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CHAPTER 1 Fundamentals of Vibration1

1.1 Preliminary Remarks2

1.2 Brief History of the Study of Vibration3

1.2.1 Origins of the Study of Vibration3

1.2.2 From Galileo to Rayleigh5

1.2.3 Recent Contributions8

1.3 Importance of the Study of Vibration10

1.3.1 Conversion of Vibrations to Sound by the Human Ear12

1.4 Basic Concepts of Vibration15

1.4.1 Vibration15

1.4.2 Elementary Parts of Vibrating Systems15

1.4.3 Number of Degrees of Freedom16

1.4.4 Discrete and Continuous Systems18

1.5 Classification of Vibration18

1.5.1 Free and Forced Vibration18

1.5.2 Undamped and Damped Vibration19

1.5.3 Linear and Nonlinear Vibration19

1.5.4 Deterministic and Random Vibration19

1.6 Vibration Analysis Procedure20

1.7 Spring Elements24

1.7.1 Nonlinear Springs25

1.7.2 Linearization of a Nonlinear Spring27

1.7.3 Spring Constants of Elastic Elements29

1.7.4 Combination of Springs32

1.7.5 Spring Constant Associated with the Restoring Force due to Gravity40

1.8 Mass or Inertia Elements41

1.8.1 Combination of Masses42

1.9 Damping Elements46

1.9.1 Construction of Viscous Dampers47

1.9.2 Linearization of a Nonlinear Damper53

1.9.3 Combination of Dampers53

1.10 Harmonic Motion55

1.10.1 Vectorial Representation of Harmonic Motion57

1.10.2 Complex-Number Representation of Harmonic Motion58

1.10.3 Complex Algebra59

1.10.4 Operations on Harmonic Functions59

1.10.5 Definitions and Terminology62

1.11 Harmonic Analysis65

1.11.1 Fourier Series Expansion65

1.11.2 Complex Fourier Series67

1.11.3 Frequency Spectrum68

1.11.4 Time- and Frequency-Domain Representations69

1.11.5 Even and Odd Functions70

1.11.6 Half-Range Expansions72

1.11.7 Numerical Computation of Coefficients73

1.12 Examples Using MATLAB77

1.13 Vibration Literature81

Chapter Summary82

References82

Review Questions84

Problems88

Design Projects121

CHAPTER 2 Free Vibration of Single-Degree-of-Freedom Systems125

2.1 Introduction127

2.2 Free Vibration of an Undamped Translational System130

2.2.1 Equation of Motion Using Newton’s Second Law of Motion130

2.2.2 Equation of Motion Using Other Methods131

2.2.3 Equation of Motion of a Spring-Mass System in Vertical Position133

2.2.4 Solution134

2.2.5 Harmonic Motion135

2.3 Free Vibration of an Undamped Torsional System148

2.3.1 Equation of Motion149

2.3.2 Solution150

2.4 Response of First-Order Systems and Time Constant153

2.5 Rayleigh’s Energy Method155

2.6 Free Vibration with Viscous Damping160

2.6.1 Equation of Motion160

2.6.2 Solution161

2.6.3 Logarithmic Decrement170

2.6.4 Energy Dissipated in Viscous Damping171

2.6.5 Torsional Systems with Viscous Damping173

2.7 Graphical Representation of Characteristic Roots and Corresponding Solutions179

2.7.1 Roots of the Characteristic Equation179

2.7.2 Graphical Representation of Roots and Corresponding Solutions180

2.8 Parameter Variations and Root Locus Representations181

2.8.1 Interpretations of ωn，ωd，ζ，and τ in the s-plane181

2.8.2 Root Locus and Parameter Variations184

2.9 Free Vibration with Coulomb Damping190

2.9.1 Equation of Motion191

2.9.2 Solution192

2.9.3 Torsional Systems with Coulomb Damping195

2.10 Free Vibration with Hysteretic Damping197

2.11 Stability of Systems203

2.12 Examples Using MATLAB207

Chapter Summary213

References214

Review Questions214

Problems219

Design Projects266

CHAPTER3 Harmonically Excited Vibration269

3.1 Introduction271

3.2 Equation of Motion271

3.3 Response of an Undamped System Under Harmonic Force273

3.3.1 Total Response277

3.3.2 Beating Phenomenon277

3.4 Response of a Damped System Under Harmonic Force281

3.4.1 Total Response284

3.4.2 Quality Factor and Bandwidth288

3.5 Response of a Damped System Under F（t） = F0eiωt289

3.6 Response of a Damped System Under the Harmonic Motion of the Base292

3.6.1 Force Transmitted294

3.6.2 Relative Motion295

3.7 Response of a Damped System Under Rotating Unbalance298

3.8 Forced Vibration with Coulomb Damping304

3.9 Forced Vibration with Hysteresis Damping309

3.10 Forced Motion with Other Types of Damping311

3.11 Self-Excitation and Stability Analysis312

3.11.1 Dynamic Stability Analysis312

3.11.2 Dynamic Instability Caused by Fluid Flow316

3.12 Transfer-Function Approach324

3.13 Solutions Using Laplace Transforms328

3.14 Frequency Transfer Functions331

3.14.1 Relation between the General Transfer Function T（s） and the Frequency Transfer Function T（iω）333

3.14.2 Representation of Frequency-Response Characteristics334

3.15 Examples Using MATLAB337

Chapter Summary343

References343

Review Questions344

Problems347

Design Projects374

CHAPTER 4 Vibration Under General Forcing Conditions375

4.1 Introduction376

4.2 Response Under a General Periodic Force377

4.2.1 First-Order Systems378

4.2.2 Second-Order Systems384

4.3 Response Under a Periodic Force of Irregular Form390

4.4 Response Under a Nonperiodic Force392

4.5 Convolution Integral393

4.5.1 Response to an Impulse394

4.5.2 Response to a General Forcing Condition397

4.5.3 Response to Base Excitation398

4.6 Response Spectrum406

4.6.1 Response Spectrum for Base Excitation408

4.6.2 Earthquake Response Spectra411

4.6.3 Design Under a Shock Environment415

4.7 Laplace Transforms418

4.7.1 Transient and Steady-State Responses418

4.7.2 Response of First-Order Systems419

4.7.3 Response of Second-Order Systems421

4.7.4 Response to Step Force426

4.7.5 Analysis of the Step Response432

4.7.6 Description of Transient Response433

4.8 Numerical Methods439

4.8.1 Runge-Kutta Methods441

4.9 Response to Irregular Forcing Conditions Using Numerical Methods443

4.10 Examples Using MATLAB448

Chapter Summary452

References452

Review Questions453

Problems456

Design Projects478

CHAPTER5 Two-Degree-of-Freedom Systems481

5.1 Introduction482

5.2 Equations of Motion for ForcedV ibration486

5.3 Free-Vibration Analysis of an Undamped System488

5.4 Torsional System497

5.5 Coordinate Coupling and Principal Coordinates502

5.6 Forced-Vibration Analysis508

5.7 Semidetinite Systems511

5.8 Self-Excitation and Stability Analysis514

5.9 Transfer-Function Approach516

5.10 Solutions Using Laplace Transform518

5.11 Solutions Using Frequency Transfer Functions526

5.12 Examples Using MATLAB529

Chapter Summary536

References537

Review Questions537

Problems540

Design Projects566

CHAPTER 6 Multidegree-of-Freedom Systems568

6.1 Introduction570

6.2 Modeling of Continuous Systems as Multidegree-of-Freedom Systems570

6.3 Using Newton’s Second Law to Derive Equations of Motion572

6.4 Influence Coefficients577

6.4.1 Stiffness Influence Coefficients577

6.4.2 Flexibility Influence Coefficients582

6.4.3 Inertia Influence Coefficients587

6.5 Potential and Kinetic Energy Expressions in Matrix Form589

6.6 Generalized Coordinates and Generalized Forces591

6.7 Using Lagrange’s Equations to Derive Equations of Motion592

6.8 Equations of Motion of Undamped Systems in Matrix Form596

6.9 Eigenvalue Problem598

6.10 Solution of the Eigenvalue Problem600

6.10.1 Solution of the Characteristic（Polynomial）Equation600

6.10.2 Orthogonality of Normal Modes606

6.10.3 Repeated Eigenvalues609

6.11 Expansion Theorem611

6.12 Unrestrained Systems611

6.13 Free Vibration of Undamped Systems616

6.14 Forced Vibration of Undamped Systems Using Modal Analysis618

6.15 Forced Vibration of Viscously Damped Systems625

6.16 Self-Excitation and Stability Analysis632

6.17 Examples Using MATLAB634

Chapter Summary642

References642

Review Questions643

Problems647

Design Projects668

CHAPTER7 Determination of Natural Frequencies and Mode Shapes671

7.1 Introduction672

7.2 Dunkerley’s Formula673

7.3 Rayleigh’s Method675

7.3.1 Properties of Rayleigh’s Quotient676

7.3.2 Computation of the Fundamental Natural Frequency678

7.3.3 Fundamental Frequency of Beams and Shafts680

7.4 Holzer’s Method683

7.4.1 Torsional Systems683

7.4.2 Spring-Mass Systems686

7.5 Matrix Iteration Method687

7.5.1 Convergence to the Highest Natural Frequency689

7.5.2 Computation of Intermediate Natural Frequencies690

7.6 Jacobi’s Method695

7.7 Standard Eigenvalue Problem697

7.7.1 Choleski Decomposition698

7.7.2 Other Solution Methods700

7.8 Examples Using MATLAB700

Chapter Summary703

References703

Review Questions705

Problems707

Design Projects716

CHAPTER 8 Continuous Systems717

8.1 Introduction718

8.2 Transverse Vibration of a String or Cable719

8.2.1 Equation of Motion719

8.2.2 Initial and Boundary Conditions721

8.2.3 Free Vibration of a Uniform String722

8.2.4 Free Vibration of a String with Both Ends Fixed723

8.2.5 Traveling-Wave Solution727

8.3 Longitudinal Vibration of a Bar or Rod728

8.3.1 Equation of Motion and Solution728

8.3.2 Orthogonality of Normal Functions731

8.4 Torsional Vibration of a Shaft or Rod736

8.5 Lateral Vibration of Beams739

8.5.1 Equation of Motion739

8.5.2 Initial Conditions741

8.5.3 Free Vibration741

8.5.4 Boundary Conditions742

8.5.5 Orthogonality of Normal Functions744

8.5.6 Forced Vibration748

8.5.7 Effect of Axial Force750

8.5.8 Effects of Rotary Inertia and Shear Deformation752

8.5.9 Beams on Elastic Foundation757

8.5.10 Other Effects760

8.6 Vibration of Membranes760

8.6.1 Equation of Motion760

8.6.2 Initial and Boundary Conditions762

8.7 Rayleigh’s Method763

8.8 The Rayleigh-Ritz Method766

8.9 Examples Using MATLAB769

Chapter Summary772

References772

Review Questions774

Problems777

Design Project790

CHAPTER 9 Vibration Control791

9.1 Introduction792

9.2 Vibration Nomograph and Vibration Criteria793

9.3 Reduction of Vibration at the Source797

9.4 Balancing of Rotating Machines798

9.4.1 Single-Plane Balancing798

9.4.2 Two-Plane Balancing801

9.5 Whirling of Rotating Shafts807

9.5.1 Equations of Motion807

9.5.2 Critical Speeds809

9.5.3 Response of the System810

9.5.4 Stability Analysis812

9.6 Balancing of Reciprocating Engines814

9.6.1 Unbalanced Forces Due to Fluctuations in Gas Pressure814

9.6.2 Unbalanced Forces Due to Inertia of the Moving Parts815

9.6.3 Balancing of Reciprocating Engines818

9.7 Control of Vibration820

9.8 Control of Natural Frequencies820

9.9 Introduction of Damping821

9.10 Vibration Isolation823

9.10.1 Vibration Isolation System with Rigid Foundation826

9.10.2 Vibration Isolation System with Base Motion836

9.10.3 Vibration Isolation System with Flexible Foundation844

9.10.4 Vibration Isolation System with Partially Flexible Foundation846

9.10.5 Shock Isolation847

9.10.6 Active Vibration Control850

9.11 Vibration Absorbers855

9.11.1 Undamped Dynamic Vibration Absorber856

9.11.2 Damped Dynamic Vibration Absorber863

9.12 Examples Using MATLAB867

Chapter Summary875

References875

Review Questions877

Problems879

Design Project894

CHAPTER 10 Vibration Measurement and Applications896

10.1 Introduction897

10.2 Transducers899

10.2.1 Variable-Resistance Transducers899

10.2.2 Piezoelectric Transducers902

10.2.3 Electrodynamic Transducers903

10.2.4 Linear Variable Differential Transformer Transducer904

10.3 Vibration Pickups905

10.3.1 Vibrometer907

10.3.2 Accelerometer908

10.3.3 Velometer912

10.3.4 Phase Distortion914

10.4 Frequency-Measuring Instruments916

10.5 Vibration Exciters918

10.5.1 Mechanical Exciters918

10.5.2 Electrodynamic Shaker919

10.6 Signal Analysis921

10.6.1 Spectrum Analyzers922

10.6.2 Bandpass Filter923

10.6.3 Constant-Percent Bandwidth and Constant-Bandwidth Analyzers924

10.7 Dynamic Testing of Machines and Structures926

10.7.1 Using Operational Deflection-Shape Measurements926

10.7.2 Using Modal Testing926

10.8 Experimental Modal Analysis926

10.8.1 The Basic Idea926

10.8.2 The Necessary Equipment926

10.8.3 Digital Signal Processing929

10.8.4 Analysis of Random Signals931

10.8.5 Determination of Modal Data from Observed Peaks933

10.8.6 Determination of Modal Data from Nyquist Plot936

10.8.7 Measurement of Mode Shapes938

10.9 Machine-Condition Monitoring and Diagnosis941

10.9.1 Vibration Severity Criteria941

10.9.2 Machine Maintenance Techniques941

10.9.3 Machine-Condition Monitoring Techniques942

10.9.4 Vibration Monitoring Techniques944

10.9.5 Instrumentation Systems950

10.9.6 Choice of Monitoring Parameter950

10.10 Examples Using MATLAB951

Chapter Summary954

References954

Review Questions956

Problems958

Design Projects964

CHAPTER 11 Numerical Integration Methods in Vibration Analysis965

11.1 Introduction966

11.2 Finite Difference Method967

11.3 Central Difference Method for Single-Degree-of-Freedom Systems968

11.4 Runge-Kutta Method for Single-Degree-of-Freedom Systems971

11.5 Central Difference Method for Multidegree-of-Freedom Systems973

11.6 Finite Difference Method for Continuous Systems977

11.6.1 Longitudinal Vibration of Bars977

11.6.2 Transverse Vibration of Beams981

11.7 Runge-Kutta Method for Multidegree-of-Freedom Systems986

11.8 Houbolt Method988

11.9 Wilson Method991

11.10 Newmark Method994

11.11 Examples Using MATLAB998

Chapter Summary1004

References1004

Review Questions1005

Problems1007

CHAPTER 12 Finite Element Method1013

12.1 Introduction1014

12.2 Equations of Motion of an Element1015

12.3 Mass Matrix，Stiffness Matrix，and Force Vector1017

12.3.1 Bar Element1017

12.3.2 Torsion Element1020

12.3.3 Beam Element1021

12.4 Transformation of Element Matrices and Vectors1024

12.5 Equations of Motion of the Complete System of Finite Elements1027

12.6 Incorporation of Boundary Conditions1029

12.7 Consistent- and Lumped-Mass Matrices1038

12.7.1 Lumped-Mass Matrix for a Bar Element1038

12.7.2 Lumped-Mass Matrix for a Beam Element1038

12.7.3 Lumped-Mass Versus Consistent-Mass Matrices1039

12.8 Examples Using MATLAB1041

Chapter Summary1045

References1045

Review Questions1046

Problems1048

Design Projects1060

APPENDIX A1064

Mathematical Relations and Material Properties1064

APPENDIX B1067

Deflection of Beams and Plates1067

APPENDIX C1069

Matrices1069

APPENDIX D1076

Laplace Transform1076

APPENDIX E1084

Units1084

APPENDIX F1087

Introduction to MATLAB1087

Answers to Selected Problems1097

Index1106