CONTENTS
Preface page xiii
1 Equivalent Single-Degree-of-Freedom System and Free
Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Degrees of Freedom 3
1.2 Elements of a Vibratory System 5
1.2.1 Mass and/or Mass-Moment of Inertia 5
Pure Translational Motion 5
Pure Rotational Motion 6
Planar Motion (Combined Rotation
and Translation) of a Rigid Body 6
Special Case: Pure Rotation about a Fixed Point 8
1.2.2 Spring 8
Pure Translational Motion 8
Pure Rotational Motion 9
1.2.3 Damper 10
Pure Translational Motion 10
Pure Rotational Motion 11
1.3 Equivalent Mass, Equivalent Stiffness, and Equivalent
Damping Constant for an SDOF System 12
1.3.1 A Rotor–Shaft System 13
1.3.2 Equivalent Mass of a Spring 14
1.3.3 Springs in Series and Parallel 16
Springs in Series 16
Springs in Parallel 17
1.3.4 An SDOF System with Two Springs and Combined
Rotational and Translational Motion 19
1.3.5 Viscous Dampers in Series and Parallel 22
Contents
Dampers in Series 22
Dampers in Parallel 23
1.4 Free Vibration of an Undamped SDOF System 25
1.4.1 Differential Equation of Motion 25
Energy Approach 27
1.4.2 Solution of the Differential Equation of Motion
Governing Free Vibration of an Undamped
Spring–Mass System 34
1.5 Free Vibration of a Viscously Damped SDOF System 40
1.5.1 Differential Equation of Motion 40
1.5.2 Solution of the Differential Equation of Motion
Governing Free Vibration of a Damped
Spring–Mass System 41
Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc) 42
Case II: Critically Damped (ξ = 1 or ceq = cc) 45
Case III: Overdamped (ξ > 1 or ceq > cc) 46
1.5.3 Logarithmic Decrement: Identification of Damping
Ratio from Free Response of an Underdamped
System (0 < ξ < 1) 51
Solution 55
1.6 Stability of an SDOF Spring–Mass–Damper System 58
Exercise Problems 63
2 Vibration of a Single-Degree-of-Freedom System Under
Constant and PurelyHarmonic Excitation . . . . . . . . . . . . . . . . . . . . . . 72
2.1 Responses of Undamped and Damped SDOF Systems
to a Constant Force 72
Case I: Undamped (ξ = 0) and Underdamped
(0 < ξ < 1) 74
Case II: Critically Damped (ξ = 1 or ceq = cc) 75
Case III: Overdamped (ξ > 1 or ceq > cc) 76
2.2 Response of an Undamped SDOF System
to a Harmonic Excitation 82
Case I: ω = ωn 83
Case II: ω = ωn (Resonance) 84
Case I: ω = ωn 87
Case II: ω = ωn 87
2.3 Response of a Damped SDOF System to a Harmonic
Excitation 88
Particular Solution 89
Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc) 92
Case II: Critically Damped (ξ = 1 or ceq = cc) 92
Case III: Overdamped (ξ > 1 or ceq > cc) 94
2.3.1 Steady State Response 95
2.3.2 Force Transmissibility 101
2.3.3 Quality Factor and Bandwidth 106
Quality Factor 106
Bandwidth 107
2.4 Rotating Unbalance 109
2.5 Base Excitation 116
2.6 Vibration Measuring Instruments 121
2.6.1 Vibrometer 123
2.6.2 Accelerometer 126
2.7 Equivalent Viscous Damping for Nonviscous Energy
Dissipation 128
Exercise Problems 132
3 Responses of an SDOF Spring–Mass–Damper System
to Periodic andArbitrary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.1 Response of an SDOF System to a Periodic Force 138
3.1.1 Periodic Function and its Fourier Series Expansion 139
3.1.2 Even and Odd Periodic Functions 142
Fourier Coefficients for Even Periodic Functions 143
Fourier Coefficients for Odd Periodic Functions 145
3.1.3 Fourier Series Expansion of a Function
with a Finite Duration 147
3.1.4 Particular Integral (Steady-State Response
with Damping) Under Periodic Excitation 151
3.2 Response to an Excitation with Arbitrary Nature 154
3.2.1 Unit Impulse Function δ(t − a) 155
3.2.2 Unit Impulse Response of an SDOF System
with Zero Initial Conditions 156
Case I: Undamped and Underdamped System
(0 ≤ ξ < 1) 158
Case II: Critically Damped (ξ = 1 or ceq = cc) 158
Case III: Overdamped (ξ >1 or ceq>cc) 159
3.2.3 Convolution Integral: Response to an Arbitrary
Excitation with Zero Initial Conditions 160
3.2.4 Convolution Integral: Response to an Arbitrary
Excitation with Nonzero Initial Conditions 165
Case I: Undamped and Underdamped
(0 ≤ ξ <1 or 0 ≤ ceq<cc) 166
Case II: Critically Damped (ξ = 1 or ceq = cc) 166
Case III: Overdamped (ξ > 1 or ceq > cc) 166
3.3 Laplace Transformation 168
3.3.1 Properties of Laplace Transformation 169
3.3.2 Response of an SDOF System via Laplace
Transformation 170
3.3.3 Transfer Function and Frequency Response
Function 173
Significance of Transfer Function 175
Poles and Zeros of Transfer Function 175
Frequency Response Function 176
Exercise Problems 179
4 Vibration of Two-Degree-of-Freedom-Systems . . . . . . . . . . . . . . . . . 186
4.1 Mass, Stiffness, and Damping Matrices 187
4.2 Natural Frequencies and Mode Shapes 192
4.2.1 Eigenvalue/Eigenvector Interpretation 197
4.3 Free Response of an Undamped 2DOF System 198
Solution 200
4.4 Forced Response of an Undamped 2DOF System Under
Sinusoidal Excitation 201
4.5 Free Vibration of a Damped 2DOF System 203
4.6 Steady-State Response of a Damped 2DOF System
Under Sinusoidal Excitation 209
4.7 Vibration Absorber 212
4.7.1 Undamped Vibration Absorber 212
4.7.2 Damped Vibration Absorber 220
Case I: Tuned Case ( f = 1 or ω22 = ω11) 224
Case II: No restriction on f (Absorber not tuned
to main system) 224
4.8 Modal Decomposition of Response 227
Case I: Undamped System (C = 0) 228
Case II: Damped System (C = 0) 228
Exercise Problems 231
5 Finite and Infinite (Continuous) Dimensional Systems . . . . . . . . . . 237
5.1 Multi-Degree-of-Freedom Systems 237
5.1.1 Natural Frequencies and Modal Vectors
(Mode Shapes) 239
5.1.2 Orthogonality of Eigenvectors for Symmetric Mass
and Symmetric Stiffness Matrices 242
5.1.3 Modal Decomposition 245
Case I: Undamped System (C = 0) 246
Case II: Proportional or Rayleigh Damping 249
5.2 Continuous Systems Governed by Wave Equations 250
5.2.1 Transverse Vibration of a String 250
Natural Frequencies and Mode Shapes 251
Computation of Response 255
5.2.2 Longitudinal Vibration of a Bar 258
5.2.3 Torsional Vibration of a Circular Shaft 261
5.3 Continuous Systems: Transverse Vibration of a Beam 265
5.3.1 Governing Partial Differential Equation of Motion 265
5.3.2 Natural Frequencies and Mode Shapes 267
Simply Supported Beam 269
Cantilever Beam 271
5.3.3 Computation of Response 273
5.4 Finite Element Analysis 279
5.4.1 Longitudinal Vibration of a Bar 279
Total Kinetic and Potential Energies of the Bar 283
5.4.2 Transverse Vibration of a Beam 286
Total Kinetic and Potential Energies of the Beam 291
Exercise Problems 295
APPENDIX A: EQUIVALENT STIFFNESSES (SPRING
CONSTANTS) OF BEAMS, TORSIONAL SHAFT, AND
LONGITUDINAL BAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
APPENDIX B: SOME MATHEMATICAL FORMULAE . . . . . . . . . . . . . . . . . 302
APPENDIX C: LAPLACE TRANSFORM TABLE . . . . . . . . . . . . . . . . . . . . . . . . 304
References 305
Index 307
ျမန္မာအင္ဂ်င္နီယာဖိုရမ္မွ ေအးေက်ာ္ထူး လင့္မွရရွိပါသည္။ ကိုေအးေက်ာ္ထူးကို ေက်းဇူးတင္ပါသည္။
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