Vibration of Mechanical Systems

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
CONTENTS......Page 9
PREFACE......Page 15
ORGANIZATION OF THE BOOK......Page 16
1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION......Page 19
1.1 DEGREES OF FREEDOM......Page 21
Pure Translational Motion......Page 23
Planar Motion (Combined Rotation and Translation) of a Rigid Body......Page 24
Pure Translational Motion......Page 26
Pure Rotational Motion......Page 27
Pure Translational Motion......Page 28
Pure Rotational Motion......Page 29
1.3 EQUIVALENT MASS, EQUIVALENT STIFFNESS, AND EQUIVALENT DAMPING CONSTANT FOR AN SDOF SYSTEM......Page 30
1.3.1 A Rotor–Shaft System......Page 31
1.3.2 Equivalent Mass of a Spring......Page 32
Springs in Series......Page 34
Springs in Parallel......Page 35
1.3.4 An SDOF System with Two Springs and Combined Rotational and Translational Motion......Page 37
Dampers in Series......Page 40
Dampers in Parallel......Page 41
1.4.1 Differential Equation of Motion......Page 43
Energy Approach......Page 45
Energy Method......Page 48
Energy Method......Page 50
1.4.2 Solution of the Differential Equation of Motion Governing Free Vibration of an Undamped Spring–Mass System......Page 52
1.5.1 Differential Equation of Motion......Page 58
1.5.2 Solution of the Differential Equation of Motion Governing Free Vibration of a Damped Spring–Mass System......Page 59
Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc)......Page 60
Case II: Critically Damped (ξ = 1 or ceq = cc)......Page 63
Case III: Overdamped (ξ > 1 or ceq > cc)......Page 64
1.5.3 Logarithmic Decrement: Identification of Damping Ratio from\n\nFree Response of an Underdamped System (0 < ξ < 1)......Page 69
Solution......Page 73
1.6 STABILITY OF AN SDOF SPRING–MASS–DAMPER SYSTEM......Page 76
EXERCISE PROBLEMS......Page 81
2.1 RESPONSES OF UNDAMPED AND DAMPED SDOF SYSTEMS TO A CONSTANT FORCE......Page 90
Case I: Undamped (ξ = 0) and Underdamped (0 < ξ < 1)......Page 92
Case II: Critically Damped (ξ = 1 or ceq = cc)......Page 93
Case III: Overdamped (ξ > 1 or ceq > cc)......Page 94
2.2 RESPONSE OF AN UNDAMPED SDOF SYSTEM TO A HARMONIC EXCITATION......Page 100
Case I: ω = ωn......Page 101
Case II: ω = ωn (Resonance)......Page 102
Case II: ω = ωn......Page 105
2.3 RESPONSE OF A DAMPED SDOF SYSTEM TO A HARMONIC EXCITATION......Page 106
Particular Solution......Page 107
Case II: Critically Damped (ξ = 1 or ceq = cc)......Page 110
Case III: Overdamped (ξ > 1 or ceq > cc)......Page 112
2.3.1 Steady State Response......Page 113
2.3.2 Force Transmissibility......Page 119
Quality Factor......Page 124
Bandwidth......Page 125
2.4 ROTATING UNBALANCE......Page 127
Solution......Page 132
2.5 BASE EXCITATION......Page 134
Solution......Page 138
2.6 VIBRATION MEASURING INSTRUMENTS......Page 139
2.6.1 Vibrometer......Page 141
Solution......Page 142
Solution......Page 144
2.7 EQUIVALENT VISCOUS DAMPING FOR NONVISCOUS ENERGY DISSIPATION......Page 146
EXERCISE PROBLEMS......Page 150
3.1 RESPONSE OF AN SDOF SYSTEM TO A PERIODIC FORCE......Page 156
3.1.1 Periodic Function and its Fourier Series Expansion......Page 157
3.1.2 Even and Odd Periodic Functions......Page 160
Fourier Coefficients for Even Periodic Functions......Page 161
Fourier Coefficients for Odd Periodic Functions......Page 163
3.1.3 Fourier Series Expansion of a Function with a Finite Duration......Page 165
3.1.4 Particular Integral (Steady-State Response with Damping) Under Periodic Excitation......Page 169
3.2 RESPONSE TO AN EXCITATION WITH ARBITRARY NATURE......Page 172
3.2.1 Unit Impulse Function d(t – a)......Page 173
3.2.2 Unit Impulse Response of an SDOF System with Zero Initial Conditions......Page 174
Case II: Critically Damped (ξ = 1 or ceq = cc)......Page 176
Case III: Overdamped (ξ >1 or ceq>cc)......Page 177
3.2.3 Convolution Integral: Response to an Arbitrary Excitation with Zero Initial Conditions......Page 178
Case II: a < t < b......Page 181
Case III: t > b......Page 182
3.2.4 Convolution Integral: Response to an Arbitrary Excitation with Nonzero Initial Conditions......Page 183
Case III: Overdamped (ξ > 1 or ceq > cc)......Page 184
3.3 LAPLACE TRANSFORMATION......Page 186
3.3.1 Properties of Laplace Transformation......Page 187
3.3.2 Response of an SDOF System via Laplace Transformation......Page 188
Poles and Zeros of Transfer Function......Page 193
Frequency Response Function......Page 194
EXERCISE PROBLEMS......Page 197
4 VIBRATION OF TWO-DEGREE-OF-FREEDOM-SYSTEMS......Page 204
4.1 MASS, STIFFNESS, AND DAMPING MATRICES......Page 205
4.2 NATURAL FREQUENCIES AND MODE SHAPES......Page 210
Case I: k1 = k2 = k3 = k......Page 212
Case II: k1 = k2 = k and k3 = 2k......Page 213
Mode shape (Modal vector) corresponding to the natural frequency ω2......Page 214
4.2.1 Eigenvalue/Eigenvector Interpretation......Page 215
4.3 FREE RESPONSE OF AN UNDAMPED 2DOF SYSTEM......Page 216
Solution......Page 218
4.4 FORCED RESPONSE OF AN UNDAMPED 2DOF SYSTEM UNDER SINUSOIDAL EXCITATION......Page 219
4.5 FREE VIBRATION OF A DAMPED 2DOF SYSTEM......Page 221
4.6 STEADY-STATE RESPONSE OF A DAMPED 2DOF SYSTEM UNDER SINUSOIDAL EXCITATION......Page 227
4.7.1 Undamped Vibration Absorber......Page 230
4.7.2 Damped Vibration Absorber......Page 238
Case II: No restriction on f (Absorber not tuned to main system)......Page 242
4.8 MODAL DECOMPOSITION OF RESPONSE......Page 245
Case II: Damped System (C = 0)......Page 246
EXERCISE PROBLEMS......Page 249
5.1 MULTI-DEGREE-OF-FREEDOM SYSTEMS......Page 255
5.1.1 Natural Frequencies and Modal Vectors (Mode Shapes)......Page 257
5.1.2 Orthogonality of Eigenvectors for Symmetric Mass and Symmetric Stiffness Matrices......Page 260
5.1.3 Modal Decomposition......Page 263
Case I: Undamped System (C = 0)......Page 264
Case II: Proportional or Rayleigh Damping......Page 267
5.2.1 Transverse Vibration of a String......Page 268
Computation of Response......Page 273
5.2.2 Longitudinal Vibration of a Bar......Page 276
5.2.3 Torsional Vibration of a Circular Shaft......Page 279
5.3.1 Governing Partial Differential Equation of Motion......Page 283
5.3.2 Natural Frequencies and Mode Shapes......Page 285
Simply Supported Beam......Page 287
Cantilever Beam......Page 289
5.3.3 Computation of Response......Page 291
5.4.1 Longitudinal Vibration of a Bar......Page 297
Total Kinetic and Potential Energies of the Bar......Page 301
Solution......Page 303
5.4.2 Transverse Vibration of a Beam......Page 304
Total Kinetic and Potential Energies of the Beam......Page 309
Solution......Page 311
EXERCISE PROBLEMS......Page 313
A.1 FIXED–FIXED BEAM......Page 317
A.4 SHAFT UNDER TORSION......Page 318
A.5 ELASTIC BAR UNDER AXIAL LOAD......Page 319
B.1 TRIGONOMETRIC IDENTITY......Page 320
B.3 BINOMIAL EXPANSION......Page 321
APPENDIX C: LAPLACE TRANSFORM TABLE......Page 322
REFERENCES......Page 323
INDEX......Page 325