Fundamentals of Applied Dynamics
by Williams Jr.
ISBN: 9780262039710  Copyright 2019
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An introductory engineering textbook by an awardwinning MIT professor that covers the history of dynamics and the dynamical analyses of mechanical, electrical, and electromechanical systems.
This introductory textbook offers a distinctive blend of the modern and the historical, seeking to encourage an appreciation for the history of dynamics while also presenting a framework for future learning. The text presents engineering mechanics as a unified field, emphasizing dynamics but integrating topics from other disciplines, including design and the humanities.
The book begins with a history of mechanics suitable for an undergraduate overview. Subsequent chapters cover such topics as threedimensional kinematics; the direct approach, also known as vectorial mechanics or the momentum approach; the indirect approach, also called Lagrangian dynamics or variational dynamics; an expansion of the momentum and Lagrangian formulations to extended bodies; lumpedparameter electrical and electromagnetic devices; and equations of motion for onedimensional continuum models. The book is unique in covering both Lagrangian dynamics and vibration analysis. The principles covered are relatively few and easy to articulate; the examples are rich and broad. Summary tables, often in the form of flowcharts, appear throughout. Endofchapter problems begin at the elementary level and become increasingly difficult. Appendixes provide theoretical and mathematical support for the main text.
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DEDICATION (pg. iii)  
BESTOWAL (pg. v)  
ABOUT THE AUTHOR (pg. vii)  
ACKNOWLEDGMENTS (pg. ix)  
PREFACE (pg. xi)  
CONTENTS (pg. xix)  
1 OUR NICHE IN THE COSMOS (pg. 1)  
11 INTRODUCTION (pg. 1)  
12 WHY HISTORY? (pg. 3)  
13 IMPORTANCE OF MATHEMATICS IN THE DEVELOPMENT OF MECHANICS (pg. 3)  
14 OUR SOURCES FROM ANTIQUITY: GETTING THE MESSAGE FROM THERE TO HERE (pg. 4)  
14.1 Invention of Writing (pg. 5)  
14.2 Hieroglyphics (pg. 6)  
14.3 Cuneiform (pg. 7)  
14.4 Ancient Egyptian Papyri (pg. 7)  
14.5 Mesopotamian Clay Tablets (pg. 9)  
15 ANCIENT EGYPTIAN ASTRONOMY AND MATHEMATICS (pg. 9)  
15.1 Ancient Egyptian Astronomy (pg. 10)  
15.2 Ancient Egyptian Mathematics (pg. 11)  
16 MESOPOTAMIAN ASTRONOMY AND MATHEMATICS (pg. 14)  
16.1 Mesopotamian Astronomy (pg. 15)  
16.2 Mesopotamian Mathematics (pg. 15)  
17 MATHEMATICS OF THE MAYANS, INDIANS, ARABS, AND CHINESE (pg. 16)  
18 THE FIRST GREAT ENGINEERING SOCIETY (pg. 19)  
19 ADVERSE CRITICISM OF ANCIENT EGYPTIAN AND MESOPOTAMIAN MATHEMATICS (pg. 24)  
110 EVOLUTION THROUGH THE HELLENIC ERA (pg. 29)  
111 THE UNIFICATION OF CELESTIAL AND TERRESTRIAL MOTION (pg. 31)  
111.1 Celestial Motion (pg. 31)  
111.2 Terrestrial Motion (pg. 44)  
111.3 Unification (pg. 45)  
112 VARIATIONAL PRICIPLES IN DYNAMICS (pg. 47)  
113 THE INTERNATIONALISM OF DYNAMICS (pg. 52)  
114 OUR NICHE IN THE COSMOS (pg. 53)  
2 DESIGN, MODELING, AND FORMULATION OF EQUATIONS OF MOTION (pg. 55)  
21 INTRODUCTION (pg. 55)  
22 DESIGN AND MODELING (pg. 56)  
22.1 The Design Process (pg. 56)  
22.2 The Modeling Process (pg. 57)  
22.3 Our More Modest Goals (pg. 58)  
23 DIRECT AND INDIRECT APPROACHES FOR FORMULATION OF EQUATIONS OF MOTION (pg. 59)  
3 KINEMATICS (pg. 68)  
31 INTRODUCTION (pg. 68)  
32 POSITION, VELOCITY, AND ACCELERATION (pg. 69)  
33 PLANE KINEMATICS OF RIGID BONES (pg. 75)  
33.1 The General Motion of a Rigid Body (pg. 75)  
33.2 Types of Plane Motion of a Rigid Body (pg. 76)  
33.3 Angular Displacement, Angular Velocity, and Angular Acceleration (pg. 77)  
33.4 A Cautionary Note about Finite Rotations (pg. 83)  
34 TIME RATE OF CHANGE OF VECTOR IN ROTATING FRAME (pg. 85)  
35 KINEMATIC ANALYSIS UTILIZING INTERMEDIATE FRAMES (pg. 90)  
36 GENERALIZATIONS OF KINEMATIC EXPRESSIONS (pg. 108)  
4 MOMENTUM FORMULATION FOR SYSTEMS OF PARTICLES (pg. 135)  
41 INTRODUCTION (pg. 135)  
42 THE FUNDAMENTAL PHYSICS (pg. 136)  
42.1 Newton’s Laws of Motion (pg. 136)  
42.2 A Particle (pg. 137)  
42.3 Linear Momentum and Force (pg. 138)  
42.4 Inertial Reference Frames (pg. 139)  
42.5 The Universal Law of Gravitation (pg. 140)  
43 TORQUE AND ANGULAR MOMENTUM FOR A PARTICLE (pg. 141)  
44 FORMULATION OF EQUATIONS OF MOTION: EXAMPLES (pg. 144)  
44.1 Problems of Particle Dynamics of the First Kind (pg. 145)  
44.2 Problems of Particle Dynamics of the Second Kind (pg. 151)  
5 VARIATIONAL FORMULATION FOR SYSTEMS OF PARTICLES (pg. 179)  
51 INTRODUCTION (pg. 179)  
52 FORMULATION OF EQUATIONS OF MOTION (pg. 180)  
53 WORK AND STATE FUNCTIONS (pg. 181)  
53.1 Work (pg. 182)  
53.2 Kinetic State Functions (pg. 183)  
53.3 Potential State Functions (pg. 185)  
53.4 Energy and Coenergy (pg. 189)  
54 GENERALIZED VARIABLES AND VARIATIONAL CONCEPTS (pg. 190)  
54.1 Generalized Coordinates (pg. 190)  
54.2 Admissible Variations, Degrees of Freedom, Geometric Contraints, and Holonomicity (pg. 195)  
54.3 Variational Principles in Mechanics (pg. 201)  
54.4 Generalized Velocities and Generalized Forces for Holonomic Systems (pg. 205)  
55 EQUATIONS OF MOTION FOR HOLONOMIC MECHANICAL SYSTEMS VIA VARIATIONAL PRINCIPLES (pg. 213)  
56 WORKENERGY RELATION (pg. 238)  
57 NATURE OF LAGRANGIAN DYNAMICS (pg. 241)  
Problems for Chapter 5 (pg. 243)  
6 DYNAMICS OF SYSTEMS CONTAINING RIGID BODIES (pg. 268)  
61 INTRODUCTION (pg. 268)  
62 MOMENTUM PRINCIPLES FOR RIGID BODIES (pg. 269)  
62.1 Review of Solids in Equilibrium and Particle Dynamics (pg. 270)  
62.2 Models of Rigid Bodies (pg. 271)  
62.3 Momentum Principles for Extended Bodies: The NewtonEuler Equations (pg. 272)  
62.4 Momentum Principles for Rigid Bodies Modeled as Systems of Particles (pg. 273)  
62.5 Momentum Principles for Rigid Bodies Modeled as Continua (pg. 275)  
63 DYNAMIC PROPERTIES OF RIGID BODIES (pg. 279)  
63.1 The Inertia Factor (pg. 279)  
63.2 ParallelAxes Theorem (pg. 290)  
63.3 Principal Directions and Principal Moments of Inertia (pg. 296)  
63.4 Uses of Mass Symmetry (pg. 298)  
64 DYNAMICS OF RIGID BODIES VIA DIRECT APPROACH (pg. 303)  
65 LAGRANGIAN FOR RIGID BODIES (pg. 308)  
65.1 Kinetic Coenergy Function for Rigid Body (pg. 308)  
65.2 Potential Energy Function for Rigid Body (pg. 310)  
66 EQUATIONS OF MOTION FOR SYSTEMS CONTAINING RIGID BODIES IN PLANE MOTION (pg. 311)  
Problems for Chapter 6 (pg. 334)  
7 DYNAMICS OF ELECTRICAL AND ELECTROMECHANICAL SYSTEMS (pg. 366)  
71 INTRODUCTION (pg. 366)  
72 FORMULATION OF EQUATIONS OF MOTION FOR ELECTRICAL NETWORKS (pg. 369)  
73 CONSTITUTIVE RELATIONS FOR CIRCUIT ELEMENTS (pg. 371)  
73.1 Passive Elements (pg. 371)  
73.2 Active Electrical Elements (pg. 376)  
74 HAMILTON’S PRINCIPLE AND LAGRANGE’S EQUATIONS FOR ELECTRICAL NETWORKS (pg. 380)  
74.1 Generalized Charge Variables (pg. 380)  
74.2 Generalized Flux Linkage Variables (pg. 382)  
74.3 Work Expressions (pg. 383)  
74.4 Summary of LumpedParameter Offering of Variational Electricity (pg. 386)  
74.5 Examples (pg. 386)  
75 CONSTITUTIVE RELATIONS FOR TRANSDUCERS (pg. 407)  
75.1 Ideal MovablePlate Capacitor (pg. 408)  
75.2 Electrically Linear MoveablePlate Capacitor (pg. 410)  
75.3 Ideal MovableCore Inductor (pg. 412)  
75.4 Magnetically Linear MovableCore Inductor (pg. 413)  
76 HAMILTON’S PRINCIPLE AND LAGRANGE’S EQUATIONS FOR ELECTROMECHANICAL SYSTEMS (pg. 415)  
76.1 DisplacementCharge Variables Formulation (pg. 416)  
76.2 DisplacementFlux Linkage Variables Formulation (pg. 417)  
76.3 Examples (pg. 419)  
77 ANOTHER LOOK AT LAGRANGIAN DYNAMICS (pg. 428)  
Problems for Chapter 7 (pg. 429)  
8 VIBRATION OF LINEAR LUMPEDPARAMETER SYSTEMS (pg. 439)  
81 INTRODUCTION (pg. 439)  
82 SINGLEDEGREEOFFREEDOM FIRSTORDER SYSTEMS (pg. 440)  
82.1 Free Response (pg. 441)  
82.2 Step Response (pg. 444)  
82.3 Ramp Response (pg. 446)  
82.4 Harmonic Response (pg. 449)  
82.5 Summary of Responses for SingleDegreeofFreedom FirstOrder Systems (pg. 459)  
83 SINGLEDEGREEOFFREEDOM SECONDORDER SYSTEMS (pg. 460)  
83.1 Free Response (pg. 461)  
83.2 Natural Frequency via Static Deflection (pg. 467)  
83.3 Logarithmic Decrement (pg. 468)  
83.4 Energy Loss of Free Vibration (pg. 471)  
83.5 Harmonic Response (pg. 472)  
83.6 Summary of Responses for SingleDegreeofFreedom SecondOrder Systems (pg. 498)  
84 TWODEGREEOFFREEDOM SECONDORDER SYSTEMS (pg. 500)  
84.1 Natural Modes of Vibration (pg. 501)  
84.2 Response to Initial Conditions (pg. 514)  
84.3 Harmonic Response (pg. 527)  
85 STABILITY OF NONLINEAR SYSTEMS (pg. 541)  
Problems for Chapter 8 (pg. 557)  
9 DYNAMICS OF CONTINUOUS SYSTEMS (pg. 576)  
91 INTRODUCTION (pg. 576)  
92 EQUATIONS OF MOTION (pg. 578)  
92.1 Longitudinal Motion of System Containing Rod (pg. 579)  
92.2 Twisting Motion of System Containing Shaft (pg. 586)  
92.3 Electric Transmission Line (pg. 589)  
92.4 Flexural Motion of System Containing Beam (pg. 594)  
92.5 Summaries (pg. 602)  
93 NATURAL MODES OF VIBRATION (pg. 607)  
93.1 Method of Separation Variables (pg. 608)  
93.2 Time Response (pg. 610)  
93.3 Eigenfunctions for SecondOrder Systems (pg. 612)  
93.4 Eigenfunctions for FourthOrder Systems (pg. 620)  
93.5 General Solutions for Free Undamped Vibration (pg. 633)  
94 RESPONSE TO INITIAL CONDITIONS (pg. 636)  
94.1 An Example: Release of Compressed Rod (pg. 636)  
94.2 An Example: Shaft Stopped after Rotation (pg. 647)  
94.3 An Example: SlidingFree Beam Initially Bent (pg. 650)  
95 RESPONSE TO HARMONIC EXCITATIONS (pg. 660)  
95.1 An Example: Specified Harmonic Motion of Boundary (pg. 660)  
95.2 An Example: Distributed Harmonic Force (pg. 662)  
95.3 An Example: Harmonic Force on Boundary (pg. 665)  
96 SUMMARIES (pg. 672)  
Problems for Chapter 9 (pg. 673)  
BIBLIOGRAPHY (pg. 684)  
1 Historical (pg. 684)  
2 Astronomy (pg. 686)  
3 Design, Systems, and Modeling (pg. 686)  
4 Elementary Dynamics (pg. 686)  
5 Intermediate/Advanced Dynamics (pg. 686)  
6 Hamilton’s Law of Varying Action and Hamilton’s Principle (pg. 687)  
7 Electric and Electromechanical Systems (pg. 687)  
8 Vibration (pg. 687)  
APPENDIX A: FINITE ROTATION (pg. 688)  
A1 CHANGE IN POSITION VECTOR DUE TO FINITE ROTATION (pg. 688)  
A2 FINITE ROTATIONS ARE NOT VECTORS (pg. 690)  
A3 DO ROTATIONS EVER BEHAVE AS VECTORS? (pg. 692)  
A3.1 Infinitesimal Rotations Are Vectors (pg. 692)  
A3.2 Consecutive Finite Rotations about a Common Axis Are Vectors (pg. 692)  
APPENDIX B: GENERAL KINEMATIC ANALYSIS (pg. 694)  
B1 ALL ANGULAR VELOCITIES DEFINED WITH RESPECT TO FIXED REFERENCE FRAME (CASE 1) (pg. 694)  
B2 EACH ANGULAR VELOCITY DEFINED WITH RESPECT TO IMMEDIATELY PRECEDING FRAME (CASE 2) (pg. 698)  
APPENDIX C: MOMENTUM PRINCIPLES FOR SYSTEMS OF PARTICLES (pg. 705)  
C1 ASSERTED MOMENTUM PRINCIPLES (pg. 705)  
C2 PRINCIPLES FOR SINGLE PARTICLE (pg. 706)  
C3 PRINCIPLES FOR SYSTEM OF PARTICLES (pg. 707)  
C3.1 Asserted System Momentum Principles (pg. 708)  
C3.2 System Momentum Principles Derived from Particle Momentum Principles (pg. 709)  
C3.3 Conditions on Internal Forces (pg. 711)  
C3.4 Relationships between Momentum Principles and Conditions on Internal Forces (pg. 712)  
C3.5 Linear Momentum Principle in Terms of Centroidal Motion (pg. 714)  
C3.6 Angular Momentum Principle about Arbitrary Point (pg. 715)  
C3.7 System of Particle Model in Continuum Limit (pg. 717)  
C4 ANGULAR MOMENTUM PRINCIPLE IN NONINERTIAL INTERMEDIATE FRAME (pg. 719)  
APPENDIX D: ELEMENTARY RESULTS OF THE CALCULUS OF VARIATIONS (pg. 728)  
D1 INTRODUCTION (pg. 728)  
D2 SUMMARY OF ELEMENTARY RESULTS (pg. 730)  
D3 EULER EQUATION: NECESSARY CONDITION FOR A VARIATIONAL INDICATOR TO VANISH (pg. 734)  
APPENDIX E: SOME FORMULATIONS OF THE PRINCIPLES OF HAMILTON (pg. 737)  
E1 MECHANICAL FORMULATIONS (pg. 737)  
E1.1 Hamilton’s Law of Varying Action (pg. 740)  
E1.2 Hamilton’s Principle (pg. 741)  
E1.3 Lagrange’s Equations (pg. 742)  
E1.4 Discussion (pg. 743)  
E2 HAMILTON’S PRINCIPLE FOR ELECTROMECHANICAL SYSTEMS USING A DISPLACEMENTCHARGE FORMULATION (pg. 744)  
E3 HAMILTON’S PRINCIPLE FOR ELECTROMECHANICAL SYSTEMS USING A DISPLACEMENTFLUX LINKAGE FORMULATION (pg. 747)  
E4 WORKENERGY RELATION DERIVED FROM LAGRANGE’S EQUATIONS (pg. 749)  
APPENDIX F: LAGRANGE’S FORM OF D’ALEMBERT’S PRINCIPLE (pg. 754)  
F1 FUNDAMENTAL CONCEPTS AND DERIVATIONS (pg. 754)  
F2 EXAMPLES (pg. 757)  
APPENDIX G: A BRIEF REVIEW OF ELECTROMAGNETIC (EM) THEORY AND APPROXIMATIONS (pg. 763)  
G1 MAXWELL’S EQUATIONS: COMPLETE FORM (pg. 763)  
G1.1 Integral Form (pg. 763)  
G1.2 Differential Form (pg. 765)  
G2 MAXWELL’S EQUATIONS: ELECTROSTATICS AND MAGNETOSTATICS (pg. 765)  
G3 MAXWELL’S EQUATIONS: ELECTROQUASISTATICS AND MAGNETOQUASISTATICS (pg. 766)  
G3.1 Electroquasistatics (pg. 766)  
G3.2 Magnetoquasistatics (pg. 768)  
G4 ENERGY STORAGE IN ELECTROQUASISTATICS AND MAGNETOQUASISTATICS (pg. 770)  
G4.1 Energy Storage in Electroquasistatics (pg. 771)  
G4.2 Energy Storage in Magnetoquasistatics (pg. 773)  
G5 KIRCHHOFF’S “LAWS” (pg. 774)  
G5.1 Kirchhoff’s Current “Law” (pg. 774)  
G5.2 Kirchhoff’s Voltage “Law” (pg. 775)  
G5.3 Summary (pg. 776)  
APPENDIX H: COMPLEX NUMBERS AND SOME USEFUL FORMULAS OF COMPLEX VARIABLES AND TRIGONOMETRY (pg. 777)  
H1 INTRODUCTION (pg. 777)  
H2 ELEMENTARY ALGEBRAIC OPERATIONS OF COMPLEX NUMBERS (pg. 780)  
H3 COMPLEX CONJUGATES (pg. 781)  
H4 A USEFUL FORMULA OF COMPLEX VARIABLES (pg. 782)  
H5 USE OF COMPLEX VARIABLES IN HARMONIC RESPONSE ANALYSES (pg. 787)  
H6 USEFUL FORMULAS OF TRIGONOMETRY (pg. 796)  
APPENDIX I: TEMPORAL FUNCITON FOR SYNCHRONOUS MOTION OF TQODEGREEOFFREEDOM SYSTEMS (pg. 800)  
I1 FREE UNDAMPED EQUATIONS OF MOTION (pg. 800)  
I2 SYNHRONUS MOTION (pg. 800)  
I3 GENERAL TEMPORAL SOLUTION (pg. 801)  
I4 SPECIAL (SEMIDEFINITE) TEMPORAL SOLUTION (pg. 803)  
I5 GENERALIZATION TO SYSTEMS HAVING MORE DEGREES OF FREEDOM (pg. 803)  
APPENDIX J: STABILITY ANALYSES OF NONLINEAR SYSTEMS (pg. 804)  
J1 STATESPACE STABILITY FORMULATION (pg. 804)  
J1.1 StateSpace Representation of Equations of Motion (pg. 804)  
J1.2 Equilibrium States (pg. 806)  
J1.3 Linearization about Equilibrium States (pg. 808)  
J1.4 Concept and Types of Stability (pg. 809)  
J1.5 Stability of Linearized Systems (pg. 811)  
J1.6 Local Stability of Nonlinear Systems (pg. 814)  
J1.7 Nonllunear Stability Analyses (pg. 815)  
J1.8 Summary of StateSpace Stability Analysis (pg. 815)  
J2 NONLINEAR STABILITY ANALYSIS FOR CONSERVATIVE SYSTEMS (pg. 816)  
APPENDIX K: STRAIN ENERGY FUNCTIONS (pg. 822)  
K1 CONCEPT (pg. 822)  
K2 STRAIN ENERGY DENSITY FUNCTION (pg. 822)  
K3 STRAIN ENERGY FUNCTION (pg. 824)  
K4 EXAMPLES (pg. 825)  
ANSWERS TO MOST OF THE ODDNUMBERED PROBLEMS (pg. 833)  
LIST OF TABLES (pg. 843)  
KEY DYNAMICAL PRINCIPLES, FORMULAS, & CONVERSION FACTORS (pg. 846)  
INDEX (pg. 851)  
DEDICATION (pg. iii)  
BESTOWAL (pg. v)  
ABOUT THE AUTHOR (pg. vii)  
ACKNOWLEDGMENTS (pg. ix)  
PREFACE (pg. xi)  
CONTENTS (pg. xix)  
1 OUR NICHE IN THE COSMOS (pg. 1)  
11 INTRODUCTION (pg. 1)  
12 WHY HISTORY? (pg. 3)  
13 IMPORTANCE OF MATHEMATICS IN THE DEVELOPMENT OF MECHANICS (pg. 3)  
14 OUR SOURCES FROM ANTIQUITY: GETTING THE MESSAGE FROM THERE TO HERE (pg. 4)  
14.1 Invention of Writing (pg. 5)  
14.2 Hieroglyphics (pg. 6)  
14.3 Cuneiform (pg. 7)  
14.4 Ancient Egyptian Papyri (pg. 7)  
14.5 Mesopotamian Clay Tablets (pg. 9)  
15 ANCIENT EGYPTIAN ASTRONOMY AND MATHEMATICS (pg. 9)  
15.1 Ancient Egyptian Astronomy (pg. 10)  
15.2 Ancient Egyptian Mathematics (pg. 11)  
16 MESOPOTAMIAN ASTRONOMY AND MATHEMATICS (pg. 14)  
16.1 Mesopotamian Astronomy (pg. 15)  
16.2 Mesopotamian Mathematics (pg. 15)  
17 MATHEMATICS OF THE MAYANS, INDIANS, ARABS, AND CHINESE (pg. 16)  
18 THE FIRST GREAT ENGINEERING SOCIETY (pg. 19)  
19 ADVERSE CRITICISM OF ANCIENT EGYPTIAN AND MESOPOTAMIAN MATHEMATICS (pg. 24)  
110 EVOLUTION THROUGH THE HELLENIC ERA (pg. 29)  
111 THE UNIFICATION OF CELESTIAL AND TERRESTRIAL MOTION (pg. 31)  
111.1 Celestial Motion (pg. 31)  
111.2 Terrestrial Motion (pg. 44)  
111.3 Unification (pg. 45)  
112 VARIATIONAL PRICIPLES IN DYNAMICS (pg. 47)  
113 THE INTERNATIONALISM OF DYNAMICS (pg. 52)  
114 OUR NICHE IN THE COSMOS (pg. 53)  
2 DESIGN, MODELING, AND FORMULATION OF EQUATIONS OF MOTION (pg. 55)  
21 INTRODUCTION (pg. 55)  
22 DESIGN AND MODELING (pg. 56)  
22.1 The Design Process (pg. 56)  
22.2 The Modeling Process (pg. 57)  
22.3 Our More Modest Goals (pg. 58)  
23 DIRECT AND INDIRECT APPROACHES FOR FORMULATION OF EQUATIONS OF MOTION (pg. 59)  
3 KINEMATICS (pg. 68)  
31 INTRODUCTION (pg. 68)  
32 POSITION, VELOCITY, AND ACCELERATION (pg. 69)  
33 PLANE KINEMATICS OF RIGID BONES (pg. 75)  
33.1 The General Motion of a Rigid Body (pg. 75)  
33.2 Types of Plane Motion of a Rigid Body (pg. 76)  
33.3 Angular Displacement, Angular Velocity, and Angular Acceleration (pg. 77)  
33.4 A Cautionary Note about Finite Rotations (pg. 83)  
34 TIME RATE OF CHANGE OF VECTOR IN ROTATING FRAME (pg. 85)  
35 KINEMATIC ANALYSIS UTILIZING INTERMEDIATE FRAMES (pg. 90)  
36 GENERALIZATIONS OF KINEMATIC EXPRESSIONS (pg. 108)  
4 MOMENTUM FORMULATION FOR SYSTEMS OF PARTICLES (pg. 135)  
41 INTRODUCTION (pg. 135)  
42 THE FUNDAMENTAL PHYSICS (pg. 136)  
42.1 Newton’s Laws of Motion (pg. 136)  
42.2 A Particle (pg. 137)  
42.3 Linear Momentum and Force (pg. 138)  
42.4 Inertial Reference Frames (pg. 139)  
42.5 The Universal Law of Gravitation (pg. 140)  
43 TORQUE AND ANGULAR MOMENTUM FOR A PARTICLE (pg. 141)  
44 FORMULATION OF EQUATIONS OF MOTION: EXAMPLES (pg. 144)  
44.1 Problems of Particle Dynamics of the First Kind (pg. 145)  
44.2 Problems of Particle Dynamics of the Second Kind (pg. 151)  
5 VARIATIONAL FORMULATION FOR SYSTEMS OF PARTICLES (pg. 179)  
51 INTRODUCTION (pg. 179)  
52 FORMULATION OF EQUATIONS OF MOTION (pg. 180)  
53 WORK AND STATE FUNCTIONS (pg. 181)  
53.1 Work (pg. 182)  
53.2 Kinetic State Functions (pg. 183)  
53.3 Potential State Functions (pg. 185)  
53.4 Energy and Coenergy (pg. 189)  
54 GENERALIZED VARIABLES AND VARIATIONAL CONCEPTS (pg. 190)  
54.1 Generalized Coordinates (pg. 190)  
54.2 Admissible Variations, Degrees of Freedom, Geometric Contraints, and Holonomicity (pg. 195)  
54.3 Variational Principles in Mechanics (pg. 201)  
54.4 Generalized Velocities and Generalized Forces for Holonomic Systems (pg. 205)  
55 EQUATIONS OF MOTION FOR HOLONOMIC MECHANICAL SYSTEMS VIA VARIATIONAL PRINCIPLES (pg. 213)  
56 WORKENERGY RELATION (pg. 238)  
57 NATURE OF LAGRANGIAN DYNAMICS (pg. 241)  
Problems for Chapter 5 (pg. 243)  
6 DYNAMICS OF SYSTEMS CONTAINING RIGID BODIES (pg. 268)  
61 INTRODUCTION (pg. 268)  
62 MOMENTUM PRINCIPLES FOR RIGID BODIES (pg. 269)  
62.1 Review of Solids in Equilibrium and Particle Dynamics (pg. 270)  
62.2 Models of Rigid Bodies (pg. 271)  
62.3 Momentum Principles for Extended Bodies: The NewtonEuler Equations (pg. 272)  
62.4 Momentum Principles for Rigid Bodies Modeled as Systems of Particles (pg. 273)  
62.5 Momentum Principles for Rigid Bodies Modeled as Continua (pg. 275)  
63 DYNAMIC PROPERTIES OF RIGID BODIES (pg. 279)  
63.1 The Inertia Factor (pg. 279)  
63.2 ParallelAxes Theorem (pg. 290)  
63.3 Principal Directions and Principal Moments of Inertia (pg. 296)  
63.4 Uses of Mass Symmetry (pg. 298)  
64 DYNAMICS OF RIGID BODIES VIA DIRECT APPROACH (pg. 303)  
65 LAGRANGIAN FOR RIGID BODIES (pg. 308)  
65.1 Kinetic Coenergy Function for Rigid Body (pg. 308)  
65.2 Potential Energy Function for Rigid Body (pg. 310)  
66 EQUATIONS OF MOTION FOR SYSTEMS CONTAINING RIGID BODIES IN PLANE MOTION (pg. 311)  
Problems for Chapter 6 (pg. 334)  
7 DYNAMICS OF ELECTRICAL AND ELECTROMECHANICAL SYSTEMS (pg. 366)  
71 INTRODUCTION (pg. 366)  
72 FORMULATION OF EQUATIONS OF MOTION FOR ELECTRICAL NETWORKS (pg. 369)  
73 CONSTITUTIVE RELATIONS FOR CIRCUIT ELEMENTS (pg. 371)  
73.1 Passive Elements (pg. 371)  
73.2 Active Electrical Elements (pg. 376)  
74 HAMILTON’S PRINCIPLE AND LAGRANGE’S EQUATIONS FOR ELECTRICAL NETWORKS (pg. 380)  
74.1 Generalized Charge Variables (pg. 380)  
74.2 Generalized Flux Linkage Variables (pg. 382)  
74.3 Work Expressions (pg. 383)  
74.4 Summary of LumpedParameter Offering of Variational Electricity (pg. 386)  
74.5 Examples (pg. 386)  
75 CONSTITUTIVE RELATIONS FOR TRANSDUCERS (pg. 407)  
75.1 Ideal MovablePlate Capacitor (pg. 408)  
75.2 Electrically Linear MoveablePlate Capacitor (pg. 410)  
75.3 Ideal MovableCore Inductor (pg. 412)  
75.4 Magnetically Linear MovableCore Inductor (pg. 413)  
76 HAMILTON’S PRINCIPLE AND LAGRANGE’S EQUATIONS FOR ELECTROMECHANICAL SYSTEMS (pg. 415)  
76.1 DisplacementCharge Variables Formulation (pg. 416)  
76.2 DisplacementFlux Linkage Variables Formulation (pg. 417)  
76.3 Examples (pg. 419)  
77 ANOTHER LOOK AT LAGRANGIAN DYNAMICS (pg. 428)  
Problems for Chapter 7 (pg. 429)  
8 VIBRATION OF LINEAR LUMPEDPARAMETER SYSTEMS (pg. 439)  
81 INTRODUCTION (pg. 439)  
82 SINGLEDEGREEOFFREEDOM FIRSTORDER SYSTEMS (pg. 440)  
82.1 Free Response (pg. 441)  
82.2 Step Response (pg. 444)  
82.3 Ramp Response (pg. 446)  
82.4 Harmonic Response (pg. 449)  
82.5 Summary of Responses for SingleDegreeofFreedom FirstOrder Systems (pg. 459)  
83 SINGLEDEGREEOFFREEDOM SECONDORDER SYSTEMS (pg. 460)  
83.1 Free Response (pg. 461)  
83.2 Natural Frequency via Static Deflection (pg. 467)  
83.3 Logarithmic Decrement (pg. 468)  
83.4 Energy Loss of Free Vibration (pg. 471)  
83.5 Harmonic Response (pg. 472)  
83.6 Summary of Responses for SingleDegreeofFreedom SecondOrder Systems (pg. 498)  
84 TWODEGREEOFFREEDOM SECONDORDER SYSTEMS (pg. 500)  
84.1 Natural Modes of Vibration (pg. 501)  
84.2 Response to Initial Conditions (pg. 514)  
84.3 Harmonic Response (pg. 527)  
85 STABILITY OF NONLINEAR SYSTEMS (pg. 541)  
Problems for Chapter 8 (pg. 557)  
9 DYNAMICS OF CONTINUOUS SYSTEMS (pg. 576)  
91 INTRODUCTION (pg. 576)  
92 EQUATIONS OF MOTION (pg. 578)  
92.1 Longitudinal Motion of System Containing Rod (pg. 579)  
92.2 Twisting Motion of System Containing Shaft (pg. 586)  
92.3 Electric Transmission Line (pg. 589)  
92.4 Flexural Motion of System Containing Beam (pg. 594)  
92.5 Summaries (pg. 602)  
93 NATURAL MODES OF VIBRATION (pg. 607)  
93.1 Method of Separation Variables (pg. 608)  
93.2 Time Response (pg. 610)  
93.3 Eigenfunctions for SecondOrder Systems (pg. 612)  
93.4 Eigenfunctions for FourthOrder Systems (pg. 620)  
93.5 General Solutions for Free Undamped Vibration (pg. 633)  
94 RESPONSE TO INITIAL CONDITIONS (pg. 636)  
94.1 An Example: Release of Compressed Rod (pg. 636)  
94.2 An Example: Shaft Stopped after Rotation (pg. 647)  
94.3 An Example: SlidingFree Beam Initially Bent (pg. 650)  
95 RESPONSE TO HARMONIC EXCITATIONS (pg. 660)  
95.1 An Example: Specified Harmonic Motion of Boundary (pg. 660)  
95.2 An Example: Distributed Harmonic Force (pg. 662)  
95.3 An Example: Harmonic Force on Boundary (pg. 665)  
96 SUMMARIES (pg. 672)  
Problems for Chapter 9 (pg. 673)  
BIBLIOGRAPHY (pg. 684)  
1 Historical (pg. 684)  
2 Astronomy (pg. 686)  
3 Design, Systems, and Modeling (pg. 686)  
4 Elementary Dynamics (pg. 686)  
5 Intermediate/Advanced Dynamics (pg. 686)  
6 Hamilton’s Law of Varying Action and Hamilton’s Principle (pg. 687)  
7 Electric and Electromechanical Systems (pg. 687)  
8 Vibration (pg. 687)  
APPENDIX A: FINITE ROTATION (pg. 688)  
A1 CHANGE IN POSITION VECTOR DUE TO FINITE ROTATION (pg. 688)  
A2 FINITE ROTATIONS ARE NOT VECTORS (pg. 690)  
A3 DO ROTATIONS EVER BEHAVE AS VECTORS? (pg. 692)  
A3.1 Infinitesimal Rotations Are Vectors (pg. 692)  
A3.2 Consecutive Finite Rotations about a Common Axis Are Vectors (pg. 692)  
APPENDIX B: GENERAL KINEMATIC ANALYSIS (pg. 694)  
B1 ALL ANGULAR VELOCITIES DEFINED WITH RESPECT TO FIXED REFERENCE FRAME (CASE 1) (pg. 694)  
B2 EACH ANGULAR VELOCITY DEFINED WITH RESPECT TO IMMEDIATELY PRECEDING FRAME (CASE 2) (pg. 698)  
APPENDIX C: MOMENTUM PRINCIPLES FOR SYSTEMS OF PARTICLES (pg. 705)  
C1 ASSERTED MOMENTUM PRINCIPLES (pg. 705)  
C2 PRINCIPLES FOR SINGLE PARTICLE (pg. 706)  
C3 PRINCIPLES FOR SYSTEM OF PARTICLES (pg. 707)  
C3.1 Asserted System Momentum Principles (pg. 708)  
C3.2 System Momentum Principles Derived from Particle Momentum Principles (pg. 709)  
C3.3 Conditions on Internal Forces (pg. 711)  
C3.4 Relationships between Momentum Principles and Conditions on Internal Forces (pg. 712)  
C3.5 Linear Momentum Principle in Terms of Centroidal Motion (pg. 714)  
C3.6 Angular Momentum Principle about Arbitrary Point (pg. 715)  
C3.7 System of Particle Model in Continuum Limit (pg. 717)  
C4 ANGULAR MOMENTUM PRINCIPLE IN NONINERTIAL INTERMEDIATE FRAME (pg. 719)  
APPENDIX D: ELEMENTARY RESULTS OF THE CALCULUS OF VARIATIONS (pg. 728)  
D1 INTRODUCTION (pg. 728)  
D2 SUMMARY OF ELEMENTARY RESULTS (pg. 730)  
D3 EULER EQUATION: NECESSARY CONDITION FOR A VARIATIONAL INDICATOR TO VANISH (pg. 734)  
APPENDIX E: SOME FORMULATIONS OF THE PRINCIPLES OF HAMILTON (pg. 737)  
E1 MECHANICAL FORMULATIONS (pg. 737)  
E1.1 Hamilton’s Law of Varying Action (pg. 740)  
E1.2 Hamilton’s Principle (pg. 741)  
E1.3 Lagrange’s Equations (pg. 742)  
E1.4 Discussion (pg. 743)  
E2 HAMILTON’S PRINCIPLE FOR ELECTROMECHANICAL SYSTEMS USING A DISPLACEMENTCHARGE FORMULATION (pg. 744)  
E3 HAMILTON’S PRINCIPLE FOR ELECTROMECHANICAL SYSTEMS USING A DISPLACEMENTFLUX LINKAGE FORMULATION (pg. 747)  
E4 WORKENERGY RELATION DERIVED FROM LAGRANGE’S EQUATIONS (pg. 749)  
APPENDIX F: LAGRANGE’S FORM OF D’ALEMBERT’S PRINCIPLE (pg. 754)  
F1 FUNDAMENTAL CONCEPTS AND DERIVATIONS (pg. 754)  
F2 EXAMPLES (pg. 757)  
APPENDIX G: A BRIEF REVIEW OF ELECTROMAGNETIC (EM) THEORY AND APPROXIMATIONS (pg. 763)  
G1 MAXWELL’S EQUATIONS: COMPLETE FORM (pg. 763)  
G1.1 Integral Form (pg. 763)  
G1.2 Differential Form (pg. 765)  
G2 MAXWELL’S EQUATIONS: ELECTROSTATICS AND MAGNETOSTATICS (pg. 765)  
G3 MAXWELL’S EQUATIONS: ELECTROQUASISTATICS AND MAGNETOQUASISTATICS (pg. 766)  
G3.1 Electroquasistatics (pg. 766)  
G3.2 Magnetoquasistatics (pg. 768)  
G4 ENERGY STORAGE IN ELECTROQUASISTATICS AND MAGNETOQUASISTATICS (pg. 770)  
G4.1 Energy Storage in Electroquasistatics (pg. 771)  
G4.2 Energy Storage in Magnetoquasistatics (pg. 773)  
G5 KIRCHHOFF’S “LAWS” (pg. 774)  
G5.1 Kirchhoff’s Current “Law” (pg. 774)  
G5.2 Kirchhoff’s Voltage “Law” (pg. 775)  
G5.3 Summary (pg. 776)  
APPENDIX H: COMPLEX NUMBERS AND SOME USEFUL FORMULAS OF COMPLEX VARIABLES AND TRIGONOMETRY (pg. 777)  
H1 INTRODUCTION (pg. 777)  
H2 ELEMENTARY ALGEBRAIC OPERATIONS OF COMPLEX NUMBERS (pg. 780)  
H3 COMPLEX CONJUGATES (pg. 781)  
H4 A USEFUL FORMULA OF COMPLEX VARIABLES (pg. 782)  
H5 USE OF COMPLEX VARIABLES IN HARMONIC RESPONSE ANALYSES (pg. 787)  
H6 USEFUL FORMULAS OF TRIGONOMETRY (pg. 796)  
APPENDIX I: TEMPORAL FUNCITON FOR SYNCHRONOUS MOTION OF TQODEGREEOFFREEDOM SYSTEMS (pg. 800)  
I1 FREE UNDAMPED EQUATIONS OF MOTION (pg. 800)  
I2 SYNHRONUS MOTION (pg. 800)  
I3 GENERAL TEMPORAL SOLUTION (pg. 801)  
I4 SPECIAL (SEMIDEFINITE) TEMPORAL SOLUTION (pg. 803)  
I5 GENERALIZATION TO SYSTEMS HAVING MORE DEGREES OF FREEDOM (pg. 803)  
APPENDIX J: STABILITY ANALYSES OF NONLINEAR SYSTEMS (pg. 804)  
J1 STATESPACE STABILITY FORMULATION (pg. 804)  
J1.1 StateSpace Representation of Equations of Motion (pg. 804)  
J1.2 Equilibrium States (pg. 806)  
J1.3 Linearization about Equilibrium States (pg. 808)  
J1.4 Concept and Types of Stability (pg. 809)  
J1.5 Stability of Linearized Systems (pg. 811)  
J1.6 Local Stability of Nonlinear Systems (pg. 814)  
J1.7 Nonllunear Stability Analyses (pg. 815)  
J1.8 Summary of StateSpace Stability Analysis (pg. 815)  
J2 NONLINEAR STABILITY ANALYSIS FOR CONSERVATIVE SYSTEMS (pg. 816)  
APPENDIX K: STRAIN ENERGY FUNCTIONS (pg. 822)  
K1 CONCEPT (pg. 822)  
K2 STRAIN ENERGY DENSITY FUNCTION (pg. 822)  
K3 STRAIN ENERGY FUNCTION (pg. 824)  
K4 EXAMPLES (pg. 825)  
ANSWERS TO MOST OF THE ODDNUMBERED PROBLEMS (pg. 833)  
LIST OF TABLES (pg. 843)  
KEY DYNAMICAL PRINCIPLES, FORMULAS, & CONVERSION FACTORS (pg. 846)  
INDEX (pg. 851) 
James H. Williams Jr.
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