Preface xi
1 Systems with One Degree of Freedom 1
1.1 Examples of One-Degree Systems 1
1.2 Undamped Free Translational Vibrations 2
1.3 Rotational Vibrations 12
1.4 Energy Method 24
1.5 Rayleigh’s Method 24
1.6 Forced Vibrations: Steady State 39
1.7 Forced Vibrations with Viscous Damping 52
1.8 Free Vibrations with Viscous Damping 61
1.9 Forced Vibrations with Viscous Damping 61
1.10 Equivalent Viscous Damping 69
1.11 General Periodic Forcing Functions 76
1.12 Arbitrary Forcing Functions 84
1.13 Arbitrary Support Motions 93
1.14 Response Spectra 99
1.15 Step-by-Step Response Calculations 107
References 113
Problems 114
2 Systems with Nonlinear Characteristics 139
2.1 Examples of Nonlinear Systems 139
2.2 Direct Integration for Velocity and Period 149
2.3 Approximate Methods for Free Vibrations 157
2.4 Forced Nonlinear Vibrations 166
2.5 Piecewise-Linear Systems 175
2.6 Numerical Solutions for Nonlinear Systems 190
References 207
Problems 208
3 Systems with Two Degrees of Freedom 217
3.1 Examples of Two-Degree Systems 217
3.2 Action Equations: Stiffness Coefficients 223
3.3 Displacement Equations: Flexibility Coefficients 225
3.4 Inertial and Gravitational Coupling 233
3.5 Undamped Free Vibrations 241
3.6 Undamped Forced Vibrations 251
3.7 Free Vibrations with Viscous Dampling 260
3.8 Forced Vibrations with Viscous Dampling 265
References 267
Problems 267
4 Systems with Multiple Degrees of Freedom 275
4.1 Introduction 275
4.2 Frequencies and Mode Shapes for Undamped Systems 276
4.3 Principal and Normal Coordinates 287
4.4 Normal-Mode Response to Initial Conditions 295
4.5 Normal-Mode Response to Applied Actions 301
4.6 Normal-Mode Response to Support Motions 309
4.7 Iteration Method for Frequencies and Mode Shapes 318
4.8 Damping in Multidegree Systems 333
4.9 Damped Response to Periodic Forcing Functions 337
4.10 Transient Response of Damped Systems 343
4.11 Step-by-Step Response Calculations 347
References 352
Problems 352
5 Continua with Infinite Degrees of Freedom 363
5.1 Introduction 363
5.2 Free Longitudinal Vibrations of Prismatic Bars 364
5.3 Forced Longitudinal Response of Prismatic Bars 373
5.4 Normal-Mode Method for Prismatic Bars 380
5.5 Prismatic Bar with a Mass or Spring at the End 387
5.6 Bars Subjected to Longitudinal Support Motions 395
5.7 Torsional Vibrations of Circular Shafts 401
5.8 Transverse Vibrations of Stretched Wires 409
5.9 Transverse Vibrations of Prismatic Beams 416
5.10 Transverse Vibrations of a Simple Beam 422
5.11 Vibrations of Beams with Other End Conditions 425
5.12 Effects of Rotary Inertia and Shearing Deformations 433
5.13 Forced Response of a Simple Beam 436
5.14 Forced Response of Beams with Other End Conditions 442
5.15 Beams Subjected to Support Motions 444
5.16 Beams Traversed by Moving Loads 448
5.17 Effect of Axial Force on Vibrations of Beams 454
5.18 Beams on Eastic Supports or Elastic Foundations 456
5.19 Ritz Method for Calculating Frequencies 461
5.20 Vibrations of Nonprismatic Beams 466
5.21 Coupled Flexural and Torsional Vibrations of Beams 474
5.22 Vibrations of Circular Rings 478
5.23 Transverse Vibrations of Membranes 495
5.24 Transverse Vibrations of Plates 495
References 505
Problems 506
6 Finite-Element Method for Discretized Continua 511
6.1 Introduction 511
6.2 Stresses and Strains in Continua 513
6.3 Equations of Motion for Finite Elements 516
6.4 One-Dimensional Elements 520
6.5 Vibrations of Beams by Finite Elements 534
References 542
Problems 543
Bibliography 551
Appendix A Systems of Units and Material Properties 553
A.1 Systems of Units 553
A.2 Material Properties 555
Appendix B Computer Programs 557
B.1 Introduction 557
B.2 Step-by-Step Solutions for Linear One-Degree Systems 558
B.3 Numerical Solutions for Nonlinear One-Degree Systems
B.4 Iteration of Eigenvalues and Eigenvectors 562
B.5 Step-by-Step Solutions for Linear Multidegree Systems 565
B.6 Program Notation 567
Flowcharts for Programs 569
Answers to Problems 587
Index 603
