International Journal of Acoustics and Vibration · the readers of this journal. Noise and vibration control of engines, automobiles, and ships Several decades ago, as the production

EDITORIAL OFFICE

EDITOR-IN-CHIEFMalcolm J. Crocker

MANAGING EDITORMarek Pawelczyk

ASSOCIATE EDITORSDariusz BismorNickolay IvanovZhuang Li

ASSISTANT EDITORSTeresa GlowkaJozef Wiora

EDITORIAL ASSISTANTAubrey Wood

EDITORIAL BOARD

Jorge P. ArenasValdivia, Chile

Jonathan D. BlotterProvo, USA

Leonid GelmanCranfield, UK

Samir GergesFlorianopolis, Brazil

Victor T. GrinchenkoKiev, Ukraine

Colin H. HansenAdelaide, Australia

Hanno HellerBraunschweig, Germany

Hugh HuntCambridge, England

Dan MarghituAuburn, USA

Manohar Lal MunjalBangalore, India

David E. NewlandCambridge, England

Kazuhide OhtaFukuoka, Japan

Goran PavicVilleurbanne, France

Subhash SinhaAuburn, USA

International Journal ofAcoustics and Vibration

A quarterly publication of the International Institute of Acoustics and Vibration

Volume 18, Number 3, September 2013

EDITORIAL

Research on Sound and Vibration in the Manufacturing IndustriesKazuhide Ohta . . . . . . . . . . . . . . . . . . . . . . . . . . 98

ARTICLES

A Novel Viscoelastic Material Modulus Function for Modifying theGolla-Hughes-McTavish Method

Luke A. Martin and Daniel J. Inman . . . . . . . . . . . . . . 102

Active Control of Radiated Sound from Stiffened Plates Using IDE-PFC Actuators

Atanu Sahu, Tirtha Banerjee, Arup Guha Niyogi and ParthaBhattacharya . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Study of the Effect of the Linear Temperature Behaviour ona Non-Homogeneous Trapezoidal Plate of Parabolically VaryingThickness

Arun Kumar Gupta and Pragati Sharma . . . . . . . . . . . . . 117

Formulation of Weighted Goal Programming Using the DataAnalysis Approach for Optimising Vehicle Acoustics Levels

Zulkifli Mohd Nopiah, Ahmad Kadri Junoh and AhmadKamal Ariffin . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Natural Frequencies and Acoustic Radiation Mode Amplitudes ofLaminated Composite Plates Based on the Layerwise FEM

JinWu Wu and LingZhi Huang . . . . . . . . . . . . . . . . . 134

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . 141

INFORMATION

New IIAV Officers and Directors . . . . . . . . . . . . . . . . . . . 99

Obituary Notices . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Book Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Editor’s Space

Research on Sound and Vibration in the ManufacturingIndustries

I began working in the field ofnoise and vibration nearly fortyyears ago. I spent the first thirty ofthose years conducting research on

vibration, noise control, and R&D management in heavy in-dustry. Eight years ago, I transferred to academia and I havebeen working at the university level ever since. In this letter, Idiscuss several of my research projects regarding sound and vi-bration, as well as the academic stimulus that has prompted thestudies. It is my hope that my experiences will be of interest tothe readers of this journal.

Noise and vibration control of engines, automobiles, and shipsSeveral decades ago, as the production of Japanese automo-biles was increasing, the control of noise and vibration be-came a critical issue that needed to be addressed in order toimprove product quality. Disk brake squeal was one seriousproblem that needed solving before the vehicles could achieveacceptance in the U.S. automotive market. As the theoreti-cal model of brake squeal—which explained the disk vibrationperpendicular to the friction force—did not exist at that time,a new theoretical model was developed. In this model, a con-tact element with distributed stiffness, damping, and frictionwas employed to generate the moment and force between thedisk and pad. The dynamic characteristics of the brake compo-nents were computed by the modal analysis technique and itscomplex eigenvalues showed us the possibility of self-excitedvibration that corresponded to the brake squeal. Using thismathematical model, measures to diminish brake squeal wereproposed and were confirmed by experiments.

I started studying engine noise and vibration thirty years agoand have continued to do so up to the present. My early workfocused on the analysis of piston slap using the dynamic char-acteristics of cylinder liners and piston profiles. The resultsof the research were presented at a SAE conference in the U.S.and enthusiastic discussion followed. At that time, I decided todevelop a theoretical method to predict the noise and vibrationof the operating engine, so that the design engineer could hearthe engine noise early on in the design process. I then spentabout twenty years completing this method. This method hasalso been applied to evaluate noise and vibration of ship struc-tures induced by the large-size main engine.

Flow-induced noise and vibration in thermal power plantsAs power plant capacity was increased, the fluid velocity inheat exchangers also increased and fluid-elastic vibration oftube arrays was induced, leading to fatigue failure. Many ex-perimental studies were carried out to determine the criticalflow velocity, and these produced two main results: (1) vi-bration suppression measures increasing the natural frequency,and (2) damping of tube array controlling the flow velocity.Our approach was to measure the fluid exciting force acting onthe tube array in the flowing water, which is proportional tothe tube displacement and the dynamic stability of the tube ar-ray in the fluid; this involved the complex eigenvalue analysisof the tube array coupled with the fluid exciting forces. Thismethod considers the distribution of flow velocity, the dynamic

characteristics of the tube array, and the difference in the nat-ural frequency and mode shape. Using this analytical method,a new vibration suppression band was invented in which fre-quency and damping of tubes does not change and only themutual displacement of tubes along the flow direction is con-strained. The results of this work were published in the ASMEPVP Journal and attracted considerable attention.

At almost the same time, the acoustic resonance in theflow direction was observed in the three-dimensional heat ex-changer. It was well known that the acoustic resonance in per-pendicular flow was caused by the vortex exciting force andwas prevented by insertion of a baffle plate to increase the nat-ural frequency in the fluid cavity. But this method could notbe applied to the acoustic resonance in the flow direction. In-tensive research on the acoustic resonance in the flow directionwas carried out and we found that acoustical treatment used in-side the heat exchanger and a change in the tube arrangementwas effective to suppress it.Underwater acoustics and a new transducerAs light consists of electromagnetic waves lasers cannot beused effectively in underwater exploration. Acoustical signalsare the only means of underwater surveillance and communica-tion. Platform silence is crucial for the success of underwatersensing systems. When using an oceanographic vessel to per-form an observation of the seafloor topography, it is necessaryto travel at as high a rate of speed as possible while still accom-modating the acoustical performance of side-scan sonar andother measurement equipment. Therefore, underwater noiseradiated by the ship hull vibration and propulsion systems wasminimized by using a two-stage isolation bed for the main en-gine, an electric propulsion system, damping treatment on theship’s hull, and a low-noise, highly-skewed propeller.

In the acoustical sensing process, new transducers were de-veloped to satisfy the requirement of the application platform.The wave height sensor equipped with the hydrofoil must havelittle self-noise caused by structural vibration. An analyticalmethod was developed to predict the dynamic response of thecoupled system of the structural vibration, the piezoelectricmaterial, and the acoustic field. A node support transducerwas developed using this method and then applied for depthdetermination on the world’s first hydrofoil driven by a dieselengine. This method was also used to develop a cylindricaltransducer for wide area surveillance.

The research work mentioned above was made feasible byadvances in experimental modal analysis techniques and nu-merical methods such as FEM, SEA and BEM. None of thiswork would have been possible without the support of an ex-cellent experiment team and numerical analysis staff.

Kazuhide OhtaEditorial Board Member, IJAV

98 International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013

News

IIAV News

New IIAV Officers and Directors

Ines Lopez Arteaga, TUE, Eindhoven, the Nether-lands (and KTH, Sweden)

Ines Lopez Arteaga is head of the Material and StructuralAcoustics group at the Marcus Wallenberg Laboratory forSound and Vibration Research (MWL) at the KTH Royal In-stitute of Technology (Sweden), where she serves as a visitingprofessor. She is also affiliated as associate professor to theMechanical Engineering Department at Eindhoven Universityof Technology (Netherlands), which she joined in 2002. InesLopez Arteaga received her degree on Mechanical Engineer-ing at the University of Navarra in San Sebastian (Spain) in1993, where she obtained her PhD in 1999 for her work titled”Theoretical and experimental analysis of ring-damped rail-way wheels”. From 1999 to 2001 she worked as a researcher,mainly in the field of railway noise, at Centro de Estudios e In-vestigaciones Tecnicas, CEIT (Spain) and in 2001 she movedto The Netherlands to work at the Technical Analysis groupof DAF Trucks on cabin acoustics until she joined EindhovenUniversity of Technology. Her research field is vibro-acousticsand her main interests are tyre/road noise and inverse acous-tics. Ines Lopez Arteaga is also editor for structural vibra-tion and elastic wave propagation for the Journal of Soundand Vibration, has organized sessions, and served on the scien-tific committees of several international conferences includingICSV18. Furthermore, she is a member of the Dutch Acousti-cal Society (NAG), the Spanish Acoustical Society (SEA) andthe IIAV.

Eleonora Carletti, IMAMOTER, Italy

Eleonora Carletti was born in Ferrara in 1955. She obtainedher MSc degree in Physics in 1979 and served as a lecturer inapplied physics at the Universities of Ferrara and Modena forseveral years. She joined the Italian National Research Coun-cil of Italy in 1984. She has served as head of the Acous-tics and Vibration Department of IMAMOTER institute since1996; she is in charge as the Scientific Responsible for severalNational and European Research Projects on machinery acous-tics and noise control. Currently she is the Chair Person of theNotified Body Noise Group of the European Commission (DGEnterprise) and is a member of the Scientific Advisory Boardof the Acoustics Research Institute (ARI) in Wien, one of theInstitutes of the Austrian Academy of Sciences. She is also oneof the noise experts for the revision process of the 2000/14/ECEuropean Directive on the reduction of the noise emitted byoutdoor machinery. She is a member of the Board of Directors

of the Acoustical Society of Italy (AIA), she serves as the Na-tional Secretary of this Society. Her areas of expertise includenoise and vibration control of complex sources; advanced mea-surement techniques for the identification of machinery noisesources; sound quality; statistics, sound power tests based onsound pressure and sound intensity measurements.

Christian Giguere, University of Ottawa, Canada

Christian Giguere received a bachelor’s degree in engineer-ing physics in 1983 from Laval University and a master’s de-gree in electrical engineering in 1986 from the University ofToronto, Canada. He obtained his PhD in Information En-gineering in 1994 from the University of Cambridge, U.K.,for research on the computational modelling of the humanauditory periphery. He joined the University of Ottawa in1995. He is currently serves as a Full Professor in Audiol-ogy and Speech-Language Pathology at the Faculty of HealthSciences and Vice-Dean Faculty Affairs. He is also hold across-appointment with the School of Electrical Engineeringand Computer Science. Professor Giguere teaches an under-graduate level course on acoustics for speech and hearing sci-ences as well as graduate level courses in speech sciences andinstrumentation in audiology. His research interests includespeech and warning sound perception in the noisy workplacewhile wearing hearing protectors or communication headsets,hearing aid signal processing, and the functional assessmentof hearing loss. He has published over 100 journal articles,conference proceedings and book chapters. Professor Giguereis also an active member of the technical committee on occu-pational hearing loss of the Canadian Standards Association(CSA). He is currently president of the Canadian AcousticalAssociation (CAA) and co-chair of the International Commis-sion on the Biological Effects of Noise (ICBEN). He has beena member of the International Institute of Acoustics and Vibra-tion (IIAV) since 2009.

Jian Kang, University of Sheffield U.K.

Professor Jian Kang, BEng MSc (Tsinghua University, Bei-jing), PhD (University of Cambridge), has been the Professorof Acoustics at the University of Sheffield, UK, since 2003.Previously he worked at the University of Cambridge and theFraunhofer Institute of Building Physics in Germany (Hum-boldt Fellow). Professor Kang is distinguished by his workin environmental acoustics, evidenced by 60 engineering-consultancy projects, 60+ funded research projects, and 600+publications. His work on acoustics theories, design guidanceand products has helped improve the noise control of under-ground stations/tunnels and soundscape design in urban areas.

International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013 99

ObituariesHe is a fellow of the UK Institute of Acoustics (IOA), a fel-low of the Acoustical Society of America (ASA), a UK char-tered Engineer, and the Editor in Environmental Noise for ActaAcustica united with Acustica (European Journal of Acous-tics). He chairs the Technical Committee for Noise of theEuropean Acoustics Association; the WUN (Worldwide Uni-versities Network) Environmental Acoustics Network; and EUCOST Action on Soundscapes of European Cities and Land-scapes. He was awarded the John Connell Award in 2011 andthe UK IOA Tyndall Medal in 2008.

Bert Roozen, KU Leuven, Belgium

Bert Roozen obtained his M.Sc. degree in mechanical en-gineering from Eindhoven University of Technology in 1986.From 1986 to 1995 he worked at Fokker Aircraft in Amster-dam, were he also performed his PhD study on the sound

Obituaries

It is with great sadness that we report the death of two col-leagues in June 2013, Finn Jacobsen and Martin Lowson. BothFinn and Martin served as two of the 13 members of the initialBoard of Directors when IIAV was incorporated in June 1995.We are grateful to colleagues at the Technical University ofDenmark and Bristol University for providing the material forthese obituaries.

Malcolm CrockerEditor in Chief IJAV

Professor Finn Jacobsen, 1949-2013

Finn Jacobsen, Associate Pro-fessor at the Technical Universityof Denmark, passed away in hishome on Sunday June 30th, 2013,at the age of 64, after sufferingfrom serious illness over the lastfew years. Finn Jacobsen wasborn on April 3rd, 1949 in Copen-hagen, Denmark, the second childof Børge and Jytte Jacobsen, in afamily of three children - his sister

Lone and younger brother Henrik.

Finn Jacobsen received his M.Sc. degree in electronic en-gineering in 1974 and his Ph.D. in Acoustics in 1981; bothdegrees were awarded by The Technical University of Den-mark (DTU). In 1985, he became an associate professor at

transmission through aircraft structures. He obtained his PhDin technical sciences from the Eindhoven University of Tech-nology in 1992. From 1995 until 2010, he worked for RoyalPhilips Electronics in Eindhoven, the Netherlands, starting atthe Philips Research Laboratories as research scientist andlater on at Philips CFT and Philips Applied Technologies aschief scientist. From 2003 until 2011, he served as a part-timeprofessor Acoustics and Noise Control at the Eindhoven Uni-versity of Technology. From 2008 until 2012, he was Presidentof the Acoustical Society of the Netherlands (NAG). Now heis board member of the Acoustical Society of the Netherlands(NAG). Since 2011, he has been affiliated to Delft Universityof Technology. In 2011, he was also working as an invitedprofessor at INSA in Lyon, France, for three months and from2012 until 2015, he has worked as an IEF ( Intra-European Fel-low) at the Acoustics and Thermal Physics laboratory (ATF) ofKU Leuven, as well as at the Production engineering, Machinedesign and Automation division (PMA) of the same university.

the Department of the Acoustic Technology, DTU, and servedas head of the Department from 1989 to 1997. In 1996, he wasawarded the degree of Doctor Technices by DTU, the highestacademic distinction within engineering and technological sci-ence in Denmark. Since 2008, he was the head of the AcousticTechnology group at the Department of Electrical Engineer-ing, DTU. Finn Jacobsen held several scientific editorial postssuch as associate editor for Acustica united with Acta Acustica(General Linear Acoustics) during 1995-2003. He also servedon the editorial board of the International Journal of Acous-tics and Vibration (IJAV.) He was the head of the organizingcommittee for the very successful 6th International Congresson Sound and Vibration (ICSV) that took place at DTU in1999. He also organized many technical sessions at interna-tional acoustics conferences.

Finn Jacobsen will be remembered for making significantscientific contributions in many areas of acoustics, but his maininterest was in general linear acoustics. He is perhaps bestknown for his contribution to the field of sound intensity; theDoctor Technices thesis contains 19 international papers onsound intensity published within a time frame of 10 years. Hehas made significant contributions to acoustics measurementtechniques and signal processing, e.g. in active noise con-trol, and the effect of band pass filters on reverberation timemeasurements. Finn Jacobsen was also very much involved intransducer technology, including microphone calibration and,in recent years, microphone array techniques. He was an ex-pert in statistical methods in acoustics, especially regarding thediffuse sound field. This was the main theme of his Ph.D. in


Obituaries1981, and he contributed his last paper in this field in 2012.Finn Jacobsen was extremely dedicated to his work and pub-lished nearly 100 papers in international journals and over 100conference papers. In 2013, he finished the book “Fundamen-tals of General Linear Acoustics” co-authored by Peter Juhl.A key point in Finn Jacobsen’s research was the experimentalvalidation of theory. After students had gone home, he wouldoften stay in the lab, performing measurements and testing newmeasurement techniques.

Finn Jacobsen was an eager and dedicated teacher, and heinsisted on deriving fundamental equations on the blackboardinstead of using power point presentations. His course on ad-vanced acoustics was very popular among the students, to agreat extent because of his lecturing style and the in-depth lab-oratory work that supplemented the theory and simulations.His door was always open to everyone, even late in the evening,and many students benefited from his advice and help. Overthe years he supervised 12 Ph.D. projects and close to 100M.Sc. projects.

Finn Jacobsen was a wonderful colleague who was ex-tremely kind and helpful. He would always take the time todiscuss any issue, scientific or otherwise and always gave hishonest opinion on any matter. Outside of work, Finn Jacobsenenjoyed good food and good wine, reading books and travel-ing, and he was very fond of going everywhere on his bicycle.

His overall contribution to the Acoustics Technology groupat DTU and the influence he has had on the students and col-leagues in acoustics is immense. His passing is a great loss andwe will miss him deeply.

Finn Jacobsen did not wish for a funeral ceremony. Instead,he set aside an amount for a gathering in his name for hisacoustics colleagues at the Technical University of Denmark.Finn Jacobsen was not married and had no children. He is sur-vived by his sister, Lone Jacobsen.

Professor Martin Lowson, 1938-2013

Professor Martin Lowson,Emeritus Professor of AdvancedTransport and former Head ofAerospace Engineering at BristolUniversity, England, died on14 June, 2013 aged 75. MartinVincent Lowson was born inTotteridge, Hertfordshire, on 5January, 1938, and was educatedat Kings School, Worcester. In1955, he started an undergraduateapprenticeship with Vickers Arm-

strong (Weybridge.) His education continued at SouthamptonUniversity, in the Department of Aeronautics and Astronau-tics, and he was awarded a BSc, Hons (first class) in 1960followed by a PhD in 1963.

As a postgraduate student he was part of the team fromSouthampton University which achieved the world’s first au-thenticated instance of human powered aircraft flight, whenthe team’s plane, the Southampton University Man PoweredAircraft, took off at Lasham Airfield in November, 1961. Hisdoctoral work in the University’s Aeronautics and AstronauticsDepartment was on ‘The Separated Flows on Slender Wings inUnsteady Motion’. Three of his 1960s papers — ‘The SoundField for Singularities in Motion’, ‘A Theoretical Study of He-licopter Rotor Noise’ and ‘Theoretical Analysis of CompressorNoise’ — are considered to be of fundamental significance inthe theoretical understanding of noise generation. His lifetimecontribution to the field of acoustics was recognised in 2011 bythe American Institute of Aeronautics and Astronautics Aeroa-coustics Award.

He went to the USA in 1964 where he held the post of Headof Applied Physics at the Wyle Laboratories, Huntsville, Al-abama until his return to the UK in 1969. Whilst in the USAhe worked on the Saturn V rocket for the Apollo Moon Pro-gramme. From 1969 to 1973, he was the Rolls-Royce Readerin Fluid Mechanics at Loughborough University. In 1973, hewas appointed Chief Scientist and later Director of CorporateDevelopment for Westland Helicopters. He was a co-patenteeof the BERP rotor system which, mounted on a Lynx heli-copter, gained the world speed record of 249.1 mph for heli-copters in 1986 which still stands today.

In 1986 he made another career change when he was ap-pointed the Sir George White Professor of Aeronautical Engi-neering at the University of Bristol. The Department was set upin conjunction with the Bristol Aeroplane Company after theSecond World War with Roderick Collar as the first Sir GeorgeWhite Professor. He set about re-invigorating the Departmentwith a number of key appointments. The Department thrivedunder Martin’s leadership and by the end of his tenure as De-partment Head in 2000, its international reputation was secure.Martin continued with his own research into very many variedthemes.

During the 1990s, his interest turned to ground based trans-port systems, attempting to understand why so few people usedpublic transport. His innovative idea was to give individuals inurban areas nearly all the freedom of movement of the car with-out the hassle of driving, ownership, parking, etc. He set up hisown company, ULTRA, in 1995 and for the period from 2000onwards spent his waking hours promoting his Personal RapidTransit scheme. In 2005, the Company was awarded a contractto provide the transport system for passengers to transfer fromthe car parks to Terminal 5 at Heathrow Airport. This has nowcarried over 600,000 paying customers.

Martin was a Fellow of Royal Academy of Engineering,Fellow of the Royal Aeronautical Society, and Fellow of theAmerican Institute of Aeronautics and Astronautics. Martinleaves a widow, Ann, and two children, Sarah and Jonathan.


A Novel Viscoelastic Material Modulus Function forModifying the Golla-Hughes-McTavish MethodLuke A. MartinNaval Surface Warfare Center Dahlgren Division, Dahlgren, Virginia, 22448, USA

Daniel J. InmanDepartment of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan, 48109, USA

(Received 28 September 2012; revised 29 November 2012; accepted 12 February 2013)

The growing popularity of finite element analysis in the 1980s and 1990s spawned new techniques for modellingdamping in complex structures. The Golla-Hughes-McTavish (GHM) method is one technique developed duringthis era. This method adds non-spatial degrees of freedom to a finite element model in order to account for aviscoelastic material’s ability to dissipate energy. In the GHM method, a material modulus function is used tocharacterize the frequency-dependent complex modulus. This paper presents a novel material modulus function,thus modifying the GHM method. The advantages of this Modified Golla-Hughes-McTavish (MGHM) approachare a reduction in the curve fitting error and a standardized approach for computing material modulus coefficients.An additional parameter is introduced for each dissipation degree of freedom. This paper will compare the originalGHM curve fit approach with this new standardized MGHM approach. Advantages of the MGHM approach andphysical insight into the model are explained.

1. INTRODUCTIONThe ability to accurately predict damping in structures is of

primary importance to mechanical and civil engineers. In theage of computers, finite element analysis has become a pop-ular tool for modelling structures ranging from automobilesand military aircrafts to bridges and buildings. The challengewhen assembling a finite element model is how to incorpo-rate damping in structures. When a structure is comprised ofmetals with few bolted joints, a lightly damped assumptioncan generally be made and applies to all the structural modes.However, structures using viscoelastic materials to passivelycontrol modes are more difficult to model because the lightlydamped assumptions are no longer a reasonable estimate. Ad-ditionally, the viscoelastic material’s damping properties varyas a function of frequency, causing the implementation fortransient- and broadband-forcing functions to be more diffi-cult. In the absence of reasonable prediction tools, the struc-tural dynamicist often builds a prototype structure in order toexperimentally determine the modal parameters. Most of to-day’s finite element solvers do allow frequency-dependent ma-terial properties to be entered and can interpolate or extrapolatemeasured material modulus data within reason. A goal of thispaper is to develop an accurate way to quantify material mod-ulus functions and report these values as material properties.These dynamic material properties can then be used in sim-ple multiple degrees of freedom simulations in MatLAB R© orMathCAD R©. The same material properties are also appropri-ate for inclusion in higher fidelity finite element codes.

In 1985, Golla and Hughes introduced an approach to mod-elling viscoelastic damping, which can be implemented infinite element codes as a prediction tool.1 McTavish andHughes expanded upon the Golla and Hughes work and de-fined the technique outlined as the Golla-Hughes-McTavish(GHM) method.2

In 1997, Friswell et al. introduced a four-parameter mini-

oscillator term instead of the traditional three-parameter term.3

The motivation for the research presented in this article wastaken from Friswell’s search for the most generalized curvefit. Their paper also draws similarities between the GHMand Anelastic Displacement Field (ADF)4 methods, ultimatelycommenting that both methods introduce new augmented finiteelement coordinates as a way to account for how a viscoelasticmaterial dissipates energy.

The past decade’s literature does not contain additionalworks to further the GHM theory. The objective of this pa-per is to present a generalized MGHM theory that will allowresearchers the advantage of having a set of equations withwhich to compute the MGHM coefficients. Note the GHMcoefficients cannot be solved in closed form. This standard ap-proach for computing MGHM coefficients could be adopted asa way to report material modulus functions for materials whoseYoung’s modulus is a function of frequency. Furthermore, theMGHM method is advantageous over the ADF techniques be-cause it casts the dynamic material modulus in terms of a ma-terial property. This approach is in contrast with the ADFtechnique that is built upon augmenting thermodynamic fields(ATFs). ATFs use the thermodynamic properties of materialsto model the energy dissipated. This paper reviews the GHMtheory and a modified GHM method using a novel materialmodulus function is presented. A standard mathematically-based curve fitting approach for the modified GHM method isalso introduced. Finally, a GHM and modified GHM curve fitfor a commonly used viscoelastic material are compared. Thefirst advantage of the MGHM method presented in this paperis that closed form solutions are derived for the MGHM pa-rameters; previously, the GHM parameters could not be solvedin closed form. An important note is that the parameters cannow be solved, in general, for as many mini-oscillator terms asneeded. The second advantage is that the curve fitting error ofthe MGHM model is reduced over the GHM model.

102 (pp. 102–108) International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013

L. A. Martin, et al.: A NOVEL VISCOELASTIC MATERIAL MODULUS FUNCTION FOR MODIFYING. . .

2. A MODIFIED GHM APPROACH

The GHM method applies the theory of linear viscoelastic-ity to a structure resulting in a relationship between dynamicstress and dynamic strain. In one dimensional static space, thisrelationship becomes Hooke’s Law. The function which re-lates the dynamic stress and the dynamic strain is defined asthe material modulus function. The novel four-parameter ap-proach presented in this article differs from the approach usedby Friswell, et al., as this new approach couples the physicalcoordinates into the dissipation equation.

2.1. Review of the TraditionalThree-Parameter GHM Method

The traditional GHM method was developing using the vis-coelastic constitutive relation from the theory of viscoelastic-ity.5 This constitutive relation in Cartesian tensor notation is

σij(t) = Gijkl(t)εkl(0) +

t∫0

Gijkl(t− τ)dεkl(t)

dτdτ. (1)

Equation (1) is a form of the more general Stieltjes integral.Equation (1) can be simplified to

σ(t) = G(t)ε(0) +

t∫0

G(t− τ)dε(t)dτ

dτ ; (2)

for a one-dimensional stress-strain system. Equation (2) isused to begin building the relationship with the traditionalGHM method. The strain ε is defined to be zero when t ≤ 0,and then Eq. (2) becomes

σ(t) =

t∫0

G(t− τ)dε(t)dτ

dτ. (3)

The Laplace transform is now taken on Eq. (3), yielding

σ(s) = sG(s)ε(s). (4)

McTavish and Hughes2 then define sG(s) to be the materialmodulus function. The GHM method defines the material mod-ulus function sG(s) to be a series of mini-oscillator terms

sG(s) = G∞

[1 +

∑n

αns2 + 2ζnωns

s2 + 2ζnωns+ ω2n

]. (5)

Equation (5) outlines a three-parameter sG(s) model which isused as the material modulus function in the traditional GHMmethod. In Eq. (5), G∞ is defined as the equilibrium valueof the modulus or the final value of the relaxation function.The material modulus function is represented as a series ofmini-oscillator terms that includes the three positive constantsαn, ζn, and ωn. These constants, or coefficients, in the mini-oscillator terms are computed by curve-fitting from experimen-tal data for a particular material.

A mechanical analogy using a single degree of freedom(SDOF) system and a single term sG(s) material modulusfunction can be used to illustrate the application of the GHMmethod. The SDOF system is comprised of a spring-mass sys-tem, where the spring is a viscoelastic material and modelled

by a relaxation function. In the SDOF, m represents the massand k represents the spring stiffness. In the case of finite ele-ment analysis, m and k would become M and K, representingthe mass and stiffness matrices. The Laplace domain equationof motion for the proposed SDOF system is

s2mq + k

[1 + α1

s2 + 2ζ1ω1s

s2 + 2ζ1ω1s+ ω21

]q = f . (6)

Next, the GHM method introduces an auxiliary coordinate z.The relationship between the physical coordinate q and theauxiliary coordinate or dissipation coordinate z is

z =ω21

s2 + 2ζ1ω1s+ ω21

q. (7)

Then, Eqs. (6) and (7) are presented in Eq. (8) as two coupledsecond-order equations of motion,

s2mq + (k + α1k)q − α1kz = f ;

s2z + 2ζ1ω1sz − ω21 q + ω2

1 z = 0. (8)

Finally, the second equation is multiplied by α1kω2

1and the two

equations are recast in matrix form. Taking the inverse Laplacetransform of Eq. (8) results in the time domain equation[

m 0

0 α1kω2

1

] [qz

]+

[0 0

0 2ζ1α1kω1

] [qz

]+[

k+α1k −α1k−α1k α1k

] [qz

]=

[f0

]. (9)

In Eq. (9), the mass, stiffness, and damping matrices are allsymmetrical. Furthermore, the mass and stiffness matrices areinvertible if m 6= 0, α1 6= 0, k 6= 0, and ω1 6= 0. However,the damping matrix is not invertible. The substitution madeinto equation Eq. (6) to arrive at the first equation in Eq. (8)is not explicitly stated in McTavish and Hughes.2 Therefore,the substitution equation is derived here by setting Eq. (6) andEq. (8) equal to each other in Eq. (10) as

s2mq + k

[1 + α1

s2 + 2ζ1ω1s

s2 + 2ζ1ω1s+ ω21

]q − f =

s2mq + (k + α1k)q − α1kz − f . (10)

Equation (10) can then be reduced to

kα1s2 + 2ζ1ω1s

s2 + 2ζ1ω1s+ ω21

q = α1kq − α1kz. (11)

Equation (11) then shows the substitution needed in Eq. (6) toarrive at the first equation in Eq. (8). For this reason, Eq. (11)is defined as the substitution equation. The full auxiliary equa-tion is

z =

[s2 + 2ζ1ω1s+ ω2

1

s2 + 2ζ1ω1s+ ω21

− s2

s2 + 2ζ1ω1s+ ω21

−

2ζ1ω1s

s2 + 2ζ1ω1s+ ω21

]q. (12)

Equation (12) is defined here as the full auxiliary equationand is useful when exploring sG(s) formulations. An im-portant note for clarification is that the substitution equation



Figure 1. The mini-oscillator mechanical analogy for the GHM method.2

(Eq. (11)), the auxiliary equation (Eq. (7)) found in McTavishand Hughes,2 and the full auxiliary equation (Eq. (12)) are al-gebraically equivalent. The substitution equation is definedhere for clarity. Equation (11) is actually the auxiliary equa-tion, relating the z and q coordinates given in Eq. (7).

The material presented in this section is the traditional GHMmethod. With the exceptions of Eqs. (3), (10), (11), and (12),all the other equations appeared in previous GHM literature.Equations (3), (10), (11), and (12) are included here to clarifythe traditional GHM method and to build a foundation for thedevelopment of the Modified GHM method in the next section.

The mechanical analogy or physical model representing theequations of motions given in Eq. (9) is shown in Fig. 1.

The mechanical analogy, shown in Fig. 1, uses the ficti-tious coordinate z to represent a dissipation degree of freedom.The primary shortcoming with this method is damping is onlycoupled into the system if the dissipation degree of freedommoves. In the case where a node of the system occurs at thedissipation degree of freedom, it is possible to have a systemwith no damping.

2.2. A Novel Four-Parameter GHM MethodIn 1997, a four-parameter model for sG(s) was investigated

as an alternative method to the traditional GHM formulation.3

Friswell et al. reported that the use of a four-parameter GHMmodel would reduce the sum of the squares’ error and pro-vide an improved curve fit. Their approach followed Golla andHughes’1 approach by assuming the physical degrees of free-dom and the dissipation degrees of freedom could be writtenin matrix form as[

M11 mm 1

][qz

]+ β

[D11 dd 1

][qz

]+

[K11 kk 1

][qz

]=

[fv0

].

(13)In Eq. (13), the mass, damping, and stiffness matrices are sym-metric. Furthermore, the mass, damping, and stiffness matri-ces are invertible if M11 6= m2, D11 6= d2, and K11 6= k2,respectfully.

In this section, an alternative four-parameter GHM methodis considered. This novel approach removes the symmetricalconstraints on the damping and stiffness matrices implied bythe model in Eq. (13). The symmetry constraint is not re-quired for the damping and stiffness matrices when develop-ing a numerical simulation using a state space approach. Therequirement in this case is that the mass matrix is invertible.Therefore, let the two-equation model used for developing theModified GHM method be[M11 mm 1

][qz

]+ β

[D11 d12d21 1

][qz

]+

[K11 k12k21 1

][qz

]=

[fv0

].

(14)

Note that although non-symmetric damping and stiffness ma-trices are considered in Eq. (14), only a non-symmetric damp-ing matrix will be explored.

2.2.1. The Modified GHM Method: Introduction of Psiinto the nth Mini-Oscillator Term

The novel approach presented here introduces the coefficientpsi, ψn, into the sG(s) material modulus function. The newcoefficient is introduced by multiplying the linear coefficient inthe mini-oscillator numerator by ψn. This new method will bereferred to as the Modified GHM method or MGHM method.The inclusion of ψn allows for a four-parameter material mod-ulus function to be developed. The effect of psi on the damp-ing matrix will be shown in this paper. Note that the traditionalGHM method is retained when ψn = 1. The inclusion of ψn inthe modified GHM method will show that the physical coordi-nate is coupled through the damping matrix into the dissipationequation. Consider a material modulus function, sG(s), of theform

sG(s) = G∞

[1 +

∑n

αns2 + 2ζnωnψns

s2 + 2ζnωns+ ω2n

]. (15)

The difference between the traditional GHM formulationshown in Eq. (5) and the MGHM formulation shown inEq. (15) is the inclusion of ψn in the numerator of the mini-oscillator term. When ψn = 1, Eq. (15) and Eq. (5) are identi-cal.

2.2.2. A Single-Term MGHM Method

In this section, a Modified GHM method is explored for asingle term material modulus function. Ultimately, the single-term MGHM method (ψn 6= 1) will be compared to the single-term GHM method (ψn = 1). Therefore, the mechanical anal-ogy to a SDOF system will be made. The same SDOF systemused to develop the traditional GHM method will be used here.When n = 1, consider the Laplace domain equation of motionfor the SDOF system where the spring is a viscoelastic materialand ψn is included

s2mq + k

[1 + α1

s2 + 2ζ1ω1ψ1s

s2 + 2ζ1ω1s+ ω21

]q = f . (16)

Next, the auxiliary equation is needed so that the system ofequations can be formulated. Therefore, similar to Eq. (11),let the substitution equation into Eq. (16) be

kα1s2 + 2ζ1ω1ψ1s

s2 + 2ζ1ω1s+ ω21

q = α1kq − α1kz. (17)

Equation (16) then becomes

s2mq + kq + kα1q − kα1z = f . (18)

Equation (18) is identical to the first equation of the systemshown in Eq. (8). Hence, the physical equation has not changeddue to ψ1. Equation (17) is the MGHM substitution equation,which will be algebraically arranged as the second equationneeded for the system. Equation (17) can also be referred to asthe dissipation equation, since it is used to remove or dissipate



Figure 2. The mini-oscillator mechanical analogy for the MGHM method.

energy from the viscoelastic spring-mass system. Starting withEq. (17) and dividing through by α1 and k yields

s2 + 2ζ1ω1ψ1s

s2 + 2ζ1ω1s+ ω21

q = q − z; (19)

and

s2z+2ζ1ω1sz+2ζ1ω1ψ1sq−2ζ1ω1sq+ω21 z−ω2

1 q = 0. (20)

Taking the inverse Laplace and grouping Eq. (20) yields

z + 2ζ1ω1z + 2ζ1ω1(ψ1 − 1)q + ω21z − ω2

1q = 0. (21)

Next, the inverse Laplace transform of Eq. (18) is taken andyields

mq + kq + kα1q − kα1z = f. (22)

Now, Eqs. (22) and (21) are used to form the time domain sys-tem of equations to represent the SDOF system with a dissipa-tion degree of freedom. Using Eqs. (22) and (21), the matrixformation of this system is[

m 00 1

] [qz

]+

[0 0

2ζ1ω1(ψ1−1) 2ζ1ω1

] [qz

]+[

k+α1k −α1k−ω2

1 ω21

] [qz

]=

[f0

]. (23)

Now the second equation in the system given in Eq. (23) ismultiplied by α1k

ω21

, resulting in[m 0

0 α1kω2

1

] [qz

]+

[0 0

2ζ1α1k(ψ1−1)ω1

2ζ1α1kω1

] [qz

]+[

k+α1k −α1k−α1k α1k

] [qz

]=

[f(t)0

]. (24)

Finally, Eq. (24) can be used to model a SDOF system withviscoelastic materials excited by an arbitrary forcing function.Again notice when ψ1 equals 1, the C21 damping matrix termin Eq. (24) vanishes and Eqs. (24) and (9) are exactly the same.Thus, the traditional GHM model is retained when ψ1 = 1.Equation (24) is a more general model than presented in thetraditional GHM literature. Notice the mass and stiffness ma-trices in Eq. (24) are symmetrically similar to the traditionalGHM method. Additional parameters could be introduced toalter the stiffness matrix but are not explored in the work pre-sented here.

The mechanical analogy or physical model representing theequations of motions given in Eq. (24) is shown in Fig. 2.

Similar to the GHM method, the mechanical analogy shownin Fig. 2 uses the fictitious coordinate z to represent a dissipa-tion degree of freedom. The primary advantage of the MGHM

method is that damping will exist even when the dissipationdegree of freedom has little to no movement. The new damp-ing mechanism or damper couples the physical degree of free-dom through the damping matrix into the dissipation equation.This new mechanical analogy implies that a damper can beadded by coupling the dissipation degree of freedom with thephysical degree of freedom through the damping matrix—butnot physically attaching to the mass, thus avoiding couplingthrough the mass matrix.

The damping matrix is no longer symmetrical in the MGHMmethod. Therefore, a stability check or pole placement checkis required for the system of equations or state matrix whenconsidering solutions.

2.2.3. A Two-Term MGHM Method with Psi

The inclusion of ψn is further explored for a two-term ma-terial modulus function. Therefore, the equation of motion forthe SDOF case with the addition of ψn when n = 2 becomes

s2mq + k

[1 + α1

s2 + 2ζ1ω1ψ1s

s2 + 2ζ1ω1s+ ω21

+

α2s2 + 2ζ2ω2ψ2s

s2 + 2ζ2ω2s+ ω22

]q = f . (25)

Next, the auxiliary equations are needed such that the systemof equations can be formulated. Therefore, let the substitutionsinto Eq. (25) be


s2 + 2ζ1ω1s+ ω21

q = α1kq − α1kz1; (26)

and


s2 + 2ζ2ω2s+ ω22

q = α2kq − α2kz2. (27)

Substituting Eqs. (26) and (27) into Eq. (25) yields

s2mq + (k + kα1 + kα2)q − kα1z1 − kα2z2 = f . (28)

Equations (26) and (27) are the substitution equations for thedissipation degrees of freedom z1 and z2. Equations (26) and(27) can be rewritten as

2ζ1ω1ψ1sq− 2ζ1ω1sq−ω21 q+ s2z1 +2ζ1ω1sz1 +ω2

1 z1 = 0;(29)

and

2ζ2ω2ψ2sq− 2ζ2ω2sq−ω22 q+ s2z2 +2ζ2ω2sz2 +ω2

2 z2 = 0.(30)

Taking the inverse Laplace and grouping Eqs. (29) and (30)yields

z1 + 2ζ1ω1z1 + 2ζ1ω1(ψ1 − 1)q + ω21z1 − ω2

1q = 0; (31)

and

z2 + 2ζ2ω2z2 + 2ζ2ω2(ψ2 − 1)q + ω22z2 − ω2

2q = 0. (32)

Equations (31) and (32) are the dissipation equations for thetwo-term sG(s) formulation. The dissipation equations andsubstitution equations are equivalent and are referred to de-pending on their form. Next, the inverse Laplace transform ofEq. (28) is taken and yields

mq + kq + kα1q − kα1z1 + kα2q − kα2z2 = f. (33)



Now Eqs. (33), (31), and (32) are used to form the time do-main system of equations to represent the SDOF system withtwo dissipation degrees of freedom. Using these equations, thematrix formation of this system ism 0 0

0 1 00 0 1

qz1z2

+ 0 0 02ζ1ω1(ψ1−1) 2ζ1ω1 02ζ2ω2(ψ2−1) 0 2ζ2ω2

qz1z2

+k+α1k+α2k −α1k −α2k

−ω21 ω2

1 0−ω2

2 0 ω22

qz1z2

=

f(t)00

. (34)

Next, the second equation in the system given in Eq. (34) ismultiplied by α1k

ω21

and the third equation in the system given

in Eq. (34) is multiplied by α2kω2

2. The resulting system of equa-

tions ism 0 0

0 α1kω2

10

0 0 α2kω2

2

qz1z2

+ 0 0 0

2ζ1α1k(ψ1−1)ω1

2ζ1α1kω1

02ζ2α2k(ψ2−1)

ω20 2ζ2α2k

ω2

qz1z2

+k+α1k+α2k −α1k −α2k

−α1k α1k 0−α2k 0 α2k

qz1z2

=

f(t)00

. (35)

Finally, Eq. (35) represents the model for a physical single de-gree of freedom whose spring is that of a viscoelastic material.Here the viscoelastic material is modelled by a two-term mod-ified GHM formulation. When ψ1 and ψ2 equal 1 in Eq. (35),the C21 and C31 damping matrix terms vanish and Eq. (35)becomes a two-term traditional GHM formulation. Again, thetraditional GHM model can be retained when ψ1 = ψ2 = 1.Equation (35) is a more general two-term model than presentedin the traditional GHM literature.

2.3. Curve Fitting for the Modified GHMMethod

The advantage of including psi in the linear numerator termis first realized in this section. The goal here is to general-ize the curve fitting of the sG(s) material modulus. The ap-proach taken is to express the sG(s) material modulus func-tion as the ratio of two polynomials. The coefficients of thesepolynomials will be expressed in terms of the modified GHMparameters. A novel formulation of the nth term sG(s) mate-rial modulus function is presented in Eq. (36). Equation (36)can be utilized when higher order sG(s) models are desired.Expressing sG(s) in this manner will allow for a direct corre-lation between the experimental coefficients and the MGHMcoefficients. In practice, experimental data is be collected and

expressed as the ratio of two polynomials or a transfer func-tion. The general expression for such a transfer function canbe expressed as

H(s) =b1s

p + b2sp−1 + . . .+ bp+1

a1sm + a2sm−1 + . . .+ am+1. (37)

The constraint p = m = 2n is applied to Eq. (37). Apply-ing this constraint equates the order of the polynomial in thenumerator and the denominator. This constraint also forces pand m to be even. The constraint defines the order of the poly-nomials in the transfer function to 2n, when n mini-oscillatorterms are given. The transfer function in Eq. (37) can then berewritten as

H(s) =b1s

2n + b2s2n−1 + . . .+ b2n+1

a1s2n + a2s2n−1 + . . .+ a2n+1. (38)

The constraint employed in Eq. (38) is desired to match theorders of the polynomials when sG(s) = H(s) is applied.Then Eqs. (36) and (38) are equated formulating Eq. (39).Equation (39) is the general expression to calculate the coef-ficients for the nth-term Modified GHM model. The coeffi-cients b1, b2, . . . , b2n+1 and a1, a2, . . . , a2n+1 on the left handside of Eq. (39) are computed by curve-fitting the frequency-dependent dynamic elastic modulus and loss factor data for aparticular viscoelastic material. The curve-fitting for the co-efficients b1, b2, . . . , b2n+1 and a1, a2, . . . , a2n+1 can be com-puted in various commercially available software packages.

2.4. Complex Modulus Curve Fittingof Viscoelastic Materials

The implementation of the traditional GHM method re-quires the analyst to use engineering judgment when evaluat-ing the goodness of the curve fit. Gibson et al.6 reported thatfrom a rigorous mathematical perspective, the curve fitting ofGHM functions is ill-posed. The Modified GHM method iswell-posed when compared with the traditional GHM method.This is because the number of natural coefficients availablein the transfer function H(s) matches the number of coeffi-cients available in the material modulus function sG(s). Thus,Eq. (39) provides the mathematical map between the MGHMcoefficients and the transfer functions coefficients.

2.4.1. A Comparison of GHM and MGHM Curve Fitsfor Sorbothane R© DURO 50

In this section, the GHM and MGHM coefficients are com-puted for Sorbothane R© DURO 50 loaded in shear. Equa-tion (39) is applied for a single-term mini-oscillator case where

sG(s) = G∞

(∏ns2 + 2ζnωns+ ω2

n

)+

(∑nαn

s2+2ζnωnψnss2+2ζnωns+ω2

n

)(∏ns2 + 2ζnωns+ ω2

n


n

) . (36)

b1s2n + b2s

2n−1 + . . .+ b2n+1

a1s2n + a2s2n−1 + . . .+ a2n+1= G∞

(∏ns2 + 2ζnωns+ ω2

n

)+

(∑nαn

s2+2ζnωnψnss2+2ζnωns+ω2

n


n


n

) .(39)



n = 1. The five nonlinear equations that result are alge-braically manipulated and the MGHM coefficients are explic-itly defined in terms of the transfer function H(s) coefficients.Expanding Eq. (39) when n = 1 yields

b1s2 + b2s

1 + b3s0

a1s2 + a2s1 + a3s0= G∞·(s2+2ζ1ω1s+ω

21

)+(α1

s2+2ζ1ω1ψ1ss2+2ζ1ω1s+ω2

1

)(s2+2ζ1ω1s+ω

21

)(s2+2ζ1ω1s+ω2

1)

;(40)

and

b1s2 + b2s

1 + b3s0

a1s2 + a2s1 + a3s0=

G∞(1+α1)s2 +G∞2ζ1ω1(1+α1ψ1)s+G∞ω2

1

s2 + 2ζ1ω1s+ ω21

. (41)

The coefficients on the left and right sides of Eq. (41) cannow be equated. In general, α1 = 1 is true for most commer-cial software packages. However, if this is not the case, thenall the coefficients can be normalized by dividing them by α1.The following coefficient relations are for a single-term Modi-fied GHM model:

b1 = G∞(1 + α1); (42)b2 = 2G∞ζ1ω1(1 + α1ψ1); (43)

b3 = G∞ω21 ; (44)

a1 = 1; (45)a2 = 2ζ1ω1; and (46)

a3 = ω21 . (47)

Equations (42), (43), (44), (45), (46), and (47) equate thesingle-term MGHM coefficients to the transfer function coef-ficients. The transfer function coefficients are determined bycurve fitting experimental complex modulus data. Many off-the-shelf software packages or algorithms are available for thecurve fitting of experimental data. The algorithm of choicehere is MatLAB’s invfreqs function.

Next, the single-term MGHM coefficients are solved interms of the transfer function coefficients using Eqs. (42), (43),(44), (45), (46), and (47). The MGHM coefficients are

G∞ =b3a1

[a1a3

] 12

; (48)

α1 = 1− b1b3

[a3a1

] 12

; (49)

ζ1 =1

2

a2a1

[a1a3

] 12

; (50)

ω1 =

[a3a1

] 12

; and (51)

ψ1 =

b2a1a2b3

[a3a1

] 12 − 1

1− b1a3

[a3a1

] 12

. (52)

Using Eqs. (48), (49), (50), (51), and (52), the MGHM coef-ficients can be directly computed from experimentally deter-mined transfer function coefficients.

Table 1. Complex modulus for Sorbothane R© DURO 50 loaded in shear.

Frequency (Hz) Storage Modulus in Shear (psi) Loss Factor5 35 0.56

15 50 0.5830 70 0.5750 90 0.50

Table 2. Transfer function coefficients from the invfreqs curve fit.

Coefficient Valueb1 133.2b2 2883b3 6277a2 54.66a3 231.4

Table 3. MGHM coefficients for n = 1.

Coefficient Value Solution 1 Value Solution 2G∞ 27.13 27.13α1 3.910 3.910ζ1 -15.21 15.21ω1 -1.797 1.797ψ1 0.2415 0.2415

Table 4. GHM coefficients for n = 1.

Coefficient ValueG∞ 3.274α1 40.99ζ1 1.127ω1 10.28

The storage and loss factor for Sorbothane R© DURO 50 canbe found at the Sorbothane R© website. The storage and lossmodulus is reported in Table 1 for DURO 50 loaded in shearover a bandwidth of 5 to 50 Hz.

The resulting transfer function coefficients are reported inTable 2. These are the coefficients from the left hand side ofEq. (41) and the left hand side of Eqs. (42)–(47).

The MGHM coefficients can be directly solved from thetransfer function coefficients reported in Table 2. The MGHMcoefficients are reported in Table 3.

In Table 3, two solution sets are reported for the MGHMcoefficients. Both of these solutions should be carried forwardwhen modelling a structure. Future papers will show when asolution set can be disregarded and is not presented here be-cause it is beyond the scope of this paper. An important noteis the MGHM coefficients ζ1 and ω1 should not be confusedwith the damping ratio or natural frequency values in the phys-ical domain. These coefficients are only used in the dissipationequations. Therefore, negative values are acceptable. Equa-tions (48)–(52) are non-linear and yield two solution sets re-sulting in the two sets of MGHM coefficients.

The GHM coefficients cannot be curve fit using the invfreqsalgorithm because the restriction limiting the linear numera-tor and denominator terms to be equal is not enforced in in-vfreqs. Therefore, the complex modulus data were curve-fittedto the GHM material modulus function using a least squaresapproach for complex data. The result of this approach is pre-sented in Table 4.

Now, the MGHM coefficients reported in Table 3 and theGHM coefficients reported in Table 4 are used to create theirrespective material modulus function. The Sorbothane com-plex modulus data from Table 1, the GHM material modulus,and the MGHM material modulus are plotted in Fig. 3.



Table 5. Curve fit comparison.

Parameter GHM MGHM GHM MGHMn = 1 n = 1 % Error % Error

Storage Modulus at 5 Hz 49.02 38.78 40.1 10.8Storage Modulus at 15 Hz 59.91 52.52 19.8 5.04Storage Modulus at 30 Hz 73.78 69.44 5.40 -0.800Storage Modulus at 50 Hz 92.35 89.97 2.61 -0.0333

Loss Factor at 5 Hz 0.5057 0.4770 -9.70 -14.8Loss Factor at 15 Hz 0.4591 0.5578 -20.8 -3.83Loss Factor at 30 Hz 0.5153 0.5897 -9.60 3.46Loss Factor at 50 Hz 0.4662 0.4984 -6.76 -0.320

Figure 3. Complex shear modulus for Sorbothane R© DURO 50 (♦), GHMcurve fit (solid), and MGHM curve fit (dashed).

Next, the GHM and MGHM curve fits are compared. Qual-itatively, Figure 3 shows the black dashed line is a better curvefit than the solid black line. Table 5 compares the percent er-ror of the GHM and MGHM curve fits at each Sorbothane R©

data point. Table 5 shows a data point by data point com-parison between the two curve fit methods and the originalSorbothane R© data reported in Table 1. The MGHM percenterror is lower than the GHM error at every data point ex-cept the loss factor at 5 Hz. The GHM storage modulus andloss factor root mean square deviations are 17.7 and 0.147, re-spectively. The MGHM storage modulus and loss factor rootmean square deviations are 4.58 and 0.0882, respectively. TheMGHM curve fit has been demonstrated as an improvement tothe GHM curve fit.

3. CONCLUSIONS

This paper presents a novel material modulus function formodelling viscoelastic materials. The new approach is coinedthe Modified Golla-Hughes-McTavish method or the MGHMmethod.

An additional parameter was introduced into the linear nu-merator term of the traditional GHM method’s material modu-lus function. The motivation was an improved material modu-lus function curve fit. The addition of psi couples the physicaldegrees of freedom into the dissipation equation through thedamping matrix.

Secondly, a general formulation for the nth mini-oscillatorterm is derived for the MGHM material modulus function.This general formulation is presented in Eq. (36). From thisformulation, the closed form curve-fitting relationships for the

n = 1 material modulus function case are presented. Thisclosed form curve-fitting relationship was not possible withthe GHM method because too few coefficients were used inthe material modulus function.

The MGHM material modulus function coefficients areequated to transfer function coefficients for the nth mini-oscillator term in Eq. (39). Given a transfer function H(s)curve fit to the complex modulus data, the MGHM coefficientscan be solved by using Eq. (39). This approach adds the math-ematical rigor previously missing for computing the GHM co-efficients.

Finally, the MGHM parameters for Sorbothane R© DURO 50loaded in shear are reported. These parameters, which de-fine the material modulus function, are needed when design-ing a structure that is going to be loaded dynamically and in-corporates viscoelastic material. Design engineers rely on re-ported values of the Young’s modulus for modelling metallicstructures. The need to report material modulus functions forviscoelastic materials has arrived, and this paper contributesby reporting a material modulus function for Sorbothane R©

DURO 50 dynamically loaded in shear.The GHM and MGHM curve fits are compared with re-

ported Sorbothane data and the MGHM curve fit has a lowerroot mean square deviation than the GHM curve fit. Thus,this paper demonstrates the improved curve fit provided by theModified GHM method.

REFERENCES1 Golla, D. F. and Hughes, P. C. Dynamics of viscoelastic

structures — a time-domain, finite element formulation,Journal of Applied Mechanics, 52, 897–906, (1985).

2 McTavish, D. J. and Hughes, P. C. Modeling of linear vis-coelastic space structures, Journal of Vibration and Acous-tics, 115, 103–110, (1993).

3 Friswell, M. I., Inman, D. J., and Lam, M. J. On the re-alisation of GHM models in viscoelasticity, Journal of In-telligent Material Systems and Structures, 8 (11), 986–993,(1997).

4 Lesieutre, G. A. and Mingori, D. L. Finite element model-ing of frequency-dependent material properties using aug-mented thermodynamic fields, AIAA Journal of GuidanceControl and Dynamics, 13, 1040–1050, (1990).

5 Christensen, R. M. Theory of Viscoelasticity: An Introduc-tion, Academic Press, New York, (1982).

6 Gibson, W. C., Smith, C. A., and McTavish, D. J. Imple-mentation of the Golla-Hughes-McTavish (GHM) methodfor viscoelastic materials using MATLAB and NASTRAN,Proc. SPIE 2445, 312, San Diego, California, (1995).

7 Sorbothane, Inc., Sorbothane Technical Data, Retrievedfrom http://www.sorbothane.com, (Accessed November 30,2009).


Active Control of Radiated Sound from StiffenedPlates Using IDE-PFC ActuatorsAtanu SahuInstitute of Composite Structures and Adaptive Systems, DLR, 38108 Braunschweig, Germany

Tirtha Banerjee, Arup Guha Niyogi and Partha BhattacharyaDepartment of Civil Engineering, Jadavpur University, Kolkata -700032, India

(Received 29 November 2011; revised 7 January 2013; accepted 6 March 2013)

It has been a practice in modern day aircraft and automobile industries to manufacture a stiffened structure forsignificantly enhancing efficiency and strength without incurring a considerable weight increase. In the presentstudy, an attempt is made to understand the effect of stiffeners on the sound radiation pattern of a vibrating plate.Subsequently, a velocity feedback control algorithm based on Linear Quadratic Regulator (LQR) methodologyis developed to attenuate the radiated sound power from the vibrating structures with surface bonded piezo fibrecomposite (PFC) patches with interdigitated electrodes (IDE) as actuators and polyvinylidene fluoride (PVDF)films as sensors. Results are obtained for different orientations of stiffeners and various locations of PFC patchesand are discussed.

1,2,3

NOMENCLATUREω Natural frequencyÛσ Radiation efficiency{σ} Stress vector{ε} Strain vector{E} Electric field vectorE Young’s modulusν Poisson’s ratioρ Material density[Q] Elastic moduli matrix{D} Electrical displacement vector[e] Piezoelectric stress/charge constant[κ] Electric permittivity or dielectric matrix[ρ] Inertia matrix[N] Shape function matrix{d},

¶d©

,¶d©

Generalized displacement, velocity, andacceleration vector

[M] Mass matrix[KUU] Mechanical stiffness matrix[Kuφ] Electro-mechanical coupling stiffness

matrix[Kφφ] Electrical stiffness matrix[Z] Position matrix of the piezo patch[B] Strain-displacement matrixQ Electrical charge{Fel} Electrical load vector{y}, {y}, {y} Displacement, velocity, and acceleration

vector in modal level

1. INTRODUCTION

In the present day automobile and aerospace industry, theobjective is to increase fuel efficiency without compromisingstructural stiffness and strength. To achieve these objectives,the present trend is to use lightweight structural elements.

However, these forms of structures have the possibility of gen-erating unacceptable levels of sound and noise due to vibration.Since interior noise has a strong effect on passenger comfortand acceptance of the vehicle, researchers and engineers arecontinually working on different methodologies to reduce thenoise level.

Classically, sound attenuation in the medium to high fre-quency acoustic range can be achieved in a passive manner byadding sound absorbing materials to the surface of the radiat-ing structure. An alternate methodology is to use destructiveinterference in the sound source path. A more advanced formof the sound control is based on altering the vibrations of thenoisy structure such that it radiates less sound. This alterationmay be achieved by introducing discrete and collocated sen-sors and actuator pairs that fall under the scope of active con-trol strategy.

The acoustic radiation problem has been addressed by var-ious researchers, including Gladwell,1 Gladwell and Zimmer-man,2 Seybert et al.,3 and others. Cunefare has shown that theradiated sound power can be expressed in terms of acousticradiation filters, surface radiation modes, and discrete surfacevelocities of the vibrating structure.4 Elliot and Johnson com-pared two different formulations for calculating the acousticpower radiated from a vibrating structure in the free field andthen implemented a feed forward control of sound power for abaffled square panel.5 Radiation efficiencies and singular ve-locity patterns were introduced by Borgiotti and Jones usingthe singular value decomposition (SVD) of a radiation resis-tance matrix.6 Gibbs et al. developed a radiation modal ex-pansion (RME) technique to reduce computational effort andthen apply a feedback control strategy with radiation filters em-ploying multiple input or output smart sensor-actuators.7 Bhat-tacharya et al. have shown that the size and geometry of the vi-brating structure influences the cut-off limit of the acoustic ra-

International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013 (pp. 109–116) 109

A. Sahu, et al.: ACTIVE CONTROL OF RADIATED SOUND FROM STIFFENED PLATES USING IDE-PFC ACTUATORS

diation filters.8 In the paper by Liu et al., laboratory tests wereperformed to validate the theoretical prediction of the soundinsulation properties of aircraft panels with ring frames andstringers.9 They used shell panels made of both isotropic aswell as composite materials. Engels et al. discussed the per-formances of different feedback controllers implemented on aplate structure using collocated velocity sensors and force ac-tuators with the objective to minimize the kinetic energy andthe radiated sound power.10 In a review paper by Fuller, dif-ferent methodologies for controlling the radiated sound from avibratory structure were discussed—e.g., active acoustic con-trol, active structural control, active structural-acoustic controlwith the help of point acoustic sources, smart foam as activeskins, piezoelectric actuator-sensor pairs, etc.11

In the present work, an attempt is made to numericallymodel and estimate the radiated sound power in the free fieldfrom flat vibrating panels subjected to external disturbances.Subsequently, the formulation is extended for stiffened plateswith stiffeners placed and oriented in different directions andpositions. The study also aims to develop an active controlstrategy based on LQR methodology to minimize the sound ra-diating into the free field where the sensor voltage correspond-ing to the structural velocity is fed back into the piezo fibrecomposite actuator patches. For the structural analysis, a fournode isoparametric quadrilateral finite element, with five me-chanical degrees of freedom (3 translational and 2 rotational)per node based on first-order shear deformation theory is em-ployed. Necessary transformation is used to incorporate thestiffeners into the FE model. A pair of electrical voltages,namely sensor voltage (φs) and actuator voltage (φa), are con-sidered as additional degree of freedom applied per elementcontaining piezo electric patches.

2. MATHEMATICAL FORMULATION

In this section, the mathematical formulation to describe theradiated acoustic power from planar baffled structures with andwithout stiffeners is presented. Thereafter, a feedback con-trol algorithm based on LQR methodology with structurallybonded piezoelectric sensor and actuator devices is formulatedand presented.

2.1. AcousticA vibrating structure consisting of an infinite number of nat-

ural mode shapes surrounded by an acoustic medium causespressure perturbation in the medium that is experienced assound. However, the structural mode shapes do not radiate in-dependently and the inter-modal coupling between structuralmodes affects the radiated sound power. Therefore, reducingdominant structural modes may have little effect on the radi-ated power. Hence, it is important to develop acoustic radiationfilters apart from identifying structural mode shapes. Acousticradiation filters describe the radiated power in terms of dis-crete surface velocities and the surface radiation resistance, asshown by Cunefare.4

In the present analysis, the vibrating structure consists of aflat plate with and without stiffeners radiating sound into a freefield. The plate is partitioned into N — numbers of equally

Figure 1. Schematic showing the baffled plate with the geometric interpreta-tion of the Reyleigh integral.

sized rectangular elements with the size very small as com-pared to the acoustic wave length under consideration. Indi-vidual elements can be assumed as a point source placed ina baffle, and following the developments by Fahy and Gardo-nio,12 the pressure at any observation point in the field can beexpressed in the form of the Rayleigh integral for the acousticHelmholtz equation. The Rayleigh integral is

p(r) =iωρ02π

∫S

vn(rs)e−ik|r−rs|

|r − rs|dS; (1)

where p(r) is the complex acoustic pressure amplitude at anylocation r, k = ω/c0 is the acoustic wave number with c0 thespeed of sound in the medium, and %0 is the density of themedium. The surface normal velocity is vn(rs) on the vibrat-ing source with a closed boundary S, as shown in Fig. (1). Thesound power generated is equal to the surface integral of thenormal component of the sound intensity

W =1

2ReÅ∮

S

p(rs)v∗n(rs)dS

ã. (2)

The Rayleigh integral is solved with the help of a numericalscheme. It is assumed that the normal velocity is constantacross each element. That makes each element behave like anelemental radiator or piston that moves with constant harmonicvelocity. For this discretization, the Rayleigh integral can alsobe written as

pf = Zfvn; (3)

where pf is the vector with pressures in a set of field points,vn is the vector with normal surface velocities of the elemen-tal radiators, and Zf is a frequency dependent transfer matrix,whose elements are given by

(Zf )ij =iωρ0Se2π

e−ikrij

rij. (4)

Se defines the area of the elemental radiator and rij is the dis-tance between the field point i and surface point j (rij =

|ri − rj |).For the same discretization, the expression for the sound

power reduces to the summation

W =Se2

Re(vHn p); (5)



where p is the vector with surface pressures, evaluated at thesame points on the surface as vn. With the substitution of p =

Zvn, the sound powerW can be obtained in terms of a discretenumber of velocity measurements as

W = vHn Rvn. (6)

In this equation, R = (Se/2)Re(Z) is called the radiation re-sistance matrix. This matrix R can be written as

R =ω2ρ0S

2e

4πc0

1 sin(kr21)

kr21· · · sin(kr1N )

kr1N

sin(kr12)kr12

1...

.... . .

...sin(krN1)krN1

. . . · · · 1

.

Since rij = rji, the radiation resistance matrix R is sym-metric and positive definite. The elements of the matrix de-pend on the properties of the acoustic medium, the frequency,and the geometry of the plate.

The radiation efficiency is defined as the ratio of the soundpower radiated per unit area by the object to the sound powerradiated by a reference source. The reference source is a baf-fled piston vibrating at a high frequency (wave number, k×area of the piston, A � 1) with a velocity equal to the spaceand time averaged, squared, normal velocity 〈v2n〉 of the object.The radiation efficiency is given asÛσ =

W

ρ0c0S〈v2n〉; (7)

where S is the total area of the object. The space and timeaveraged normal velocity is given by

〈v2n〉 =1

2S

∫S

|vn(rs)|2dS = vHn Nvn. (8)

The radiation modes and the radiation efficiencies can be ob-tained by carrying out a singular value decomposition (SVD)on the matrix N−1R. As the radiation resistance matrix R

depends upon the frequency, the SVD must be performed atall the frequency steps. As a result, both the radiation ef-ficiency and the radiation mode shapes are functions of fre-quency. In order to predict the total power radiated over a cer-tain frequency bandwidth, the characteristics of the R matrixare modelled over that bandwidth.

2.2. Constitutive RelationshipThe structural form considered here consists of an isotropic

stiffened plate with IDE-PFC patches bonded as actuators andpiezo monolithic patches as sensors. The constitutive relation-ship for an isotropic material can be expressed as

σxxσyyτyzτzxτxy

=

E

(1−ν2)Eν

(1−ν2) 0 0 0Eν

(1−ν2)E

(1−ν2) 0 0 0

0 0 E2(1+ν) 0 0

0 0 0 E2(1+ν) 0

0 0 0 0 E2(1+ν)

εxxεyyγyzγzxγxy

.

(9)

Figure 2. Actuation mechanism of piezo fiber composite (PFC) with interdig-itated electrode (IDE).

In the case of piezo patches, the standard IEEE norm is fol-lowed and presented below as

{σ} = [Q]{ε} − [e]T{E} (10)

and{D} = [e]{ε}+ [κ]{E}; (11)

where the notations have their usual meanings.The electric field vector {E} is expressed as

{E} = −∇φ; (12)

where φ is the applied electric potential.In case of IDE-PFC patches (Fig. (2)), the electric field

along the length of the patch between two consecutive elec-trode fingers is assumed to be linearly distributed with alter-nate electrodes being suitably grounded. There is a perfectbond between the piezo layer and the elastic substrate. Hence,the electric field for the actuator layer is expressed as

Ea = −φa

hs; (13)

where hs is the distance between IDE.In case of sensor elements, the electric field is assumed to

be linearly distributed through the thickness of the piezo layerwith the electrode in contact with the substrate being suitablygrounded.

2.3. Finite Element FormulationThe governing equation for the structural motion is devel-

oped using the Hamiltonian formulation and is expressed as

d

dt

Å∂T

∂q′i

ã− ∂

∂q′i(T − U) = Pi; (14)

where Pi is the generalized force and q′i are the generalizedcoordinates.

The kinetic energy T is expressed as

T =1

2

∫V

{u}T{ρ}{u}dv. (15)

The potential energy U consists of potential due to mechanicalstrain UM and electrical strain energy UE, where UM is givenas

UM =1

2

∫V

{ε}T {σ} dv. (16)

With the inclusion of a piezoelectric patch as an actuator inthe host structure, the potential energy due to the electricfield needs to be taken care of in the potential energy term inEq. (14) as:

UE =1

2

∫V

{E} {D} dv. (17)



The sensor is separately modelled and is explained in the nextsection.

The structure is discretized using a four-node isoparamet-ric element. Five mechanical degrees of freedoms (3 trans-lational and 2 rotational) are assumed per node. As alreadymentioned, one actuator voltage per element is incorporated inthe FE model. The actuator modelling is done as per the workgiven in the paper by Azzouz et al.13 Therefore, following theisoparametric concept, the mechanical degrees of freedom canbe expressed in terms of the linear shape functions as

{u} = [N]{de}; (18)

where {u} is the displacement vector at any point of the finiteelement and {de} is the nodal displacement vector.

The finite element form of the governing equations of thesystem are written as

[M]¶d©+ [KUU] {d}+ [KUφ] {φa} = {Fm} (19)

and[KφU]{d} − [Kφφ]{φa} = {Fel}; (20)

where Fm is the mechanical load vector and[M] =

∫A[N]T[ρ][N]dA;

[KUU] =∫A[B]T[D][B]dA;

[Kφφ] =∫V

Ä1h2s

ä[Bφ]

T[κ][Bφ]dV ; and

[KUφ] =∫V

ÄVfhs

ä[B]T[Z]T[e]T[Bφ]dV = [KφU]

T.

2.4. Sensor ModellingThe direct piezoelectric equation is used to calculate the

output charge from the piezo sensors induced by mechanicalstrains. The electric displacement developed on the sensor sur-face is directly proportional to the mechanical strain acting onthe sensor. By using a zero-input-impedance circuit, it is pos-sible to show that the first term in the right-hand side of thepiezoelectric constitutive Eq. (10) contributes less than the sec-ond term on the same side of the equation, and the second termin the right-hand side of the piezoelectric constitutive Eq. (11)is less significant than the first term on the same side of theequation. Thus, the sensor behaves like a pure current source.For such a configuration, the charge developed for the ith sen-sor patch at location z = hi+1 (distance from the mid-plane ofthe structure) can be expressed as

Qi(t) =

∫Ai

D(x, y, hn+1, t)dA; (21)

where D(x, y, hn+1, t) = [e]T{ε} and {ε} = [Z][B]{de}.The sensor current is proportional to the rate of charge de-

veloped. Therefore,

isensor =˙Q; (22)

and the sensor output voltage becomes

φs = Rf isensor (23)

orφs = Rf [K

es]

T¶de©; (24)

where [Kes]T =

∫Ai[e]T[Z][B]dA is called the sensor stiffness

and Rf is the resistance offered by the piezo patch.In the present formulation, the voltage produced in the sen-

sor patches (Eq. (24)) mounted on the vibratory structure isfed back to the actuator patches after multiplying with the gainvalue G obtained from control law and thus the third term inEq. (19) represents the equivalent piezoelectric load. Hence,the final governing equation takes the form

[M]¶d©+ [KUφ] ·G · [Ks]T{d}+ [KUU] {d} = {Fm}. (25)

The final governing equation in the element level when assem-bled gives the global equation for the entire domain

[Mgl]¶dgl

©+ [Cgl]

¶dgl

©+ [Kgl] {dgl} = {Fgl} . (26)

The FE model is developed in the MATLAB environmentfor response calculation and is simultaneously used to imple-ment the control algorithm.

After carrying out the response analysis (Eq. (26)), one canobtain the surface velocities with and without the control volt-ages and obtain the radiated sound power from the vibratingplate from Eq. (6).

3. LQR CONTROL ALGORITHM

In the present section, the control algorithm implemented forthe sound power reduction is discussed briefly. For the presentstudy, a linear quadratic regulator (LQR) is considered, whichis based on the quadratic performance index. The state-spacemodel of the system in modal level is given byß

yiyi

™=

ï0 1

−ω2i 0

òßyiyi

™+

ï0

−KUφ

ò{φa} ,

i = 1, 2, 3 (number of modes considered);

orx = Ax+Bu. (27)

The linear feedback control law for the system may be writtenas

φa = −Gφs. (28)

The elements of the gain matrix G are found by minimizingthe performance index J ,14

J =

∫ ∞0

L(x, u)dt; (29)

where L is a quadratic function of x and u; the final form isexpressed as follows:

J =

∫ ∞t0

(xTQix+ uTRiu)dt (30)

i = 1, 2, 3 (number of modes considered).Here, both Q and R matrices are real symmetric and Q con-

sists of sensor stiffness at modal level while R can be chosenby the control designer. Again Q and R denote the relativeimportance of error and expenditure of control effort. The op-timization of the performance index J results in the continuoustime Riccati equation, which is solved in the MATLAB plat-form.



Figure 3. Schematic diagrams of unstiffened and stiffened plates.

4. RESULTS AND DISCUSSION

The present work on the behaviour of the vibrating structureon the radiated acoustic energy into the free field is carriedout in two phases. The effect of the stiffeners on the radiatingsound energy from the simply supported plate structure sub-jected to external mechanical excitations is investigated in thefirst phase of the work. In the next phase, an active controlmethodology is implemented to attenuate the emitted energyin some selected cases. The different structural forms consid-ered for the present study are presented below:

Case I: Simply-supported unstiffened plate (Fig. (3a));

Case II: Simply-supported stiffened plate with singleunidirectional stiffener parallel to x-axis (Fig. (3b));

Case III: Simply-supported stiffened plate with bidirec-tional cross stiffeners (Fig. (3c));

Case IV: Simply-supported stiffened plate with doubleunidirectional stiffeners parallel to x-axis (Fig. (3d)).

A square aluminium plate of size 1.0m ×1.0m with thick-ness 0.004m (Young’s modulus, E = 70.0GPa, Poisson’s ra-tio, ν = 0.3 and density, ρ = 2700 kg/m3) is considered forthe analysis. For a stiffened plate, the depth of the stiffener istaken as 0.1m with the thickness maintained the same as thatfor the plate. For validating the developed formulation, a freevibration study is carried out initially on a stiffened plate andis discussed in the next subsection.

4.1. Validation of the Stiffened PlateFormulation

Free vibration frequencies are obtained for a simply-supported stiffened plate with unidirectional stiffener (Case II)from the present finite element model and from the ANSYSver. 11.0 model using the SHELL 63 element. They are pre-sented in Table 1. The dimensions and the material propertiesfor the structure are kept the same as mentioned in the previ-ous section. The flat portion of the plate is discretized into a20×20 FE mesh while the stiffener is discretized using a 20×2mesh. From Table 1 it is observed that the natural frequenciesobtained from the two models compare well.

As stated earlier, the objective of the present study is toobtain radiated acoustic power from the vibrating plates. To

Table 1. Comparison of non-dimensional frequencies (λi) of an isotropic (alu-minium) stiffened plate (Case II).

Mode Number Present Finite Element Code ( λi) ANSYS ( λi)1 0.0092 0.00912 0.0125 0.01233 0.0148 0.0147

Note:λi = ωi

√ρ(1−ν2)

E [Niyogi et al.15]

Figure 4. Radiation efficiency plots for the first 4 modes of the plate.

achieve this objective, initially a singular value decomposition(SVD) to obtain the eigenvalues and the associated eigenvec-tors is performed on the radiation resistance matrix R, givenin Eq. (6). Using the radiation modal expansion (RME) tech-nique, as explained in Gibbs et al.,7 with the cut-off frequencyat 250Hz, the radiation plots are obtained and presented inFig. (4), which according to Gibbs et al.7 are termed as theradiation efficiency plots. The eigenvectors represent the radi-ation mode shapes. It is important to note that the radiation re-sistance matrix is dependent only on the plate geometry and isindependent of the boundary condition and structural stiffness.In the present study, for all the cases considered, the radiatingsurface has the same geometry, and hence obtaining the eigen-vectors and singular values for one case is sufficient and is thenused for all the cases. The obtained radiation efficiency plotswith the radiation mode shapes are subsequently in conjunc-tion with the structural velocities used to calculate the radiatedenergy (Eq. (6)). In the next sections, acoustic response resultsfor various cases are obtained and discussed.

4.2. Radiated Acoustic Power fromUnstiffened and Stiffened Plates

In the present section, radiated sound power from the fourdifferent types of vibrating structures (Case I to IV), due tothe application of an external transient excitation of 1 kN, isobtained. The load (Fig. (5)) is applied for 0.01 seconds overan area of 0.05m ×0.05m with the centre of the loading pointlocated at 0.125m along the x- and y-axes from the origin ofthe global coordinate system (Fig. (3)). The radiated sound

power, obtained using Eq. (7), is calculated as log10( Ûσ

10−12

)10in dB and is plotted in Fig. (6).

It is observed from Fig. (6) that the reduction of radiatedacoustic power in each mode is not assured with the additionof stiffeners. The sound power level increases for Case III ascompared to Case I. For stiffened plates, the arrangement of thestiffener plays an important role in the reduction of the radiatedsound power level. Though in both Case III and Case IV thenumber of stiffeners is the same, in Case IV the radiated power



Figure 5. Transient load applied at 0.125m along the x- and y-axes of platestructure.

Figure 6. Radiated acoustic power of the stiffened plates (Case I to IV) sub-jected to a transient excitation of 1.0 kN (Fig. (5)).

is less in all the modes due to the arrangement of the stiffeners(Figs. (3c) and (3d)). It is also observed that with the additionof stiffeners the system frequency shifts; hence, the placing orarrangement of the stiffener needs some optimization study sothat it does not interfere with the loading frequency. Moreover,stiffening the structure with the addition of a stiffener at anyarbitrary position is not the sole solution to reduce the emittedacoustic energy, which necessitates the need for active control.Furthermore, the proper optimization technique should be im-plemented for the placement of stiffeners.

Additionally, the output to input pressure ratio (transmissionratio) is calculated for various cases and presented in Fig. (7).It is observed that the ratio is always less than 1. This suggeststhat only a part of the input energy is radiated as sound.

Figure 7. Output pressure to input pressure (transmission ratio) for the stiff-ened plates subjected to a transient excitation of 1.0 kN (Fig. (5)).

Figure 8. Piezoelectric patch locations for (a) a unidirectional and (b) a bidi-rectional stiffened plate.

Table 2. Mechanical and electrical properties of the actuator and sensor(Guennam and Luccioni,16 and Liew et al.17

IDE-PFC Actuator (PZT-5H) PVDF Film SensorGeometric Properties

Length, L 0.05m 0.05mWidth,B 0.05m 0.05mThickness 0.001m 40× 10−6 m

Elastic and Material PropertiesE — 2.0GPaν — 0.28

ρ 7740 kg/m3 1800 kg/m3

Electrical Propertiese22 22.9C/m2 —

e31= e32 — 0.046 C/m2

κ11= κ22 1.27× 10−8 F m−1 106.2F m−1

κ33 1.51× 10−8 F m−1 —

Note: Spacing of interdigitated electrode hs = 0.0005m and fibre volume fraction

Vf = 0.2

4.3. Active Control of Radiated AcousticPower

It is already seen in the earlier section that the attenuation ofthe acoustic energy is not always promised with the stiffeningof the structure. Hence, a piezo-based active control strategybased on velocity feedback is attempted in the second phase.Piezo fibre composite with interdigitated electrode (IDE-PFC;PZT 5H) patches are used as actuators while PVDF films areemployed as sensors for which the properties are mentionedin Tables 2 and 3.16, 17 Among the four cases (I to IV), studieshave been conducted on two typical stiffened plates with unidi-rectional (Case II) and bidirectional stiffeners (Case III). Eightpairs of collocated actuator-sensor patches are mounted on thetop of the stiffened plate. The schematic is shown in Fig. (8).

Prior to the control of radiated sound, the piezoelectric for-mulation is validated with standard data available in open lit-erature and with the results obtained from ANSYS software.An IDE-PFC patch (PZT-5H) measuring 0.05m long, 0.02mwide, and having a thickness of 0.001m with a fibre-volumeratio of unity, with a pair of electrodes placed at a distanceof 0.05m along the length, is modelled using the present FEformulation and in the ANSYS ver. 11.0 model using theSOLID 5 element. An electrical potential of unit (1) volt-age is applied across the electrode. The longitudinal strainobtained from the ANSYS model is 0.350 × 10−8 and thatfrom the present FE model is found to be 0.350 69 × 10−8,which compares extremely well. Moreover, the in-plane blockforce obtained from the actuator patch is calculated to be0.4580N/m which compares very well with the formulationgiven by Bent18 in his work.



Table 3. Elastic Properties of the IDE-PFC actuator (PZT-5H) (Guennam andLuccioni16)

Q11 Q12 Q22 Q31 Q32 Q33 Q44 Q55 Q66

130.6 85.66 135.8 88.3 90.42 121.3 23.47 22.99 22.99

Figure 9. Radiated acoustic power (uncontrolled and controlled) from thestiffened plate with the unidirectional stiffener (Case II) subjected to a transientexcitation of 1.0 kN (Fig. (5)).

4.4. Control of Radiated Acoustic Powerfrom the Stiffened Plate withUnidirectional Stiffener (Case II)

As has been previously discussed in implementing the activecontrol scheme, the sensor signals obtained from the vibrat-ing structure are fed back to the actuator with proper feedbackgains derived from the LQR algorithm. These gain values areobtained by minimizing J (Eq. (29)), which is again depen-dent on R (Eq. (30)). As R is correlated with expenditure ofcontrol effort, it is chosen in such a manner that the capacityof the actuator patches should not be exceeded. In the presentstudy, two different principal orientations of the IDE-PFC ac-tuators are considered—(i) along the x-axis and (ii) along they-axis. The dimension, material properties, and the nature ofthe applied external loading are already mentioned in Section4 and Section 4.2. The piezoelectric properties of the actuatorand sensor are presented in Tables 2 and 3.

The plots of the uncontrolled and controlled sound power ra-diating into the free field are shown in Fig. (9). As observed inFig. (9), the radiated sound power can be reduced significantlyby considering the structural velocities, as only the feedbackstate which effectively increases the damping. Moreover, thisactive control scheme is capable of reducing the radiated en-ergy, which is not always assured with the passive strategy likearbitrarily stiffening the structure. It is also seen that for thePFC actuators having their principal actuation direction alongthe x-axis, the reduction of radiated acoustic power in the freefield is almost 7.7 dB for the first mode; whereas, for the samemode almost 18 dB reduction is achieved with the principalactuation direction along the y-axis. From the plot, it is clearthat for the present structural configuration, the d22 actuationis much more pronounced in attenuating the radiated energy inthe first and second modes when compared with the d11 actu-ation for the same control effort. In the first case, maximumvoltage in the actuator patch is approximately 48 volts (peakto peak), whereas in the next case it is approximately 46 volts.It is imperative to note that in both cases, control in the thirdmode is not very effective, as the actuator patches lay on the

Figure 10. Mode shape plots for the stiffened plates (Case II and III).

Figure 11. Radiated acoustic power (uncontrolled and controlled) from thestiffened plate with the bidirectional stiffener (Case III) subjected to a transientexcitation of 1.0 kN (Fig. (5)).

inflexion line (Fig. (10)). Hence, it is evident from Fig. (10)(3rd mode of Case II) that the actuator patches are incapableof controlling the energy radiated in the third mode with thegiven actuator location.

4.5. Control of Radiated Acoustic Powerfrom the Stiffened Plate with a Bidirec-tional Stiffener (Case III)

In the present section, the attempt is to reduce the freefield sound power radiated from the bi-directionally stiffenedplate (Case III) with the same number of actuator-sensor pairs(Fig. (8b)) as in the previous case, but with a different orienta-tion of the patches. The results are plotted in Fig. (11).

In the first case, i.e., when the principal actuation directionis along the x-axis, a reduction of sound power amounting to11.9 dB in the 1st mode and 10.5 dB in the 2nd mode is achievedwith 43 volts as maximum input voltage to the actuator patch,but it reduces to 42 volts when the actuation direction is ro-tated by 90°. It is also seen that for d22 actuation, the reductionachieved in the two modes is 12 dB and 10.7 dB, respectively.Hence, it can be concluded that in this particular case the ori-entation of the patches does not have much influence, which isalso shown in the mode shapes for this case (Fig. (10)).



5. CONCLUSION

In the present work, a study has been carried out on thesound radiation characteristics of the un-stiffened as well as thestiffened plates in the free field. As it is shown that arbitrarilystiffening the structure is not the sole solution to attenuatingthe radiated sound power, one should either go for optimiz-ing the stiffener locations or implementing an active controlscheme for a reasonable reduction in the radiated energy. Inthe present work, an active control scheme is implemented byproviding sensor voltages as feedback signals multiplied withthe proper gain values obtained from the LQR algorithm tothe actuator patches collocated on the structure. Two caseshave been considered to elucidate the effectiveness of the con-trol algorithm; however, it is found that for maximum utiliza-tion of the active control system, an optimization study shouldbe carried out for positioning the piezo patches. The workcan be extended further to optimize the stiffener locations ofthe structure and also in optimizing the positions of actuator-sensor pairs towards achieving the maximum reduction in theradiated sound.

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9 Liu, B., Feng, L., and Nilsson, A. Sound transmissionthrough curved aircraft panels with stringer and ring frameattachments, J. Sound Vib., 300 (3-5), 949–973, (2007).

10 Engels, W. P., Baumann, O. N., Elliot, S. J., and Fraanje,R. Centralized and decentralized control of structural vibra-tion and sound radiation, J. Acoust. Soc. Am., 119 (3), 1487–1495, (2006).

11 Fuller, C. R. Active control of sound radiation fromstructures: progress and future directions, Active 2002,Southampton, United Kingdom, (2002).

12 Fahy, F. and Gardonio, P. Sound and structural vibration:radiation, transmission and response, Oxford UniversityPress, Oxford, United Kingdom, (2007), 2nd Ed.

13 Azzouz, M. S., Mei, C., Bevan, J. S., and Ro, J. J. Finiteelement modeling of MFC/AFC actuators and performanceof MFC, J. Intel. Mat. Syst. Str., 12 (9), 601–612, (2001).

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Study of the Effect of the Linear Temperature Be-haviour on a Non-Homogeneous Trapezoidal Plateof Parabolically Varying ThicknessArun Kumar GuptaM. S. College, Saharanpur, U.P., India

Pragati SharmaH.C.T.M., Kaithal, Haryana, India

(Received 8 February 2012; revised 16 October 2012; accepted 6 February 2013)

An analysis is presented for studying the effect of the linear temperature behaviour on the transverse vibrationof a non-homogeneous trapezoidal plate of varying thickness on the basis of classical plate theory. The non-homogeneity of the plate material is assumed to arise due to the variation in density, which is assumed to varyparabolically. The thickness of the plate also varies parabolically. A two-term deflection function is performed tosolve the equation of motions using the Rayleigh-Ritz method. The frequency equation is derived when two edgesof the plate are simply supported and two are clamped, which is called clamped simply-supported clamped simply-supported. Effects of the non-homogeneity with other plate parameters—such as aspect ratio, taper constant, andthermal gradient on the first two modes of vibration—have been analysed. Results are presented in graphical form.

1

NOMENCLATURE

ω Angular frequencya Length of the plateb Width of the plate at left edgec Width of the plate at right edgex Longitudinal coordinatey Vertical coordinateα1 Non-homogeneity constanth Plate thicknessE Young’s modulusυ Poisson’s ratioτ Temperature distributionξ Non-dimensional coordinate = x

a

η Non-dimensional coordinate = yb

T Kinetic energyV Strain energyλ Frequency parameterz Transverse coordinatew Deflection functionD(ξ) Flexural rigidityβ Thermal gradientα Taper constant

1. INTRODUCTION

Plates of varying shape and thickness are found in many en-gineering structures and are used extensively in machine de-sign, nuclear reactor technology, naval structures, and acous-tical components. The practical importance of dynamic anal-ysis, in addition to the classical static analysis and the numer-ical evaluation of the vibrational characteristics of structural

elements such as plates, has become an important part of thedesign process.

Vibration phenomenon, common in mechanical devices andstructures, is undesirable in many cases, such as machine tools.But this phenomenon is not always unwanted; for example, vi-bration is needed in the operation of vibration screens. Overtime, engineers have become increasingly conscious of the im-portance of the elastic behaviour of plates as well as the naturalfrequencies and mode shapes of the plates, which, from a tech-nical point of view, is indispensable information. For reasonsof both practical and academic interests, numerous publica-tions concerned with the vibration of plates have been pub-lished.

Gupta and Sharma estimated the effect of thermal gradi-ent on transverse vibrations of an orthotropic trapezoidal plateof linearly varying thickness.1 Gurses et al. did the analysisof shear deformable laminated composite trapezoidal plates.2

Kang and Lee gave the free vibration analysis of an unsymmet-ric trapezoidal membrane.3 Khani and Aziz observed the ther-mal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and the heat transfer coeffi-cient.4 Leung et al. calculated the free vibration of lami-nated composite plates subjected to in-plane stresses using thetrapezoidal p-element.5 Liew and Lim discussed a global con-tinuum Ritz formulation for flexural vibration of pre-twistedtrapezoidal plates with one edge built in.6 Liew used a hybridenergy approach for the vibrational modelling of laminatedtrapezoidal plates with point supports.7 Liew also evaluatedthe vibration of symmetrically laminated cantilever trapezoidalcomposite plates.8 Maruyama et al. carried out an experimen-tal study of the free vibration of clamped trapezoidal plates.9

International Journal of Acoustics and Vibration, Vol. 18, No. 3, 2013 (pp. 117–123) 117

A. K. Gupta, et al.: STUDY OF THE EFFECT OF THE LINEAR TEMPERATURE BEHAVIOUR ON A NON-HOMOGENEOUS TRAPEZOIDAL. . .

McGee et al. studied the vibrations of cantilevered skewedtrapezoidal and triangular plates with corner stress singulari-ties.10 McGee and Butalia estimated the natural vibrations ofshear deformable cantilevered skewed trapezoidal and trian-gular thick plates.11 Qatu studied the vibrations of laminatedcomposite-free triangular and trapezoidal plates.12

Gupta and Singhal analysed the effect of non-homogeneityon thermally induced vibrations of an orthotropic visco-elasticrectangular plate of linearly varying thickness.13 Gupta etal. observed the thermal gradient effect on the vibration ofa non-homogeneous orthotropic rectangular plate having a bi-direction of linearly varying thickness.14 Gupta and Kumarstudied the thermal effect on vibration of an orthotropic rect-angular plate with parabolic thickness variations.15 Gupta et al.analysed the vibrations of a visco-elastic orthotropic parallelo-gram plate with linear thickness variation in both directions.16

Gupta et al. observed the non-homogeneity effect on the vi-bration of an orthotropic visco-elastic rectangular plate of lin-early varying thickness.17 Gupta et al. did the vibration anal-ysis of a non-homogeneous circular plate of nonlinear thick-ness variation by the differential quadrature method.18 Lal etal. calculated the transverse vibrations of non-homogeneousrectangular plates of uniform thickness using boundary char-acteristic orthogonal polynomials.19 Lal and Dhanpati consid-ered the transverse vibrations of non-homogeneous orthotropicrectangular plates of variable thickness.20 Gupta et al. anal-ysed the vibration of a visco-elastic rectangular plate wherethe thickness varies linearly in one direction and parabolicallyin the other.21 Gupta et al. discussed the vibration of a visco-elastic orthotropic parallelogram plate with parabolic thicknessvariations.22 Feng and Min investigated the vibrations of anaxially moving visco-elastic plate with parabolically varyingthickness.23 Gupta and Sharma analysed the forced axisym-metric response of an annular plate of parabolically varyingthickness.24

A recent survey of the literature shows that none of theauthors deal with the vibration study of a non-homogeneoustrapezoidal plate with parabolically-varying thickness with athermal effect. Therefore, a study is being presented to deter-mine the natural frequencies for the first two modes of vibra-tion on the basis of classical plate theory. The Rayleigh-Ritzmethod is used to calculate the natural frequencies with two-term deflection functions.

Frequency parameters for the first two modes of vibra-tion are calculated for a clamped simply-supported clampedsimply-supported (C-S-C-S) non-homogeneous trapezoidalplate with parabolically-varying thickness and linearly dis-tributed temperature. Both the modes of vibration are evalu-ated for various values of taper constant α, thermal gradientβ, aspect ratios c/b and a/b, and non-homogeneity constantα1. A comparison has been done between the results obtainedand a non-homogeneous trapezoidal plate whose thickness anddensity vary linearly.25

2. GEOMETRY OF THE PLATE

Consider a non-homogeneous thin symmetric trapezoidalplate. The geometry of the plate is as shown in Fig. 1.

Figure 1. Geometry of the trapezoidal plate with various thicknesses.

3. MATHEMATICAL FORMULATION

If the plate is subjected to a one dimensional linear temper-ature distribution along the x-axis, then τ can be expressed as

τ = τ0

(1

2− x

a

); (1)

where τ denotes the temperature excess above the referencetemperature at any point at distance x/a and τ0 denotes thetemperature excess above the reference temperature at the endx = −a/2.

When the non-dimensional variables ξ = xa and η = y

b areintroduced, Eq. (1) becomes

τ = τ0

(1

2− ξ

). (2)

The temperature dependence of the modulus of elasticity formost engineering materials as is given by Nowacki26 is

E = E0 (1− γτ) ; (3)

where E0 is the value of Young’s modulus along the referencetemperature, i.e., when τ = 0 and γ is the slope of the variationofE with τ . Placing Eq. (2) into Eq. (3), the modulus variationbecomes

E = E0

[1− β

(1

2− ξ

)]; (4)

where β = γτ0 (0 ≤ β ≤ 1). As stated earlier, plates of vary-ing thicknesses are often encountered in engineering applica-tions in comparison to the uniform thickness. Therefore, thethickness h of the plate is assumed to be varying parabolicallyin x direction, i.e.,

h(ξ) = h0

[1− (1− α)

(ξ +

1

2

)2]; (5)



where h0 is the maximum plate thickness that occurs at the leftedge and αh0 is the minimum plate thickness that occurs at theright edge.

The non-homogeneity of the plate material is assumed toarise due to the variation in density, which is assumed to varyparabolically. Therefore, the density of the plate material canbe taken as

ρ = ρ0

[1− (1− α1)

(ξ +

1

2

)2]; (6)

where ρ0 is the mass density at ξ = − 12 .

The expression for the maximum strain energy and themaximum kinetic energy for a trapezoidal plate with aparabolically-varying thickness is considered here as

V =ab

2

∫A

D(ξ)

{(1

a2∂2w

∂ξ2+

1

b2∂2w

∂η2

)2

−

2(1− υ)

[1

a2b2∂2w

∂ξ2∂2w

∂η2−(

1

ab

∂2w

∂ξ∂η

)2]}

dA; (7)

and

T =ab

2ω2

∫A

ρh(ξ)w2dA; (8)

where A is the area of the plate and ω is the angular frequencyof vibration.

Also, flexural rigidity D(ξ) is given by

D(ξ) = D0

[1− (1− α)

(ξ +

1

2

)2]3

; (9)

where

D0 =Eh30

12(1− υ2); (10)

is the flexural rigidity of the plate at the side ξ = − 12 . From

Eqs. (10) and (4), one gets

D0 =E0

[1− β

(12 − ξ

)]h30

12(1− υ2). (11)

Using Eq. (11) in Eq. (9) results in

D(ξ) =E0h

30

12(1− υ2)

[1−(1−α)

(ξ+

1

2

)2]3[1−β

(1

2−ξ)].

(12)After applying Eqs. (5), (6), and (12), Eqs. (8) and (7) comeout as

T =ab

2ω2h0ρ0

∫A

[1− (1− α)

(ξ +

1

2

)2]·

[1− (1− α1)

(ξ +

1

2

)2]w2dA; (13)

Figure 2. C-S-C-S boundary condition of a trapezoidal plate.

and

V =ab

2

E0h30

12(1− υ2)

∫A

[1− (1− α)

(ξ +

1

2

)2]3

·

[1− β

(1

2− ξ

)]{(1

a2∂2w

∂ξ2+

1

b2∂2w

∂η2

)2

−

2(1− υ)

[1

a2b2∂2w

∂ξ2∂2w

∂η2−(

1

ab

∂2w

∂ξ∂η

)2]}

dA.

(14)

4. METHOD OF SOLUTION

The Rayleigh-Ritz method is used to find the solution of thepresent problem. To apply this method, the maximum strainenergy must be equal to the maximum kinetic energy,

δ(V − T ) = 0. (15)

To satisfy the boundary condition of the plate, the two termdeflection functions are considered as

w = A1

{(ξ+

1

2

)(ξ− 1

2

)}2{η −

(b−c2

)ξ +

b+c

4

}·{

η +

(b−c2

)ξ − b+c

4

}+ A2

{(ξ+

1

2

)(ξ− 1

2

)}3

·{η −

(b−c2

)ξ +

b+c

4

}2{η +

(b−c2

)ξ − b+c

4

}2

;

(16)

where A1 and A2 are constants.

The boundary condition for the plate considered hereis clamped simply-supported clamped simply-supported, asshown in Fig. 2. Also for the plate considered here, the bound-



aries are defined by four straight lines:

η2 =c

4b− ξ

2+

1

4+cξ

2b;

η1 = − c

4b+ξ

2− 1

4− cξ

2b;

ξ1 = −1

2;

and ξ2 =1

2. (17)

Using Eq. (17) in Eqs. (14) and (13), strain energy and kineticenergy can be expressed as

V =ab

2

E0h30

12(1− υ2)

12∫

− 12

η2∫η1

[1− (1− α)

(ξ +

1

2

)2]3

·

[1− β

(1

2− ξ

)]{(1

a2∂2w

∂ξ2+

1

b2∂2w

∂η2

)2

−

2(1− υ)

[1

a2b2∂2w

∂ξ2∂2w

∂η2−(

1

ab

∂2w

∂ξ∂η

)2]}

dη dξ;

(18)

and

T =ab

2ω2h0ρ0

12∫

− 12

η2∫η1

[1− (1− α)

(ξ +

1

2

)2]·

[1− (1− α1)

(ξ +

1

2

)2]w2dη dξ. (19)

Now substituting the values of V and T from the Eqs. (18) and(19) in Eq. (15), we have

δ(V1 − λ2T1) = 0; (20)

where

λ2 =12ω2ρ0a

4(1− υ2)

E0h20; (21)

is a frequency parameter,

V1 =

12∫

− 12

η2∫η1

[1− (1− α)

(ξ +

1

2

)2]3

·

[1− β

(1

2− ξ

)]{(1

a2∂2w

∂ξ2+

1

b2∂2w

∂η2

)2

−

2(1− υ)

[1

a2b2∂2w

∂ξ2∂2w

∂η2−(

1

ab

∂2w

∂ξ∂η

)2]}

dη dξ;

(22)

and

T1 =

12∫

− 12

η2∫η1

[1− (1− α)

(ξ +

1

2

)2]·

[1− (1− α1)

(ξ +

1

2

)2]w2dη dξ. (23)

In Eq. (20), A1 and A2 are two unknowns to be evaluated inthe following manner:

∂(V1 − λ2T1)

∂Am= 0; m = 1, 2. (24)

On simplifying Eq. (24), the following form is obtained:

bm1A1 + bm2A2 = 0, m = 1, 2; (25)

where bm1, bm2 (m = 1, 2) involves the parametric constantsand the frequency parameter.

For a non-zero solution, the determinant of the co-efficientof Eq. (25) must vanish. In this way, one can get frequencyequation as ∣∣∣∣b11 b12

b21 b22

∣∣∣∣ = 0. (26)

5. RESULTS AND DISCUSSION

The frequency equation (26) provides values of thefrequency parameter for various values of plate parame-ters. In the present investigation, vibration behaviour ofa clamped simply-supported clamped simply-supported non-homogeneous plate with parabolic thickness variation is con-sidered for different values of taper constant, thermal gradi-ent, non-homogeneity constant, and aspect ratios c/b and a/b.The frequency equation (26), being quadratic in λ2, will givetwo roots that correspond to the first and second modes of vi-bration, respectively. Poisson’s ratio is taken as 0.33. Theseresults are presented in Figs. 3 through 10.

Figures 3 and 4 contain graphs of the frequency parameterλ with the taper constant α for the first and second mode, re-spectively, for

(a) aspect ratio a/b = 1.0,

(b) aspect ratio c/b = 0.5, 1.0,

(c) non-homogeneity constant α1 = 0.4,

(d) thermal gradient β = 0.0, 0.4, and

(e) taper constant α varying from 0.0 to 1.0.

It can be observed from the figures that as the taper constantincreases, the frequency parameter also increases for both themodes of vibration. It can be easily shown that as the thermalgradient β increases, the frequency parameter decreases. Inaddition, it can be seen from Figs. 3 and 4 that on increasingthe value of the aspect ratio c/b from 0.5 to 1.0, the frequencyparameter decreases.

Figures 5 and 6 show the variation of the frequency param-eter with thermal gradient β for the first and second mode, re-spectively, for

(a) aspect ratio a/b = 1.0,

(b) aspect ratio c/b = 0.5, 1.0,

(c) non-homogeneity constant α1 = 0.4,

(d) taper constant α = 0.0, 0.4, and



Figure 3. Variation of the frequency parameter λ with the taper constant α forthe first mode.

Figure 4. Variation of the frequency parameter λ with the taper constant α forthe second mode.

(e) thermal gradient β varying from 0.0 to 1.0.

It can be observed from the figures that as the thermal gra-dient increases, the frequency parameter decreases for both themodes of vibration. It can be easily shown that as the taperconstant increases, the frequency parameter increases. In ad-dition, it can be seen from Figs. 5 and 6 that on increasing thevalue of the aspect ratio c/b from 0.5 to 1.0, the frequencyparameter decreases.

Figures 7 and 8 contain graphs for different combinations ofthe taper constant and the thermal gradient as

(i) α = 0.0, β = 0.0,

(ii) α = 0.0, β = 0.4,

(iii) α = 0.4, β = 0.0, and

(iv) α = 0.4, β = 0.4.

Two values of the a/b aspect ratio (0.75, 1.0) and four valuesof the c/b aspect ratio (0.25, 0.50, 0.75, 1.0) have been usedwith the non-homogeneity constant α1 as 0.4.

It is evident from the figures that when the aspect ratio a/bis constant, i.e., 0.75 or 1.0, then an increase in the aspect ratioc/b will decrease the frequency for both the modes of vibra-tion. It is clear that when the taper constant α increases, thefrequency parameter also increases. The frequency parameterdecreases when the non-homogeneity constant α1 increases.

Figure 5. Variation of the frequency parameter λ with the thermal gradient βfor the first mode.

Figure 6. Variation of the frequency parameter λ with the thermal gradient βfor the second mode.

In Figs. 9 and 10, the variation of the frequency parameterwith different values of the non-homogeneity constant α1 anddifferent combinations of the thermal gradient and taper con-stant are shown. Two values of the aspect ratio c/b (0.5, 1.0)and one value of the aspect ratio a/b (1.0) have been consid-ered.

It is evident from the figures that when the non-homogeneityconstant α1 increases, the frequency parameter decreases.Also when α increases (keeping β constant), the frequency pa-rameter increases, and when β increases (keeping α constant),the frequency parameter decreases.

It is also clear from the figures that when the aspect ratio c/bincreases from 0.5 to 1.0, the frequency parameter decreases.

6. CONCLUSIONS

The effect of non-homogeneity, which is assumed to arisedue to the variation in density of the plate material on the fre-quencies of a trapezoidal plate of parabolically varying thick-ness, has been studied on the basis of classical plate theory.Mass density was also varied parabolically.

It is observed that the frequency parameter decreases withthe increasing of the value of the non-homogeneity constant;it also decreases with the increasing values of thermal gradi-ent, while all other plate parameters are kept fixed for all theboundary conditions. The behaviour of the frequency parame-ter is found to increase with the increasing values of the taper



Figure 7. Variation of the frequency parameter λ with the aspect ratio c/b forthe first mode.

Figure 8. Variation of the frequency parameter λ with the aspect ratio c/b forthe second mode.

constant and with the increasing values of the aspect ratio c/b,the frequency parameter decreases for both the modes of vi-bration.

A comparison has been done with the non-homogeneousplate having linearly-varying thickness and density.25 Onecan see that the plate with linearly-varying thickness has alesser value of frequency parameter rather than a parabolically-varying thickness. Therefore, a non-homogeneous plate witha linear variation in thickness shows more stability. Thus, achange in the natural frequencies of a plate can be achievedby the proper choice of various plate parameters consideredhere, which will be of great practical importance to design en-gineers.

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of orthotropic trapezoidal plate of linearly varying thick-ness, Journal of Vibration and Control, 17, 1591–1598,(2011).

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Figure 9. Variation of the frequency parameter λ with the non-homogeneityconstant α1 for the first mode.

Figure 10. Variation of the frequency parameter λ with the non-homogeneityconstant α1 for the second mode.

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Formulation of Weighted Goal Programming Usingthe Data Analysis Approach for Optimising VehicleAcoustics LevelsZulkifli Mohd NopiahDepartment of Mechanical and Materials Engineering, Universiti Kebangsaan Malaysia, 43000, Bangi, Selangor,Malaysia

Ahmad Kadri JunohInstitute of Engineering Mathematics, Universiti Malaysia Perlis, 026 00, Arau, Perlis, Malaysia

Ahmad Kamal AriffinDepartment of Mechanical and Materials Engineering, Universiti Kebangsaan Malaysia, 43000, Bangi, Selangor,Malaysia

(Received 25 March 2012; revised 13 August 2012; accepted 14 November 2012)

Two factors for customers to consider when purchasing a vehicle are the vehicle acoustical comfort and the vi-bration in the vehicle cabin, both of which contribute to a comfortable driving environment. The amount ofdiscomfort is included by frequency, magnitude, direction, and the period during which the noise is experiencedand also where the vibration is experienced in the cabin. The main sources of vibration that have been identifiedpreviously are the vibration transmitted from engine and the vibration transmitted from the interaction of the tireswith the road surface. In this study, we investigate the effect of the vibration caused by the tire interaction withthe road surface by estimating the amount of noise produced due to this phenomenon. The methodology focuseson the trends which occur in the noise exposure and on the vibration exposure that has been generated throughoutthe engine operating rpm range in both stationary and non-stationary conditions. The vibration dose value (VDV)fwas used to assess the amount of vibration exposure that is transmitted to the driver’s body in the cabin. Throughthe study, we have proved that the vibration caused by the tire-road surface contact is a major contributor to thecabin’s interior noise. Based on the results, a goal programming method was developed to optimise the noise lev-elin the cabin by considering the vibration as an input in the model. Finally, a multi-objective goal programmingwas developed successfully which could be used to optimize the noise level in the cabin by looking at the value ofthe VDV required at particular engine speeds (rpm).

1. INTRODUCTION

Vibration in vehicles produces noise that affects the emo-tions and focus of the drivers. The exposure to the vibrationwill affect visual performance by producing blurred visual im-ages, since the vibration can cause relative movement betweenthe retina and the objects viewed. This scenario can affect driv-ing comfort in that it can reduce the focus of a driver and ithas the potential to cause road accidents. Based on previousresearch which examined the vehicle cabin acoustical environ-ment, the main sources of noise that have been identified maybe categorized into a few main sources, namely the transmis-sion of the engine, the exhaust exit, and the noise from thetires that is produced by the tire-road contact. However, in thisstudy, we focused on the main source of the vibration causedby tire interaction with the road surface.1, 2 Based on previousstudies concerning vehicles moving at a constant rate of speed,the dominant source of noise was found to be the vibrationcaused by tire-road contact.

In order to evaluate the noise level in the cabin, various stan-

dards and methodologies were considered and used, to predictthe changes in sound quality while the vehicle was operated inboth stationary and non-stationary conditions.3, 4 The findingsfrom the results can be used particularly by vehicle manufac-turers to modify and improve the construction of vehicle to re-duce the vibration exposure along with the reduction of noisein the cabin.5, 6

Various studies have been conducted by researchers to iden-tify and estimate the generation of noise and vibration in thevehicle cabin in both stationary and non-stationary conditions.Generally, the transmission of noise into the vehicle is causedby two main sources: (1) airborne noise and (2) structure bornenoise. One of the sources that contribute to the vehicle’s inte-rior noise is the vibration caused by the tire-road contact. Withregard to this, some studies have been carried out to investigatethe effect of the tire tread rubber on the noise generated. As aresult, various characteristics of the tread rubber will producedifferent levels of noise that are caused by different levels ofvibration.

Following the research carried out previously, it has been


Z. M. Nopiah, et al.: FORMULATION OF WEIGHTED GOAL PROGRAMMING USING THE DATA ANALYSIS APPROACH FOR OPTIMISING. . .

shown that there are two main sources of vehicle interior vi-bration: (1) the transmission of the engine vibration which iscaused during the acceleration and deceleration of the vehi-cle,7 and (2) the vibration caused by the tire interaction withthe road surface. Generally, both the noise and the vibrationin the vehicle cabin directly correspond to the changes in theengine speed, since the changes are in direct proportion to theengine rpm.8, 9 The types of vibrations that are normally pro-duced by the engine surface are the vibrations brought aboutby the combustion process and also the vibrations caused bythe mechanical movements.

Normally, the vibration is not only caused by the tires, butalso by the vibration of the structure, where the vibration istransmitted to the vehicle body and other parts. While the ve-hicle is in motion, the main source of vibration is that pro-duced by the contact between the tires and the road. As theengine speed increases, the vibration velocities become higheras well, and more vibration is generated. Besides depending onvelocity, the vibration also depends on the type of road surface,as rougher road surfaces increase vibration levels.

1.1. The Effects of Noise in the VehicleCabin

Noise is caused by a vibrating or moving object, where thevibration is produced at the original location and transferredto the listener’s ear. This phenomenon will cause a listenerto hear the event that occurs along with experiencing the ac-companying vibration. The pitch of the sound depends on howrapidly the object is vibrating. The noise level depends on themagnitude of the force used to vibrate the object.

The effects of noise on health have much to do with the highlevel of sound. In general, noise problems can lead to hearingimpairment,10, 11 hypertension, sleep disturbance, and listenerannoyance. Thus, these effects are stressful for listeners andcan also cause road accidents. Many interpretations have beenmade about the annoyance and the correlation between noiselevels and health effects, where annoyance can influence thequality of the driving as well as increase the stress level of thelisteners. While driving, the major effect of noise is to make adriver’s or passenger’s speech more difficult to hear. Generally,the human brain compensates for a noisy background so thatspeech becomes louder with clearer, more audible syllables.

Noise and vibration levels can be measured by using dedi-cated equipment. However, harshness involves the evaluationof noise and vibration by subjective impressions that is usingpsychoacoustics.12 In the automotive world, we are concernedwith sound and vibration in order to produce vehicles withbetter acoustics environments especially in the cabin.13 Thus,sound and vibration are critical elements which are worthy ofconsideration, since current vehicles and transportation systemare strongly influenced by the properties of the sound and vi-bration.14 Hence, by knowing the sources of sound, there is apossibility for researchers to manipulate these and reduce thenumber of unpleasant sounds that are generated in the cabin.

Table 1. The states of noise annoyance.

VACI scale Annoyance state1 Most annoying2 Medium annoying3 Marginal4 Medium pleasant5 Most pleasant

1.2. Evaluation of NoiseGenerally, it is important for automotive engineers to de-

termine and classify noise levels by providing a noise levelbenchmark in order to improve the current noise level.15 Todecrease the noise annoyance level, it is essential for automo-tive researchers to be able to assess the level of acoustics nu-merically and quantitatively. Since drivers and passengers canonly evaluate and categorize the level of noise subjectively,16

the evaluation is mainly based on sensations and perceptions.However, for engineering purposes, the assessment related tothe noise level must be determined and measured in order toallow researchers to obtain the amount of noise in a numeri-cal form. As a solution, researchers have formed an acousticalindex by taking into account the sound quality.

A methodology has been introduced to develop andform an acoustical index in order to assess the annoyanceof the noise generated by referring to the parameters ofsound quality of the noise measured in the vehicle cabin.17

The details of the sound quality parameters are as follows.Loudness: the main metric is designed referred

to represent the human perception to-wards the strength of the noise

Sharpness: high-frequency content in the fre-quency spectrum of the signal

Roughness: low-frequency modulation —approximately 70 Hz

Fluctuation strength: low-frequency modulation —approximately 4 Hz (the amplitudemodulation in the time sample)

The Vehicle Acoustical Comfort Index (VACI) is introducedto assess the quality of the acoustics in the vehicle cabin byclassifying the level of noise following the degree of annoy-ance experienced by the listener. The states of noise annoy-ance are shown in Table 1. The VACI may be expressed as alinear combination formed by the metrics of sound quality anda constant coefficient produced from the multiple regressionanalysis.18 The equation which represents the VACI is givenby Eq. (1):

VACI =∑

cj ·Qij + k; (1)

where VACI = vehicle acoustical comfort index, cj = the co-efficient for the jth sound quality metric, Qij = the jth soundquality metric for the ith condition, and k = the constant coef-ficient.

To obtain the vehicle comfort coefficients, Eq. (2) is solvedto produce a new equation to represent the level of annoyanceof the noise experienced in the VACI form.∑

cj ·Qij = Pi (i = 1, n j = 1, 4); (2)

where cj = the coefficient for the jth sound quality metric,Qij = the jth sound quality metric for the ith condition, and



Pi = the vehicle acoustical comfort factor for the ith condi-tion.

For the equations above, the j index is restricted from 1to 4 because in the formation of the new equation for VACI,we only consider four types of sound quality in our calcula-tions. The details of these parameters will be discussed in thenext section. Meanwhile, the vehicle comfort factor, P , is thevalue obtained from the average value derived from the resultsof subjective evaluations, which are performed based on thescales determined in Table 1.

1.3. Evaluation of VibrationSince vibration is a major contributor to noise, this study

has investigated the correlation between vibration and noise inorder to determine the trends of changes involved in factorsthat influence driving comfort. The measurement of vibrationmust be carried out following a central coordinated system atthe interface of the human body.19, 20

The evaluation method that has normally been used to as-sess whole body vibration is based on British Standard 6841(BS 6841) and International Standard 2631 (ISO 2631). In thisstudy, we have chosen to use the BS 6841 where the standarduses the root mean square (r.m.s.) acceleration to evaluate themagnitude of vibration. In our case, the measurement of the vi-bration is performed with the assumption that the accelerationis in the vertical direction.

In this study, we have considered the vibration dose value(VDV) as most suitable to assess the vibration level sincevehicle motion is influenced by shocks and impulsive veloc-ity changes. The VDV assessment method uses the time-integrated fourth power of the acceleration that is measuredin the cabin. The VDV results show the total amount of vibra-tion experienced by humans within a period of time. In otherwords, VDV (ms−1.75) can be considered as a measure of thetotal exposure to vibration, which takes into account the mag-nitude, frequency, and the period of exposure. The definitionof the VDV is given in Eq. (3) as follows:

VDV =

Ç∫ T

0

a(t)4dt

å 14

; (3)

where VDV = vibration dose value (ms−1.75), a(t) =frequency-weighted acceleration, and T = total period in sec-onds during which the vibration occurs.

1.4. Goal ProgrammingAs discussed previously, noise generation corresponds to the

vibration in the car cabin. Thus, we have focused on lineartrends because we believe that the noise and vibration directlycorrespond to the changes in the engine rpm. The tool thatis normally used to solve the optimization problem that in-volves linear correlation is linear programming (LP). The LPis a mathematical modelling method that has been used to ob-tain the best outcome where it involves problems which in-clude many constraints. Generally, to formulate the LP model,three important elements must be obtained in order for the op-timization model to be developed successfully: (1) decision

variables, (2) objectives, and (3) constraints. All of these ele-ments are in the linear inequality form and they are combinedin the mathematical model to solve the optimization problem.Even though the LP model is widely used in optimization prob-lems, it is only applicable when the problems involve a linearrelationship, and the objective function and other constraintsmust be strictly satisfied. In certain cases, researchers some-times have to compromise and provide some flexibility in theconstraints to obtain reliable and acceptable results: thus, goalprogramming (GP) is used as an alternative method instead ofthe LP model.

Generally, the GP is quite similar to the LP. The differencebetween them is that the GP is normally used for optimizationthat involves multi-objective problems in the model, while theLP is limited to one objective only. In the GP, while the tar-get is satisfied, it depends on the fact that the goals that needtobe satisfied are already fulfilled, and it shows that the require-ments of the decision-maker have been fulfilled. Generally,to measure the achievement of the objective in the GP model,the aspiration level is used. In the GP, the model attains themost optimal solution after each goal of the constraints hasbeen reached as closely as possible to the aspiration level. Inthe GP, there are two methods that are normally used for op-timization purposes: the weight method and the pre-emptivemethod. The weight method is a GP that solves a single objec-tive model consisting of the weighted sum of the goals, and thepre-emptive method optimizes the goals in the model one at atime, based on the priority from highest to lowest. However,in this study, our focus will only be on the weight method.

2. METHODOLOGY

The Proton Perdana V6 Sedan automatic car was used asthe test vehicle. Sound quality measurements were performedon the test vehicle cabin in two conditions: stationary (idlewithout excitation from road roughness) and n on-stationary(moving). Tests were carried out in two stages: objective andsubjective.

2.1. The Objective TestIn these objective tests, portable Bruel & Kjaer and multi-

channel PULSE 3560 devices were used to record the noise.Here, the Head & Torso (HAT) was used to measure the noiseand was located on the left side of the driver’s seat. Since theHAT has both right and left channels, only the recorded noisefrom the right channel was analyzed as the input data. Thesound quality software B&K type 7698 was used to analyzethe measured noise in order to obtain the required parameters.In order to evaluate the acoustics in the VACI, the noise fromthe right channel (parameters of sound quality) was substitutedin the equation that has been proposed previously.17 The noiseand vibration measurements were conducted as function of theengine speed (rpm) of the car. This is because we assume thatbesides transmission from the engine is the main contributorof the noise and vibration, another primary contributor is thevibration caused by the tire interaction with the road surface.Thus, if we perform the measurement based on the car veloc-



Table 2. Locations and characteristics of the tested roads.

Road type Location Characteristichighway Bangi (Kajang Highway) smooth road surfacepavement Putrajaya (urban road) pavement type

Table 3. Tested engine speeds.

Stationary Highway Pavement1500 rpm 1600 rpm 1200 rpm2000 rpm 1900 rpm 1500 rpm2500 rpm 2500 rpm 1600 rpm3000 rpm 2800 rpm 1800 rpm3500 rpm — 1900 rpm

ities, it can only be done while the car is moving. However,we need to do the measurement for both the stationary andnon-stationary conditions. In this case, we have relied on twoassumptions:

i. By making a comparison of the noises at the same en-gine speed in both the stationary and non-stationary con-ditions, the stationary engine noise when the car is station-ary (parked) is almost similar to the noise of the engine ofa moving car without the wind noise.

ii. No noise caused by tire interaction with the road surface isproduced while the car is parked in a stationary condition.

The sensor employed as the vibration detector was a B&Kisotron accelerometer 751-100, installed on the front floor atthe HAT side (next to the driver). The sensitivity of the isotronmodel was recorded to be 10mV/ms−2 and the dynamic fre-quency response was noted to be up to 5.5 kHz for the x-axisand 3.0 kHz for the y- and z-axes. The software used to mea-sure the noise was a B&K Pulse Labshop. To evaluate the levelof the vibration on the car’s floor, Eq. (3) was used to obtain thevibration dose value. Previous research has shown that varia-tions in road surface roughness will influence the generation ofvibration and noise, so for this study, two types of roads weretested: highway and paved road. The locations and character-istics of the roads are presented in Table 2 and Fig. (1). Thehighway has a flat and smooth surface reputed to have minimaloccasional unevenness, which results in minimal disturbances.Meanwhile, the paved road is originally a cobbled street struc-tured by smooth stones similar to one another, with an averagethickness of 5 mm.

The experiments were performed by two members. Onemember took the role of driver by maintaining specific enginespeeds (rpm) and the member worked with the notebook com-puter to record the noise and vibration levels. The period ofeach measurement was 10 seconds and it was repeated fourtimes to ensure the data were reliability. The engine speedsfollow the condition and type of road as shown in Table 3.

2.2. Subjective TestIn the subjective test, the acoustics in the car cabin was eval-

uated by ten people. They were given the subjective assess-ment based on the scales that had been determined previouslyin Table 1. The vehicle acoustical comfort factor provided theoutputs to be used as inputs to materialise the development ofa new index of the VACI. The details of the procedures areshown in Fig. (2).

(a)

(b)

Figure 1. Highway road surface (a) and pavement road surface (b).

3. DATA ANALYSIS

Table 4 provides the results of the measurement of soundquality, whereas Table 5 gives the results of measurement ofthe exposed vibration in the VDV. From the results, we havefound that certain sound quality parameters correspond withthe changes of the rpm of the engine. Obviously, we havefound that the parameters of loudness increase, and the param-eter of sharpness metrics decreases with the increase in the en-gine rpm. However, no particular trend has been identified con-cerning the changes of the roughness and fluctuation strengthvalue for different engine speeds. From Table 4, we see thathigher engine speeds will produce higher velocities in the carand will automatically increase the amount of the vibration atthe car floor, especially while the car is moving. From Table 5,we see that the vibration in the VDV directly corresponds withthe increase in the engine rpm.

To determine the significance for each of the sound qual-ity parameters that corresponds with the changes in the enginespeed, we must satisfy the following two criteria:

i. TheR-square value—which is the regression indicator be-tween each parameter of sound quality—and the rpm mustbe higher than 50%,

ii. The P -value has to be smaller than 0.05.

By carrying out regression analysis, the trend with the enginerpm for each sound quality parameter can be observed. Later,we will only choose the parameter that has satisfied the abovecriteria for the computation to be made in the next stage. As



Figure 2. Optimisation process flow.

Table 4. The average VDV versus the average of sound quality parameters.

ConditionVDV

AverageLoudness Sharpness Roughness Fluct. str.

(ms−1.75) (Sone) (Acum) (Asper) (Vacil)

Stationary

0.0141 4.8 1.362 1.44 1.220.0249 10.4 1.173 1.46 1.210.0334 11.8 1.061 1.48 1.070.0223 13.4 0.964 1.50 0.940.0270 19.3 0.762 1.57 0.91

Highway

0.3657 21.1 0.861 1.68 1.660.4115 24.8 0.781 1.72 1.450.4490 29.8 0.674 1.88 1.500.4367 30.8 0.867 1.92 1.97

Pavement

0.6906 22.6 0.981 2.01 1.310.8377 29.8 0.973 2.21 1.370.9038 33.0 0.969 2.34 1.450.9152 34.5 0.948 2.42 1.301.1661 43.6 0.937 2.56 1.42

a result, we have found that only the loudness parameter di-rectly corresponds with the engine rpm in both the stationaryand non-stationary conditions (as shown in Table 6).

From the equation produced in the regression analysis, thegeneration of the sound quality loudness parameter as a func-tion of the engine speed can be expressed as seen in Eq. (4).Table 7 gives the amount of the noise that is assumed to havebeen generated by the vibration due to the tire-road contact.From the table, we clustered the noise into two sources (1)noise produced only by the transmission of the engine and (2)the noise that is influenced by both of the sources—the trans-mission of the engine and also the vibration that is caused bythe interaction between the tires and the road surface. Observ-ing the trends, we can make a prediction about the noise that isgenerated only by the tires through the proposed Eq. (5) below.

SQ = civ + α; (4)

where SQ = the value of the parameter of the sound quality

Table 5. The average VDV (ms−1.75) versus engine speed (rpm).

ConditionEngine Read Average

speed (rpm) 1 2 3 4 (ms−1.75)

Stationary

1600 0.0164 0.0162 0.0180 0.0165 0.01412000 0.0295 0.0241 0.0220 0.0201 0.02492500 0.0105 0.0184 0.0125 0.0174 0.03343000 0.0245 0.0259 0.0243 0.0326 0.02233500 0.0278 0.0280 0.0334 0.0238 0.0270

Highway

1600 0.3627 0.3435 0.3273 0.3183 0.36571900 0.3573 0.3700 0.3628 0.3484 0.41152500 0.5727 0.5510 0.5601 0.6221 0.44902800 0.7509 0.6230 0.6431 0.6612 0.4367

Pavement

1200 0.6228 0.6588 0.7076 0.6141 0.69061500 0.8283 0.8692 0.8853 0.9255 0.83771600 0.7329 0.7886 0.8002 0.7177 0.90381800 1.0243 1.0732 1.3573 0.9960 0.91521900 1.3631 1.2668 1.0392 1.4986 1.1661

Table 6. R-square, multiple R- and P -values from the regression analysis.

Condition/Metrics Regression (R2) MultipleR P -value

road type

Stationary

Loudness 0.897 0.947 0.00678Sharpness 0.803 0.896 5.5E-14Roughness 0.178 0.422 1.6E-12

Fluctuation strength 0.423 0.179 4.4E-06

Non-stationary(highway)

Loudness 0.846 0.920 0.00117Sharpness 0.011 0.106 0.00013Roughness 0.647 0.418 4.8E-07

Fluctuation strength 0.018 0.136 0.25484

Non-stationary(pavement)

Loudness 0.854 0.924 5.7E-05Sharpness 0.032 0.178 3.6E-07Roughness 0.755 0.869 3.3E-06

Fluctuation strength 0.002 0.046 0.093

Table 7. The amount of noise due to tire-road contact for both road types.

Engine speeds Stationary Highway Pavement(rpm) L L θL L θL

1200 3.62 18.39 14.77 22.03 18.411300 4.26 19.21 14.95 24.68 20.421400 4.90 20.03 15.13 27.33 22.431500 5.55 20.86 15.31 29.98 24.441600 6.19 21.68 15.49 32.63 26.441700 6.83 22.50 15.67 35.28 28.451800 7.47 23.32 15.85 37.93 30.461900 8.11 24.14 16.03 40.58 32.472000 8.75 24.96 16.21 43.23 34.482100 9.39 25.78 16.39 45.88 36.492200 10.03 26.60 16.57 48.53 38.502300 10.67 27.42 16.75 51.18 40.512400 11.31 28.24 16.93 53.83 42.522500 11.96 29.07 17.11 56.48 44.532600 12.60 29.89 17.29 59.13 46.532700 13.24 30.71 17.47 61.78 48.542800 13.88 31.53 17.65 64.43 50.552900 14.52 32.35 17.83 67.08 52.563000 15.16 33.17 18.01 69.73 54.57

metric, ci = the coefficient for each sound quality metric, v =

the engine speed in rpm, and α = the constant variable.

θSQ = |SQi − SQ0| ; (5)

where θSQ = the amount of noise caused by tires, SQi = thevalue of the parameter of the sound quality metric in the non-stationary condition (i = highway, pavement), and SQ0 = thevalue of the parameter of the sound quality metric in the sta-tionary condition.

Based on previous research, to evaluate the acoustical envi-ronment, the VACI index was used to evaluate the annoyanceof the noise experienced in the cabin. However, previous re-search on noise annoyance levels in moving vehicles only pro-vides a formula to predict the noise annoyance level for a fewtypes of roads. Thus, in this study we develop a formula to pre-dict the acoustical by considering the noise annoyance when



(a)

(b)

Figure 3. The trends in the VACF values for sound quality parameters: loud-ness (a) and sharpness (b).

the vehicle is in a stationary condition. By taking into accountthe factors of significance for the parameter of sound quality,we are able to create a new formula to evaluate the noise an-noyance through the multiple regression analysis. The formuladeveloped is shown in Eq. (6) below. In this case, the param-eters of roughness and fluctuation strength were ignored sincetheir significance was small, in reference to Table 6. However,based on the plots in Fig. (3), during the regression analysis,the values of the vehicle acoustical comfort factor was foundto correspond in direct proportion to the parameters of loud-ness and sharpness.

VACI = −0.245L+ 0.18S + 5.07; (6)

where L = the input of the loudness parameter, and S = theinput of the sharpness parameter.

Based on Table 7, we adopted the t-test and analysis of vari-ance (ANOVA), in order to compare the generation of noisebased on the roughness of the different roads (highway andpaved road) and also to identify and to determine the differencebetween the two conditions (stationary and non-stationary)(Fig. (4) and Fig. (5)).

4. MATHEMATICAL MODELLING

In our previous experiments and data analysis, we used twotypes of road—highway and pavement—but in our optimiza-tion problem we had to choose either the highway or pavementdata as the inputs for our model, since the concepts of mod-elling that were used are similar to one another. FollowingTable 8, the vibration level increases directly with increases inthe rpm of the engine (Fig. (6a)). Thus, to recognize the pat-tern of the trends, Fig. (6b) was plotted based on Table 8 inorder to obtain the correlation between the values of the VACI

(a)

(b)

Figure 4. Comparison between the noise generated for the stationary-highway(a) and highway-pavement (b).

Table 8. Correlation between vibration and noise annoyance levels.

Engine speeds Vibration dose value VACI(rpm) (VDV) [ms−1.75] Stationary Highway1200 0.238 — —1500 0.287 5 —1600 0.336 — 4.141800 0.380 — —1900 0.434 — 3.362000 — 4 —2500 0.532 3 2.302800 — — 1.213000 — 2 —3500 — 1 —

— No test carried out

with the changes in the rpm. So in order to form an equationto represent the above correlation, we used a regression anal-ysis approach by considering the criteria mentioned in Table6 for the interpretation of data. Meanwhile, in order to obtainthe correlation between the noise experienced and the vibra-tion, we plotted Fig. (6c) by mapping the value of the VACIwith the value of the VDV from Table 8. The trend that wasobserved in Fig. (6) can be expressed in Eq. (7) and (9). Us-ing regression analysis, we obtained Eq. (8) to represent thecorrelation between both of the factors. These three equationsshow the linearity trends for both stationary and non-stationary(highway) conditions and they are used as input constraints inthe optimisation model.

Highway

VACI = −2.4× 10−3v + 7.9 (7)

VACI + 8.83V DV = 7.2 (8)

Stationary

VACI = −2× 10−3v + 8 (9)

where v = engine speed in rpm, and VDV is given in ms−1.75.



(a)

(b)

(c)

(d)

Figure 5. The ANOVA results using the normal probability plot (a), versusfits (b), histogram (c) and box plot (d).

By using the equations formed previously, goal programmewas developed to optimize the vibration level in order to pro-vide the required acoustical quality in the car cabin. To de-velop the model, the range for the VACI was specified to befrom 2.5 to 5 due to the fact that the level of annoyance thatis required in the cabin is from scale 3 (marginal state) up toscale 5 (the most pleasant state). Meanwhile, when the vehi-cle is in the non-stationary condition, normally the maximumrpm that could be achieved is close to 3000 rpm. The model ofoptimisation that was developed was based on weighted goalprogramming in which the model considers the function’s lin-earity archived and it varies the preference of weights to allowfor trade-offs between goals in the model. The model devel-

(a)

(b)

(c)

Figure 6. The trends of vibration and acoustical comfort a) VDV-rpm, b)VACI-rpm and c) VACI-VDV.

oped is expressed as follows:

Min a =

Q∑q=1

Åuqnqkq

+vqpqkq

ã;

subject to

fq(x) + nq − pq = bq q = 1, ... , Q;

x ∈ F ;

nq, pq > 0 q = 1, ... , Q;

where uq = preferential weight-minimisation of nQ, vq =

preferential weight-minimisation of pq , and kq = normalisa-tion contact of the q-th goal.

Based on the previous section, the GP is considered to be themost important model that is capable of solving the multipledecision criteria. Therefore, the GP is widely used as a multi-objective technique that is able to solve optimisation problemsin various fields, including engineering fields. In this case, ourobjective in the GP model is to obtain the optimal acousticslevel (VACI) by fulfilling the constraints of the GP that are ex-pressed below. Two goals are identified and are to be used in



the model: the acoustics level required—which is noise an-noyance following the value of the VACI—must be within therange of 2.5 to 5, and the engine rpm must be between 1200and 3000.

Through the model, a few goals have been identified andachieved.The model has six goals where it aims to have maxi-mum acoustical comfort in line with the maximum engine rpm.Based on previous trends through the data analysis, the max-imum engine speeds will generate the maximum level of vi-bration in the cabin. Thus, we have come to understand thatfor a certain rpm it will potentially generate different valuesof vibration and noise. In order to define the weights wi ofthese six goals in our model, they were developed based ontheir level of priority and importance in the equations or con-straints. The goals to be satisfied are listed as follows. Theselection of weight:

1) The values of the VACI correspond to the rpm of the engine– w1.

2) The values of the VACI correspond to the value of VDV –w2.

3) The values of the VACI must be greater than 2.5 (better thanthe marginal state) – w3.

4) The rpm must at least achieve a particular level of enginespeed, k [rpm] – w4.

5) The values of the VACI must be within the range and cannotbe more than 5 – w5.

6) The rpm must be less than 3000 rpm (since for normaldriving conditions, the range of the engine speed normallyachieved is between 1200 and 3000 rpm) – w6

(wi = the weight for particular ith equation or constraint).

In order to develop a reliable model, the importance of theweights must be set following the importance of each one thatis decided by the researcher. In our case, based on the abovegoals, we had set the weight for w1 to be of similar impor-tance with other weights of w2 and w5. The other scheme ofweight selection can be expressed as below. In general, if thetotal value of the combined weights is equal to 1 or is equiv-alent to 100%. The weight selection serves as the main rulein order to influence the required results from the model. Theequations formed and the constraints developed from the LPare converted to the GP following the stages below:

w1 = w2 = w5 = 0.3; w1, w2, w5 ≥ w3;

w3 = 0.05; w3 ≥ w4, w6;

w4, w6 = 0.025.

1) Deviational variables are introduced to minimize the goalunderachievement and overachievement.

2) Cost weight penalties for underachievement and over-achievement are introduced to represent the priority of themanagement.

3) Four additional sets of cell-goal equations are used toachieve the goal.

4) The goal equation cell captures the target’s derivation ofthe LP model and defines the target clearly after being con-verted with the deviational variable.

5) The set of cells will form the input cells for any target valuefor the goals.

6) The weighted penalty cost is derived by multiplying andsumming the combination between the value of the devia-tional variables and the penalty cost in a single cell whichrepresents the target cell.

7) The target cell-weighted penalty cost cell is minimized.

8) Minimizing the combination value of the weighted penaltycost cell to achieve the solution.

Non-negative deviational variables η−i and η+i (i =

1, 2, 3, . . . , 6) are used to represent the deviations of the con-straints.

Minimise

0.3(g1)

7.9=

0.3(η−1 + η+1 )

7.9

0.3(g2)

7.2=

0.3(η+2 + η+2 )

7.2

(Satisfying the trends of noise annoyance level);

Minimise

0.05(g3)

2.5=

0.05(η−3 )

2.5

0.025(g4)

5=

0.025(η+4 )

5

(Satisfying the goal of noise annoyance level);

Minimise

0.3(g5)

k=

0.3(η−5 )

k

0.025(g6)

3000=

0.025(η+6 )

3000

(Satisfying the goal of engine speeds).The combination of deviation variables in the objective

function can be expressed as

Minimise

z =

0.3η−17.9

+0.3η+17.9

+0.3η−27.2

+0.3η+27.2

+

+0.15η−32.5

+0.05η+4

5+

0.15η−5k

+0.05η+63000

.

This is subject to the trends of noise

VACI + 2.4× 10−3v + η−1 + η+1 = 7.9

andVACI + 8.33 VDV + η−2 + η+2 = 7.2;

the goal of noise

VACI + η−3 + η+3 = 2.5



Table 9. Goal programming results.

Engine Objective Vibration Vehicle Acoustical

η+3 η−3 η−4 η−6speeds function Dose Value Comfort Indexv z (VDV) (VACI)

(rpm) [ms−1.75] Index Scale1200 0 0.2510 4.984 5 2.484 0 0.016 18001300 0 0.2785 4.741 5 2.241 0 0.259 17001400 0 0.3060 4.498 5 1.998 0 0.502 16001500 0 0.3335 4.255 4 1.755 0 0.745 15001600 0 0.3610 4.012 4 1.512 0 0.988 14001700 0 0.3885 3.769 4 1.269 0 1.231 13001800 0 0.4161 3.526 4 1.026 0 1.474 12001900 0 0.4436 3.283 3 0.783 0 1.717 11002000 0 0.4711 3.040 3 0.540 0 1.960 10002100 0 0.4986 2.797 3 0.297 0 2.203 9002200 0 0.5262 2.554 3 0.054 0 2.446 800

2300 0.0378 0.5537 2.311 2 0 0.189 2.689 7002400 0.0864 0.5812 2.068 2 0 0.432 2.932 6002500 0.0135 0.6087 1.825 2 0 0.675 3.175 5002600 0.0184 0.6362 1.582 2 0 0.918 3.418 4002700 0.0232 0.6638 1.339 1 0 1.161 3.661 3002800 0.0280 0.6912 1.096 1 0 1.404 3.904 2002900 0.0329 0.7188 0.853 1 0 1.647 4.147 1003000 0.0378 0.7463 0.610 1 0 1.890 4.390 0

η−1 , η+1 , η

−2 , η

+2 , η

+4 , η

−5 , η

+5 , η

−6 = 0

andVACI + η−4 + η+4 = 5;

the goal of the engine speed

v + η−5 + η+5 = k

andv + η−6 + η+6 = 3000;

the additional constraint

VDV− 0.184 ≥ 0

andthe no-negativity constraint

VACI, VDV, v ≥ 0

andη−i , η

+i ≥ 0 (i = 1, 2, 3, . . . , 6).

5. RESULTS

Table 9 gives the results of the goal programme developed.The objective function value is not overly important, but in ourcase, the deviation variable values play a major role in find-ing whether or not the goals in the model have been satisfied.From the results, the optimal solutions do not produce zero forthe objective function value since certain goals of the VACI orengine speed are not satisfied in the model. However, in orderto achieve certain levels of noise annoyance, we have success-fully obtained the required value of the vibration dose valuein the results. The results also show that the changes of therpm engine affect the level of the vibration and which producesmore noise in the cabin.

6. CONCLUSIONS

The first objective of the study was to find the amount ofnoise caused by the tire interaction with the road surface. Inorder to obtain some amount of noise, we carried out an ex-periment and performed the data analysis from the first to the

last stage, based on the procedures explained in the previoussection. As seen in the data analysis, the roughness of the roadserves as a major contributor to the vibration which exist in thecabin. Since different road surfaces affect the noise level that isgenerated, it was very important to us to consider and take noteof the specifications and patterns of the tires surfaces. Hence,the equations proposed through this study cannot be applieddirectly by other researchers since different models of vehicle,different types of tires, and different types of roads contributeto the different noise levels in the cabin.

In this study, we have also successfully clustered the amountof noise into two main sources: (1) that caused by the trans-mission of the engine noise and (2) noise caused by the tire-road contact. The results also prove that the main contributorof the noise in the cabin is the noise produced from the in-teraction between the tires and the road surface. Through theobservation of the results, we can conclude that the changesin the rpm affect the vibration experienced in the cabin. Thevibration might be transmitted to the car body through the me-chanical vibration due to the transmission of the engine andthrough the tire-road contact. Thus, we can also conclude thathigher rpm can produce more vibration and at the same timegenerate more noise in the cabin. Besides that, we have alsoproven that by considering the loudness parameter in the noise,a comparison between both of the conditions would be a cor-rect measure, since the results show that this parameter canrepresent human perceptions more accurately.

The study has also shown that the use of the GP is able tosolve the optimization problem related to linearity-based con-straints. In this case, the GP can assist the researcher to gen-erate accurate results by taking into account the priority andflexibility of each constraint in the problems modeled.

The proposed model serves as a technical method to min-imize the noise generated by decreasing the vibration level inthe cabin. Based on the results, this model is able to predict themaximum level of exposed vibration in order to obtain a morepleasant acoustical environment in the cabin. Thus, we believethat by modifying a particular structure in the car system thelevel of vibration may be reduced and one will potentially beable to reduce the generation of noise and increase the value ofthe VACI.

The advantage of this model is that it is useful to automotiveresearchers in developing a better acoustical environment inthe cabin by following the benchmark of the maximum vibra-tion experienced proposed by the model. By reducing the levelof vibration further below the benchmark, it will provide amore pleasant acoustical environment in the cabin. The modeldeveloped by the GP is the most appropriate attempt since itconsiders the constraints and flexibilities in the computationand takes into account that the output results will be more re-liable and more accurate. To enable this model to be used inthe future, some changes and additional information may beworth contemplation, such as considering a different vehicle, adifferent type of tire, or a different road surface. These may beable to serve to include additional effects on the sound qualityand to influence the essential aspects of the cabin acousticalcomfort.



Bibliography1 O’Boy, D. J. and Dowling A. P. Tire/road interaction

noise—a 3D viscoelastic multilayer model of a tire belt,Journal of Sound and Vibration, 322 (4–5), 829–850,(2009).

2 O’Boy, D. J. and Dowling A. P. Tire/road interactionnoise—numerical noise prediction of a patterned tire on arough road surface, Journal of Sound and Vibration, 323(1–2), 270–291, (2009).

3 Wang, Y. S., Lee, C.-M., Kim, D.-G., and Xu, Y. Sound-quality prediction for nonstationary vehicle interior noisebased on wavelet pre-procesing neural network model,Journal of Sound and Vibration, 299 (4–5), 933–947,(2007).

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9 Gonzalez, A., Ferrer, M. D. D., Pinero, G., and Bonito,J. J. G. Sound quality of low-frequency and car enginenoises after active noise control, Journal of Sound and Vi-bration, 265 (3), 663–679, (2003).

10 Tung, C. Y. and Chao K. P. Effect of recreational noiseexposure on hearing impairment among teenage students,Research in Developmental Disabilities, 34 (1), 126–132,(2013).

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19 Daruis, D. D. I., Nor, M. J. M., Deros, B. M., and Fouladi,M. H. Whole-body vibration and sound quality of Malay-isan cars, Proc. 9th Asia Pacific Industrial Engineering andManagement Systems Conference, Indonesia, (2008).

20 Nahvi, H., Fouladi M. H., and Nor M. J. M. Evaluation ofWhole-Body Vibration and Ride Comfort in a PassengerCar, International Journal of Acoustics and Vibration, 14(3), 143–149, (2009).


Natural Frequencies and Acoustic Radiation ModeAmplitudes of Laminated Composite Plates Basedon the Layerwise FEMJinWu Wu and LingZhi HuangSchool of Aircraft Engineering, Nanchang Hangkong University, Nanchang, People’s Republic of China

(Received 21 April 2012; revised 27 December 2012; accepted 7 February 2013)

In this paper, the natural frequencies and acoustic radiation mode amplitudes of laminated composite plates arestudied. The layerwise finite element model is imposed to determine the natural frequencies and velocity distri-butions of laminated composite plates. The amplitude of the laminated composite plates are then discussed basedon the acoustic radiation mode, the effects of the panel orientation angle, the elastic modulus ratio, the width-depth ratio, and the damping ratio on the first acoustic radiation mode. A sixteen-layer laminated plate was usedas an example, and the numerical simulations and experimental results show that the natural frequencies of thelaminated composite plate can be analysed accurately using the proposed model. Furthermore, it is found thatthe effects of the panel orientation angle and width-depth ratio on the acoustic radiation mode amplitude of thelaminated composite plates are significant.

1. INTRODUCTION

Laminated composite plates have been widely used inaerospace vehicles, maritime carriers, and wind turbine blades,where high strength, high stiffness, and low weight are im-portant properties.1 When these composite structures are usedin dynamic environments, vibration control and noise reduc-tion become of great technical significance. The vibration oflaminated composite structures is generally in low-frequencyranges. Thus, noise reduction of laminated composite struc-tures in low-frequency ranges is very important. Althoughactive vibration control could be used to decrease structuralnoise,1 low noise design is the most reliable means of reducingradiated noise. The most suitable function of a low noise de-sign is in sound radiation power.2, 3 A low noise design optimi-sation is considered with the goal of minimising the total soundpower of the structure. The sound radiation power is related tothe characteristics of the structure.2–4 Therefore, it is importantto study the relation of the laminated composite structure pa-rameters to the sound radiation power in low-frequency ranges.

For planar radiators, it is well known that the low-frequencysound radiation is directly related to the velocity distributionof the structure’s surface. Different velocity distributions havedifferent contributions to the radiation sound power. Velocitydistributions corresponding to certain mode shapes are moreefficient radiators. It has been widely accepted that structuralacoustic problems can be analysed based on so-called radiationmodes.1 Radiation modes are sets of independent radiatingvelocity distributions. The dominant radiation modes are thefirst few order modes at low-frequency ranges. Controlling thefirst few order radiation modes’ amplitudes has been shownto reduce the total sound power efficiently in low-frequencyranges.1, 5, 6

For laminated composite plates, the dynamic response ofthe structures must first be analysed in order to study the firstfew order radiation modes’ amplitudes of the plates in low-frequency ranges. Laminated plate theories are essential to

providing an accurate analysis of laminated composite plates,and a variety of laminated plate theories have been developedand reported in the literature. A review of the various equiva-lent single layer and layerwise laminated plate theories can befound in Reddy’s work.7 Because of their complex behaviourin the analysis of laminated composite plates, some technicalaspects must be taken into consideration. For example, theclassical laminate plate theory (CLPT) is based on the Kirch-hoff plate theory. It is the simplest theory, but the shear de-formation effects are neglected.8 Furthermore, it results in anunderestimate of the deflection and an overestimate of the nat-ural frequencies. The first and higher order shear deformationtheories are improvements over classical theories. For the firstand higher order shear deformation theories, transverse sheardeformation through the thickness of the structure is not ig-nored.9

Another aspect in the analysis of composite structures is theexistence of couplings among stretching, shearing, bending,and twisting. These couplings can significantly change the re-sponse of composite structures and need to be considered. Thelayerwise lamination theory assumes a displacement represen-tation formula in each layer.10 The layerwise finite elementtheory can be seen as a three-dimensional theory. The assumedlayerwise displacement field uses a linear Lagrange polyno-mial approximation for the thickness of each lamina and a con-stant transverse displacement for the entire thickness. The in-terlaminar stresses can be predicted accurately, and the layer-wise finite element theory can better adapt to the combinationof boundary conditions. In this paper, the dynamic responsesof the laminated composite plates are examined, based on alayerwise finite element theory.

In recent years, there has been much research on the vi-bration and acoustical characteristics of laminated compositeplates. For example, Li et al.11, 12 deal with the structural vi-bration and acoustic radiation of a fluid-loaded laminated platebased on the first order shear deformation theory (FSDT) andthe classical Kirchhoff-Love thin plate theory. Cao et al.13


JinWu Wu, et al.: NATURAL FREQUENCIES AND ACOUSTIC RADIATION MODE AMPLITUDES OF LAMINATED COMPOSITE PLATES. . .

present the structural and acoustical characteristics of stiffenedplates in the form of the transverse displacement spectra andsound pressure. The equations of motion for the compositelaminated plate are derived on the basis of the first-order sheardeformation plate theory. Acoustic radiation is analysed usingthe Fourier wave number transform and the stationary phasemethod. The numerical study of the vibration and acoustic re-sponse characteristics of a fibre-reinforced composite plate in athermal environment is present by Jeyaraj et al.,14 and the crit-ical buckling temperature and vibration response are obtainedusing the finite element method based on the classical lami-nate plate theory (CLPT), while the sound radiation character-istics are obtained using a coupled FEM/BEM technique. Niuet al.2–4 propose optimizing the vibrating laminated compositeplates for minimum sound radiation. However, little has beenreported on the relation of the laminated composite structureparameters to the sound radiation in the low-frequency range.This study aims to disclose the vibrating features and acous-tic radiation mode amplitudes of laminated composite platesso that effective composite structure parameter approaches canbe implemented to address the low noise design problem ofsuch structures in low-frequency ranges.

In this paper, the structure vibration and acoustic radia-tion mode amplitudes of laminated composite plates based onlayerwise finite element models are presented. Furthermore,based on the acoustic radiation mode, the effects of the orien-tation angle, the elastic modulus ratio, the width-depth ratio,and the damping ratio on the amplitude of the first acousticradiation mode for laminated composite plates are discussed.

2. RADIATION MODES THEORY

Consider a planar structure vibrating with an angular fre-quency ω and radiating sound into the upper half space V ex-terior to the panel surface S for z > 0. The density of themedium is ρ, and the sound speed is c. The surface area of thevibrating plate is S. The plate is divided into J elements withequal areas. Assume that the elemental source is small com-pared to the acoustic wavelength. The vector of the normalvelocities of each element is denoted as U(ω). The acousticpower can be expressed as1

W(ω) =ρcS

2U(ω)HR ·U(ω); (1)

where the superscript H denotes the complex conjugate trans-pose. The matrix R is a J × J matrix of the real partof the acoustic transfer impedance between each pair of el-ements. Because the matrix R is real, symmetric and pos-itive definite,1 the following eigenvalue decomposition is al-ways possible: R = QΛQT, where the superscript T de-notes the transpose. The matrix Λ is a diagonal matrix witheigenvalues λi, which are related to the radiation efficiency.Q = [Q1, Q2, . . . , Qi, . . . , QJ ] is a group of orthogonal sur-face vibration patterns, and Qi is a real vector representingthe ith radiation mode shape. Because the matrix R is relatedto the frequency, the radiation mode shapes Qi are also fre-quency dependent. It has been demonstrated that the radiationmode shapes can be chosen to be independent of the frequencyat low-frequency ranges.15, 16

Any vibration velocity distribution U(ω) has a representa-

Figure 1. Laminated plate geometry and coordinate system.

tion in terms of the radiation modes Qi:

U(ω) =J∑i=1

vi(ω)Qi; (2)

wherevi(ω) = QT

iU(ω) (3)

and vi(ω) is called the ith radiation mode’s amplitude.1

Because R = QΛQT, the total radiation acoustic power canbe rewritten as

W (ω) =ρcS

2

J∑i=1

λi|vi(ω)|2. (4)

From Eq. (4), it can be demonstrated that each radiationmode contributes to the sound power independently. Equa-tion (4) shows that the total radiation acoustic power W (ω) isthe sum of the individual mode amplitudes squared, multipliedby their corresponding eigenvalues. The relative contributionof the first radiation mode to the radiated sound power is onlydominant for low frequencies.1, 17 At low-frequency ranges,the sound power of the first radiation mode is the main part ofthe total sound power. Moreover, after cancelling the soundpower of the first radiation mode, the total acoustic power canbe reduced significantly.1 Therefore, acquiring informationon the amplitudes of the first radiation modes is important tocontrol the acoustic radiation power of the vibrating structure.However, for higher frequencies, all radiation modes have sim-ilar radiation efficiencies, so more radiation modes have to betaken into account to reduce the radiated noise. Moreover, inorder to obtain reduction at higher frequencies, i.e., ka > 1,where k is the wave number and a is the characteristic radiusof the structure, more radiation modes are needed.

In this paper, the vibration and sound radiations of a lami-nated composite plate at low-frequency ranges are presented.This will be discussed in the next section using the layerwiseFEM.

3. THE NORMAL VELOCITY DISTRIBUTION

Based on the layerwise theory, each laminate in the direc-tion of the thickness needs to be interpolated twice, and thelaminated plate geometry and coordinate system are shown inFig. (1). The displacement of the laminated plate can be writ-ten as:18

U(x, y, z, t) =2n+1∑i=1

µi(x, y, t)Ψi(z); (5)



V (x, y, z, t) =2n+1∑i=1

vi(x, y, t)Ψi(z); (6)

W (x, y, z, t) =2n+1∑i=1

wi(x, y, t)Ψi(z). (7)

U(x, y, z, t), V (x, y, z, t), and W (x, y, z, t) are the displace-ments in the x, y, z directions, respectively; n denotes thelayer number; 2n + 1 denotes the interpolation surface num-ber; ui(x, y, t), vi(x, y, t). and wi(x, y, t) denote the displace-ments of the ith interpolation surface in the x, y, z directions,respectively. Ψi(z) is the interpolation coefficient, where:18

Ψ1(z) = φ11(z) (z1 ≤ z ≤ z3); (8)Ψ2i(z) = φ2i(z) (z2j−1 ≤ z ≤ z2j+1); (9)

Ψ2i+1(z) =

ßφ3i(z) (z2j−1 ≤ z ≤ z2j+1)φ1(i+1)(z) (z2j+1 ≤ z ≤ z2j+3)

; (10)

Ψn(z) = φ3n(z) (z2n−1 ≤ z ≤ z2n+1); (11)

where j denotes the layer number of the laminated plate, zkdenotes the ordinate of the kth interpolation layer, and φ1i, φ2i,and φ3i denote:

φ1j(z) = (1− zz/hj)(1− 2zz/hj); (12)φ2j(z) = 4zz/hj(1− zz/hj); (13)φ3j(z) = −zz/hj(1− 2zz/hj); (14)

where j = (1, 2.....n), hj is the jth layer thickness, and zz isthe local coordinate of the plate thickness.

According to finite element theory, ui(x, y, t), vi(x, y, t)and wi(x, y, t) in Eqs. (5–7) can be rewritten as:

ui(x, y, t) =m∑k=1

Nk(x, y)ukT (t); (15)

vi(x, y, t) =m∑k=1

Nk(x, y)vkT (t); (16)

wi(x, y, t) =m∑k=1

Nk(x, y)wkT (t). (17)

wherem denotes the finite element node. Nk(x, y) is the shapefunction expression, uk, vk, wk are the x, y, z coordinates, re-spectively, of the finite element kth node, and T (t) is the timefunction. In this paper, a finite element unit is eight noderectangular elements, and the length by width of each unit is2a× 2b. The shape function expression of the eight-node rect-angular elements can be written as:

N1 = (1− x/a)(1− y/b)(−x/a− y/b− 1)/4; (18)

N2 = (1− x/a)2(1− y/b)/2; (19)N3 = (1 + x/a)(1− y/b)(x/a− y/b− 1)/4; (20)

N4 = (1− (y/b)2)(1 + x/a)/2; (21)N5 = (1 + x/a)(1 + y/b)(x/a+ y/b− 1)/4; (22)

N6 = (1− (x/a)2)(1 + y/b)/2; (23)N7 = (1− x/a)(1 + y/b)(−x/a+ y/b− 1)/4; (24)

N8 = (1− (y/b)2)(1− x/a)/2. (25)

Substituting Eqs. (15–17) and (18–25) into Eqs. (5–7) yieldsthe unit shape matrix N:

N = [N11,N12, . . . ,Nlk, . . . ,N38]

(l = 1, 2, 3; k = 1, 2, . . . , 8); (26)

where

Nlk =

NkφikNkφik

Nkφik

. (27)

According to the elastic displacement strain, the unit strain ma-trix can be written as

B = [B11,B12, · · · ,Blk, · · · ,B38]; (28)

where

Blk =

∂Nk

∂x φik∂Nk

∂y φik

Nk∂φik

∂z

Nk∂φik

∂z∂Nk

∂y φik

Nk∂φik

∂z∂Nk

∂x φik∂Nk

∂y φik∂Nk

∂x φik

. (29)

The stress-strain relationship of orthotropic materials is

[ε]i = [S]i × [σ]i; (30)

where the matrix S can be written as

S =

1

E1

−v12E2

−v13E3

−v12E2

1

E2

−v23E3

−v31E1

−v23E3

1

E3

1

G231

G311

G12

j

.

(31)If there is a viscoelastic laminate plate, the elastic modulus

ratio of each viscoelastic layer can be denoted as E = E · (1 +ηl), where l =

√−1 and η is the loss factor. According to the

shaft formula, when the x, y, z coordinate axis is inconsistentwith the layer material coordinate axis, the displacement-strainrelationship can be expressed as:

[σ]i = [D]i × [ε]i; (32)

where D denotes the stiffness matrix.From Eqs. (28) and (32), we obtain the B matrix and D

matrix, and the unit stiffness matrix K can be expressed as

K =

∫∫∫V

BTDBdxdydz. (33)

The unit mass matrix M can be expressed as

M =

∫∫∫V

ρNTNdxdydz. (34)



Table 1. Material properties of the viscoelastic laminate plate.

Property ValueLength (x direction) 0.3048 mLength (y direction) 0.3480 m

Thickness h1 = h3 = 0.007 62mYoung’s modulus E1 = E3 = 68.9GPa

Poisson ratio υ1 = υ3 = 0.3, υ2 = 0.49Shear modulus G2 = 0.869MPa

Loss factor η = 0.5

Density ρ1 = 2 740 kg/m3, ρ2 = 999 kg/m3

Table 2. Natural frequencies and the loss factor of the viscoelastic laminatedplate.

Mode Natural frequency (Hz) Loss factorRef.17 Ref.18 Present Ref.15 Ref.16 Present

1 60.3 60.24 58.3 0.190 0.1901 0.17182 115.4 115.22 114.5 0.203 0.2034 0.19243 130.6 130.43 130.0 0.199 0.1991 0.17204 178.7 178.46 178.3 0.181 0.1806 0.17005 195.7 195.42 196.6 0.174 0.1737 0.1562

Generally, the damping value of the composite laminateplate in the low-modal range is related to the degree ofanisotropy and each mode of the composite laminate plate. TheRayleigh damping model is used in this paper, and the dampingmatrix C can be written as

C = αM + βK; (35)

where α, β are the mass and stiffness factors, respectively. Theelement equivalent nodal force can be expressed as

F =

∫∫∫V

NTpvdxdydz. (36)

From the unit stiffness matrix K, mass matrix M, dampingmatrix C, and element nodal force F, the total power equationcan be obtained:

MX + CX + KX = F. (37)

From Eq. (37), the surface normal displacement vector Xcan be determined, and the normal velocity vector UX(x, y, ω)and natural frequency ω can be obtained from

∣∣K− ω2M∣∣ =

0.

4. NUMERICAL EXAMPLES

In order to check the quality of the layerwise FEM method,the natural frequencies of a viscoelastic laminate plate and aT300 laminated plate are studied. The first acoustic radiationmode amplitude of a 16-layer laminate plate with a simply-supported boundary condition is presented.

4.1. Natural Frequencies of a Laminate PlateThe physical properties of a viscoelastic laminate plate un-

der a simply-supported boundary condition are summarized inTable 1. The natural frequencies and loss factor of the vis-coelastic laminate plate are determined: ω′ =

√re(ω)2 and

η′ = im(ω)2/re(ω)2. We used regular grids of 60×60 ele-ments in the x- and y-directions.

The result is obtained from Eq. (37) and is compared withthe result in references19 and.20 A comparison of the results isshown in Table 2. The results show good agreement for boththe frequencies and loss factors in this paper. Reference19 takesa first-order shear deformation theory to describe the defor-mation of the faces. The first-order shear deformation theory

Table 3. Material properties of the T300 laminated plate.

Property ValueComposite material T300Length (x direction) 0.27 mLength (y direction) 0.27 m

Thickness h = 0.002mYoung’s modulus E1 = 1.24GPa,E2 = 7.5GPa,E3 = E2

Shear modulus G12 = 4.6GPa,G13 = G12 = G23Poisson ratio υ12 = υ13 = 0.31

(a)

(b)

Figure 2. Experimental set-up of the T300 laminated plate.

(FSDT) provides a balance between computational efficiencyand accuracy for the global structural behaviour of thin andmoderately thick laminated composite plates. Reference20 alsotakes an analytic solution based on the layerwise theory anduses eigenfunctions as the plate boundary conditions.

Consider sixteen anti-symmetric 0-degree angle-ply T300laminated plates with four sides clamped; the physical prop-erties of the plate are summarized in Table 3. In order to checkthe accuracy of the method in this paper, the numerical simu-lations and experimental results of the natural frequency of theT300 laminated plate are analysed.

The experimental set-up is shown in Fig. (2). In the sim-ulations, clamped boundary conditions are assumed. In theexperiments, the boundary conditions approximate clampedboundary conditions. A T300 laminated plate is mounted ona cast-iron box with a B&K4808 hammer attached at the topof the plate to provide the primary driving force, and a pointforce located at x = 0.25 · Lx, y = 0, 75 · Ly is applied toexcite the plate. The accelerometer output signal of the plateis measured using a B&K4370 accelerometer. Sixteen points,which are uniformly distributed on the vibrating plate, are se-lected for measurement in the experiments. When the relatedparameters of the modal analysis software in the host PC havebeen well adjusted, the output signal and the input force ham-



Figure 3. Transfer function of the T300 laminated plate.

Table 4. Natural frequencies of the T300 laminated plate.

Result/mode 1 2 3 4 5 6Simulation result (Hz) 169 226 364 513 552 648

Experimental result (Hz) 153 220 360 502 531 661Error (%) 10.05 2.65 1.09 2.14 3.80 2.01

mer signal are simultaneously measured by the modal analysissoftware. The transfer functions of the sixteen points are anal-ysed in the modal analysis software. Curve fitting the sixteentransfer functions give the natural frequencies of the structure.A typical measured transfer function is shown in Fig. (3). Allexperiment apparatuses are demarcated, and all measurementsignals are transformed by the engineering unit and limited toa frequency range below 800 Hz. The numerical simulationsand experimental results of the natural frequencies are listed inTable 4. Table 4 shows that there are errors between the ex-perimental and numerical simulation results. The first reasonis because the boundary condition is slightly different in theanalytical model and in the experimental set-up. The secondreason is that the density and Poisson ratio of the laminatedcomposite plate are not constant.

4.2. Factors of the Acoustic Radiation ModeAmplitude

The physical properties of the 16-layer laminate simply-supported plate are as follows: Lx = 0.40 m, Ly = 0.25 m, thesingle thickness is 0.125 mm, the total thickness is h = 2 mm,Young’s modulus is E1 = 181 GPa, and v12 = v13 = v23 =0.28. One point force (force amplitude of 10 N), located atx = 0.5 · Lx, y = 0.5 · Ly , is applied to excite the plate. As-sume that the bonding layer materials damping ratio is ignored.

4.2.1. The Effect of the Laying Angle

Assume that the laminated plate damping ratio is 0.2, theshear modulus is G23 = G21 = G13 = 7.17 GPa, andE2 = E3 = 10.3 GPa. The four orientation angles areα = (−15/15)8, (−30/30)8, (−45/45)8 and (−0/90)8. Thenatural frequencies and the first acoustic radiation mode ampli-tude of the laminated plates with different orientation anglesare shown in Table 5 and Fig. (4). It is found that the firstacoustic radiation mode amplitude of the laminated simply-supported plate is only related to the (odd, odd) order struc-ture mode. Frequencies corresponding to the maximum am-plitude are equal to frequencies of the (odd, odd) order struc-ture mode. These characteristics are the same as those of theacoustic radiation of the general single-layer structure. In ad-dition, Table 5 and Fig. (4) show that the acoustic radiationmode is only related to the geometrical shape of the radiating

Table 5. Natural frequencies of the laminated plate with different orientationangles.

Mode Natural frequencies (Hz)α = 15◦ α = 30◦ α = 45◦ α = (0/90)◦

1 96.3 127 152 1302 187 291 288 2303 278 301 427 4424 327 507 480 4705 383 544 604 5226 520 571 735 6617 538 790 844 7568 557 806 876 7939 577 852 885 917

Figure 4. Effect of the orientation angle on radiation mode amplitude.

structure and is independent of the material properties. Fur-thermore, besides (−0/90)8, the smaller the orientation angleof a laminated plate is, the smaller is the natural frequencyof the corresponding structure mode values. This means thatthe more natural the frequency number is, the greater the first-order acoustic radiation mode amplitude is and the greater thecorresponding radiated power value is at the same external ex-citation frequency. Therefore, if a 45 degree orientation anglefor the laminated plate is used, the laminated plate noise canbe reduced significantly.

4.2.2. The Effect of the Young’s Modulus Ratio

Assuming that the orientation angle of the laminated plateis α = (−45/45)8 and the damping is 0.2, the shear modu-lus is G23 = G21 = G13 = 0.6E2 and Young’s modulus isE2 = E3. In addition, the elastic modulus ratio E1/E2 is, re-spectively, 5, 10, 15, and 20 for the four orientation angles. Thenatural frequency and the first order acoustic radiation modeamplitude of the laminated plates with different Young’s mod-ulus ratios are shown in Table 6 and Fig. (5).

Table 6 and Fig. (5) show that the natural frequency of thesame structural mode is almost equal at different Young’s mod-ulus ratios, which means that the corresponding frequency ofthe maximum amplitude of the first order acoustic radiationmode is almost identical. This illustrates that the Young’s mod-ulus ratio has little effect on the acoustic radiation mode am-plitude.

Table 6. Natural frequencies of the laminated plate with different E1/E2.

Mode Natural frequencies (Hz)E1/E2 = 5 E1/E2 = 10 E1/E2 = 15 E1/E2 = 20

1 159 154 152 1512 298 291 288 2873 476 442 429 4424 510 489 481 4775 635 614 605 6016 803 755 737 7287 871 853 845 8248 1001 912 880 8419 1152 945 887 862



Figure 5. Effect of E1/E2 on the radiation mode amplitude.

Table 7. Natural frequencies of the laminated plate with different b/h.

Mode Natural frequencies (Hz)b/h = 75 b/h = 100 b/h = 125 b/h = 150

1 251 190 152 1272 476 359 288 2403 476 534 428 3574 793 598 480 4015 884 753 605 5056 997 885 735 6057 1186 916 844 7068 1214 1050 886 7329 1389 1092 885 885

4.2.3. The Effect of the Thickness Ratio

Assuming that the orientation angle of the laminated plateis α = (−45/45)8 and the Young’s modulus is E2 = E3 =10.3 GPa, the shear modulus is G23 = G21 = G13 = 0.6E2.In addition, b/h is, respectively, 150, 125, 100 and 75 for thefour orientation angles, where b is the defined side length ofthe plate. The natural frequency and the first order acousticradiation mode amplitude of the laminated plates with differentb/h ratios are shown in Table 7 and Fig. (6).

Table 7 and Fig. (6) show that the b/h ratio has a great effecton the acoustic radiation mode amplitude. The correspondingnatural frequencies of the same order structure mode vary withthe b/h ratio, and the larger the b/h ratio is, the smaller thenatural frequencies of the same order structure mode are. Theacoustic radiation mode amplitude decreases as the peak cor-responding to the frequency increases, and the total radiatedsound power decreases accordingly when the thickness of thelaminated plate increases. Therefore, a smaller b/h ratio isbetter in order to reduce the laminated plate noise. However,increasing the thickness is limited by the material and process.

Figure 6. Effect of b/h on the acoustic radiation mode amplitude.

(a)

(b)

Figure 7. Effects of the damping ratio on acoustic radiation mode amplitudeand the sound radiation power.

4.2.4. The Effect of Damping

Assuming that the Young’s modulus of the laminated plateis E2 = E3 = 10.3 GPa, the shear modulus is G23 = G21

= G13 = 0.6E2 and the orientation angle of the laminatedplate is α = (−45/45)8. The first-order acoustic radiationmode amplitude and the acoustic power of the laminated plateswith different dampings are shown in Fig. (7).

It is well known that the damping does not affect the struc-ture of the natural frequency, which is also reflected in Fig. (7).In addition, the figure shows that the damping ratio is largerand that the first order acoustic radiation mode amplitude andthe acoustic power of the laminated plate are smaller at thesame external excitation frequency.

5. CONCLUSIONS

In this paper, the natural frequency and acoustic radiationmode amplitude of laminated composite plates have been in-vestigated. Based on the layerwise finite element theory, thenatural frequency and dynamic response of laminated com-posite plates are obtained. The effects of the laminated com-posite plate’s parameters on the first-order acoustic radiationmode amplitude of the laminated composite plates are dis-cussed when the excitation point is located at the centre ofthe plate. The numerical calculation and experimental resultsshow that the natural frequencies of the laminated compositeplate can be analysed accurately using the layerwise finite el-ement model. In addition, the layerwise finite element modelhas been successfully used to analyse the acoustic radiation ofclamped laminated composite plates. The proposed methodcan provide the structural parameters of the optimal design oflow-noise laminated plates.

There is still a need to further investigate the influence of thedifferent excitation points on the results. Furthermore, the case



in which the multi-excitation points simultaneously impact thelaminated composite plate may be considered in future work.

ACKNOWLEDGMENTS

This work is supported by the National Natural ScienceFoundation of China (No: 51265038) and the AeronauticalScience Foundation of China (No: 2011ZA56002).

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About the AuthorsDaniel J. Inman received his PhD in mechanical engineering from Michigan State Univer-sity in 1980 and spent 14 years at the University of Buffalo, followed by 19 years at VirginiaTech. He is currently the Chair of the Department of Aerospace Engineering at the Univer-sity of Michigan. He has published eight books, a large number of journal and conferencepapers, and given 56 keynote or plenary lectures. He has graduated 57 PhD students and su-pervised more than 75 MS degrees. He is a Fellow of the American Academy of Mechanics(AAM), the American Society of Mechanical Engineers (ASME), the International Instituteof Acoustics and Vibration (IIAV), and the American Institute of Aeronautics and Astronau-tics (AIAA). Since 1999 he has served as the Technical Editor of the Journal of IntelligentMaterial Systems and Structures. He was awarded the ASME Adaptive Structures Award inApril 2000, the ASME/AIAA SDM Best Paper Award in April 2001, the SPIE Smart Struc-tures and Materials Life Time Achievement Award in March of 2003, the ASME/Boeing BestPaper Award by the ASME Aerospace Structures and Materials Technical Committee 2007,the ASME Den Hartog Award in 2007, and the Life Time Achievement Award in StructuralHealth Monitoring in 2009. He has served as a Member-at-Large on the Society of Experi-mental Mechanics Executive Board (2008–2010) and is a former Chair of the ASME AppliedMechanics Division.

Luke A. Martin earned his BS in mechanical engineering from West Virginia UniversityInstitute of Technology in 2002. He earned his MS in mechanical engineering in 2004 andhis PhD in mechanical engineering in 2011, both from Virginia Tech and under the tutelageof Dr. Daniel Inman. He is currently a Senior Mechanical Engineer at the Naval SurfaceWarfare Center in Dahlgren, Virginia, where he has worked since 2004. At Dahlgren, he is thetechnical lead for shock and vibration testing and analysis of military systems. He is activein the development of MIL-STD-810 and is constantly pursuing new technical approachesfor performing shock and vibration tests that reduce cost and increase test fidelity. He hasmentored seven engineering interns and trained more than ten new hire engineers. Martin wasawarded the Dr. Charles J. Cohen Science and Technology Excellence Award in 2012 for hiswork developing a custom, multiple-axis vibration test for the US Navy. Additionally, he isa Dahlgren Academic Fellow (2006) and received the Distinguished Service Award from theTau Beta Pi Engineering Honor Society in 2008.

Atanu Sahu received his undergraduate Civil Engineering degree in 2007 from Bengal Engi-neering & Science University in Shibpur, India, and completed post-graduation in StructuralEngineering from Jadavpur University, India, in 2010. Currently, he is a PhD scholar inthe Department of Civil Engineering at Jadavpur University. Sahu is involved in the vibro-acoustics research of the Institute of Composite Structures & Adaptive Systems of the Ger-man Aerospace Center (DLR) in Braunschweig, Germany, as a DAAD scholar. His researchinterests are in exterior and interior structural acoustics involving finite element method(FEM), boundary element method (BEM), active structural acoustic control, and analysisof laminated composite structures.

Arup Guha Niyogi received his undergraduate degree in Civil Engineering in 1984 fromBengal Engineering College, Shibpur, Howrah, India, and completed his graduate studies inStructural Engineering at Jadavpur University in 1987. He received his Ph.D. from the IndianInstitute of Technology, Kharagpur, India, in 2001. His research interests regard the analy-sis of laminated composites and interior structural acoustics involving FEM and BEM. Hehas been involved in teaching in the Department of Civil Engineering at Jadavpur Univer-sity, Kolkata, India, since 1990, after a stint of about 4 years in the construction and designindustry.


About the Authors

Tirtha Banerjee completed his undergraduate degree in Civil Engineering from JadavpurUniversity in Kolkata, India, in 2011. During his undergraduate study, he participated in re-search projects in the areas of active vibration control and mathematical modelling of dynam-ical systems as a member of the structural dynamics research group in the Civil Engineeringdepartment of Jadavpur University. He spent the summer of 2009 at the National AerospaceLaboratory in India, conducting research on structural dynamics and aeroelasticity. In thesummer of 2010, he was awarded the DAAD (WISE) fellowship to conduct research onactive structural acoustic control at the Institute of Composite Structures and Adaptive Sys-tems at the German Aerospace Center (DLR) in Braunschweig, Germany. After completinghis undergraduate studies in 2011, he attended Duke University for graduate studies and ispresently a PhD student in the Nicholas School of Environment and Earth Sciences at DukeUniversity in NC, USA. His current research interest lies in mathematical modelling of turbu-lence in the atmospheric boundary layer—specifically vegetation canopies—and in generalphenomenological theory of atmospheric turbulence.

Partha Bhattacharya obtained his BE in 1991 from the Civil Engineering Department atJadavpur University and his MTech from the Department of Aerospace Engineering at IIT-Kharagpur, India, in 1996. He received his PhD in 2001 from IIT-Kharagpur. Bhattacharyareceived a DAAD fellowship during 2001–2002 and worked as a visiting scientist at theDLR, Braunschweig. He worked with the National Aerospace Laboratories in India from1999–2006 and is presently working as an associate professor with Jadavpur University inIndia. His present research interests involve numerical vibroacoustics and smart structuresapplication. He has to his credit about 20 publications in international journals and confer-ences.

Arun Kumar Gupta currently works in the Department of Mathematics at M.S. College,Saharanpur, U.P., India, and has been teaching undergraduate and graduate courses in math-ematics since 1983. His main duties include teaching, evaluating, and demonstrating. Otherthan teaching, his research interests are in vibrations of plates. He has participated in manynational and international seminars and conferences, has published more than 65 researchpapers in international journals of repute, and has reviewed several international research pa-pers. He is also on the editorial board of different international journals. Gupta has guidedmore than 17 PhD students and also worked as Professor in Mathematics at the University ofTikreet in Tikreet, Iraq.

Pragati Sharma received a Master of Science degree in mathematics with first division fromJ. V. Jain College, Saharanpur, U.P., India, in 2001. She then obtained a Master of Philosophydegree with first division from Alagappa University, Karaikudi, Tamil Nadu, India, in 2006.In December of 2012, she was awarded a Doctorate in Mathematics under the supervisionof Dr. Arun Kumar Gupta of Maharaj Singh College, Saharanpur, U.P., India. Since Marchof 2004, Sharma has worked as an assistant professor in the Department of Mathematicsat the Haryana College of Technology and Management, Kaithal, Haryana, which is underKurukshetra University, Kurukshetra, Haryana, India. Sharma has been actively involved inteaching for the past nine years, has participated in various workshops and conferences, haspresented many papers, and has published seven research papers in journals of high repute.


About the Authors

Zulkifli Mohd Nopiah is the Coordinator and an Associate Professor at the FundamentalEngineering Studies Unit (UPAK) from the Faculty of Engineering and Built Environment,Universiti Kebangsaan Malaysia (UKM). He graduated with a Bachelor in Mathematics fromthe University of Minnesota, USA. He received his Master in Operational Research from theUniversity of Southampton, UK, in 1994, and then obtained his PhD in Operational Researchfrom the University of Portsmouth, UK, in 1998. His specialty is in operational research,optimisation, and multi-objective optimisation evolutionary algorithms.

Ahmad Kadri Junoh is currently a PhD student at the Fundamental Engineering StudiesUnit (UPAK) at Faculty of Engineering and Built Environment, UKM. He received his firstdegree from the University of Akita, Japan, in 2002, with a Bachelors degree in MechanicalEngineering. In 2008, he received his Masters degree from the Faculty of Science and Tech-nology, UKM, in mathematics. His specialty is in statistical studies, optimisation, artificialintelligence, and operational research.

Ahmad Kamal Ariffin is a professor at the Faculty of Engineering and Built Environment,Universiti Kebangsaan, Malaysia (UKM), and is head of the Department of Mechanical andMaterials Engineering. He graduated from UKM in 1990 with a Bachelors degree in Me-chanical Engineering. Later, he received his PhD in 1995 under the Mechanical EngineeringDepartment and Institute of Numerical Methods in Engineering from the University of Wales,Swansea. Currently, he is aggressively involved in various field research, such as the com-putational method in engineering under the area of powder mechanics, friction, corrosion,fracture mechanics, finite element/discrete element, and parallel computations.

Jinwu Wu is an associate professor and vice dean of the School of Aircraft Engineering,Nanchang HangKong University, Peoples Republic of China. He received his bachelors de-gree in 1999 and Masters degrees in power machinery and engineering in 2003 from JiangsuUniversity, Peoples Republic of China. He also completed his PhD at Jiangsu Universityfrom 2003 to 2006. Wu has authored papers on the theory of acoustical radiation and the de-sign of modal sensors using piezoelectric (smart) materials for noise control. In addition, hispresent research field is vibration control and low noise design of laminated composite plates.With graduate students as his research assistants, he works with his colleagues to completeresearch objectives related to these interests.

Lingzhi Huang is currently a PhD candidate in the mechanical engineering at the NationalUniversity of Defense Technology, Peoples Republic of China. He received his bachelorsdegree in aircraft power engineering in 2006, and under the guidance of Associate ProfessorJinwu Wu, he was granted a Masters degree in aircraft power engineering from NanchangHangKong University, Peoples Republic of China, in 2012. His main research fields arenoise and vibration control of structures and signal processing. Huang has authored paperson vibration control and low noise design of laminated composite plates.


Book ReviewsComputational Aspects of StructuralAcoustics and Vibration

Edited by: Göran Sandberg and Roger OhayonSpringer Wien, New York, 2008, 276 p., 113 illus.ISBN: 978-3-211-89650-1Price: US $ 179

This book, published in 2008,is based on the course materi-als given at CISM InternationalCentre for Mechanical Sciencesin Udine during the summer of2006. The editors are ProfessorGöran Sandberg of Lund Univer-sity (Sweden) and Roger Ohayon,emeritus professor of StructuralMechanics and Coupled SystemsLaboratory (Paris, France). Thebook contains five lectures:

1. Variational Formulations of Interior Structural-AcousticVibration Problems (by J.F. Deü, W. Larbi and R.Ohayon)

2. Fundamentals of Fluid-Structure Interaction (by G. Sand-berg, P.A. Wernberg and P. Davidsson)

3. Sound in Vibrating Cabins: Physical Effects, Mathemat-ical Description, Computation Simulation with FEM (byF. Ihlenburg, University of Applied Sciences, Hamburg,Germany)

4. Model Based Partitioned Simulation of Coupled Systems(by C.A. Felippa and K.C. Park, University of Colorado,USA)

5. On Topological Design Optimization of StructuresAgainst Vibration and Noise Emission (by N. Olhoff andJ. Du, Aalborg University, Denmark)

The computational methods are very useful tools when in-vestigating acoustic and structural acoustic problems. Thebook is addressed to graduate students and researchers in fieldsof structural dynamics, acoustics, and vibration. The booksummarizes the European (France, Sweden, Germany, Den-mark) and American (USA) experiences.

Lecture One deals with variational formulations for linearvibration of elastic structure coupled with internal inviscid,homogenous compressive fluid (liquid or gas). The authorsexplain the local equations, variational formulations, and fi-nite element discretization. They also give numerical examples(Vibrational Analysis of a Plate/ Acoustic Cavity with Damp-ing Interface and Free Vibration of a Piezoelectric CylindricalShell Filled with Fluid). At the end of this lecture, a list ofreferences is given with citations up to the year of 2008.

The topic of Lecture Two is the acoustic and structuralacoustic analysis of passenger compartments in automobilesand aircrafts. Because of increased use of lightweight mate-rials, it is hard to ensure passenger comfort in terms of lowlevel interior noise and vibration. Similar problems occur inlightweight constructions of buildings. It is a great advantageof this part that educational software routines and elementswith their theoretical backgrounds are included. They in-troduce the basic formulations of structural acoustic systems

based on different independent variables. The programmingcodes of 15 pages are given in MATLAB language in appen-dices. The reference list contains 66 items.

In Lecture Three the sound in vibrating cabins and the prob-lems are handled with Teutonic thoroughness. The interactionbetween the fluid particles and the oscillations at the structure-fluid interface spread through the cavity in the form of waveswhich are reflected at the boundaries. The interference ofincoming and reflected waves may lead to resonant standingwaves. This effect can be the cause of booming noise in ve-hicle cabins. The use of vibroacoustic simulation and investi-gating more design variants would be a great help to solve theproblem. The lecture discusses the modelling problem in onedimension and 3-D and provides an elementary FEM solution(also with damping and porous material). It evaluates the vi-broacoustic comfort, gives computational examples, and has a41-item bibliography.

Lecture Four is the American contribution to the topic. It is“extracted from a set of graduate lectures on the time-domainsimulation of structural dynamics and coupled systems.” “Forthe treatment of coupled systems, emphasis is placed on par-titioned analysis procedures.” New approaches are presentedin the reduction of model equations and the use of computeralgebra systems. It gives a detailed discussion of coupled sys-tems, the partitioned analysis, the preliminary method design,coupled parabolic equations, and fluid-structure interaction forunderwater shock. The material has two appendixes: “Sta-bility analysis tools” and “The Routh-Hurwitz criterion.” TheBibliography has 42 items. The computer analysis of coupledsystems is at the beginning of its development. It developsslowly because of the complexity of problems and approaches.Important steps ahead are the availability and applicationof computer algebra systems and the development of modeltest systems.

Lecture Five is a tutorial paper dealing with the topologi-cal design optimization of structures. With this aim it drivesthe structural eigenfrequencies of vibration as far as possiblefrom an external exitation frequency. The proposed solutionsinclude (1) maximizing the fundamental eigenfrequency, (2)maximizing the distance gap between two consecutive eigen-frequencies, (3) maximizing the dynamic stiffness of the struc-ture, and (4) minimizing the sound power radiated from thestructural surface. All of these possibilities are discussedtheoretically through mathematical formulations and illustra-tions of the numerical results. The first method works cost-efficiently, while the multiplicity of optimum eigenfrequenciesneeds more care and attention. The paper has a 41-item bibli-ography.

One may ask: is it worth reading a book on a quickly devel-oping area, which contains a knowledge level ending in 2008?The definite answer is yes. The book gives a lot of useful the-oretical, computational, and practical information which is es-sential also in 2013. The book is recommended for all whointend to use computation tools for design against noise andvibration of structures and have some undergraduate knowl-edge of the field.

Béla BunaFRAMA01dBH Ltd, Budapest


Documents

International Journal of Acoustics and Vibration · the readers of this journal. Noise and vibration control of engines, automobiles, and ships Several decades ago, as the production