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Dynamics for Engineers
Springer-Science+Business Media, LLC
B.B. Muvdi A.W. Al-Khafaji J.W. McNabb Bradley University
Dynamics for Engineers With 1118 Illustrations
Springer
Bichara B. Muvdi Amir W. Al-Khafaji J.W. McNabb Civil Engineering and Construction Bradley University Peoria, IL 61625 USA
The original illustration depicting an object impacting the Earth (Chapter 16) courtesy of William K. Hartmann of the Planetary Science Institute, Tucson, AZ.
Library of Congress Cataloging-in-Publication Data Muvdi, B.B.
Dynamics for engineers / B.B. Muvdi, A.W. Al-Khafaji, J.W. McNabb. p. cm.
Includes bibliographical references and index. ISBN 978-1-4612-7336-3 ISBN 978-1-4612-1914-9 (eBook) DOI 10.1007/978-1-4612-1914-9 1. Dynamics. 2. Mechanics, Applied. 3. Kinematics. I. Al-Khafaji,
AmirWadi. II. Title. TA352.M88 1996 620.1'04—dc20 96-13159
Printed on acid-free paper.
© 1997 Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1997 Softcover reprint of the hardcover 1st edition 1997 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Lesley Poliner; manufacturing supervised by Jeffrey Taub. Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 6 5 4 3 2 1
ISBN 978-1-4612-7336-3
To my wife, Gladys, children, B. Charles, Diane, Katherine, Patti, andGeorge, and grandchildren, Valerie, Christopher, and Richard, as weIlas my dose friends, for their patience and continued support andencouragement during the preparation of the manuscripts.
B.B. Muvdi
To my children, Ali, Laith, and Elise, for keeping me young and happy.
A.W. AI-Khafaji
To Ada Spring McNabb, our children, their spouses, and youngwriters: Amy ChIoe, Nicholas, and Phoebe.
J.W. McNabb
Preface
Approach
"Mechanics is one of the branches of physics in which the number of principles is at oncevery few and very rich in useful consequences. On the other hand, there are few scienceswhich have required so much thought-the conquest of a few axioms has taken morethan 2000 years."-Rene Dugas, A History 0/ Mechanics
Introductory courses in engineering mechanics (statics and dynamics)are generally found very early in engineering curricula. As such, theyshould provide the student with a thorough background in the basicfundamentals that form the foundation for subsequent work in engineering analysis and design. Consequently, our primary goal in writingStatics for Engineers and Dynamics for Engineers has been to developthe fundamental principles of engineering mechanics in a manner thatthe student can readily comprehend. With this comprehension, thestudent thus acquires the tools that would enable him/her to thinkthrough the solution ofmany types of engineering problems using logicand sound judgment based upon fundamental principles.
We have made every effort to present the material in a concise butclear manner. Each subject is presented in one or more sections followed by one or more examples, the solutions for which are presentedin a detailed fashion with frequent reference to the basic underlyingprinciples. A set of problems is provided for use in homework assignments. Great care was taken in the selection of these problems toensure that all of the basic fundamentals discussed in the precedingsection(s) have been given adequate coverage. The problems in a givenset are organized so that the set begins with the simplest and ends withthe most difficult ones. A sufficient number of problems is provided ineach set to allow the use of both books for several semesters withouthaving to assign the same problem more than once. Also, each andevery one of the twenty three chapters in both books contains a set ofreview problems which, in general, are a little more challenging thanthe homework set of problems. This feature allows the teacher thefreedom to choose homework assignments from either or from bothsets. The two books have a total number of problems in excess of 2,600.All of the examples and problems were selected to reflect realistic situations. However, because these two books were written for beginningcourses in statics and dynamics, the principal objective of these exam-
viii Preface
MathBackground
Free-BodyDiagram
Organization
NontraditionalTopics
pies and problems remains to demonstrate the subject matter and toillustrate how the fundamental principles of mechanics may be used inthe solution of practical problems.
The prerequisite mathematical background needed for mastery of thematerial in both books consists primarily of high school courses inalgebra and trigonometry and a beginning course in differential andintegral calculus. We have made occasional use of vector analysis inthe development of some concepts and basic principles and in thesolution of some problems. The needed background in vector operations, however, is introduced and developed as needed throughout thebooks, particularly, in Chapters 3, 5, and 13. It should be emphasized,however, that the vector approach is used only when it is judged to offerdistinct advantages over the scalar method. Such is the case, for example,in the solution of three-dimensional problems.
The very important concept of the free-body diagram is introduced inChapters 2 and 4 but is used extensively throughout. It is our firm beliefthat the free-body diagram greatly enhances the understanding of thefundamental principles of mechanics. Thus, the free-body diagram isused not only in the solution of statics problems but also in the solution of dynamics problems whether Newton's second law, the energymethod, or the impulse-momentum technique is used in their solution.
We have organized the material in both books to enable us to presentthe simple concepts before embarking on more difficult ones. Thus, thetreatment of the mechanics of particles is dealt with before consideringthe mechanics of rigid bodies. Also, two-dimensional mechanics is presented separately, and before, three-dimensional mechanics. This approach allows the instructor to focus on simple concepts early in thesemester and to postpone the more complex concepts until a later datewhen the student has had a chance to develop some maturity in theprinciples ofmechanics. Furthermore, the separation between two- andthree-dimensional mechanics in both Statics for Engineers and Dynamicsfor Engineers, provides the instructor flexibility in selecting topics toteach during a given semester.
In addition to the traditional topics found in existing books, we haveincluded several new topics that we feit may be of interest to someteachers. These include: Axial Force and Torque Diagrams, GeneralTheorem for Cables, a brief treatment of the Six Fundamental Machinesand the use of Lagrange's Equations in the formulation of the equations of motion. It should be stated, however, that among the topicscovered (both traditional and new) are some that are judged to be notessential for an understanding of the basic concepts of mechanics of
Units
SpecialFeatures
Preface Ix
rigid bodieso These topics are identified by asterisks in the Table ofContents.
In view of the fact that the international system of units, referred to asSI (System International), is now beginning to gain acceptance in thiscountry, it was decided to use it in this book. However, it is realizedthat a complete transition from the U.So Customary to SI units will bea slow and costly process that may last as long as 20 years and possiblylonger. Some factors that will playa significant role in slowing downthe transformation process are the existing literature of engineeringresearch and development, plans, and calculations, as weIl as structuresand production machinery, that have been conceived and built usinglargely the U.So Customary system of unitso Thus, the decision wasmade to use both systems of units in this book. Approximately one-halfof the exampies and one-half of the problems are stated in terms of theU.So Customary system whereas the remainder are given in terms of theemerging SI system of unitso
In addition to the features described under APPROACH, these twotextbooks contain some special features that may be summarized asfollows:
1. Many of the example and homework problems are designed toobtain general symbolie solutions that allow the student to viewengineering problems from a broad point of view before assigningspecific numerical data.
2. Each of the twenty-three chapters in both books is prefaced by acarefully written vignette designed to motivate the student prior toundertaking the study of the chapter.
3. The two books contain over 350 examples carefully worked outin sufficient detail to make the solutions easily understood by thestudent.
40 The two textbooks are characterized by the extensive use of freebody diagrams as weIl as impulse-momentia and inertia-force diagrams. Each of these diagrams is accompanied by a right-handedcoordinate system that establishes the sign convention being usedoWith a few exceptions, three-dimensional, right-handed, x-y-z coordinate systems are shown with the x axis coming out of the paper.However, all of the two-dimensional, right-handed, x-y coordinatesystems are shown with the x axis pointing to the right.
5. Each of the two volumes, Staties for Engineers and Dynamies forEngineers have a companion Solutions Manualo
In addition to providing complete solutions to all of the homeworkproblems, each Solutions Manual contains suggested outlines for coursesin statics and in dynamics.
x Preface
Appendices
Acknowledgments
Six Appendices containing information useful in the solution of manyproblems have been included at the end of each of the two books.Appendix A contains information about a selected set of areas, Appendix B, information about a selected set of masses, Appendix C, usefulmathematical relations, Appendix 0, selected derivatives, Appendix E,selected integrals, and Appendix F, information about supports andconnections.
We acknowledge with much gratitude the assistance we received in thetyping of the manuscript by Ms. Sharon McBride, Ms. Janet Maclean,and Wilma AI-Khafaji. We also acknowledge with thanks the helpgiven by Dr. Farzad Shahbodaghlou and Dr. Akthem AI-Manaseer inthe typing of the Solutions Manuals. The authors are grateful to themany colleagues and students who have contributed significantly andoften indirectly to their understanding of statics and dynamics. Contributions by many individuals are given credit by reference to theirpublished work and by quotations. The source of photographs is indicated in each case. These books have been written on the propositionthat good judgment comes from experience and that experience comesfrom poor judgment. We certainly feel that the books are areal contribution to our profession, but we have miscalculated the enormoussacrifices they required. The quality of these books has been and willcontinue to be judged by our students and colleagues whose commentsand suggestions have contributed greatly to the successful completionof the final manuscript. We would not have been able to complete thisproject without the help and support of our families. We apologize forany omissions.
The authors appreciate the efforts of the reviewers, who, by their criticisms and helpful comments, have encouraged us in the preparationand completion of the manuscript.
Our thanks to our editor Mr. Thomas von Foerster. All the people atSpringer-Verlag who were involved with the production of this bookdeserve special acknowledgment for their dedication and hard work.
B.B.M.A.W.A.J.W.M.
Contents: Dynamics for EngineersPreface vii
13 Kinematics of Particles 1
Rectillnear Motion 2
13.1 Position, Velocity and Acceleration 213.2 Integral Analysis of Rectilinear Motion 1613.4 Rectilinear Motion at Constant Acceleration 3313.5 Relative Motion of Two Particles 39
Curvilinear Motion 50
13.6 Position, Velocity, and Acceleration 5013.7 Curvilinear Motion-Rectangular Coordinates 5213.8 Curvilinear Motion-Tangential and
Normal Coordinates 6413.9 Curvilinear Motion-Cylindrical Coordinates 7813.10 Relative Motion-Translating Frame of Reference 93
14 Particle Kinetics: Force and Acceleration 110
14.1 Newton's Laws of Motion 11114.2 Newton's Second Law Applied to a System of Particles 11414.3 Newton's Second Law in Rectangular Components 11614.4 Newton's Second Law in Normal and
Tangential Components 13514.5 Newton's Second Law in Cylindrical Components 15314.6* Motion of a Particle Under a Central Force 16614.7* Space Mechanics 168
14.7.1 Governing Equation 16814.7.2 Conic Sections 16914.7.3 Determination of K and C 17114.7.4 Initial Velocities 17214.7.5 Determination of Orbital Period 17414.7.6 Kepler's Law 175
xii Contents: Dynomlcs tor Englneers
15
16
17
18
Particle Kinetics: Energy 191
15.1 Definition ofWork 19215.2 Kinetic Energy-The Energy of Motion 21015.3 Work-Energy Principle for a Single Particle 21015.4 Work-Energy Principle for a System of Particles 22015.5 Work of Conservative Forces 22715.6 Gravitational and Elastic Potential Energy 23215.7 Principle of Conservation of Mechanical Energy 235
Particle Kinetics: Impulse-Momentum 250
16.1 Definition of Linear Momentum and Linear Impulse 25216.2 The Principle of Linear Impulse and Momentum 26116.3 System of Particles-Principle of Linear Impulse
and Momentum 26916.4 Conservation of Linear Momentum 27016.5 Impulsive Forces and Impact 27116.6 Definition of Angular Momentum and Angular Impulse 28916.7 The Principle of Angular Impulse and Momentum 29116.8 Systems of Particles: The Angular Impulse-Momentum
Principle 29916.9 Conservation of Angular Momentum 30016.10 Applications of Energy and Impulse-Momentum
Principles 30916.11* Steady Fluid Flow 32316.12* Systems with Variable Mass 33116.13* Space Mechanics 341
Two-Dimensional Kinematics ofRigid Bodies 356
17.1 Rectilinear and Curvilinear Translations 35817.2 Rotation About a Fixed Axis 36917.3* Absolute Motion Forrnulation-General Plane Motion 38517.4 Relative Velocity-Translating Nonrotating Axes 39617.5 Instantaneous Center of Rotation 40917.6 Relative Acceleration-Translating Nonrotating Axes 41617.7* Relative Plane Motion-Rotating Axes 425
Two-Dimensional Kinetics of Rigid Bodies:Force and Acceleration 452
18.1 Mass Moments of Inertia18.2 General Equations of Motion
454474
Contents: Dynamlcs for Engineers xiii
18.3 Rectilinear and Curvilinear Translation 47518.4 Rotation About a Fixed Axis 49618.5 General Plane Motion 51518.6 Systems of Rigid Bodies 527
19 Two-Dimensional Kinetics of RigidBodies-Energy 551
19.1 Definition ofWork 55219.2 Kinetic Energy 55419.3 The Work-Energy Principle 55719.4 The Conservation of Mechanical Energy Principle 57619.5 Power and Efficiency 588
20 Two-Dimensional Kinetics of Rigid Bodies:Impulse-Momentum 603
20.1 Linear and Angular Momentum 60520.2 Principles of Impulse and Momentum 60920.3 Principles of Conservation of Momentum 62720.4 Eccentric Impact 641
21 Three-Dimensional Kinematics ofRigid Bodies 657
21.1 * Motion About a Fixed Point 65921.2* General 3-D Motion (Translating, Nonrotating Axes) 66421.3* Time Derivative of a Vector with Respect
to Rotating Axes 67021.4* General 3-D Motion (Translating and Rotating Axes) 683
22 Three-Dimensional Kinetics ofRigid Bodies 699
22.1* Moments of Inertia of Composite Masses 70022.2* Mass Principal Axes and Principal Moments of Inertia 70222.3* The Work-Energy Principle 71822.4* Principles of Linear and Angular Momentum 72322.5* General Equations of Motion 73522.6* General Gyroscopic Motion 74922.6* Gyroscopic Motion with Steady Precession 75222.7* Gyroscopic Motion with Zero Centroidal Moment 755
xlv Contents: Dynamlcs tor Englneers
23 Vibrations 769
23.1* Free Vibrations of Particles-Force and Acceleration 77023.2* Free Vibrations of Rigid Bodies-Force and Acceleration 78723.3* Free Vibrations of Rigid Bodies-Energy 79823.4* Lagrange's Method-Conservative Forces 80823.5* Forced Vibrations-Force and Acceleration 81423.6* Damped Free Vibrations-Force and Acceleration 82523.7* Damped Forced Vibrations-Force and Acceleration 83223.8* Lagrange's Method-Nonconservative Forces
and MDOF 838
AppendicesAppendix A. Properties of Selected Lines and AreasAppendix B. Properties of Selected MassesAppendix C. Useful Mathematical RelationsAppendix D. Selected DerivativesAppendix E. Selected IntegralsAppendix F. Supports and Connections
Answers
Index
852855859861863866
872
895
Contents: Statics for Engineers1
2
3
4
Introductory Principles1.1 Review of Mechanics1.2 Idealizations and Mathematical Models1.3 Newton's Laws1.4 Newton's Law of Universal Gravitation1.5 Systems of Units and Conversion Factors1.6 Dimensional Analysis1.7 Problem Solving Techniques1.8 Accuracy of Data and Solutions
Equilibrium of a Particle in Two Dimensions2.1 Scalar and Vector Quantities2.2 Elementary Vector Operations2.3 Force Expressed in Vector Form2.4 Addition of Forces Using Rectangular Components2.5 Supports and Connections2.6 The Free-Body Diagram2.7 Equilibrium Conditions and Applications
Equilibrium of Particles in Three Dimensions3.1 Force in Terms of Rectangular Components3.2 Force in Terms of Magnitude and Unit Vector3.3 Dot (Scalar) Product3.4 Addition of Forces Using Rectangular Components3.5 Equilibrium Conditions and Applications
Equilibrium of Rigid Bodies in Two Dimensions4.1 Concept of the Moment-Scalar Approach4.2 Internal and External Forces-Force Transmissibility
Principle4.3 Replacement of a Single Force by a Force and a Couple4.4 Replacement of a Force System by a Force and a Couple
xvi Contents: Statlcs tor Engineers
4.5 Replacement of a Force System by a Single Force4.6 Replacement of a Distributed Force System by a Single Force4.7 Supports and Connections4.8 The Free-Body Diagram4.9 Equilibrium Conditions and Applications
5
6
7
8
Equilibrium of Rigid Bodies in Three Dimensions5.1 Definition of the Cross (Vector) Product5.2 The Cross-Product in Terms of Rectangular Components5.3 Vector Representation ofthe Moment ofa Force5.4 Varignon's Theorem5.5 Moment of a Force about a Specific Axis5.6 Vector Representation of a Couple5.7 Replacement of a Single Force by a Force and a Couple5.8 Replacement of a General Force System by a Force and
a Couple5.9 Equilibrium Conditions and Applications5.10 Determinacy and Constraints
Truss Analysis6.1 Analysis of Simple Trusses6.2 Member Forces Using the Method of Joints6.3 Members Carrying No Forces6.4 Member Forces Using the Method of Sections6.5* Determinacy and Constraints6.6 Compound Trusses6.7* Three-Dimensional Trusses-Member Forces Using the
Method of Joints
Frames and Machines7.1 Multiforce Members7.2 Frame Analysis7.3 Machine Analysis
Internal Forces in Members8.1 Internal Forces8.2 Sign Conventions8.3 Axial Force and Torque Diagrams8.4 Shear and Moment at Specified Cross-Sections8.5 Shear and Moment Equations
9
10
11
12
Contents: stattes for Englneers xvii
8.6 Load, Shear, and Moment Relationships8.7 Shear and Moment Diagrams8.8* Cables under Concentrated Loads8.9* General Cable Theorem8.10* Cables under Uniform Loads8.11 Frames-Internal Forces at Specified Sections8.12 Internal Force Diagrams for Two-Dimensional Frames
Friction9.1 Nature and Characteristics of Dry Friction9.2 Angles of Static and Kinetic Friction9.3 Applications of the Fundamental Equations9.4 The Six Fundamental Machines9.5* Friction on V-Belts and Flat Belts9.6* Friction on Pivot and Collar Bearings and Disks9.7* Friction on Journal Bearings9.8 Problems in Which Motion May Not Be Predetermined
Centers of Gravity, Centers of Mass,and Centroids10.1 Centers of Gravity and of Mass10.2 Centroid ofVolume, Area, or Line10.3 Composite Objects10.4 Centroids by Integration10.5* Theorems of Pappus and Guldinus10.6* Fluid Statics
Moments and Products of Inertia11.1 Concepts and Definitions11.2 Parallel Axis Theorems11.3 Moments of Inertia by Integration11.4 Moments of Inertia of Composite Areas and Masses11.5* Area Product of Inertia11.6* Area Principal Axes and Principal Moments of Inertia11.7* Mohr's Circle for Area Moments and Products ofInertia11.8* Mass Principal Axes and Principal Moments ofInertia
Virtual Work and Stationary Potential Energy12.1 Differential Work of a Force12.2 Differential Work of a Couple
xviii Contents: Statics tor Englneers
12.3 The Concept of Finite Work12.4 The Concept ofVirtual Work12.5* Work of Conservative Forces12.6* The Concept of Potential Energy12.7* The Principle of Stationary Potential Energy12.8* States of Equilibrium
