s 26~36" actuators 79-81cross-axis sensitivityand 1121- damping and 28-33,342-351 active isolation and 47-48

1122 fuzzy logic and 43-45 active materials 58-61, 60F, 61Fenvironmental testing and 491 groundhook 41 configurations 759-760location 495 hybrid 26-28, 27F, 42 distributed 1134-1138MEMS and 772-774 isolation 46-48 effectiveimplementation of 68-70,rotation 1080-1081 optimal 55 68F, 69F, 70F, 71F

acceptance testing 490 passive suspensions 37, 37F electrical input 71-72, 72F, 73F,acoustics semiactive 26-28, 27F, 37F, 38-44, 74F

aerodynamics and 93 40F, 41F, 42F, 43F, 44F electrostriction and 482-490

actuators (continued) algorithms (continued) autocorrelation function (continued)energy extraction 70 differentiation/integration and rain-on-the-roof excitation and 241magnetostrictive materials 64-65, 1193-1199 signal processing and 1200-1204

753-762 DYNA3D and 305-312 spectral density and 296MEMS and 771-779 Gauss-Seidel995 AutoMAC 270piezoelectricity and 61-64, 62F, hourglassing 305, 308 automobiles

63F least mean squares 82-84, 86 bridges and 202-207proof-mass 653 Levenberg-Marquart 866 ground transportation systems andRMA 48 model updating 834 603-620sensitivity and 1140-1141 neural networks and 871-872 tire vibrations and 1369-1379shape memory alloys and 65-68, recursive least square 84 vehicular vibrations 37-45

65F, 66F, 67F, 1147 steepest descent 81-82 Seea/so crashstructonic cylindrical shells and aliasing errors 670 AutoRegressivemodel 674, 675, 902

1138-1141 AID converters and 368 AutoRegressiveModel withSeea/so MEMS; smart structures antialiasing filters and 367-368 eXogenous input (ARX)model

Actuators and Smart Structures 58-81 data acquisition and 365-369 677, 679, 684Adaptive Filters 81-87 differentiation/integration and 1195 model-based identification and 675,

convergence coefficients 86-87 parameters and 369 677, 679-680, 682, 684-685,LMS algorithm 82, 83F, 84-86 American Beauty 1064 685FRLS algorithm 84 American National Standards Institute AutoRegressiveMoving Averagesteepest descent algorithm 81-82, (ANSI)245, 1224 (ARMA)model 675

82F American Petroleum Institute (API) model-based identification and 674Seea/so optimal filters 1081, 1224 AutoRegressiveMoving Averagewith

adaptive resonance theory (ART) 866 Ampere's law 755 eXogenous input (ARMAX)ADAPTxAutomated System analytic signals 643F model 675 679,684

Identification Hilbert transform and 643-646 model-based identification and 675,Software 682 Anderson, PW 744 679-680 682 684-685 684F

ND converters 368-373 Andronov, AA 431 685F " "additive noise 666-668 animation 415,416 Averaging98-110Ader, Clement 1065 annular flmd flow 1023 basic operations 98-101, 98F, 99F,ADINA 305 ANSYS247,305 100F 101FADXL50 a~celerometer 772, 773F a p~st~riori ve~ification 463 exponential 103aerodynamics a pnonreasonmg 135, 632, 637,1266, frequency domain 108-109, 110F

ACROBAT and, 78 1268 jitter effects 102F 103-108 108Fblades and 174-178 Arabs 112 109F ' "buffeting and, 92 arbitrary Eulerian Lagrangian (ALE) moving average (MA) 99-100 99Fcontinuous turbulence and, 91 formulation 251T, 252 100F ' ,flutter and 553-565, 565-577 DYNA3D and 306 PSD 108~109gusts and, 89-91, 92 meshes and 281-282 rec rsion 100-101 100F 102maneuvering and, 89-93 Aristotle 126 " TD~ 101-108, 102F, 103F, 104F,s~>und,878 , , ARMA (autoregressivemovmg 105F, 106F, 107F, 108F, 109FvibratIOnongms 1191 average) process 1202-1203, . II d', aXla oa mgaeroelastlc effects 1584-1586 1203F

I d 236 243' . co umns an -aeroservoelastlclty 95, 95F Arnold VI 432 crash and 310Fcontrol surface buzz 96, 96F ARTeMISExtractor 682 S / I d'dynamic maneuvers 89-93, 90F, ASCE7-98 tabular values 1583, ee a so oa mg

91F, 92F, 93F 1583T, 1584flight loads 88-89, 88F, 89F asymptotic modal analysis (AMA) Bground loads 93-95 1269 . ,limit cycle oscillations 95, 95F asymptotic techniques back propagatIOnalgonthm 873-874negative damping 97 nonlinear systems and 957-962 ba~fl,ed plate 889Fpanel flutter 96, 96F attenuation 1558-1559 Baja], I: 928-943, 952-966stall flutter 96 97F attractors Balancmg 111-124vortex sheddi~g 97 chaos and 227-236 calculations o,f88

Aeroelastic Response, 87-97 austenite Seeshape memory alloys coupled rotation and 111F, 112-aerospace Austrian Standards Organization 113, 112F, 113F

acoustic tests and 494 (ON) 1224 flexible state and 118-119SNDT and 905 auto-associative neural network influence coefficients and 119-123,

Agnes, G 1-9 (ANN) 864-865 120F, 122F, 124TAhmadian, M 37-45 autocorrelation function 977-978, rigid states and 113-118, 114F,airbags 283F, 285 1592 115F, 116F, 117F, 118FAkaike Information Criterion (AIC) adaptive filters and 81-87 rotor-stator interactions 1107-1121

682, 1205 cepstral analysis and 217-218 tire vibrations 1370algorithms columns and 241 torsion and 112-113

back propagation 873-874 model-based identification and 675 Banks, HT 658-664contact 308 PSD 297F bar plots 413

INDEX I iii

bars 1125-1127 bearings (continued) bilinear transformation 393-394Barton, J 971 fault 1083 time frequency and 1360-1361basic linear algebra subprograms fluids and 153-155, 153F, 154F, binary representation 232

(BLAS)991 155F biodynamics 1571Basic Principles 124-137 functions and 152-153 mechanical impedance and 1571

calculus of variations 126 HFRT 146, 146F models for 1572Dunkerley's method 135-136, 136F journal bearing 150F, 151 transmissibility and 1571Euler's equations 126-130, 129F localized defects 144-151, 144T, bioengineeringflexural motion 130, 131F 145F, 146F, 148F, 149F, 150T, crashworthiness and 308-311Galerkin's method 135 150F biorthogonality 1074generalized system coordinates magnetic links 158-161, 159F, biotechnology 778

130-131 160F, 161F birdstrike 93Hamilton's principle 131 misalignment and 1116-1118, 1186 Birkhoff, George 228, 233Lagrange equations 131 property comparison 155F, 161T, Blades and Bladed Disks 174-180,415Maxwell's theorem of reciprocity 161 breakdowns and 176

136, 136F roller bearing 143, 157-158 instability and 175parameters 125, 125F rolling faults and 1187-1188 localization and 741F, 745, 750F,Rayleigh's Principle 131F, 132-134, sealing systems 161-162, 162F, 751F

133F, 134F shock pulse counting 146 propellers 1170Ritz method 134 signature generation 144-145, pulsations and 177, 177Fvectors 125-126, 127F, 128F 144T,145F resonance 179F

Bauchau, 0 461-467 statistical parameters 145-146 rotation vs. rest 177-178Bayesian Information Criterion (BIC) synchronized averaging 147 signal generation and 1191

682 time-frequency distribution 147 strain and 175T, 178FBaz, A 351-364,1144-1155 wavelet transform 147 strains and 175Beams 137-143, 1329-1330, 1330F, See also rotor dynamics; standards technology and 176

1331F Bearing Vibrations 152-165 See also disks; helicopter damping;Bernoulli-Euler 748F Bellville spring 755, 1181, 1181F rotationboundary conditions and 181F, Belts 165-174 blanching 622

183-185, 184F, 184T, 185T drives 170-174, 171F, 172F, 173F, block diagrams 686F, 687Fcontinuous systems and 1312-1317 moving 166-170, 166F, 167F, block-Lanczos algorithm 695-696cross-axis contamination and 1122 168F,170F Blue Wave Ultrasonics 759Duffing's equation and 233-235 stationary 166, 166F Bode plots 122,418, 421F, 757, 758FDunkerley method and 135-136 Belytschko-Tsy shell 305 Boltzmann superposition model 661-Euler-Bernoulli theory and 137- Belytshco-Shwer beam 305 662

141, 140F bending moment sensitivity 1125- Bond number 739, 739Ffeedforward control 517 1127 Booch, G 969, 975Fflexural radiation and 1458, 1468 bending strains 1136 Book, W 1055-1063Gelerkin method and 135 Bendixson's theorem 593 Boolean matrices 997-1000localization and 741-751, 748F Benson, DJ 278-286 Borel measure 664magnetically buckled 234F Bernoulli, John 124, 1344 bounce transmissibility 610MEMS 787-789 Bernoulli-Euler beams 748F Boundary Conditions 180-191natural frequencies and 414-415 Bert, CW 236-243, 286-294 beams 181F, 183-185, 184F, 184T,piezoelectric damping and 354-360 Bessel function 729 185T, 1329-1330shape memory alloys and 1147- Bessel functions 288 cables and 211

1148 Betti's principle 197-198 continuous methods and 288shells and 1155-1167 bias error 670 coupled systems and 186, 187F,ship vibrations 1167-1173 bicoherence 145F, 147-149, 149F 188F, 189Fsound and 885-886 bifurcation DYNA3D and 306time frequency and 1366F center manifold theory and 962 equations of motion and 1329,Timoshenko theory and 142 eigenvalues and 818 1329Ftransmissibility and 1523-1527 Hopf 435 external problem and 1279transverse vibrations and 137-142, local 963 FEA software and 253

138T, 139T, 140F, 140T, 141F, nonlinear analysis and 962-965 field dynamics and 492-493, 495141T, 142T, 143F normal form theory 963 finite element methods 531-533

vibration intensity and 1483-1484, parametric excitation and 1003, guided waves and 794-8051486-1487 1006 integral formulation 1278-1279,

Bearing Diagnostics 143-152 perturbation methods and 1009- 1282-1285bearings 1078 1011 internal problem and 1279-1282

bicoherence 148 vibro-impact systems and 1533- liquid sloshing and 726-740cage fault 149 1536, 1539, 1543 longitudinal waves 181-183, 181Fcepstral analysis and 147, 218-220 Bigret, R 111-124, 152-165, 174- membranes and plates 1331coupling 163, 163F, 164F 180, 1064-1069, 1069-1077, moving 190distributed defects 149-151 1078-1084,1085-1106,1107- noise and 888-889diversity 154F, 156-157, 156F 1120 nonlinear analysis and 945failure modes 143 biharmonic operator 1331 nonreflecting 190

I iv INDEX

Boundary Conditions (continued) Cables (continued) Chaos (continued)radiation efficiency 894F, 895F, suspended 209-210, 210F symbolic dynamics and 229-232

896F, 897F tangential displacement and 211 vibro-impact systems and 1531-semidefinite systems 191 cage fault 145,149 1548shells 1159 Cai, GQ 1238-1246 chatter 589sound pressure and 888-889 calculus of variations 126, 1344-1346, chemical reactions 1440three-dimensional 186-188 1345T Chladni figures 414tire vibration and 1378 Euler's equations 126-130 Choi-Williams distribution (CWD)torsion 183 finite element methods 533 147, 148F, 598, 600F, 1364-waves and 1559-1564 flexural motion 130, 131F 1366, 1365F

Boundary Element Methods 192-202, calibration 818 time-frequency and 600201F, 1279-1282 Campbell diagram 414, 415F Cholesky factorization 52, 712-713

eigenvalues and 198 cantilevers circle-fitting method 822-823elasticity and 197 beam model 692F, 693F, 694F circular plates 416F, 1026-1027Fourier transforms 201 Duffing's equation and 233-235 civil engineeringfundamental solutions 192-195, pipes 1021-1022 SNDT and 905

192F, 193F plates 1028, 1028F clamped-clamped plate 1025Fharmonic oscillations and 195-198 time frequency and 1367F classes 969Helmholtz equation and 195 Cantor set 233 diagram 970FKirchhoff plates and 197 capacitive displacement sensors 1399- hierarchy and 974-976Laplace transforms 201 1400 closed loop control 50symmetry and 194-195 capacitor sensors 1398F, 1399F Coad, P 969, 971transient problems and 198-201 Cardona, A 967-976 coaxiality fault 1083, 1099

Box-Jenkins method 677 Cartwright, ML 228 cognitionBragg cell 702, 1404 cascade spectrum plot 385, 428F whole-body vibration and 1575Branca, Giovanni 1064 CATIA 305 Cohen's class of distributions 1362-Braun, S 98-110, 294-302, 665-672, Cauchy principal value 642 1366

1208-1223, 1406-1419, 1587- cavities 1078, 1170 collaboration diagrams 9701595 dimensions 1265 Columns 236-243

Breguet, Louis 1065 SEA and 1268 dimensionless frequency 237T

Bresse-Timoshenko theory 239 center manifold theory 432-435, 962 end conditions 237T

Bridges 202-207 center of gravity 1490-1493 free lateral vibration and 237-239dynamic response of 203 Centre Technique des Industries nonlinear vibration of 240-241frequencies of 202-203, 203F, 204F Mecaniques 1078 random vibration of 241-242

railroad tracks and 206 Cepstrum Analysis 216-227,747-748 stepped 239

traveling load and 204F, 205, 205F, ballpass frequency and 220F tapered 239

206F bearing diagnostics and 145F, 147 thin-walled 239-240

British Standards Institution (BSI) bearing outer race fault 219F transverse shear flexibility and 239complex applications 222-227 Commercial Software 243-2561224 definitions for 216-218 dynamic applications and 244-2456841 860,1572 echo removal 225F history of 2446842 625 editing effects 223F quality assurance and 245-246

broad-band random excitations 1266Broyden -Fletcher-Goldfarb-Shanno forcing function 226F Comparison of Vibrational Properties

FRFs 226F modal properties 265-272algorithm 637 liftering and 224, 226F response properties 272-277Bubnov-Galerkin method 240 power spectrum and 218-222, 221F spatial properties and 256-264buckling 238 terminology of 216 compass plots 418Fcolumns and 241 unwrapped phase 218, 218F compensators 55-56buffeting 92 zoom spectra 224F complementary energy method 289,buildings 1578 ceramic actuators 482-488 290, 1320See also active control CFRF matrix 276F, 277F complex envelope displacement

bulk waves 908 chains 213 analysis (CEDA) 1269, 1271-Burg's method 1203 Chaos 227-236 1274, 1273FButterworth filters 703 bifurcation and 435 complex exponential method 821

CDuffing's equation 233-235 complex exponential model class 675history of 228 complex mode indicator functioninitial conditions 229-232, 1097- (CMIF) 425, 429, 429F

C++ 971-972 1098 component diagrams 971, 972Fnumerical efficiency and 972-973 invariant manifolds and 228-229 component mode synthesis 1334-

Cables 209-216, 209F Melnikov's method and 232 1335chains and 213 nonlinearity sources of 235 composite FRF 425linear theory and 210-215, 211F Poincare maps and 228 compound FRF matrices 274, 275, 278modal analysis of 211-215 rotor dynamics 1097-1098 compound modes 417noise 1133 rotor-stator interactions 1112-1115 compression 887nonlinear model of 209-210, 214F Smale's horseshoe and 232-233 columns and 236-243shallow sag 210-213 strange attractors and 233-235 packaging and 983-988

INDEX Iv

compression (continued) Correlation Functions (continued) cycle limits 1095-1097plate vibration and 1029 stochastic processes and 296 rotor-stator interactions 1107-1110

computational methods COSMOS/M 247-248 cyclostationary phenomenon 602averaging and 98-110 cost function 977 cylinderscorrelation functions and 298-299 helicopter damping and 633 bearings vibrations and 153-155eigenproblems and 463 neural networks 870 flexural radiation and 1460-1463,

computational model updating (CMU) Coulomb forces 636 1465-1468851-854 damping 337

Computation for Transient and Impact energy 475 DDynamics 278-286 friction 582-583

computers shock isolation and 1180-1183 d'Albans, Marquis de Jouffroy 1064classes and 969 coupled analyses 254T, 255 d'Alembert 1099, 1344crashworthiness and 304 coupled systems 1080 principle 126, 131, 607DYNA3D and 305-312, 305T balancing and 111-124 rotation and 1070improvement of 968T bearing vibration and 163 rotor dynamics 1085localization and 749 boundary conditions 186, 187F, variational methods and 1355nonlinear systems and 953 188F, 189F Dalpiaz, G 1184-1193object oriented programming and differentiation/integration and 1194 D'Ambrogio, W 1253-1264

967-976 equations of motion and 1326 damping 344F, 413, 414F, 424F, 426Fparallel processing and 990-1001 gyroscopic 1097, 1099-1101, absolute motion transducers andtasks 968T 1099F, 1100F 1383See also MEMS isolation theory and 1490-1494 active 351-364

cone kernel distribution (CKD) 1364- magnetostrictive materials 753-762 active constrained layer (ACLD)1366 power balance and 1267 353-360, 355F, 356F, 359T,

Constantinides, AG 380-395 rotation and 112-113, 1068-1069 359F, 361F, 362F, 363Fconstitutional white finger 621 SEA and 1266 active mass driver (AMD) systemcontact algorithms 280-281, 308 standards and 1231-1232 30, 33FContinuous Methods 286-294 variational methods and 1354 active/semi-active 347-349

complementary energy method couple unbalance 1185 advanced concepts 640-641289-290 crack propagation 509-512, 1083 augmentation 630

differential transformation method Craig, RR Jr. 691-698 chaos and 228291-292 Crash 302-314 classical 723-725

formulation and solution 287-288 application examples 312 complex modulus 338-339Galerkin method 290, 292-293 axial loading 310F Coulomb 337lower-bound approximations 291 bioengineering and 310-311 critical 314-319Rayleigh method and 288-289 component roles and 308 directly-coupled 344FRitz method 290 contact algorithms and 308

continuous systems 1312-1317, DYNA3d 305-312, 305T, 307T discrete elements and 395-402

1327-1332 economic elements and 308 dual frequency 638-639

variational methods and 1350-1354 human head impact 312F Duffing's equation and 233

continuum mechanics 973-974 impact crushing 309F Duhamel's Principle and 1308

contour maps 416 safety standards 303T earthquakes and 442F, 449-460,

control See active control solid mechanics and 304-305 451F,460F

controlled numerical center (CNC) supercomputers and 304 elastically coupled 345-346, 346F

tools 1379-1380 vehicle collision 304, 310F, 311F, elastomeric testing and 631-632

control surface buzz 96 312F equations of motion and 1324-

convergence 466F CRAY supercomputer 304-312, 305T 1332convolution 381,1304-1308 Critical Damping 314-319 equivalent viscous 340-341, 341F,

Cooper, JE 87-97 definition of 315-317 721-722coordinate measuring machines distributed parameters 318-321 in FE models 321-327

(CMMs) 1490 initial conditions and 315F fluids and 467-475See also isolation theory lumped parameters and 317-318 fluid-structure interactions and 551

Coordinate Modal Assurance critical moment theorem 1070-1071 flutter and 553-565Criterion (COMAC) 270-271, cross-axis sensitivity 1131-1132 fractional derivative 325270F, 275, 276 cross flow 1023 friction and 582-589, 590-592

coordinate orthogonality check cross-generalized mass (CGM) matrix fundamental theory and 1290-(CORTHOG) 271 267 1299

coordinates 130-131 cross-orthogonality (XOR) matrices helicopter 629-642Co-Quad plots 418 267 hybrid 28-30, 30F, 31F, 32F, 353-Coriolis effects 409-410, 857, 860 crystalline growth 754 360Correlation Functions 294-302, 680 Curie brothers 1011, 1014 hysteretic 323-324, 646, 658-664

computational aspects of 298-299 Curie-Weiss law 476,478 impulse response function 1338-flow propagation 299F, 300F, current analysis 378 1339matrices and 298, 299-302 current sensor 1399-1400, l400F isochrones and 420nonstationary signals and 296 curse of dimensionality 902, 902F isolation theory and 1487-1506,random signals and 297-298 CuZnAI 660 1507-1521

I vi INDEX

damping (continued) damping ratios 413 degrees of freedom (DOF) (continued)Krylov-Lanczos methods and 691- Danish Standards Association (DS) FRF data and 418, 420

697 1224 ground transportation and 605-613linear matrix methods 721-726 dashpot impulse response function 1335-liquid sloshing and 734-740 damping measurement and 332- 1343localization and 747 333 inverse problem and 1259-1265magnetic constrained layer (MCLD) Duhamel's Principle and 1308 isolation theory and 1488-1490,

352, 353F parameters 636, 661 1494-1501mass-spring system and 687F, 689, viscous damping and 336-337 Krylov-Lanczos methods and 691-

689F data 365F 697materials 327-331, 329F, 331F, acquisition 364-376 modal analysis and 828

337-338 aliasing 365-369, 365F, 366F, modal properties and 269-271matrices 360-363 367F, 369F model updating and 852, 854maximum control voltage and 360, basic diagnostics and 377 NNM and 919-920, 919F

360T cleansing and 901 nonlinear system resonance andmeasurement of 332-335 discretization errors and 369-371, 928-943membranes and 762-770 370F, Nyquist plot of 424FMEMS 779-781, 780F, 783-784, displays and 413-431 parallel processing and 992, 995

784T environmental testing and 491-492 piezoelectric damping and 358modal analysis and 820-824 external sampling and 373-375, Rayleigh method and 1309-1312modal properties and 324-325 373F residuals and 848-851modulus 338-339 feature processing and 901-904 resonance and 1047-1055, 1048F,mounts and 342-349 modal 413-417 1049F, 105OF, 1051T, 1053Fnegative 97 neural networks and 863-868 robots 1060nonclassical 725-726 periodic 491 Schur method and 995-1000nonlinearity and 420 random 491 signal representation and 645nonlinear resonance and 932-934 resampling schemas 374F, 375, SNDT and 899, 899Fpassive 343-345, 351-352 375F spatial properties and 257, 258physical mechanisms of 321F sigma-delta converters 371-372, stochastic analysis and 1240-1242,piezoelectric 352, 352F, 353F, 354- 371F, 372F 1250-1252

360 Simpson's rule 1197-1198 structure-acoustic interaction andplate vibration and 889-890 SNDT and 900-904 1265proportional 723 transient 491 substructuring and 1333-1335,residuals and 848-850 triggering and 373, 373F 1333Fresonance and 1046-1055 See also modal analysis superposition and 1300-1301robots and 1056-1057, 1063 Data Acquisition 364-376 translation and 1325Frotor dynamics 1085-1088, 1092 data set 674, 676F variational methods and 1357-1359rotor-stator interactions 1107-1121 David, A 1001-1009 viscous damping and 324-325semiactive 30-33, 33F, 34F, 35F, da Vinci, Leonardo 125-126 See also boundary conditions

36F deformable mirrors 483-485, 485F De Laval, Carl Gustav 112, 1064,shape memory alloys and 352, degrees of freedom (DOF) 1326F, 1065

353F, 1146, 1149-1151 1327F, 1332 De Laval model 1070-1071, 1072,ship vibrations 1168 absorbers and 15-18 1085-1088,1092,1102-1103,signal generation and 1184-1185 active suppression and 49-51 1110,1112single frequency 632-637 averaging method and 960-961 chaos and 1112-1115structural 323-324 cepstral analysis and 217F Den-Hartog's implementation 2suspension and 37-38, 616 chaos and 227-236 deployment diagrams 970, 972Ftires and 618 critical damping and 315F design optimization tools (DOT) 637treatment types 351-354, 352F, damping measurement and 332- deterioration 376, 376F

353F, 354F, 333 deterministic models 1200, 1203-vehicular vibration and, 38-44 damping mounts and 342-351 1204viscoelastic 325-326, 326F, 339- direct problem and 1254-1259, detuning 404

340, 649, 656-658 1257F, 1258F, 1259F Deutches lnstitut fur Normung (DIN)viscous 321-323, 324-325, 336- Duhamel's Principle and 1308 1224

337, 340-341, 344F, 633-634, earthquakes and 444, 447-460, development testing 490636-637, 1294-1295, 1496- 447F Devloo, P 967-9761501, 1548-1551 environmental testing and 498 Devonshire theory 476-477

wind and 1583 equations of motion and 1291- Diagnostics and ConditionSee also absorbers 1293, 1324-1327 Monitoring, Basic Concepts 376-

Damping, Active 351-364 FEA software and 245 380Damping in FE models 321-327 finite difference methods and 524- detection and 381

viscous 321-323 525 deterioration time and 376FDamping Materials 327-331 flutter 567-570 general principles of 376-378, 377FDamping Measurement 332-335 forced vibration and 1295-1299 rotation and 379, 379TDamping Models 335-342 force transducers and 1123, 1125, signal generation and 1184-1193Damping Mounts 342-351 1126, 1130 vibration signatures and 379

INDEX I vii

differential transformation method displays 413-431 dynamic systems291-292,291 T frequency response data 417-420 basic principles of 125-126

differentiation 1195-1196, 1198 frequency spectra 420-421 blades and 174-178Digital Filters 380-395, 392F modal data 413-417 coordinates for 130-131

acausal 1196 model order indication and 425- fundamental theory and 1290-1299bilinear transformation and 393- 429 nonlinear systems and 952-966

394, 393T, 394T, 395F Displays of Vibration Properties 413- object oriented programming andcanonic realization and 385, 385F 431 974-975differentiation/integration and distributed parameter systems (DPSs) packaging and 983-989

1193-1199 318-321 parametric excitation and 1001-discrete-time systems and 381-382, sensor/actuators and 1134-1143 1009

381F Doebling, S 898-906 rotor-stator interactions 1107-1120finite impulse response 385-387, domains Seeparallel processing tire vibrations 1369-13 79

387T, 387F Dongarra, JJ 973 variational formulations in 1322-frequency sampling and 389-390, Donnell-Mushtari-Vlasovequations 1324

390F, 393F 1159-1160,1161-1162 vibro-impact systems and 1531-infinite impulse response 391-394 Donnel's theory 353 1548minimax design and 390, 391T, Doppler effect 700, 700F dynamic unbalance 1186

392F laser vibrometers and 700-706 Dyne, S 1193-1199seven-point 1196 double modes 416, 419Fsignal flow and 383-385, 383F, chaos and 229-232 E384F, 390F doubling map 230Fsuppression and 385 Drew, SJ 1443-1455, 1456-1480 Earthquake Excitation and Responsewindows and 387-389, 388F drop test 617 of Buildings 439-461, 441FSeea/so signals Dubois-Pelerin principle 973 attenuation and 440Tdigital signal processing (DSP)367 Dubuisson, B 869-877 damping and 451T, 451F, 460Fdimension reduction 432-435 Duffing's equation 227,228,233-235,

Dimentberg, MF 1033-1039, 1040- 235F,1110 elastic MDOF systems 449-453

1046, 1246-1252 Duhamel's Principle 444, 1305-1308, elastic SDOF system and 447-448dipole sound 880-882 1306F Fourier amplitude spectrum andDirac's delta function 1336, 1336F Duncan, Dowson 1070 444F,445F

direct problem 1254-1259, 1257F, Dunkerley method 112, 135-136 ground motion and 439-446, 441F,

1258F, 1259F Durbin's method 1203 442F,443F, 444F,445F, 446F

direct solvers 992 DYNA3D 305-312, 305T MDOF systems and 455-459Dirichlet preconditioner 999-1000 codes for 305-307 SDOF systems 444, 447-455, 447F,

Discrete Elements 395-404 dynamic analysis 452F

damping and 402 bearings 143-152 shape functions and 453F

mass/inertia 396-397, 397F bifurcation theory and 435 spectra smoothing and 450Fmodeling of 396 cables and 209 tripartite response and 449F

springs 396F, 398, 398F, 400T classification and unfolding 435 eccentricity 112-113torsional systems and 396 dimension reduction and 432-435 rigid states and 115

discrete Fourier transform (DFf) displacement fields 200 echo removal 225FSeeFourier transforms displays and 413-431 economic elements 308

discrete systems 1309-1311 earthquakes and 439-461 eddy current sensors 1400time 381-382 FEA software and 243-256 efficiencyvariational methods and 1355-1359 finite element methods 535 modal radiation 892-894

discretization 369-371, 973 fluids and 467-475 Eigensystem Realization AlgorithmD~ks404-413,405F impact 278-286 (ERA) 673, 677-682

asymmetric 408 isolation theory and 1487-1506, Eigenvalue Analysis 461-467axisymmetric 404-407 1507-1521 computational methods for 463gyroscopic couple and 1099-1101 laboratory vs. field 492-493 inverse problems 686-691nodal diameter 405-411 MEMS 779-781 Rayleigh quotient and 462rotating 409-410 normal form simplification and Rayleigh-Ritz analysis and 462-463symmetric 407-408 435 similarity transformation methodstypes of 404 robots 1055-1063 461,464vibration response and 405F, 410- rotors 1085-1106 Sturm sequence property and 463

413 shape memory alloys and 1144- vector iteration methods and 464-See a/so rotation 1155 466

displacement stability and 431-438, 433F, 435F, eigenvalues 424Fbeam vibration and 1330 436F,437F 4 DOF pitch plane model 608equations of motion and 1324- structural modifications and 1253- bifurcations and 818

1332 1264 boundary element methods 198t&splacementsensors transient 278-286 chaos and 228

capacitive 1399-1400 Seea/so active control; rotation; continuous systems and 1312-1317eddy current 1400 structural analysis damping and 322-323LVDT 1400-1401 Dynamic Stability 431-438 FEA software and 255T, 256

I viii INDEX

eigenvalues (continued) electrorheological (ER) fluids 640 environmental testing (continued)finite element methods 530-533 actuators and 58-72 input-output relationships 497-498,hysteretic damping and 323-324 new applications of 474-475 497Flocalization and 744 semiactive control and 467-468 laboratory dynamic and 492-493,model validation and 851-854 smart fluids and 468-474, 468F, 500-502nonlinear systems and 953 469F,470F, 471F, 472F, 473F LDVs and 1406normal form simplification and electrostatic field measurement locations and 494

435 MEMS 785-787, 785F, 791T motion control and 493plate vibration and 889 Electrostrictive Materials 475-490, seismic instruments and 1121-1134residuals and 848-851 476F,488F test methods and 490-491, 494-rotation and 1071-1072 actuator classification and 484F 495structural-acoustic interactions applications of 482-490 use identification and 491-492

1281 deformable mirrors and 483-485, Environmental Testing,See also rotation 485F, 486F, 487F Implementation 496-504

elasticity flapper 489F Environmental Testing, Overviewaeroelastic effects and 87-97, 1584- interferograms and 486F 490-496

1586 ion rattling and 480F EPROM circuits 779boundary element methods 196, microscopic origins of 475-476 equations

196F multimorph mirror and 485F 2SLSmethod 679bulk waves and 908 oxide perovskites and 477-482, absolute motion 1382-1395cables and 210, 213 479T, 479F, 481F active absorbers 1-6columns and 238 phenomenology of 476-477 active damping 356-359, 360-364damping materials and 327-330 servo valve 489F active isolation 46discrete elements and 398, 398F, temperature and 477, 478F, 480- active suppression 49-57

399T, 400T 482, 482F, 483F,484F actuators and smart structures 62-FEA software and 251T elementary run out 112 63, 65, 66, 70-72fluid/structural interaction and element technology 282-283 adaptive filters 82-86

545-550 Elishakoff, I 236-243 aeroelasticity 89flutter and 553-565, 565-577 Elliott, SJ 81-87, 977-982 ARX models 677hysteretic damping and 659-660 encapsulation 969 autocorrelation function 977-978,isolation theory and 1487-1506 Energenics, Inc. 759 1592lag dampers and 629-642 energy averaging methods 98-110, 1244-noise and 887-898 absorption 308 1246nonlinear analysis and 948F, 1110- actuators and 70-72 balancing 112-117, 118-121

1111 basic principles of 124, 125-126 basic principles 124-136piezoelectricity and 1013 FEA 1320-1322 bearings 145, 147, 148, 155-156,potential energy and 1346-1347 flow 1266-1267, 1267F, 1269F 160sensor/actuators and 1134-1143 fundamental theory and 1290-1299 belts 166-169, 170-172shape memory alloys and 1144- Hamilton's Principle and 131 bending strains 1136

1155 hybrid control and 649-658 biharmonic differential operatorshock isolation and 1180-1184 magnetostrictive materials 754- 1024transmissibility 1522-1527 755 blades 175, 177, 178vibration intensity and 1480-1487 operator 601 boundary conditions 181-188viscoelastic dampers and 649, 656- Rayleigh's method 1309-1317 boundary element methods 192-

658 resonance and 1047-1048 201waves and 1566-1568 Ritz method 1318-1319 bounded waves 1560-1562See also boundary conditions; SEA and 1266 Bresse-Timoshenko 239

damping shape memory alloys and 1148 bridges 203-206elastic modulus 1329 stochastic systems and 1250-1252 cables 209-213Electricite de France (EDF)220 time-average 1268 calculus of variations 126, 130Electric Power Research Institute 1078 variational formulations 1322- capacitance 1399electrodynamic shaker 494 1324 CEDA 1271electromagnetic acoustic transducers See also spectra cepstral analysis 217, 222

(EMATS) 913 engineering units (EU) 1209-1211 chaos 228-234electromagnetic damping composites ensemble average response 1268 circle-fitting method 822

(EMDC) 354, 354F envelopes 493-494 circular plates 1026electromagnetic sensors 1401-1402, environmental testing 490-496, 492F, civil structures 28

1402F 496-504 classical damping 723electronic speckle pattern correlation and 497F, 499F, 500F, columns 237-238, 239-242

interferometry (ESPI)699, 705, 503F complex exponential method 821706F, 709F envelopes and 493-494 continuous methods 287-293

full field measurement and 707-709 field dynamic 492-493, 498-500, correlation functions 295-301electronic vibration origins 1191 499F correlation methods 680Electrorheological and fixture design and 495-496 cost function 977

Magnetorheological Fluids 467- force identification and 502-503 crack propagation 510-511475 frequency domain and 497-498 crashworthiness 306-308

INDEX I ix

equations (continued) equations (continued) equations (continued)critical damping 315-316, 317-321 generalized inverses 720 model based identification 674-683Curie-Weiss 476, 478 Green's 879, 888 model class 674-676D'Alembert 1322 ground transportation 603, 608- model order selection 682damping in FE models 321-325 621 model updating 845, 847-851damping materials 327-329 guided wave 1552, 1553, 1554 mode of vibration 838, 840-843damping models 336-341 Hamilton's principle 131, 1322- Moore-Penrose generalized inversedamping mounts 343-348 1323, 1328 717-718data acquisition 370-372 hand-transmitted vibration 626 motion 1291-1293, 1325, 1326,data set 674 Hanning window 1590 1327, 1328, 1329, 1331digital filters 381-382, 385-393 helicopter damping 631-635, 636- NM 601Dirac's delta function 1336 637 narrow-band demodulation 602direct problem 1254-1259 Helmholtz 195, 878 Navier's 911discrete elements 396-402 Hilbert transforms 642, 643 neural networks 869-876disks 404-405, 413 Hohenemser/Prager 238 NNM 918-920, 921-922dissipated power 1266 Hooke's 1012 noise 878-886, 888-895distributed actuation 1136-1137 Hu-Washizu stationary principle nonclassical damping 725Donnell-Mushtari-Vlasov 1159- 1321 nonlinear analysis 945, 947-949,

1162 hybrid control 650-654 953-956, 957-964Doppler frequency 1404 hysteretic damping 659-660, 661- nonlinear system resonance andDuffing 228 664 929-941Duhamel's Principle 1305-1308 Ibrahim time domain method 821 nonsingular linear systems 711-713Dunkerley's method 135-136 impulse response function 1336- nullspace 320DYNA3D and 306-307 1343 optimal filters 977-981dynamics 279-281, 283 input power vector 1267 packaging 984dynamic stability 432-435 internal problem 1279-1282 parallel processing 990, 992-999earthquake excitation 444-459 inverse iteration of rigid body parametric excitation 1002, 1003-eigenvalue analysis 461-463, 464- modes 718-719 1006

465,466 inverse problems 686-688, 1259- Parseval's 388, 445electromagnetic sensors 1402 1264 perturbations 232electrostriction 475-479, 484 isolation theory 1491-1505, 1509- piezoelectric materials 1012-1013,energy flow exchange 1266 1512, 1514, 1517-1518, 1521 1013T,1015-1017energy operator 601 Kaiser window 1590 pipes 1019-1022environmental testing 492-493, Kirchhoff-Helmholtz 888, 1457 plate vibration 889-890, 892, 1024

497-502 Kolomogorov 1249 Poisson's ratio 327ERA 680-681 Krylov-Lanczos methods 691-696 principle of least action 1323Euler-Bernoulli beam theory 137- Lagrange 131,240, 721, 1323 production error 676

142 Laplace transforms 1336-1338, Prony method 678-679Euler's 126-130, 878 1407-1409 proportional damping 723exponential window 1594 laser vibrometry 700, 701, 702, random processes 1033-1037,external problem 1277 703, 704, 707 1040-1041, 1043-1045FEA software and 255T Lenz's law 1402 rational fraction polynomialfeedforward control 513-516 liquid sloshing 727-735, 736, 737- method 823finite difference methods 520-528 740 Rayleigh integral 889finite element methods 531-543, LMS method 679 Rayleigh method 1309-1317

1322 Love-type 1155-1159, 1159T, Rayleigh quotient 462FIR filters 1594 1160-1161 Rayleigh-Ritz analysis 462flexural motion 130 magnetostrictive materials 754-755 Rayleigh's Principle 132-134flexural radiation 1457-1465 Mathieu 730, 1111 Rayleigh wave 1553fluids 470, 473 Maxwell 136, 1135 reciprocity 1267fluid/structure interaction 545-551 Melnikov's method 232 rectangular plates 1026flutter 556-559, 566-570 membranes 763-769, 1135 rectangular window 1588FMO 601 MEMS 780, 782-783, 785, 787- response properties 274-278FM4 601 789 Ritz method 134, 1322-1319Fokker-Planck-Kolmogorov 1238, minimum total complementary robot vibrations 1060

1248-1249 energy 1320 rotation 1070-1077, 1083-1084forced problem 1281-1282 minimum total potential energy rotor dynamics 1085-1095, 1097-forced response 579-581 1320 1106forced vibration 894-896 modal analysis 824-828 rotor-stator interactions 1107,force window 1593 modal density 1266 1110-1119Fourier-based identification 666- modal directivity 890 running sum 1196-1197

667, 668-671 modal parameters 683 Scruton number 1584free vibration 1029, 1293-1295 modal properties 266-269, 270- SEAT 1577friction 583-587, 591-593 271 seismic instruments 1121-1132Galerkin's method 135 modal radiation efficiency 892-894 shape memory alloys 1145-1153gear diagnostics 597-598, 600-602 mode acceleration 719 shells 1155-1165

[ x INDEX

equations (continued) equations (continued) excitationship vibrations 1168 virtual work principle 1320 acoustic 897shock 1173-1174 viscous damping 721, 722, 1548- aerodynamics and 1191signal generation 1185, 1188-1191 1549 aliasing error and 670signal integration/differentiation wavelet transformation 600, 1420- averaging methods 1244-1246

1193-1198 1428 balancing and 112-123signal processing, model based wave number 891 bias error and 670

1199-1205 Weiner-Khintchine theorem 1361 blades and 174-178similarity transformations 461, 464, wind 1579, 1580, 1581-1583, capacitance and 1399

471 1584 complex stiffness and 338-339Simpson's rule 1197-1198 windows 1587-1590, 1592-1594 Duffing equation and 228singular systems 714-716 Z transforms 1409-1411 Duhamel's Principle and 1304-Smale's horseshoe 233 See also Fourier transforms 1308SNDT 899 equations of motion (EOM) 1324 earthquakes and 439-461sound 1444-1446, 1451-1452 beams and 1329-1330 environmental testing and 497-498sound power 890-891 continuous system models and fluid/structural interaction andsound pressure with boundaries 1327-1331 545-551

888-889 membranes and 1331 forced response 579-581spatial properties 257-258, 260- multiple DOF 1326-1327 Fourier-based identification and

262 plates and 1331 665-671spectral analysis and 1209-1211, rods and 1328-1329 ground transportation and 605-621

1213-1221 shafts and 1328-1329 hydraulics and 1191spectral coherence function (SCF) single DOF 1324-1325 identification and 673-685

602 strings and 1328-1329 impulse response function 1335-standard wave 1329 equilibrium 1343statistical moments 1243-1244 averaging method and 960-962 isolation theory and 1507-1521steady-state power balance 1267 basic principles of 125-126 leakage errors and 671stochastic analysis 1238-1246 cables and 209-210 localization and 747stochastic differential calculus chaos and 227-236 MEMS 779-7811246-1252 curvature 209 modal 814F, 815, 816-817stochastic systems 1246-1252 discrete elements and 396 model-based identification andstrain-life method 507 eigenspaces and 953 673-685Stribeck 1188 environmental testing and 498- noise and 887-898Strouhal number 1584 500 nonlinear resonance and 928-943structonic cylindrical shells 1139 equations of motion and 1327 nonlinear systems and 956structural-acoustic interactions invariant manifolds and 955-956

1275-1282 nonlinear systems and 952-966 oil film 1191structural dynamic modifications point classification and 953 parametric 1001-1009

1254-1264 rotor dynamics 1096F, 1097-1098 periodic 1296-1297structural system parameters 823 variational methods and 1355 random errors and 671Sturm sequence property 463 See also chaos; parametric random processes 1040-1046substructuring 1333-1335 excitation; stability resonance and 1046-1055Succi method 1281 equipartition 1266 rotor dynamics 1102superposition 1300-1304 error rotor-stator interactions 1107-1121SVD 716-717 Fourier-based identification and ship vibrations 1167-1173synchronized averaging 598 666-671 stochastic analysis and 1238-1246,system coordinates 130-131 modal analysis and 834 1247-1252Taylor series 210 model-based identification and time frequency and 1366-1368time frequency methods 1360-1364 676-677 transverse vibration and 169Timoshenko beam theory 142 model updating and 845 vibration isolation and 1501-1504tire vibrations 1373-1377 neural networks 870 viscous damping and 1548-1551transmissibility 343, 348, 1522- spectral analysis and 1222-1223 See also active control; damping

1527 ETREMA Products, Inc. 759 exponential model class 676trapezium rule 1197 Euler 1099 external problem 1276F, 1279ultrasonics 212-213, 908-909, Bernoulli beam theory 137-142,

910-911, 1437-1438 166, 338 Funbound waves 1566-1567, 1569 field equations 878undamped vibration 722-723 Lagrange equations 1344-1346, failure models 143, 252Tvariational methods 1344-1359 1345T Faraday-Lenz law 755, 761vectorial approach 125-126 relations 316 far-field approach 890vector iteration 464-466 ED Machinery Safety Directive 626, Farhat, C 710-720vehicular vibration 38-42 1576 Farrar, C 898-906vibration absorbers 10-23 ED Physical Agents Directive 627- Fassois, SD 673-685vibration dose value 1575 628, 1576 fast oscillating function 1273Fvibration intensity 1480-1487 Ewins, DJ 332-335, 404-413, 805- Fatigue 505-512vibro-impact 1532-1540 813,829-838,838-844 classical approach to 508

INDEX I xi

Fatigue (continued) finite element analysis (FEA) flexural motion (continued)crack propagation analysis 509- (continued) sound and 1456-1480

512, 509T, 510F, 511F meshing and 250 source ratios 1463-1468, 1464F,estimation 647 MSC/NASTRAN 248-249 1466F, 1467F, 1468F, 1469F,strain-life method 505-508, 506F, nonlinear analyses 254 1470F, 1471F, 1472F, 1473F,

507T,508F,509T nonstandard elements 250 1474F, 1475F, 1476F, 1477F,faults object oriented programming and 1478F, 1479F, 1480F

bearings and 1083, 1187-1188 967-976 transmissibility 1523-1527coaxiality 1083, 1099 SAMCEF 249-250 Floquet theory 338, 960, 1002gears and 1189-1191, 1191F, software quality assurance and shape memory alloys and 1151-

1192T 245-246 1152FDAC matrix 275F solution methods and 255-256 fluids 887, 949Federal Aviation Administration standard elements 250 acoustic radiation and 545-550,

(FAA)245 Finite Element Methods 243-244, 545F, 546F, 547F, 548F, 549F,feedback 3-5, 28 250-253,530-544,1332 551F, 552Ffeedforward control 28, 46-47 basic approach to 531-533, 541T, bearings and 152-165

applications of 516-519, 518F, 542T damping modes 469, 469F, 470F,519F, 520F crash and 302-314 551

beams and 517 damping in 321-327 electrorheological (ER) 467-475,description of 513-516, 513F, 514F dynamic problems 535 640networks 866 eigenvalues 533 excitation and 473F

Feedforward Control of Vibration, error analysis 540-543, 543F force feedback and 474, 474F513-520 modal properties and 265, 266- force/velocity form 472F

Feeny, BF 924-928 267,269 ideal Bingham 469FFeigenbaum's cascade 1543-1544 model updating and 844-847, 852 lag dampers and 629-641Feldman, M 642-648 nonlinear analysis 536-538, 536F, liquid sloshing 726-740FETI (dual Schur complement method) 538F, 539F lumped model 472F

997-1000 nonstructural problems 539, 541T magnetorheological (MR) 467-475,field dynamic piezoelectric damping and 352-358 640-641

boundary conditions and 492-493, propagation 534 particle orientation and 468F495 spatial properties and 256-264 pipes and 1019-1024

environment 492-493, 498-500 static problems 534 plate vibration and 1033measurement location and 494 structural-acoustic interactions Reynolds number and 472F

filters 1277-1278 semiactive dampers and 31-33acausal 1196 structural problems 536-539, 536F, shock isolation and 1183adaptive 81-87 538F, 539F signal generation and 1191antialiasing 367-368 substructuring and 1333-1334 smart 467-475Butterworth 703 finite impulse response (FIR) filters sound and 1443-1455digital 380-395 385-387, 977, 978F, 1594 structure interaction and 544-553FIR 385-387, 977-978, 978F, 1594 time domain formulation and 977- ultrasonics and 1437-1441IIR 1594 978 whirl and 1191minimax, 390 Wiener 978-981 Fluid/Structure Interaction 544-553optimal 977-982 finite spherical monopole 1447 Flutter 553-565recursion 100-101 Fisher's discriminant 903 active pylons 574, 575Fspectral analysis and 1216-1218 fixed references 1398, 1398F aeroelasticity and 553-560, 553F,Weiner, 978-981 fixture design 495 554F, 555F, 556F, 557F, 558F,

final prediction error (FPE) 1205 Flanagan-Belytschko constant 305 559F, 560FFinite Difference Methods 520-530 flap 629-630 binary 561

central 525-527, 526F flappers 485-488 modal properties 566-567, 567Fformulas 521, 521F, 522T, 523T Flatau, A 753-762 model of 555-559, 567forward 524-525 flexibility 118-119, 209 panel 96operators 522, 524T, columns and 239 simulation 570-571, 571F, 572Fpartial differential equations 528, isolation theory and 1487-1506 smart wings 575-576, 576F

529F plate vibration and 1033 solution 559-560finite element analysis (FEA) robots 1056-1057 suppression 572-573, 573F, 574F

ABAQUS246-247 sound and 1443-1455 test validation 561-563, 562F,ANSYS 247 spatial properties and 260-264 563Fboundary conditions 253 tire vibration and 1374-1375 torsion 561COSMOS/M 247-248 flexural motion 130, 1456-1480 transfer function 569coupled analyses 255 basic theory of 1456-1463, 1456F, types of 560-561dynamic applications and 244-245 1458T, 1459F, 1459T, 1460F, Flutter, Active Control 565-577energy methods and 1320-1322 1461F, 1461T, 1462T, 1463F FMO601geometric nonlinearity 252 boundary conditions and 186, FM4 601history of 244 187F, 188F, 189F foam blocks 1182linear analyses 253-255 MEMS 787-789 Fokker-Planck-Kolmogorovequationsmaterials and 252-253 Ritz method and 134 1238, 1248-1249

I xii INDEX

forced problem 1281-1282 &equency(conunued) frequency (continued)Forced Response 578-582 bearing diagnostics and 143-152 plate vibration and 414-415,889-

base excitation 580-581, 580F, Bragg cell 1404 890, 1024-1031581F bridges and 202-207 plotting of 413-414

harmonic excitation 579-580, cables and 211-215 residuals 850579F,580F capacitance and 1399 resonance 1046-1055, 1285

resonance 581-582, 581F columns and 236-243, 237T response function (FRF) 646, 1366forced vibration 1295-1299 convergence coefficients and 86-87 ride natural 614

friction damping and 585-586 critical damping and 314-319 Ritz method and 290modes of 843-844 cycle limits and 1107-1110 rotor dynamics and 1088-1090

force transducers 1123 damping in FE models and 321-327 rotor-stator interactions 1107-1121bending moment sensitivity and damping materials and 327-331 SEA and 1265-1272

1125-1127 damping models and 335-342 shape memory alloys and 1148-hammer attachment and 1124 damping mounts and 342-351 1153loading effects and 1128-1130 digital filters and 380-395 shells and 1160-1163rigid foundation 1123 disks and 404-413 shifting devices 702-703stinger and 1124-1125 Domain Assurance Criterion sigma-delta converters and 371-372

Forde, BWR 972 (FDAC) 275-276 signal generation and 1185-1192Fortran 973 domain methods 822-823 signal integrationJdifferentiationFourier, Joseph 1065 earthquakes and 439-461 1193-1199Fourier analysis energy methods and 1308-1324 spectra 420-421, 428F, 1208-1223

earthquakes and 443-445 environmental testing 496-504 stationary belts and 166helicopter damping and 634, 639 equations of motion and 1324- structure-acoustic interaction andtime frequency and 1360-1361 1332 1265-1274

Fourier transforms 1411-1419, Euler-Bernoulli beam theory 137- time methods and 1360-13691412F, 1414F, 1415F, 1416F, 141 tire vibrations 1369-13791418F excitation 414 transduction and 755-759, 755F,

adaptive filters 84-86 force transducers and 1123-1132 756F, 757F, 758Faveraging and 109 Fourier-based identification and ultrasonic 906-918, 1437-1441boundary element methods 201 665-671 vibro-impact systems and 1531-bridges and 203 FRF data and 417-420, 421F 1548CEDA and 1271, 1272F fundamental theory and 1290-1299 weighting 625-626, 626F, 627Fcorrelation function and 296 gear diagnostics and 741-751 weightings 1572, 1572T, 1573TdifferentiationJintegration and ground transportation 603-620 whole-body vibrations and 1570-

1193-1199 hand-transmitted vibration and 1578Duhamel's Principle 1305 623F windows and 1587-1595impulse response function 1339- helicopter damping and 629-642 See also rotation

1343 Hilbert transforms and 642-648 frequency response functions (FRFs)influence coefficients and 122 identification and 673-685 820linearity 1413 index 274F, 275F absolute motion transducers andMDL criterion and 1205 input-output domain 497-498 1382modal analysis 815 instantaneous 644 Assurance Criterion (FRAC) 274,nonlinear testing and 1287 isolation theory and 1487-1506, 276properties of 1412-1419 1507-1521 averaging and 100spectral analysis and 1208-1223 LDVs and 1403-1406 cepstral analysis and 217, 218, 222-wavelets and 1420-1423 liquid sloshing and 736-739 227windows and 1588-1592, 1588F, localization and 741-751 compound matrix 274,278

1589F membranes and 763-767 cross-axis contamination and 1122See also cepstral analysis MEMS 779, 783-784, 784T direct problem 1254-1259, 1257F,

fractals 1191 misalignment and 1186 1258F, 1259Ffractional derivative models 325 MMIF and 425-429 environmental testing and 299-302,free-free planar truss 420F modal analysis and 820-824 497-498free vibration modal indicators and 425-429 flutter and 563

complex 842-843 model-based analysis and 673-685, forced response and 579continuous methods and 287-293 1204-1205 force transducers and 1128-1130friction damping and 584-585 motion sickness and 856-861, 859F Fourier-based identification andorigins of 840-841 moving belts and 166-170 665-671orthogonality and 841-842 neural networks and 867-868 impulse response function 1341-structonic shell systems and 1139- noise and 877-887, 887-898 1342

1140 nondimensionall026T, 1027T, inverseproblem 1259-1265, 1259F,frequency 1028T, 1029T 1262F, 1263F, 1264F

actuator sensitivity and 1140-1141 nonlinear stiffness and 425F modal analysis 813-820, 824-828aliasing and 365-369 nonlinear system resonance and model updating and 852averaging and 99-100, 108-109 928-943 neural networks and 864basic diagnostics and 379-381 nonlinear testing and 1286F residuals and 848-850basic principles of 124 Nyquist 445 resonance and 1053-1054

INDEX I xiii

frequency response functions (FRFs) General Problem of the Stability of Ground Transportation Systems(continued) Motion (Lyapnuov) 1097 (continued)

response properties and 272-277 geometry suspension springs 615, 616F,shape memory alloys and 1150- eccentricity and 112-113 tires 617F, 618

1151, 1152F isolation theory and 1490 weight 615, 615F, 615TFresnel equations 182 membranes and 763-767 Guckel rings 788friction 589-590, 950 nonlinear systems and 251T, 252, guided waves 1551-1559

base isolators 587-589, 587F, 588F 947, 953 attenuation of 1558-1559counter-clockwise rotation 595F piezoelectricity and 1015 bars and 1556damping and 335, 582-589 rotating line 1067, 1068F cylinders and 1556dry 583-584, 583F, 584F, 586F rotation and 112 definition of 1551element acceleration 594F run out and 112-113 engineering and 1559forced vibration and 585-586 shape memory alloys and 1151 flat boundary and 1553free vibration and 584-585, 584F structure-acoustic interaction and plates and 1552F, 1553-1556limit cycle and 592-593 1265 theory 1552-1553links and 153-161 Gem, FH 565-576 Guyan reduction method 258, 1333misalignment and 1186 Gibbs energy 476 Gyration 239negative damping and 590-592 Giurgiutiu, V 58-80 gyroscopes 19-21, 1332PDF 594F Gladwell, GML 691 gyroscopic couple 1097,1099-1101,rotor dynamics 1095 glass transition region 329 1099F, 1100Fsnubber 1181, 1182F global error indicator 273Spurr's sprag-slip and 593-597 global positioning systems (GPS) 774 Hstatic vs. kinetic 582 global sonic nondestructive testingstochastic systems and 1250-1252 (GSNDT) 899,906 Haddow, A 1285-1289time history 594F aerospace and 905 Hagg's number 1088basis of 899velocity curve 591F

civil engineering and 905 Hall probe 761Friction Damping 582-589

data processing and 900-904 Hallquist, J 278-286Friction Induced Vibrations 589-596 Hamiltonian mechanics 228Froude scale 630 history of 904-905 bifurcation and 435Fuller, CR 513-520 operational evaluation and 900

distributed actuation and 1136full-field measurement 707-708 rotation and 905

energy methods and 1322-1323Goldman, Paul 1112fuzzy logic 43-45Golubitsky, M 432 nonlinearity and 235Graeffe's root-squaring method 463 piezoelectricity and 1014, 1015-

G Gram-Schmidt orthogonalization 691 1017Ritz method and 290

Galerkin method 135,239,240,290,graphical comparison 266

Hamilton's principle 131,358, 1328gravimetric calibration 1131292-293, 1332 gravity variational methods 1344, 1348-dimension reduction and 432-435 equations of motion and 1325 1349, 1355

Galilean referential 1085, 1086, 1106 liquid sloshing and 739-740 hammers 1121, 1124Galilei, Galileo 124-125 Green function 879, 888, 889, 1281 Han, RPS 972galloping 1585 Griffin, MJ 621-629,856-861,1570- Hand-Transmitted Vibration 621-629Gamma, E 969 1578 band spectra 623FGandhi, F 1548-1551 Griffin, S 46-48 blanching 624F, 626TGaussian classifiers 872 Groundhook control 41 dangerous processes 622TGaussian elimination 711, 711F, 715 Ground Transportation Systems 603- disorders of 621TGaussian theory 1135 620 effects of 621-625, 624T

time frequency and 1362 2 DOF pitch plane 609-610, 611F, evaluation standards for 625-628Gaussian white noise 1238-1240, 612F frequency-weighted 626F, 627F

1241-1246 2 DOF quarter car 612-613, 614F preventive measures 625, 625TGauss-Newton method 677 4 DOF pitch plane ride 608 sources of 621Gauss-Seidel algorithm 995 7 DOF 606F Stockholm Workshop scale 622TGauss's law 755 analysis models 605-621, 607F, threshold level 628FGabor, Denis 642 608F white finger 627FGear Diagnostics 597-602, 1080 damping 616, 617F whole-body vibration 1574

algorithms for 598-603 deflection ratio 613, 613F Hann, F 1578-1587cepstral analysis and 218-220 driver/passenger sensitivity 603- Hanning window 389, 672F, 1216-Choi-Williams distribution 600F 604, 605F, 619T 1219, 1218F, 1590-1591, 1590Fdisks and 404-413 fatigue time 604F harmonics 1270failure modes and 597 guidelines 614 boundary element methods 195-planetary system 597F loading 616F 198, 1279-1282signal generation and 1189-1191, mass transmissibility 613, 613F bridges and 202-207

1191F, 1192T parameter determination 614-621, chaos and 227-236standards and 1232-1233 619T,620T continuous systems and 1312-1317tooth averaging 599F PSD 618-621, 618F flexural radiation and 1456-1480vibration model of 597-598 ride natural frequency 614 forced problem 1281-1282

I xiv INDEX

harmonics (continued) Hooke's equations 660, 754, 1012 Identification, Fourier-Based Methodsforced response 579-581 Hopf bifurcation 435 665-672forced vibration and 1291-1296 horseshoe maps 232-234 additive noise 667F, 668Ffree vibration and 1293 hourglassing 305, 308 error mechanisms and 669-671friction and 589-596 Householder's method 467 frequency domain and 666, 666Fimpulse response function 1341- Hubble Space Telescope 485, 488F MIMO systems 669, 669F

1342 Hughes shells 305 MISO systems 668-669, 669Fnonlinear resonance and 928-943 hull wake 1169 noise-corrupted signals and 666-resonance and 1046-1055, 1288 human body 668response and 1286 crashworthiness and 308-311 response decomposition 667F, 669Frotor dynamics 1085-1088, 1097 energy absorption and 308 schema for 668Frotor-stator interactions 1107- force transducer and 1124 test 665F

1121 ground transportation 603-620 windows 672Fsignal integration/differentiation See also neural networks Identification, Model Based Methods

1193-1199 Hu-Washizu stationary principle 1321 673-685, 673F, 674Fsonic 898-906 Hybrid Control 649-658 ARMAX responses 684F, 685Fsuperposition and 1301-1304 active-passive devices 653-658, AR order 684Fultrasonic 906-918 654F,655F, 656F, 657F Blackman-Tukey method 685Fwindows and 1216-1218 design strategies 650-652, 650F, classification and 673-674See also excitation; oscillation; 651F continuous line response 684F,

sound piezoelectric network 653-655 685FHartmann, F 192-202 proof-mass actuator 652F, 653 data set and 674Hayek,S544-553, 1480-1487, 1522- stiffness and 658, 658F elements of 673

1531 viscoelastic layer 656-658 estimation criterion and 676-682Heckl, M 1265 hybrid damping 353-354 example of 684Helicopter Damping 629-642 characteristics of 354-360 frequency response 685F

advanced concepts 640-641 control law and 358 frequency stabilization and 684Faugmentation and 630 motion equations and 358 least squares methods 678-680dual frequency characterization hybrid-discrete-continuous model modal parameters and 683

638-639, 638F, 639F, 640F 172 model class and 674-676elastomeric testing 630F, 631-632, Hybrid III Family 311 order and 682-683, 684F

631F Hydraulic Institute 1225 parameter extraction and 683hysteresis modeling and 633F, 636- hydraulics 1191, 1232 schema for 674F, 676F

637, 638F, 639, 640F hyperstaticity 1116-1118 SISO system, 676Frotary hub 629F, 630F hyper-surfaces 228 validation and 683single frequency characterization hysteresis impacts 1531-1548, 1532F, 1533F

632-637, 634F, 635F, 636F double frequency 639 bearing faults and 1187-1188stiffness and 633-635, 633F, 636F, magnetostrictive materials 759 bifurcation and 1533-1536, 1535F,

637F, 638F single frequency 636-637 1539, 1539Fviscous 633-634, 641F transduction and 757 chaos and 1542-1548, 1542F,

Helmholtz equation 878 Hysteretic Damping 323, 325, 658- 1545F, 1546F, 1547Fboundary element methods 195 664 classification and 1536, 1537Fparallel processing and 1000 Boltzmann superposition model dynamics 278-286, 953

Hessenberg form 467 examples of 1533, 1533F, 1534F,heteroclinic point 229 661-662

1535Fdefinition of 659-660high frequency resonance technique frequency-dependent 323 noise 1449-1452(HFRT) periodic stability and 1535F,

bearing diagnostics and 146, 146F, isolation theory and 1494-1501 1538F, 1539-1542, 1539F,1188 Kelvin model 660-661, 660F 1541F

Hilbert Transforms 642-648, 642F, loading 659F stability regions and 1536, 1538F648F material complex modulus 323- subharmonic regions and 1536-

analytic signals and 643-646 324 1539, 1538FCEDA and 1271 nested loops and 662F impedance 1444-1445, 1444Tcepstral analysis and 218, 222 nonlinear models 662-664 bearings vibrations and 153-155notation 642 relay operators 663F, direct problem and 1254-1259properties of 642-643 resonance and 1050 influence coefficients and 120-123transformers and 647-648 inverse problem and 1259-1265vibration systems and 646-647 I rotor dynamics 1093

Hodges, CH 744 improved reduced system (IRS)258Hohenemser-Prager equations 238, Ibn-al-Razzaz 112 impulse response functions (IRFs) 820

241 Ibrahim, R 582-589, 589-596, 726- cepstral analysis and 217-218Holmes, PH 431 740 time frequency and 1366-1368,Holmes, P] 227-236 Ibrahim time domain method 821- 1366F, 1367Fhomoclinic loop 231F 822,823 vibration isolation and 1504-1505homoclinic points 229 IC technology See MEMS inchworm motor 759

Smale's horseshoe and 232-233 IDEAS-MS305 indicator function plot 430F

INDEX I xv

inertia 1325 Inverse Problems in Vibration Kelvin model 660-661, 662basic principles of 124, 125-126 (Gladwell) 691 Kelvin-Voigt damping 338-339belt drives and 170-172 ion propulsion 771 kernel design 1363columns and 239-240 isochrones 420, 426F, 1093 Kijewski, T 1578-1587coupling and 1326 isolation theory 1507 Kimball-Love observation 338discrete elements and 396-397, ambient vibrations 1507-1508, kinematic conditions

396F, 397F, 1508F equations of motion and 1324-fluid-structure interactions and coupled systems 1490-1493, 1332

549-550 1494F, 1496F, 1497F, 1498F, fundamental theory and 1290-1299force transducers and 1123-1132 1499F kinetic energy 1266isolation theory and 1490, 1507- criteria of 1508F, 1509F, 1510- columns and 240

1521 1512, 1511 T discrete elements and 395-397model updating and 845 degrees of freedom (DOF) and equations of motion and 1327plate vibration and 1033 1494-1501, 1494F, 1496F, friction and 589-596rotor dynamics 1085-1106, 1329 1497F, 1498F, 1499F, 1500F, Hamilton's Principle and 131shear flexibility and 239 1501F piezoelectric damping and 357shells and 1159 detrimental effects and 1508-1509 resonance and 1047unbalance and 1185-1186 dynamic systems 1512, 1513F rotor dynamics 1102vector iteration and 464-466 elastic mounts 1490 time-average energy and 1268vibration isolation and 1487-1506 experimental selection 1521 variational methods and 1350, 1353See also damping; mass; rotation general purpose machines 1517- See also energy

infinite impulse response (IIR) 391- 1519, 1518F, 1519F Kirchhoff-Helmholtz equation 888394 geometric properties 1490 Kirchhoff plates 197

infinitesimal cube 1566F impacts 1515 Klapka, I 967-976influence coefficients 119-123 impulse excitation 1502-1504, Kobayashi, AS 505-512infrared analysis 378 1503F, 1504F Kolmogorov-Arnold-Moser theoreminheritance diagrams 969, 975F inclined mounts 1493 235initial peak 1098 inertia and 1490, 1515-1517, Kolmogorovequation 1249Inman, D 278-286, 314-319 1516T, 1516F Krasnosel'skii-Pokrovskii kernelsinstability diagram 414 mounting conditions 1517-1518, 664instantaneous frequency 644-645 1518F Krishnan, R 629-642instantaneous phase 643, 644F nonlinearity 1504-1505 Kronecker delta function 186, 1275Institute of Electrical and Electronic nonrigid structures and 1520 Krousgrill, CM 928-943

Engineers (IEEE) 245 polyharmonic excitations 1515 Krylov-Lanczos Methods 691-698Instrumental Variable (IV) method precision and 1513-1514 block-Lanczos algorithm and 695-

675-676,680 random excitation 1501-1502 696integration 1196-1198 single frequency excitations and modes of 692-694intelligent transportation systems (ITS) 1514-1515 other applications 697

771 transmission model 1509-1510, physical meaning of vectors 691-intelligent vehicle systems (IVS) 773 1509F, 1510F 695interaction diagrams 968F wave effects 1505, 1506F Krylov subspaces 467interface force 1124 isotropic links 1093-1097 Kunaporn, S 512interference diagrams 414 iteration Kyobashi Seiwa Building 30interferometers 701-702 eigenproblems and 463See also laser based measurements

interlaminar stress 787-789inverse 465-466, 466F L

internal problem 1276F, 1279-1282solvers 993-994

International Electro-technicalsubspace 466 laboratory dynamic 492-493, 500-

Commission (1EC) 1224vector methods and 464-466 501,500F

International Road Traffic Accident See also chaos LADWP Receiving Station 442

Database (IRTAD) 302 lag 629-630

International Standards Organization J See also helicopter damping

(ISO) 245, 604, 1081-1082, Lagrangian mechanics

1224-1225 Jacobi method 464, 467 DYNA3D and 305

2631 1572, 1575 jitter 102F, 103-108, 108F, 109F energy methods and 290, 1323

5349 625, 626T Joule effect 754 FETI and 997-1000

nonrotating parts and 1226-1232 formulation 131, 1324, 1344-1346,

standards and 1228-1232 K 1345T

invariant manifolds 228-229, 955- interpolation 389

956 Kaiser window 1590 motion 1327inverse iteration 718-719 Kajima Shizuoka Building 30, 34F, Lame coefficients 186, 1567Inverse Problems 686-690 35F Lame parameter 1136, 1137

classical 686-688 Kane's method 1060 Lanczos methodstructural dyanmics and 1259- Kapania, RK 1335-1343 FEA software and 256

1264, 1259F, 1262F, 1263F, Kareem, A 1578-1587 vectors 692-6941264F Karman-type nonlinearity 1015 See also Krylov-Lanczos methods

I xvi INDEX

Laplace transforms 1407-1409 links 152-153 Love-type equations of motion 1155-boundary element methods 201 bearings 153-165 1159, 1159T, 1160-1161identification and 673 fluid 153-155 Lowe, MJS 1437-1441, 1551-1559,impulse response function 1336- isotropic 1093-1097 1559-1564, 1565-1570

1338 kinematic 1060 low oscillating function 1273Fmembranes and plates 1331 magnetic 158-161 LQG/LTR controller 1147

Laser Based Measurements 698-710, misalignment and 1116-1118 LV factorization 711-712703F, 1403F, 1403-1404 rotation and 1068-1069 lumped parameter approach 887

applications of 705-706 Liquid Sloshing 726-740 classification and 1290-1291Doppler vibrometer techniques Bond number 739F critical damping and 317-318

700-706, 704F, 705F coordinates 728F sensor/actuators and 1134-1143full-field measurement and 707-709 damping 851 superposition and 1300-1304geometric properties and 707F free and forced 727-729 Lyapunov 1097holographs and 699, 708F gravitational field 739-740 Lyon, RH 1265non-Doppler techniques 706-707 mechanical models 731-734scanning systems 703F, 704F modeling 731F, 732F, 733F Mspeckle noise and 705 parametric 730

laser Doppler vibrometer (LDV) 1403- rigid moment of inertia 733F, Ma, F 721-7261406 734F McConnell, KG 1121-1134, 1381-LDL super T factorization 712-713 road tankers 735-738, 737F, 738F, 1397,1398-1406lead magnesium niobates (PMNs) 739F Mach numbers 563660 surface motion 731F Mach-Zehnder interferometer 701-leakage 671, 1588-1591 tank shapes 735T 702, 702F, 1404, 1404Fwindows and 1216-1218 liquid spring 1182 Macjkie, RI 972learning algorithm 871-872 Littlewood, JE 228 McKee, K 143-152least-squares method 677, 678-680, Liu shells 305 MACSYMA 10101203 loading 253T MAC values 852-854, 853TLeissa, AW 762-770,1024-1031 bending moment sensitivity 1125- Maddux, GE 1398-1406Lenz's law 1402 1127 MADYMO 305, 311Lesieutre, GA 321-327 bridges and 202-207 magnetorheological (MR) fluids 467-Levenberg-Marquart (LM) algorithm columns and 236-243 475,640-641866 crash and 310F actuators and, 58-72Levinson recursion 1203 earthquakes and 446-460 new applications of 474-475Li, CJ 143-152,597-603 flight 88-89 semiactive control and 467-468Liapunov-Floquet transformations force transducers and 1123-1132 smart fluids and 468-474, 468F,1004-1005 ground 93-95

469F, 470F, 471F, 472F, 473FLieven, NAJ 578-582 gunfire 93 magnetic constrained layer dampinglimit cycle 592-593, 592F hysteretic damping 659-660, 659FLin, YK 1238-1246 impulse response function 1335- (MCLD) 352, 353F

Linear Algebra 710-720 1343 magnetic links 158-161BLAS 991 isolation theory and 1518-1520 magnetic systems 1088Moore-Penrose generalized inverse noise and 887-898 Magnetostrictive Materials 660, 753-

762717-719 packaging and 983-989 actuation configurations 58-72,nonsingular systems 710-714 shape memory alloys and 1147-singular systems 714-716 1148 759, 760F

SVD 716-717 signature generation and 144-145, magnetism and 753-755, 754T

linear analyses 252-255, 253T, 254T 144T sensing configurations 760-761,cables and 210-215 suspension 37-38 761FFourier-based identification and Localization 741-751 transduction 755-759, 756F, 757F,

665-672 bladed disk assembly 741F, 745, 758FLinear Damping Matrix Methods 750F, 751F Maia, NMM 820-824, 824-829

721-726 coupled oscillators and 746F, 747F MA (moving average) modeling 1199,linear dependence 295F engineering significance of 748-749 1203F, 1203linear interpolation function 282 forced response and 750F manifolds 229F, 955-956linear least squares method 677 gear faults and 598 center 962Linear Multi Stage (LMS) method harmonic frequency and 745F Duffing's equation and 233-235

674-680 history of 744-745 invariant 228-229linear systems key results of 745-748 Melnikov's method and 232

Duhamel's Principle and 1304- mode 742-744, 743F, 746F, 748F, Maple 1010, 13401308 749F Marcondes, J 983-989,1173-1180

stochastic analysis and 1247-1252 NN and 922 Markov theory 677superposition and 1299-1304 truss beams and 741F, 749 stochastic systems and 1246-1252

linear variable differential transformer logarithmic decrement method 617 martensite See shape memory alloys(LVDT) 1193,1398,1400-1401, logarithms See cepstral analysis mass1401F looping balancing and 111-124

Link, M 844-856 hybrid control and 650-652 basic principles of 124, 125-126

INDEX I xvii

mass (continued) mathematics (continued) mathematics (continued)continuous systems and 1312-1317 Cantor set 233 windows 1587-1595discrete elements and 396-397, chaos theory 227-235 Young's modulus 889, 1567

396F, 397F, classes 975 Mathieu equations 228, 730, 1110,Dunkerley's method and 135-136 continuum mechanics 973-974 1111eigenvalue analysis and 461-467 cost function 977 MathWorks 682equations of motion and 1324- Dirac's delta function 1336 MATLAB 682, 1343

1332 Duffing's equation 233-235 matrices 1267forced response and 581-582 eigenvalue analysis 461-467 active suppression 51-55force transducers and 1123-1132 equations of motion 1324-1332 adaptive filters 81-87isolation theory and 1507-1521 expectation operator 345 CFRF 274, 276F, 277F, 278MEMS 781-782 external problem 1279 CGM 267model updating and 845 finite difference methods 520-528 classes 975FRayleigh method and 1309-1312 Floquet theory 338, 1002 correlation 298, 299-302Ritz method and 1318-1319 fundamental equations of motion critical damping and 317-318robots and 1062 1291-1293 cross-sensitivity 1126sensor/actuators and 1134-1143 fuzzy logic 43-45 damping 321-324, 346-347, 356-transmissibility 1522-1527 Gaussian theory 1135 358, 721-726unbalance and 1185-1186 gear diagnostics and 598-602 direct problem and 1254-1259variational methods and 1355 Hilbert transforms 642-648 elemental 363vehicular vibration and, 37-45 Jacobi method 464, 467 environmental testing and 496-504vibration isolation and 1487-1506 Kolmogorov-Arnold- Moser FDAC 275FSee also inertia; rotation theorem 235 FETI and 997-1000

mass-spring model 687F, 689, 689F, Krylov-Lanczos methods and 691- flutter 567-570733F, 734 697 forced problem 1281-1282

dashpot system 591F Krylov subspaces 467 FRF 274-278discrete elements and 398, 398F, Lame' constants 1567 influence coefficients and 120-123

399T, 400T Lenz's law 1402 inverse problems 465-466, 686-resonance and 1047-1048 linear algebra 710-720 688shock isolation and 1180-1184 Mathieu equations 228, 730, 1110, laboratory vs. field 492-493superposition and 1300-1301 1111 Lanczos 694-695See also boundary conditions; Maxwell equation 1135 linear algebra and 710-720

damping Melnikov's method 232 linear damping 721-726material anistropy 251T model updating 847 lumped mass 279-280

acoustic impedance and 1561T, numerical efficiency and 972-973 MAC 268F, 269, 269F1562F parallel processing 990-1001 mass 260-264

damping and 327-331 Poisson ratio 889, 1331, 1568 model-based identification and 673,effects of 1569 QR algorithm 467 673Fevanescent waves and 1563F random processes 1033-1039 piezoelectric damping and 356-358facilities 252T Rayleigh quotient 462 Poincare' method and 957-959guided waves and 1551-1559 Rayleigh-Ritz analysis 462-463 random processes 1041, 1043plate vibration and 1029 root mean square 245 Rayleigh method and 1309-1312Rayleigh waves and 1554F running sum 1196-1197 rigidity 363robots and 1063 Schur method and 995-1000 rotation 1070-1077unbound waves and 1565-1570 Scruton number 1584 rotor dynamics 1085-1088, 1091-

material complex modulus 323-324 SEA 1265-1272 1092, 1102material damping 337-338 segment averaging 1219-1221 RVAC 274F, 275-277material properties sensor/actuators and 1141 sensitivity 847

actuators and 58-72 signal integration/differentiation shape memory alloys and 1147-MEMS and 797, 798T 1193-1199 1148shape memory alloys 1144-1155 similarity transformation methods signal processing 1201-1203shock absorption and 1174 461,464 singular 714-715, 720ultrasonics and 1439 Simpson's rule 1197-1198 skyline storage and 714F

Mathematica 1010, 1340, 1343 Smale-Birkhoff homoclinc theorem sparse 713-714mathematics 233 stiffness error 260-264

autocorrelation functions 977-978, Smale's horseshoe 232-233 stochastic systems 1247-12521592 SRSS 245 structural dynamic modifications

averaging methods 98-110, 1244- statistical moments 1243-1244 and 1253-12641246 Strouhal number 1584 Sturm sequence property 463

basic principles 124-137 Sturm sequence property 463, 467 submatrices 360Bendixson's theorem 593 Succi method 1281-1285 subspace iteration and 466Bessel function 729 Taylor series 1325 TOR 257block-Lanczos algorithm 695-696 trapezium rule 1197 transformation 360Bond number 739F, 739 van der Pol equations 227-228 XOR 257, 267Broyden- Fletcher-Goldfarb-Shanno vector iteration methods 464-466 See also eigenvalue analysis

algorithm 637 wind 1586 MATRIXx 682

I xviii INDEX

Maxwell, James Clerk 1065 meshing 249T, 250, 601 Modal Analysis, Experimental,equation of 1135 mapping solutions from 281-282 Parameter extraction methodsmodel of 337 See also gear diagnostics 820-824theorem of reciprocity 136, 819 metglas amorphous ribbons 761 Modal Assurance Criterion (MAC)

mean time before failure (MTBF) Michelson interferometer 701F, 701- 268F, 269F, 274-2761249 702, 1404F, 1404 COMAC and 267-271

mean-value method 1269 microelectromechanical systems See correlation functions and 296measurement MEMS numerical comparison and 266

damping 332-335 MILSpees 1225 SNDT and 903full field 707-709 MIMO (multi-input, multi-output) vector correlation and 267-270influence coefficients and 122 method 821-823 modal properties 889-890packaging 984-987, 984F, 984T, Mindlin-Reissner plates 197 active suppression and 51-52

985F, 985T, 986F, minimax filters 390 actuator sensitivity and 1140-1141rotation 1080-1081 minimum complementary energy balancing and 111-124standards for 1224-1238 1347 blades and 177See also Laser-Based Measurements; minimum description length (MDL) BLAS 991

modal analysis criterion 1205-1206, 1206F boundary element methods 192-mechanical impedance 1571 misalignment 1116-1118, 1186 202mechanical shock 1228 Modal Amplitude Coherence (MAC) cables and 211-215Melnikov's method 232, 233 683 COMAC 275, 276Membranes 762-770 modal analysis 805-813, 820-823 comparison of 265-272

circular 765, 765F, 765T, 766T applications 829-838 complex plot 429Fcomplicating effects 769, 769F, calibration 818 continuous methods and 286-294

770F damping and 826-828 continuum mechanics and 973-974other shapes 764F, 765T, 767, data processing 815-816, 816F damping and 321-327, 721-726

767F, 768F error location 834 density 1266, 1268rectangular 763, 764F excitation 808-809, 809F, 814F, direct graphical comparison 266shells and 1155-1167 815,816-817 direct numerical comparison 266strain 1135-1136 frequency-domain methods 822- disks and 404-413vibration 1331 823 displays and 413-417See also cables history of 806 DOF correlation 270-271

MEMS, Applications 771-779 mathematical construction 811- double 419FMEMS, Dynamic Response 779-794 813 earthquakes and 458-460MEMS, General Properties 794-805 measurement 813-820 energy 1266-1267MEMS (microelectromechanical method classification and 820 equations of motion and 1326

systems) 771, 1011, 1017, 1142 model construction 824-829 Euler-Bernoulli theory and 137-141acoustic microsensors 774, 775F multipoint testing 817-818 experimental analysis and 805-813applications of 760F, 771, 772T, NN and 922 FEA software and 244

777-778 parameter quality and 823 F-F-F-F square plates 1027Fdamping 783-784, 784T pretesting 819 finite element methods 530-544design 803-804, 803T, 804F procedures of 806 flutter and 566-567dynamic response 779-794 property comparison 830-832, Fokker- Planck -Kolmogorovelectrostatic field 785-787, 785F, 831F, 833F equations 1248-1249

791T response properties 824-826, 824F, forced problem 1281-1282, 1298-fabrication technology 798-801, 837-838 1299

799T, 801T, 802T, 803F, 803T sensing mechanism 815, 815F free-free planar truss 420Fgeneral properties of 794-805 signal processing 810, 811F free vibration frequencies andinterlaminar stress 787-789, 788F, spatial models 824-826, 824F 1026T

789F structural analysis 834-837, 836F gear diagnostics and 597-603machining 795F, 796, 797 support conditions 818-819 geometric nonlinearity 252mass 781, 782F, 782T theory of 807-808 indicator function 425-429, 430Fmaterial choice 797, 798T time-domain methods 821-822 influence coefficients and 119-123measurement 796, 796T transducers 809 inverse problem 1259-1264, 1259F,microaccelerometers 772-774, troubleshooting 829 1262F, 1263F, 1264F

772T, 773F, 774F updating 832-834 Krylov-Lanczos methods and 691-micro actuators 775-777, 776F, validation 819, 830 697

777T Modal Analysis, Experimental, localization and 741-751micropumps 777, 777F, 778F Applications 829-838 membranes and 762-770microsensors 771-772, 773F, 774, Modal Analysis, Experimental, Basic model-based identification and

775F principles 805-813 673-685microvalves 777, 777F, 778F Modal Analysis, Experimental, multivariate 429Fpackaging 801 Construction of models from tests near-field approach 891signal conditioning 789-790 824-829 noise and 887-898stiffness 782-783, 783T Modal Analysis, Experimental, nonlinear normal 918-924voltage conversions 786T, 790-793, Measurement techniques 813- nonlinear system resonance and

793F 820 928-943

INDEX I xix

modal properties (continued) modeling (continued) mounts (continued)nonlinear testing and 1286, 1289 pendulum 732-738, 732F fluid-elastic 347Foptimal filters and 977-982 pseudoresponse 849-850 multidirectional 346order selection and 682-683 quality and 847 multiple DOF 348Forthogonality criteria 266-267 robots 1055-1063 passive damping and 343-345overlap and 1265 rotor-stator interactions 1107-1121 SDOF 345F, 346Fparameter extraction and 683-685 selection and 1205-1206, 1206F See also dampingplanar flexural shapes 418F sequential methods 1203 MSC/NASTRAN 248-249plate vibration and 889-890 ship vibrations 1167-1173 Mucino, VH 302-314radiation efficiency 892-894, 893F signals and 1184-1193, 1199-1208 Mullins effect 631Rayleigh's Principle and 132-134 spectra and 1204-1205, 1205F multi degrees of freedom (MDOF)Rayleigh waves and 1554F stochastic 1199-1200 820rotation 1071-1072, 1075-1076 superposition and 1300-1301 multilayer feed forward (MLFF)rotor dynamics 419F, 1085-1106, symptom-based 1185 networks 866, 867

1090 test/analysis residuals 848-851 multilayer neural networks (MLNN)scale factor (MSF) 266 tool wear and 1380 864,872-875SEA and 1266-1267 transverse vibration and 169 multipath propagation 299-301,shape memory alloys and 1144- updating of 844-856 299F,300F

1155 validation of 851-855, 852F, 853F, multiple input/single output (MISO)shells and 1155-1167 853T, 854F, 855F systems 665, 668-669, 669Fsound radiation and 891-892 Model Updating and Validating 844- multiple input/multiple outputstochastic differential calculus 856 (MIMO) systems 665, 669, 669F

1246-1252 Mode of Vibration 838-844 feedforward control 513-520structural dynamics and 1253-1264 complex 842-843 multiple instructions/multiple datastructural modes and 683 definition of 838, 839F (MIMD) processors 990superposition and 1302-1304 essential features of 839-840, 839F, multivariate mode indicator functionthree-dimensional plot of 423F 840F (MMIF) 425-429, 429Fvalidation and 683 forced 843-844 MUMPS 795vector correlation 267-270 free 840-843 Muszynska, Agnes 1112wave number and 891 orthogonality 841-842weak elements 418F types of 839 Nwind and 1578-1587 mode shape plots 415-417

mode acceleration method 719 mode synthesis 1334-1335 NA4 601Mode Indicator Functions (MIFs) modulus damping 338-339 Naeim, F 439-461

425-429 modulus difference 271 Nanjing Communication Tower 30modeling momentum conservation principle 307 narrow-band demodulation 602

absolute motion and 1382-1395 monopole sound 880 NASTRAN 405additional methods for 1207 Monte Carlo simulation 740, 1046 National Agency for Finite Elementsantiresonances 848 localization and 746 and Standards (NAFEMS)245AR 1201-1203, 1202F Moon, FC 233 National Highway Traffic SafetyARMA 1201, 1203 Moore-Penrose generalized inverse Administration (NHTSA) 303T,basic principles 124-137 717-719 304biodynamics and 1572 Moors 112 Natori, MC 1011-1018classes of 674-676 motion natural frequenciescontinuous system 1327-1331 control 493 of plates and beams 414-415damping 321-327, 332-335, 335- envelopes and 493-494 plotting of 413-414

342, 850-851 friction and 589-596 Navier's equations 911deterministic 1200 sprag-slip 593-597 NB4 601discrete elements 396 transducers for 1398-1406 near-field approach 891equations of motion and 1324- See also oscillation

1332 Motion Sickness856-861 Neumann preconditioner 1000

finite element method and 844-845 causes of 857 neural networks

flutter and 555-565 dose value (MSDV) 859, 860 architecture and 870, 871F

gear vibration 597-598, 599,1189- nonvertical oscillatory motion 859- assessment and 867-868error and 870

1191, 1190F, 1191F 860 faults and 865-867hysteresis 636-637, 639 vertical oscillatory motion and 857-identification methods and 673-685 859 learning and 871-872, 874

liquid sloshing and 731-734 Motor Vehicle Safety Standard monitoring and 864-867MA 1199, 1203, 1203F (MVSS)303T, 304 multilayer 872-875, 873F, 875F

mass-spring 733F, 733-734 mounts 343F, 344T neurons 869-870, 869F

modal analysis and 824-829, 829- active/semi-active damping and patterns and 874838 347-349 perceptron 871-872

normal mode 848-849 basic concepts of 342-343 processing tasks and 863Tparameter identification and 847- dynamically coupled 347F radial basis function 875-877

848 elastically coupled damping and Neural Networks, diagnosticparametric methods 1203 345-346 applications 863-868

I xx INDEX

Neural Networks, general principles nonlinear analysis (continued) Nuclear Regulatory commission869-877 finite element methods 536-538 (NRC) 245

neurological disorders 622 Hilbert transforms and 646 nullspace 714-715, 720Newkirk effect 1107-1110 neural networks and 869-877 Nyquist plots 418-420Newton, Isaac 124, 1099, 1344 overview of 944-951 CEDA and 1271

law of motion 1324, 1325, 1329 parametric excitation and 1003- damping measurement and 333rotation and 1070, 1085 1009 DOF 424F

Newton-D'Alembert principle 1112 perturbation and 957-962 frequency 445Niemkiewicz, J 1224-1238 Poincare' method and 957-959 influence coefficients and 122Nitinol 660 resonance and 1054-1055 modal analysis and 822FSee also shape memory alloys rotor dynamics 1094F, 1095-1097, receptance 423F

nodes 1095F sigma-delta converters and 371-372circular plates and 1026-1027 sources and 945-947diameter (ND) 405-413 stability and 953-957 0laboratory vs. field testing 492-493 stochastic analysis and 1238-1246line indices and 414-415 vortices and 949F object oriented programmingmodal shape plots and 415-417 See also chaos class diagram 969rotation and 1073 Nonlinear Normal Modes (NNM) collaboration diagrams 970noise 887-898, 887-888, 896-898 918-924 component diagrams 971baffled plates and 889F, 891-896, applications of 921-924 continuum mechanics and 973-974896F, 897F cyclic assembly and 921F criteria of 968-969boundary conditions and 888-889 definition of 918-921 deployment diagrams 970cable 1133 degrees of freedom (DOF) andcorrupted signals 666-668 919F, 920 dynamic systems and 974-975dipole sources 882F resonance and 921-922, 923F language of 971-972Green's functions and 879 stiffness and 425F, 920F methodology of 969Huygen's source 884-885, 885F Nonlinear System Identification 924- numerical efficiency and 972-973monopole sources 885F, 886F, 928 sequence diagrams 970

887F Nonlinear System Resonance software design and 967-968neural networks and 867 Phenomena 928-943 state transition diagram 970plate vibration and 889-891, 893F, cubic 940-943 vs. procedural programming 968F

894F, 895F quadratic 937-939 Object Oriented Programming in FEpower and 883-884 response curves 932F, 933F, 935F, Analysis 967-976quadrupole sources 883F, 884T 936F, 938F, 939F, 940F, 941F, odd-odd plate mode 892-893radiating field and 879-882 942F, 943F oil analysis 378sigma-delta converters and 372 single degree of freedom and 930- oil film excitation 1191sound power and 890-891 937 oil-pressure servo valves 485-488sound pressure with boundaries two degrees of freedom and 937- oil whip 1085

888-889 943 online systems 1082, 1083spherical harmonics and 881 T nonlinear systems 924-925, 928,1332 open loop control 49-50superposition and 884-886 active states and 925-926 Operating Deflection Shapes (ODSs)waves and 878-879 elastic recall 1110-1111 272,275

Noise Radiated by Baffled Plates 887- hysteretic damping 662-664 optical heterodyning 1404-1405898 misalignment and 1116 Optimal Filters 977-982, 978F, 981F

Noise Radiated from Elementary nonparametric identification and time domain formulation and 977-Sources 877-887 926-927 978

nonconservative forces 1332 overview of 944-951 Wiener 978-981Nondestructive Testing, Sonic 898- parametric identification and 927 optimization

906 perturbation methods and 1010 hybrid control and 650-652Nondestructive Testing, Ultrasonic piezoelectricity and 1015-1017 rotor dynamics 1095

906-918,1439 transverse vibration and 167-169 structural modifications and 1262-nondestructive testing (NDT) 378, von Karman 1136 1264

1559, 1563-1564, 1568 See also chaos orthogonality 266-267, 270, 271nonlinear analysis 252, 254T, 255T, Nonlinear Systems Analysis 952-966 basic principles of 124

953 Nonlinear Systems, Overview 944- biorthogonality 1074asymptotic techniques and 957-962 951 forced vibration and 1298-1299averaging method and 960-962 nonsingular linear systems 710-714 inverse problems 688backlash and 951F nonstationary signals 296 isolation theory and 1514bifurcations and 963-965, 964F, nonvertical oscillatory motion 859- iterative solvers and 994

965F 860 mode of vibration and 841-842classes and 947-953 normal form theory 963, 1005 nonlinear systems and 926Coulomb friction law and 950F normalized cross-orthogonality Rayleigh's method and 1317dimension reduction and 962-965 (NCO) 270, 271 shells and 1155effects of 944 Norton, MP 877-887, 887-898, TAM and 259-260equilibrium and 953-957, 954F, 1443-1455, 1456-1480 Zernlike polynomials 484

955F, 956F nuclear radiation 1133 orthonormality 461

INDEX I xxi

oscillation Parallel Processing (continued) PATRAN 305averaging method and 960-962 FETI method 997-999 pendulum model 732, 732F, 734,bearings vibrations and 152-161 gradient iteration and 994, 994T 736boundary element methods 195- iterative methods and 993-994 absorbers and, 22

198 network topology and 990, 991F chaos and 228chaos and 227-236 Schur methods 995-1000, 997T, equations of motion and 1325F,critical damping and 314-319 998F 1325fast function 1273F parameters localization and 743F, 745FEA software and 245 4 DOF pitch plane model 608 road tanker and 735-738flutter and 553-565 analog inputs and 369 perceptron 871friction and 589-596 averaging method and 960-962 perfect inviscid fluid 1567fundamentals of 1290-1299 basic principles of 124 periodic data 491gear diagnostics and 597-603 bearing diagnostics and 145-151 periodic orbits 230Thelicopter damping and 629-642 bifurcation and 435 chaos and 227-236liquid sloshing and 726-740 chaos and 1112-1115 periodic rezoning 282limit cycle 95 critical damping and 314-319 periodic truss beam 741Flocalization and 741-751, 746F damping materials and 327-331 Perkins, NC 209-216,944-951low function 1273F damping measurement and 332- perovskites 477-482motion sickness and 856-861, 335 perturbation methods, 232, 957-962

857F, 858F, 859F distributed 318-321 bifurcation and 1003F

NNM and 918-924 earthquakes and 439-461 nonlinear systems and 1010noise and 877-887, 887-898 equations of motion and 1292- parametric excitation and 1003-nonlinear analysis and 952-966 1293 1004, 1009nonlinear resonance and 928-943 extraction of 683 rotation and 1002F

nonvertical motion and 859-860 feedforward control 513-520 See also chaosPoincare' method and 957-959 ground transportation 603-620 Perturbation Techniques for

power balance and 1267 inverse problems 686-688 Nonlinear Systems 1010

resonance and 1046-1055 Lame 1014, 1015 Peterka, IF 1531-1548

SEA and 1266-1267 lumped 317-318 phase space 228

stick-slip 589, 590F Markov 677 phase unwrapping 218

time frequency and 1360-1369 modal analysis and 821F phasors 1270

tire vibrations 1369-1379 model-based identification and physical classes 975

vector iteration and 464-466673-685 Pierre, C 741-751

vertical motion and 857-859model updating and 845-848 piezoelectric actuators 482-488

vibro-impact systems and 1531-neural networks and 864 absolute motion sensing and 1385-NNM and 919 1387

1548 noise and 887-898 hybrid control and 653-656viscous damping and 1548-1551 nonlinear systems and 952-966 viscoelastic layer and 656-658whole-body vibrations and 1570- signal processing and 1199-1208 piezoelectric materials 1011-1012,

1578 SNDT and 899 1012T,1017See also absorbers; boundary stochastic analysis and 1238-1246 active absorbers and, 1-3,2

conditions structural dynamic modifications active damping and 352, 352FOutput Error method 677 1253-1264 active isolation and, 47, 48overdamped system 316 updating 847-848 actuators and 58-72, 61Toverlap processing 1593 Parametric Excitation 1001-1009, advanced theory 1014-1015oxide perovskites 477-482 1332 applications 1013

cables and 214F damping and 353F, 354-360P Floquet theory and 1002, 1004- feedforward control 519

1005 flappers 485-488Packaging 983-989 nonlinear analysis and 1003-1009 four fundamental equations of

loading and 988F perturbation methods and 1010 1013, 1013Tmeasurement and 984-987, 984F, point-mapping and 1006 hysteretic damping and 660

984T, 985F, 985T, 986F problem formulation and 1002 linearity and 1012MEMS and 801 resonance and 1054 sensor/actuators and 1134-1143structural design and 983-987, transverse vibration and 169 shells and 358-360

983F, 983T See also excitation thermoelasticity and 1013testing of 988 parametric identification 673 thermoelectromechanical couplingtransmissibility and 987F, cables and 212F, 213F 1015-1017

Pade matching 1200, 1204 Fourier-based 665-672 Piezoelectric Materials and ContinuaPAM-CRASH 305 modal analysis and 820-824 1011-1018Pan, J 877-887, 887-898 nonlinear systems and 926-927 Pipes 1019-1024parallelograms 126 nonlinear testing and 1288 annular fluid flow and 1023Parallel Processing 990-1001 Parseval's theorem 388, 445 cross flow 1023

direct solvers and 992 Pascal, Blaise 1064 external fluid flow and 1022domain decomposition 992F, 994- passband patterns 745F, 748F internal fluid flow and 1019-1022

1000 passive damping 351-352 pistons 1445-1445

I xxii INDEX

pitch 629-630 potenti~l energy (continued)excitation 610

radial based function (RBF) 864 875-

p!tch-catch setup 906, 907FHamilton's Principle and 131 877 '

pitchfork bifurcation 215 930minimum 1346-1347 Radiation by Flexural Elements 130

pitch plane model 'minimum total 1320 1456-1480 '

2 DOF 609-610piezoelectric damping and 357 basic theory of 1456-1463, 1456F,

2 DOF quarter car model 612-613Rayleigh method and 132-134, 288 1458T, 1459F, 1459T, 1460F,

4 DOF 608resonance and 1047 1461F, 1461T, 1462T, 1463F

road excitation and 606-607 618-viscous damping and 336 boundary conditions and 186

621 'See also energy 187F, 188F, 189F '

suspension and 615-616potential mode identifier 429 MEMS 787-789

weight and 615power balance Ritz method and 134

plasticity 252Tmodal groups and 1266-1267 sound and 1456-1480

Plates 1024-1031,1331oscillators and 1266, 1267 source ratios 1463-1468 1464F

acoustic radiation and 545-550WIA and 1269 1466F, 1467F, 1468F, 1469F,

baffled 887-898power spectral density (PSD) 245 1470F, 1471F, 1472F, 1473F,

cantilevered skew 1028F1581 ' 1474F, 1475F, 1476F, 1477F,

circular 1026-1027, 1027T, 1027Fcepstral analysis and 216-227 1478F, 1479F, 1480F

1028T, 1028F 'classical analysis and 595F, 1210- transmissibility 1523-1527

complicating effects 1029-10331211, 1212F, 1214, 1219-1221 rahmonics 218-220

continuous systems and 1316-1317 correlation functions and 296-299 complex cepstrum and 222-227

flexural radiation and 1459 1463-297F ' rain-on-the-roof excitation 241

1465 ' random signals and 1219-1223 Ram, YM 686-690

forced vibration and 894-896 road excitation and 618-621 Randall, RB 216-227,364-376

Kirchhoff 197 windows and 1591 random data 491

MEMS 787 Pozo, R 973random error 671

modal properties and 889-890 Prasad, MG 1299-1304spectral analysis and 1222-1223

modal radiation efficiency 892-894 prediction error method 676-677Random Processes 1033-1039

896F, 897F ' Preisach plane 664autocorrelation function 1037

model updating and 845 prescribed conditions 252T1037F '

natural frequencies and 414-415 Principal Response Functions (PRFs)damping 1037, 1038F

rectangular 1026 272,278Fourier transforms and 1033

SEA and 1267 Principia Mathematica (Newton)normalization and 1033

sensor/actuators and 1134-1143 124singularities 1034, 1035F

sound power and 890 principle of least action 1323random signals 297-298, 644F, 1185

triangular 1030F probability 1238-1242generation of 1185-1192

ultrasonic waves and 910 Prony method 678-679, 682 1200spectral analysis and 1219-1223

vibration intensity and 1484-1485 1204 " stochastic analysis and 1238-1246

1487 ' proof-mass actuators 653Random Vibration, Basic Theory

plotting propellers 1170-11721040-1046

bar plots 413 proper orthogonal decompositioncentral limit theorem and 1044

Campbell diagrams 414 (POD) 9261044F '

compass 416 proper orthogonal values (POVs) 926degrees of freedom (DOF) 1040-

FRF data and 417-420 pseudoresponse residual 849-8501042

indicator function 430F pseudo-velocity (PSV)444Gaussian noise and 1045-1046

instability diagram 414 pulleys See beltsmean square response 1042, 1042F

interference diagrams 414 P-waves 913, 913Frange charts 1405F

mode shape 415-417 pylons 574Rankine, William 112

root-locus diagrams 413 pyrotechnic shock 495Rao, SS202-207, 395-404, 520-530,

singular value 425, 428F Pythagoras' theorem 579530-544, 1019-1024, 1308-

spectrum 420-421, 428F1324, 1344-1360

pneumatic spring 1183 QRateau, Edmond 1065

Poincare, Henri 228 431rational fraction polynomial method

Po!ncare map 228, 232, 235F QR algorithm 467823

POIncare' method 957-959 quadrupole sound 882, 883FRayleigh's method

point-mapping 1006 qualification testing 490basics of 132-134

Poisson's ratio 327,889, 1331, 1567 quasi harmonic motion 590Fcolumns and 238, 239

shells and 1156 quefrency 216continuous systems 1312-1317

polar notation 643 bearings and 218-220discrete systems 1309-1311

pole placement 54finite element methods 533

polymorphism 969 Rintegral 889, 890

polynomial model classes 675MEMS 787

positive position feedback 3-5 Rade, D 9-26noninteger power 289

potential energy 1266 Rades, M 256-264, 265-272, 272-ordinary 288-289

columns and 239quotient 462

equations of motion and 1327277,413-431,1046-1055, random processes 10381180-1184 substructuring 1332

INDEX I xxiii

Rayleigh-Ritz analysis 290,462-463, response properties (continued) rotation (continued)691 RVAC matrices 274F bearing faults and 1187-1188

iterative solvers and 993 sets 274-277 bearing vibration and 152-163Rayleigh waves 1553, 1554F transduction and 755F, 755-759, biorthogonality 1074, 1074F

ultrasonic testing and 908 756F, 757F, 758F blades and 174-180Raynaud's disease 621 See also active control; modal columns and 239-240receptance 421F, 423F analysis coupled 112-113, 1068-1069reciprocity 136, 1267 Response Vector Assurance Criterion data collection 1082

modal analysis and 819 (RVAC) 274F, 275-277 definitions 1069rectangular plates 1026 Reynolds number 1585 disks and 409-410recurrent neural networks (RNN) 867 ride natural frequency 614 eccentricity and 112-113recursion 100-101, 108 rigidity 113-118, 166, 1373-1374 equations of motion and 1325Reissner energy 1344, 1347 RISK processors 990 flexible state and 118-119residuals 848-850 Ritz method 290, 1332 fluid-structure interactions andresonance basics of 134 549-550

actuators and 482-489 energy and 1318-1320 forced responses 1071-1076antiresonance and 1046-1055 finite element methods 533 formulations 1070-1071bearing diagnostics and 146, 146F MEMS 787 free responses 1072-1073cubic systems 940-943 substructuring 1332 friction and 589-596degrees of freedom (DOF) and 930- vectors 691 helicopter damping and 629-642

943, 1047-1055, 1048F, Rivin, E 1487-1506, 1507-1521 historical studies of 1064-10661049F, 1050F, 1051T, 1053F Rixen, Daniel 710-720, 990-1001 influence coefficients and 119-

electrostriction and 475-490 RMS acceleration 604 123forced response and 581-582 RMS value 1582, 1583 limits 1081-1082, 1081Fforced vibration and 1296 windows and 1593 links and 1068-1069location 1051 road excitations 618-620 liquid sloshing and 726-740mass-spring damping 1047-1048, Robert, G 243-256 localization and 741-751

1048F, 1049F, 1050F Robot Vibrations 1055-1063 machines and 1064-1069, 1067F,MEMS 779 actuated joints 1059 1068F,NNM and 921-922 altered operational strategies 1063 maintenance 1078, 1079Fnonlinear systems and 928-943, arm improvement 1062 misalignment and 1186

944, 1054-1055 commanded motion 1057, 1058F, monitoring 1078-1084parameter sweep through 1288 1059F, 1063 motion sickness and 856-861phase method 1052-1053 components 1059 natural frequencies 1071primary 931-934, 938-939 damping 1063 natural modes 1071-1072quadratic systems 937-939 degrees of freedom (DOF) 1060, nodal points 1073, 1073Fresiduals and 848 1060F, 1061F nonlinear systems and 956rotor dynamics 1097 flexibility and 1056F, 1057 online systems 1082secondary 934-937 kinematic linkages 1060, 1060F resetting sensitivity 1077sharpness of 1051 mass allocation 1062 rigid states and 113-118ship vibrations 1170-1171 material selection 1063 sensing and measurement 1080-sub-harmonic 1288 modeling 1057 1081, 1080Ftesting 1052 See also MEMS shells and 1159transduction and 755-759, 755F, Rochelle salt 1013 signal generation and 1184-1193

756F, 757F, 758F rockets 1265 signals and 1192Tviscous damping and 1548-1551 rod vibration 1328-1329, 1328F, SNDT and 905whole-body vibrations and 1570- 1329F, 1330F spectral analysis and 1218

1578 rigid state 118-119 speed and 118-119See also frequency; harmonics rolling bearing faults 1187-1188 sprag-slip 593-597

Resonance and Antiresonance 1046- root-finding methods 463 standards 1082, 1224-12381055 root-locus diagrams 413 transient analysis 1083-1084

response properties 272-277 root mean square (RMS) 145 unbalance and 1185-1186aeroelastic 87-97 averaging and 98 rotor-bearing systems 413, 414F, 417,bridges and 202-207 Rosenhouse, G 124-137, 180-191, 420FCFRF matrices 276F, 277F 1304-1308 helicopters 629-642FDAC matrices 275F Rotating Machinery, Essential unbalance plot 420forced 1075-1076 Features 1064-1069 Rotor Dynamics 1085-1106free 1072-1073 Rotating Machinery, Modal chaos 1096F, 1097-1098FRFs 273F Characteristics 1069-1077 coaxiality fault 1098F, 1099individual 272-273 Rotating Machinery, Monitoring De Laval's model 1085-1088MEMS 779-794 1078-1084 formulation 1102modal analysis 824-826, 837-838 rotation 414 gyroscopic couple 1099-1101,parameters 1267-1268 abnormal situations 1078-1080, 1099F, 1100Fprincipal 277-278 1082-1083 initial peak 1098pseudoresponse 849-850 balancing and 111-124 isotropic links 1093-1097, 1094Fresiduals 848-850 basic diagnostics and 379, 379T modal properties 1090

I xxiv INDEX

Rotor Dynamics (continued) SeismicInstruments, Environmental shear (continued)modeling 1085-1088, 1086F, Factors 1121-1134 shells and 1159, 1161-1162

1087F absolute motion transducers and sheetmetal 286natural frequencies 1088-1090, 1381-1396 Shells1155-1167

1089F, 1090T bending moment sensitivity 1125- curvilinear surface coordinatestemporary rates of flow 1102-1103, 1127 1155, 1156T

1103F, 1104F, 1105F calibration 1130-1132 damping and 1165torsion 1105-1106, 1106F cross-axis sensitivity and 1121- Donnell-Mushtari-Vlasovequationsunbalance 1086F, 1091-1092, 1122, 1122F, 1123F 1159-1162

1091F, 1098 environmental factors 1132-1133 forced vibrations and 1163-1165Rotor-Stator Interactions 1107-1120 force transducers and 1123-1132, frequency and 1160-1163, 1163F,

chaos 1112-1115, 1113F, 1114F, 1123F, 1124F, 1125F, 1126F, 1164F1117T 1127F, 1128F, 1129F, 1131F, Love-type equations of motion

contact at reducing speed 1107 1132F 1155-1159, 1159T, 1160-1161

cycle limits 1107-1110, 1108F, loading effects 1128-1130 membranes and 762-770, 1159-1109F seizure 151 1160

deterministic contacts 1109F, self-organizing feature maps (SOFM) piezoelectricity and 358-360,1110-1112, 1110F, 11l1F 865 1014-1015

misalignment 1116T, 1116-1118, semi-iterative solvers 994-1000 rotary inertia and 1159, 1161T1117F,1118F Sendagay INTES building 30, 30F sensor/actuators and 1134-1143

support deterioration 1118 Sensors and Actuators 1134-1144 shear and 1159tremors 1118-1121, 1119F distributed 1134-1138 vibration intensity and 1485

roughness index (RI) 618-621 magnetostrictive materials 753-762 shimmy 358

Routh-Hurwitz stability 23 MEMS and 771-779 Ship Vibrations 1167-1173

Rubbing Shafts Above and Below the sensitivity and 1140-1141 background of 1168, 1168F

Resonance (Critical Speed) (Taylor) structonic cylindrical shells and damping and 1171

1107 1138-1141 excitation and 1168-1170

Rumbaugh, J 969, 971 See also MEMS resonance and 1170-1171

Runge-Kutta-Gill method 1119 separatrices 232 Shock 1173-1180

running sum 1196-1197 sequence diagrams 970, 971F absorption 1174, 1175F, 1176F,

run out 112-113, 1099-1101, 1099F, servo valves 485-488 1177-1180, 1177F, 1178F,

1100F Sestieri,A 1253-1264, 1275-1283 1179F

RVAC matrices 274F, 275-277, 278 shaft encoder 375 bearing diagnostics and 146Shaft Rubbing (Newkirk) 1107 distribution 1174-1176, 1174F,

Sshaft vibration 1191, 1328-1329, 1175F

1330F fragility and 1177, 1177F

Saddles229, 1544-1548standards and 1228-1232 hammershock 93

shallow sag cable 210-213 packaging and 983-989SAMCEF248-249 Shape Memory Alloys (SMAs)660, springs and 1180-1181sampling 1144-1155, 1146T viscosity 283

data acquisition and 364-376 active damping augmentation Shock Isolation Systems1180-1184digital filters and 389-391 1149-1150, 1150F, 1151F, short term Fourier transform (STFT)external 373-375 1152F 1364sigma-delta converters and 371-372 active impedance and 1151-1153, Shteinhauz, GD 1369-1379Simpson's rule 1197-1198 1153F, 1154F, 1155F Sidahmed, M 376-380,1184-1193,software resampling and 375 actuators and 65-68 1379-1380

satellite ultraquiet isolation control actuators 1147, 1147F side force coefficient 738Ftechnology experiment (SUITE) fibers 352, 353F Sieg,T 629-64248 passive damping 1146, 1147F sigma-delta converters 371-372

scanning laser Doppler vibrometer properties of 1145F, 1145-1146, signal envelope 644(SLDV)700, 703, 704F, 1403 1146F Signal Generation Models for

full-field measurement and 707- smart structures and 75-77 Diagnostics 1184-1193709 stiffness and 1147-1148, 1147F, Signal Integration and Differentiation

scattering 1568 1148F, 1149F, 1150F 1193-1199SCELucerne Valley Station 443 Shaw, S 10to SignalProcessing, Model BasedSchmerr, LW Jr. 906-918 shear 887, 1135 Methods 1199-1208Schur method beam vibration and 1330 signals

FETI997-1000, 998F, 999F bending moment sensitivity and aerodynamics and 1191primal complement 995-997 1125-1127 bearings and 146, 146F, 1187-1188

Schwarz method 995 boundary conditions and 183 continuous 1193-1194, 1194F,Sciulli,D 46-48 columns and 239-240 1195FScott, RZ 137-143 damping materials and 327-331 decomposition and 647Scruton number 1584 flexibility 239 demodulation and 647sealing 161-162, 1088 fluid-structure interactions and deterministic 1203-1204seating dynamics 1577 545-552 digital filters 380-395segment averaging 1219-1221 piezoelectic damping and 354-360 DSP 367

I xxvi INDEX

spectra (continued) state diagrams 970, 971F stochastic analysis (continued)zero padding and 1213-1214, state space model class 676 averaging methods 1244-1246

1214F static condensation 1333 differential calculus (SDE) 1246-zoom 222, 224F static loading 209 1252See also frequency static unbalance 1185 exact probability solutions 1238-

Spectral Analysis, Classical Methods stationary probability 1239-1242 1240, 1241F1208-1223 statistical energy analysis (SEA) 1265- models 1199, 1201-1203

Spencer, BFJr. 26-34 1266 processes 296spillover 53-54 alternatives to 1269-1272 statistical moments 1243-1244sprag-slip mechanism 593-597, 593F description of 1268 Stochastic Analysis of Non-Linearspnngs modal groups and 1266-1267 Systems 1238-1246

belt drives and 170-172 stators See rotor-stator interactions Stochastic Systems 1246-1252,model updating and 845 steady states 1285 1248F, 1249F, 1251Fshock isolation and 1180-1184 Steffen, V Jr. 9-26 Stoneley wave 1553superposition and 1300-1301 Steiglitz-McBride method 1203 stopband patterns 745Fsuspension 37-38 Steindland, A 431-438 storage modulus 327transmissibility and 1523-1527 stick-slip motion 583, 590F strainSee also mass-spring model stiffness columns and 239

sprung mass 604, 605 columns and 241 damping models and 335-3422 DOF pitch plane model 609-610 complex 338-339, 634-635 finite element methods 536-5382 DOF quarter car model 612-613 continuous systems and 1312-1317 flutter and 553-5654 DOF pitch plane model 608 damping and 321-325, 327-331, motion sensors and 1388-1389excitation and 605-607 632-641, 638F piezoelectic damping and 353suspension and 37-38, 615-616 eigenvalue analysis and 461-467 shape memory alloys 1145weight and 615 equations of motion and 1292- variational methods and 1350,

Spurr's sprag-slip mechanism 593- 1293, 1324-1332 1350F, 1353597, 593F ground transportation and 607 See also actuators

Square Root of Sum Squares (SRSS) gyroscopic couple and 1099-1101 strain-life method 507-508245 helicopter damping and 632-641, strength of materials theory 1328,

squeal 589 638F 1330,1331squeeze flow devices 473 hybrid control and 658, 658F stressstability isolation theory and 1493-1506, Boltzmann model and 661-662

active suppression and 56-57 1507-1521 damping models and 335-342misalignment and 1116-1118 Krylov-Lanczos methods and 691 finite element methods 536-538nonlinear systems and 953-956 lag dampers and 629-641 flutter and 553-565rotor dynamics 1088-1090, 1093- matrix 712, 715 interlaminar 787-789

1095, 1097-1098 MEMS 782-783, 783T Kelvin model and 660-661, 662rotor-stator interactions 1107-1121 model updating and 845 piezoelectricity and 1012-1013Routh-Hurwitz 23 NNM and 920F screening 490tremors 1118-1121 plate vibration and 1029 shape memory alloys and 1145vibro-impact systems and 1536- Rayleigh method and 1309-1312 stress-life method 505-507

1548 robots and 1062 Stribech, Richard 153stability, dynamic 431-438 rotor dynamics 1085-1088, 1092, Stribeck equation 1188stabilization diagrams 429 1102 string vibration 591, 1328-1329,stall flutter 96 rotor-stator interactions 1107-1121 1330Fstandard finite elements 250T Schur method and 995-1000 continuous systems and 1315standard linear model 660-661, 662 sensor/actuators and 1134-1143 taut 213standards 1224 shape memory alloys 1144-1155 transverse vibration and 167

balancing and 117-118, 117F shells and 1155-1167 Strouhal number 1584broadband 1226 signal generation and 1184-1185 Stroustrup, B 972condition monitoring 1233-1237, SNDT and 899 structonic shell systems 1135F, 1137F,

1234F, 1235T, 1235F, 1236T spatial properties and 260-262, 1138-1141, 1138F, 1139F,gears 1232-1233, 1234F 263F 1140F, 1141F, 1142F, 1143F,machinery 1225 structural modifications and 1262- 1144Fmeasurement 1226, 1226F 1264 structural analysisnon-rotating parts 1226-1228, superposition and 1300-1301 boundary integral formulation and

1227T, 1228T, 1229T, 1229F tire vibration and 617-618, 1374- 1278-1279organizations for 1224-1225 1375 external problem and 1279rotating parts 1081, 1082, 1228- viscosity-elasto-slide (SVES) 636- FEM and 1277-1278

1232, 1230T, 1231F, 1231T, 637 internal problem 1279-12821232F See also absorbers seismic instruments 1121-1134

Standards for Vibrations of Machines Stiharu, I 771-779, 779-794, 794-805 shape memory alloys and 1144-and Measurement Procedures stingers 1124-1125 1155

1224-1238 stochastic analysis 744, 1238 substructuring 1332-1335standard wave equation 1329 approximate probability solutions Structural Dynamic Modificationsstate condensation 258 1241-1242 1253-1264

INDEX I xxvii

structural dynamics 256T, 490 structure-acoustic interaction Taylor series 1325acoustic radiation and 545-550, (continued) beam vibration and 1329

545F, 546F, 547F, 548F, 549F, statistical energy analysis and 1265- cables 210551F, 552F 1266, 1268-1274 chaos and 228

actuators and, 58-72 structural interactions and 1275- friction and 591bounded waves and 1563-1564 1283 normal form theory and 1007bridges and 202-207 thermal methods 1270-1271 Technical Committees 1225damping and 323-324, 551 WIA and 1269 temperaturedirect problem and 1254-1259, Structure-Acoustic Interaction, High damping materials and 329-330

1257F, 1258F, 1259F Frequencies 1265-1274 electro stricti on and 475, 477, 478F,disks and 404-413 Structure-Acoustic Interaction, Low 480-482, 482F, 483F, 484Fearthquakes and 439-461 Frequencies 1275-1283 frequency equivalence conceptfinite element methods 534-539 structure under test (SUT) 1124-1125 329fluid interaction and 544-553, 949 Sturm sequence property 463, 467 piezoelectricity and 1013, 1014-flutter and 553-565, 565-577 Su, Tsu-Jeng 697 1017forced response 578-582 subharmonic instability 421 shape memory alloys and 1144-ground transportation and 603- substructuring 256,1332-1335 1155

620 Succi method 1280F, 1281-1285, transducers and 1133guided waves and 1551-1559 1281F,1282F tensile stress 1029impulse response function 1335- Sun, J-Q 342-351 tension 166

1343 Sunar, M 1332-1335 belt drives and 170-172inverse problem 1259-1265, 1259F, supercomputers 304-312, 305T shallow sag and 210-213

1262F, 1263F, 1264F superposition 1286, 1300 suspension 209-210isolation theory and 1487-1506, applications of 1300F, 1301-1304, Terfenol-D 47, 48, 660, 753, 759

1507-1521 1301F, 1302F, 1303F, 1304F test-analysis model (TAM) 265, 269,Krylov-Lanczos methods and 691- linearity and 1300-1301, 1300F, 272

697 1301F hybrid 258, 262Flocalization and 741-751 suspended cable 209-210 IRS 260FMEMS and 771-779, 779-794, suspension 615 modal 258, 261F

794-805 2 DOF pitch plane model 609-610 orthogonality and 259-260model-based identification and 2 DOF quarter car model 612-613 spatial properties and 256-264

673-685 4 DOF pitch plane model 608 static 259Fmodel updating and 844-856 active 38 Testing, Non-Linear Systems 1285-modification 1253-1264 adjustable 38 1289neural networks and 869-877 damping and 616 linear/nonlinear response 1285-noise and 877-887, 887-898 passive 37 1287packaging and 983-989 semiactive 38 procedures for 1287-1289periodic truss beam 741F springs and 615 testing methodsresponse properties 272 tire vibrations 1371-1372 accelerated 495robots 1055-1063 weight and 615

environmental 494-495smart structures and 73-80 See also ground transportation

packaging and 988-989SNDT and 898-906 Theory of Vibration, Duhamel'ssound 1443-1455

systems Principle and Convolution 1304-vehicular vibration 37-45

symbolic dynamics 229-232 1308vibration intensity and 682, 1480-

symmetry Theory of Vibration, Equations ofbifurcation and 435

1487 boundary element methods 194-Motion 1324-1332

wind and 1578-1587, 1579F, 195Theory of Vibration, Energy Methods

1580F, 1583T cables and 212-2131308-1324

See also boundary conditions; continuous systems 1312-1317,modal analysis columns and 239, 241 1316F

structural reduction See Krylov- disks and 404-413 discrete systems 1309-1311,Lanczos methods inverse problems 686-688 1311F, 1312F

structure-acoustic interaction 1265 isolation theory and 1490 FEA 1320-1322boundary integral formulation and iterative solvers and 994 Rayleigh's method and 1309-1317

1278-1279 modal shape plots and 416 Ritz method and 1318-1319energy balance with two oscillators NNM and 919 variational formulations 1322-

1266 stochastic analysis and 1238-1240 1324envelope method 1271-1272 synchronized averaging 147, 598 Theory of Vibration, Fundamentalsexternal problem and 1279 system coordinates 130-131 1290-1299FEM and 1277-1278 classification and 1290-1291internal problem 1279-1282 T equations of motion 1291-1293,power balance in two modal groups 1291F, 1292F, 1293F

1266-1267 tandem systems 696F forced vibration and 1296F,power balance with oscillators 1267 tangential displacement 211 1297Fresponse parameters 1267-1268 taut string 213 free vibration and 1293-1299,SPL 1268F, 1268 Taylor, HD 1107 1294F, 1295F

I xxviii INDEX

Theory of Vibration, Impulse Tire Vibrations 1369-1379, 1370F Transform Methods (continued)Response Function 1335-1343 frequency influences 1378-1379 Laplace 1407-1409

Fourier transforms 1339-1343 responses 1372-1377, 1374F, spectral analysis and 1208-1223initial condition response 1339 1375F, 1376F Z-transforms 1409-1411, 1410F,Laplace transformation method sources of 1370-1371 1412F

1336-1338, 1337T, 1338F suspension and 1371-1372, 1371T, See also Fourier transformsTheory of Vibration, Substructuring 1372F Transforms, Wavelets 1419-1435

1332-1335 tire construction and 1371, 1371F continuous wavelet 1423-1425,Theory of Vibration, Superposition Toeplitz structure 1203 1424F, 1425F

1299-1304 Tomasini, EP 698-710 discrete wavelet 1427-1433, 1429F,applications of 1300F, 1301-1302, Tonpilz 759 1430F, 1431F, 1432F

1301F, 1302F, 1303F, 1304F tools classes 974 distribution sampling 1425-1427,linearity and 1300-1301, 1300F, Tool Wear Monitoring 1379-1380 1426F

1301F tooth averaging 598, 599F Fourier 1420-1422, 1421F,Theory of Vibration, Variational See also gear diagnostics 1422F

Methods 1344-1360 Tordan, MJ 364-376 performance of 1433, 1433F,applications 1350-1358 torsion 112-113, 396 1434Fcalculus 126, 1344-1346 belts and 169 transient analysis 1209, 1213-1214,Hamilton's principle 1348-1349, boundary conditions and 183 1223

1355 columns and 239-240 absolute motion and 1393-1395solid mechanics 1346-1348 continuous systems and 1315 boundary element methods 198-

thermal analogy methods 1269, 1270- isolation theory and 1493 2011271 MEMS 782-783, 783T data 491

thermal equilibrium 631 moving belts and 166-170 dynamics 278-286thermographic analysis 378 rotor dynamics 1105-1106 rotation 1083-1084thickness 1033 standards and 117-118 translation 1325F, 1326Thom, R 432 transmissibility 1523 motion sickness and 856-861three-node bending mode 417 transducers 755-759, 755F, 756F, whole body vibration and 1574time 757F, 758F, 1398 transmissibility 343, 1522

average energy 1268, 1270 accelerometers 1383, 1390-1392, active/semi-active damping anddomain analysis displays 429 1390F, 1391F, 1392F, 1395- 347-349domain formulation 977-978 1396, 1396T, 1397T biodynamics and 1571domain methods 821-822 calibration of 1130-1132 bounce 610Duhamel's Principle and 1308 capacitive 1390, 1390F elastically coupled damping andDYNA3D and 306 data acquisition and 364-376 345-346dynamics and 278-286 displacement sensors and 1399- flexural 1523-1531, 1525F, 1526F,earthquake history and 447-453 1401 1527F, 1528F, 1529F, 1530F,frequency distribution 147 electromagnetic sensors 1402 1531Fgear diagnostics and 600 environmental factors and 1132- gear diagnostics and 597-603Hilbert transforms and 648 1133 longitudinal 1522integration 256, 256T force 1123-1132 multidirectional mounts and 346normal form theory and 1005, LDVs 1403-1406 passive damping and 343-345

1007 magnetostrictive materials 755-761 seat dynamics and 1577periodic center manifold reduction modal properties and 809 torsion and 1523

1005 mounting and 1133 See also boundary conditionsperturbation methods and 1003- piezoelectric 1385-1387, 1385F, transport speed 166

1004, 1009 1386F, 1387F, 1388F, 1389T transverse shear flexibility 239transient dynamics and 278 pressure 1384 transverse vibrationwindows and 1587-1595 seismic displacement 1383 continuous systems and 1315-1317See also parametric excitation seismic force 1384 moving belts and 166-170

time domain averaging (TDA) 101 seismic velocity 1383 nonlinear effects of 167-169performance of 103-108 standards and 1226F, 1232-1238 variational methods and 1352-periodic extraction and 109 strain gauge 1388-1389, 1389F 1353, 1352Frecursion 102 transient response 1393-1395, trapezium rule 1197regular 101 1393F, 1394F, 1395T tremors 1118-1121

Time Frequency Methods 1360-1369, ultrasonic 912-917 Trevithick, Richard 10641368F velocity and 1402-1406 triangular plates 1030F

applications of 1366-1368 Transducers for Absolute Motion triboelectric effects 1133bilinear 1367F 1381-1397 triggering 373Cohen's class and 1362-1366 Transducers for Relative Motion Trigger Scuba, Inc. 759Fourier analysis and 1360-1361 1398-1406 Troger, H 431-438spectrogram and 1364 Transform Methods 1406-1419 truncation 1256Wigner-Ville distribution 1361- differentiation/integration and tuned mass dampers See absorbers

1363 1193-1199 turbulence See windTimoshenko beam model 142, 1329 Fourier 1411-1419, 1414F, 1415F, turbines 1191, 1229tire properties 617-618 1416F, 1418F turbomachines 220, 1191

INDEX I xxix

two-state least squares (2SLS)method vectors vibrational properties (continued)679 basic principles of 125-126, 127F energy methods 1308-1324

Tzou, HS 1011-1017, 1134-1153 critical damping and 317-318 environmental testing 490-496,iteration methods and 464-466 496-504

U Krylov-Lanczos methods and 691- equations of motion and 1324-695, 697 1332

Uchino, K 475-490 modal correlation and 267-270 fatigue and 505-512Ueda, Y 228 neural networks and 871-872 feedforward control of 513-520ultrasonic nondestructive testing 906- nonlinear systems and 953-956 finite difference methods 520-530

918,1439-1440 Poincare' method and 957-959 finite element methods 530-544acoustic impedance and 911T Rayleigh quotient 462 flexural radiation and 1456-1480A-scan 906 Rayleigh-Ritz analysis 462-463, fluids and 467-475, 544-553attenuation and 911 691 flutter and 553-565B-scan 906, 907F Veldvizen, T 973 forced response and 578-582bulk waves and 908 velocity Fourier-based identification 665-

C-scan 907, 907F actuator sensitivity and 1140-1141 672

displacement and 909F electromagnetic sensors and 1402 friction 582-589, 589-596

Lamb waves and 909F friction and 583-584, 589-597 fundamentals of 1290-1299

material density and 908T LDVs and 1403-1404, 1405-1406 gear diagnostics 597-603

plate waves and 909 optical heterodyning and 1404- ground transportation and 603-620

pulse-echo inspection 906, 907F 1405 guided waves 1551-1559

Rayleigh waves and 908 vertical oscillatory motion 857-859, hand-transmitted 621-629

reflection/refraction and 910-911, 860F helicopter damping 629-642

910F vestibular system 857 Hilbert transforms and 642-648

testing system of 907F vibrational properties hybrid control and 649-658

transducers and 912-916, 912F,absorbers 1-9, 9-26 hysteretic damping 658-664

913F, 914F, 915F, 916F, 917F,active damping and 351-364 identification methods and 673-685

1439 actuators and smart structures 58- impact and 1531-1548

Ultrasonics 1437-1441 81 impulse response function 1335-

acoustic emission and 1440 adaptive filters 81-87 1343

chemical reactions and 1440active isolation 46-48 intensity and 1480-1487

cleaning and 1439active suppression of 48-58 inverse problems 686-690

definition of 1437aeroelasticity 87-97 isolation theory 1487-1506, 1507-

electronics and 1440averaging and 98-110 1521

features of 1437-1439, 1438Tbalancing and 111-124 linear algebra and 710-720basic diagnostics and 376-380 linear damping matrix methods

generation of 1440-1443 basic principles of 124-137 721-726imaging and 1439 beams and 137-143 liquid sloshing 726-740magnetostrictive materials 759 bearings and 143-152, 152-165 localization and 741-751material properties and 1439 belts and 165-174 magnetostrictive materials 753-welding and 1440 boundary conditions and 180-191 762

U-Mode Indicator Function (UMIF) boundary element methods and magnitude 1572278,429 192-202 membranes and 762-770

unbalance 112-113, 1185-1186 bounded waves 1559-1564 MEMS 771-779, 779-794flutter and 553-565 bridges and 202-207 modal analysis 265-272, 805-813,rotor dynamics 1091-1092, 1095, cables and 209-216 820-824, 824-829, 829-838

1098-1098 civil structures and 26-36 model-based identification androtor-stator interactions 1107- columns and 236-243 673-685

1121 continuous methods 286-294 model updating/validating and 844-uncertainty principle 1212-1213 damping measurement and 332- 856Ungar, EE 327-331 335, 342-351, 1548-1551 mode of 838-844unified matrix polynomial approach damping models and 321-327, motion sickness and 856-861

(UMPA) 823 335-342 neural networks and 869-877unified modeling language (UML) data acquisition 364-376 noise and 877-887, 887-898

969 digital filters and 380-395 nonlinear normal modes and 918-unperturbed phase portrait 235F discrete elements and 395-404 924unsprung mass See sprung mass disks and 404-413 nonlinear systems and 928-943,

displays and 413-431 952-966V dose value (VDV) 1575-1576, nonlinear testing and 1285-1289

1576F, 1577F optimal filters and 977-982Vakakis, AF 918-924 Duhamel's Principle 1304-1308 packaging and 983-989van der Pol equations 227-228 dynamic systems and 431-438, parallel processing and 990-1001variational principle 433F, 435F, 436F, 437F parametric excitation and 1001-

columns and 240 earthquakes and 439-461 1009Varoto, PS496-504 eigenvalue analysis 461-467 perturbation methods and 1009-Vecelic's system 689F electrostriction 475-490 1011

I xxx INDEX

vibrational properties (continued) Vibration Isolation, Applications and viscoelastic damping (continued)piezoelectric materials and 1011- Criteria 1507-1521 materials and 327-331

1018 Vibration Isolation Theory 1487- measurement and 334pipes and 1019-1024 1506 models of 339-340plates and 1024-1031 ambient vibrations 1507-1508, passive 351-352random processes 1033-1039, 1508F transverse vibration and 169

1040-1046 coupled systems 1490-1493, viscoplasticity 252Tresonance and 1046-1055 1494F, 1496F, 1497F, 1498F, viscosityresponse property comparison and 1499F damping 321-323, 344F, 633-634

272-277 criteria of 1508F, 1509F, 1510- SVES 636-637robots and 1055-1063 1512, 1511T Viscous Damping 1548-1551, 1549F,rotation 1064-1069, 1069-1077, degrees of freedom (DOF) and 1550F

1078-1084 1494-1501, 1494F, 1496F, equivalent 340-341rotor dynamics 1085-1106 1497F, 1498F, 1499F, 1500F, isolation theory and 1496-1501rotor-stator interactions 1107-1120 1501F models of 336-337seismic instruments and 1121-1134 detrimental effects and 1508-1509 See also dampingsensors and actuators 1134-1144 dynamic systems 1512, 1513F visual system 857shape memory alloys and 1144- elastic mounts 1490 motion sickness and 859, 860

1155 experimental selection 1521 whole body vibration and 1574shells and 1155-1167 general purpose machines 1517- Voigt model 660ship vibrations 1167-1173 1519, 1518F, 1519F voltage conversionshock and 1173-1180 geometric properties 1490 MEMS 790-793signal generation 1184-1193 impacts 1515 Volterra series 926signal integration/differentiation impulse excitation 1502-1504, volute spring 1181, 1182F

and 1193-1199 1503F, 1504F von Karman geometric nonlinearitysignal processing and 1199-1208 inclined mounts 1493 1136sound and 1443-1455 inertia and 1490, 1515-1517, vorticesspatial comparison and 256-264 1516T, 1516F shedding 97spectral analysis and 1208-1223 mounting conditions 1517-1518, wind and 1584-1585standards for 1224-1238 1518F Vorus, WS 1167-1173stochastic analysis and 1238-1246, nonlinearity 1504-1505

1246-1252 nonrigid structures and 1520Wstructural-acoustic interactions polyharmonic excitations 1515

1265-1274, 1275-1283 precision and 1513-1514Wagner function 558structural dynamics and 1253- random excitation 1501-1502

1264 single frequency excitations and Waizuddin Ahmed, 603-620substructuring 1332-1335 1514-1515 Wang, KW 649-658superposition and 1299-1304 transmission model 1509-1510, Watt, James 112, 1064, 1064time frequency and 1360-1369 1509F, 1510F wave intensity analysis (WIA) 1269tires and 1369-1379 wave effects 1505, 1506F wavelet transformations (WT) 598,tool wear and 1379-1380 Vibration Transmission 1522-1527 600transducers and 1381-1397, 1398- See also transmissibility bearing diagnostics and 147, 148F

1406 Vibro-Impact Systems 1531-1548, continuous 1423-1425, 1427transmission 1522-1531 1532F, 1533F discrete 1427-1433ultrasonic 906-918, 1437-1441 bifurcation and 1533-1536, 1535F, performance of 1433unbounded waves 1565-1570 1539F, 1539 signal generation and 1191variational methods 1344-1360 chaos and 1542-1548, 1542F, STFT 1420-1423vehicular 37-45 1545F, 1546F, 1547F wave number 895Fwhole body 1570-1578 classification and 1536, 1537F boundary conditions and 894Fwind induced 1578-1587 examples of 1533, 1533F, 1534F, plate vibration and 893windows and 1587-1595 1535F Wave Propagation, Guided Waves in

vibration exciter 1124-1125 periodic stability and 1535F, Structures 1551-1559Vibration Generated Sound, 1538F, 1539-1542, 1539F, Wave Propagation, Interaction of

Fundamentals 1443-1455 1541F Waves with Boundaries 1559-Vibration Generated sound, Radiation stability regions and 1536, 1538F 1564

by Flexural Elements 1456-1480 subharmonic regions and 1536- Wave Propagation, Waves in anvibration-induced white finger (VWF) 1539, 1538F Unbounded Medium 1565-1570

621-622,625 vibrometers 700-706, 704F, 705F wavesVibration Intensity 1480-1487 in-plane 703-704 acoustic impedance and 1561T,

beams 1484, 1486 rotational 704-705 1562Fcomplex 1482 SLDV 703, 1403 with boundaries 1559-1564flexural waves 1483-1485 Villari effect 754, 761 boundary element methods 200plates 1484-1485, 1487 virtual work principle 1320 bulk 908rods 1486 variational methods and 1353 cables and 210-215shells 1485 viscoelastic damping 325-326, 649, cylindrical 1557F, 1570waves in elastic media 1482-1483 656-658 discrete structures and 1563-1564