Mechanical vibrations : theory and applications

Cover
......Page 1
Title Page
......Page 5
Copyright
......Page 6
About the Author......Page 7
Preface to the SI Edition......Page 8
Preface......Page 9
Contents......Page 11
1.1 The Study of Vibrations......Page 21
1.2 Mathematical Modeling......Page 24
1.3 Generalized Coordinates......Page 27
1.5 Dimensional Analysis......Page 31
1.6 Simple Harmonic Motion......Page 34
1.7 Review of Dynamics......Page 36
1.8 Two Benchmark Examples......Page 47
1.9 Further Examples......Page 49
1.10 Summary......Page 54
Problems......Page 57
2.1 Introduction......Page 75
2.2 Springs......Page 76
2.3 Springs in Combination......Page 82
2.4 Other Sources of Potential Energy......Page 88
2.5 Viscous Damping......Page 91
2.6 Energy Dissipated by Viscous Damping......Page 94
2.7 Inertia Elements......Page 96
2.8 External Sources......Page 104
2.9 Free-Body Diagram Method......Page 107
2.10 Static Deflections and Gravity......Page 114
2.11 Small Angle or Displacement Assumption......Page 117
2.12 Equivalent Systems Method......Page 120
2.13 Benchmark Examples......Page 126
2.14 Further Examples......Page 128
2.15 Chapter Summary......Page 136
Problems......Page 139
3.1 Introduction......Page 157
3.2 Standard Form of Differential Equation......Page 158
3.3 Free Vibrations of an Undamped System......Page 160
3.4 Underdamped Free Vibrations......Page 167
3.5 Critically Damped Free Vibrations......Page 174
3.6 Overdamped Free Vibrations......Page 176
3.7 Coulomb Damping......Page 180
3.8 Hysteretic Damping......Page 187
3.9 Other Forms of Damping......Page 191
3.10 Benchmark Examples......Page 194
3.11 Further Examples......Page 198
3.12 Chapter Summary......Page 205
Problems......Page 208
4.1 Introduction......Page 225
4.2 Forced Response of an Undamped System Due to a Single-Frequency Excitation......Page 228
4.3 Forced Response of a Viscously Damped System Subject to a Single-Frequency Harmonic Excitation......Page 234
4.4 Frequency-Squared Excitations......Page 240
4.5 Response due to Harmonic Excitation of Support......Page 248
4.6 Vibration Isolation......Page 254
4.7 Vibration Isolation from Frequency-Squared Excitations......Page 258
4.8 Practical Aspects of Vibration Isolation......Page 261
4.9 Multifrequency Excitations......Page 264
4.10 General Periodic Excitations......Page 266
4.11 Seismic Vibration Measuring Instruments......Page 275
4.12 Complex Representations......Page 279
4.13 Systems with Coulomb Damping......Page 280
4.14 Systems with Hysteretic Damping......Page 285
4.15 Energy Harvesting......Page 288
4.16 Benchmark Examples......Page 293
4.17 Further Examples......Page 301
4.18 Chapter Summary......Page 309
Problems......Page 313
5.1 Introduction......Page 333
5.2 Derivation of Convolution Integral......Page 335
5.3 Response due to a General Excitation......Page 338
5.4 Excitations Whose Forms Change at Discrete Times......Page 343
5.5 Transient Motion due to Base Excitation......Page 350
5.6 Laplace Transform Solutions......Page 352
5.7 Transfer Functions......Page 357
5.8 Numerical Methods......Page 360
5.9 Shock Spectrum......Page 370
5.10 Vibration Isolation for Short Duration Pulses......Page 377
5.11 Benchmark Examples......Page 381
5.12 Further Examples......Page 385
5.13 Chapter Summary......Page 390
Problems......Page 392
6.1 Introduction......Page 403
6.2 Derivation of the Equations of Motion......Page 404
6.3 Natural Frequencies and Mode Shapes......Page 408
6.4 Free Response of Undamped Systems......Page 413
6.5 Free Vibrations of a System with Viscous Damping......Page 416
6.6 Principal Coordinates......Page 418
6.7 Harmonic Response of Two Degree-Of-Freedom Systems......Page 421
6.8 Transfer Functions......Page 424
6.9 Sinusoidal Transfer Function......Page 428
6.10 Frequency Response......Page 431
6.11 Dynamic Vibration Absorbers......Page 434
6.12 Damped Vibration Absorbers......Page 440
6.13 Vibration Dampers......Page 444
6.14 Benchmark Examples......Page 445
6.15 Further Examples......Page 452
6.16 Chapter Summary......Page 462
Problems......Page 464
7.1 Introduction......Page 479
7.2 Derivation of Differential Equations Using the Free-Body Diagram Method......Page 481
7.3 Lagrange'
s Equations......Page 487
7.4 Matrix Formulation of Differential Equations for Linear Systems......Page 498
7.5 Stiffness Influence Coefficients......Page 503
7.6 Flexibility Influence Coefficients......Page 511
7.7 Inertia Influence Coefficients......Page 517
7.8 Lumped-Mass Modeling of Continuous Systems......Page 519
7.9 Benchmark Examples......Page 522
7.10 Further Examples......Page 528
7.11 Summary......Page 537
Problems......Page 539
8.1 Introduction......Page 553
8.2 Normal-Mode Solution......Page 554
8.3 Natural Frequencies and Mode Shapes......Page 556
8.4 General Solution......Page 563
8.5 Special Cases......Page 565
8.6 Energy Scalar Products......Page 572
8.7 Properties of Natural Frequencies and Mode Shapes......Page 575
8.8 Normalized Mode Shapes......Page 578
8.9 Rayleigh's Quotient......Page 580
8.10 Principal Coordinates......Page 582
8.11 Determination of Natural Frequencies and Mode Shapes......Page 585
8.12 Proportional Damping......Page 588
8.13 General Viscous Damping......Page 591
8.14 Benchmark Examples......Page 594
8.15 Further Examples......Page 598
8.16 Summary......Page 603
Problems......Page 605
9.1 Introduction......Page 613
9.2 Harmonic Excitations......Page 614
9.3 Laplace Transform Solutions......Page 619
9.4 Modal Analysis for Undamped Systems and Systems with Proportional Damping......Page 623
9.5 Modal Analysis for Systems with General Damping......Page 631
9.6 Numerical Solutions......Page 634
9.7 Benchmark Examples......Page 635
9.8 Further Examples......Page 640
9.9 Chapter Summary......Page 643
Problems......Page 645
10.1 Introduction......Page 653
10.2 General Method......Page 656
10.3 Second-Order Systems: Torsional Oscillations of a Circular Shaft......Page 659
10.4 Transverse Beam Vibrations......Page 671
10.5 Energy Methods......Page 687
10.6 Benchmark Examples......Page 692
10.7 Chapter Summary......Page 696
Problems......Page 698
11.1 Introduction......Page 709
11.2 Assumed Modes Method......Page 710
11.3 General Method......Page 713
11.4 The Bar Element......Page 716
11.5 Beam Element......Page 720
11.6 Global Matrices......Page 725
11.7 Benchmark Example......Page 729
11.8 Further Examples......Page 734
11.9 Summary......Page 746
Problems......Page 748
12.1 Introduction......Page 757
12.2 Sources of Nonlinearity......Page 758
12.3 Qualitative Analysis of Nonlinear Systems......Page 763
12.4 Quantitative Methods of Analysis......Page 767
12.5 Free Vibrations of SDOF Systems......Page 769
12.6 Forced Vibrations of SDOF Systems with Cubic Nonlinearities......Page 773
12.7 MDOF Systems......Page 779
12.8 Continuous Systems......Page 780
12.9 Chaos......Page 781
12.10 Chapter Summary......Page 789
Problems......Page 790
13.1 Introduction......Page 801
13.2 Behavior of a Random Variable......Page 802
13.3 Functions of a Random Variable......Page 804
13.4 Joint Probability Distributions......Page 813
13.5 Fourier Transforms......Page 817
13.6 Power Spectral Density......Page 823
13.7 Mean Square Value of the Response......Page 828
13.8 Benchmark Example......Page 832
13.9 Summary......Page 834
Problems......Page 837
Appendix A Unit Impulse Function and Unit Step Function......Page 845
Appendix B Laplace Transforms......Page 847
Appendix C Linear Algebra......Page 853
Appendix D Deflection of Beams Subject to Concentrated Loads......Page 862
Appendix E Integrals Used in Random Vibrations......Page 866
Appendix F Vibes......Page 867
References......Page 871
Index......Page 873