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978-1-107-69417-0 - Vibration of Mechanical Systems
Alok Sinha
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CONTENTS
Preface
page xiii
1 Equivalent Single-Degree-of-Freedom System and Free
Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Degrees of Freedom
1.2 Elements of a Vibratory System
1.2.1 Mass and/or Mass-Moment of Inertia
Pure Translational Motion
Pure Rotational Motion
Planar Motion (Combined Rotation
and Translation) of a Rigid Body
Special Case: Pure Rotation about a Fixed Point
1.2.2 Spring
Pure Translational Motion
Pure Rotational Motion
1.2.3 Damper
Pure Translational Motion
Pure Rotational Motion
1.3 Equivalent Mass, Equivalent Stiffness, and Equivalent
Damping Constant for an SDOF System
1.3.1 A Rotor–Shaft System
1.3.2 Equivalent Mass of a Spring
1.3.3 Springs in Series and Parallel
Springs in Series
Springs in Parallel
1.3.4 An SDOF System with Two Springs and Combined
Rotational and Translational Motion
1.3.5 Viscous Dampers in Series and Parallel
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Alok Sinha
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Contents
Dampers in Series
Dampers in Parallel
1.4 Free Vibration of an Undamped SDOF System
1.4.1 Differential Equation of Motion
Energy Approach
1.4.2 Solution of the Differential Equation of Motion
Governing Free Vibration of an Undamped
Spring–Mass System
1.5 Free Vibration of a Viscously Damped SDOF System
1.5.1 Differential Equation of Motion
1.5.2 Solution of the Differential Equation of Motion
Governing Free Vibration of a Damped
Spring–Mass System
Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc )
Case II: Critically Damped (ξ = 1 or ceq = cc )
Case III: Overdamped (ξ > 1 or ceq > cc )
1.5.3 Logarithmic Decrement: Identification of Damping
Ratio from Free Response of an Underdamped
System (0 < ξ < 1)
Solution
1.6 Stability of an SDOF Spring–Mass–Damper System
Exercise Problems
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2 Vibration of a Single-Degree-of-Freedom System Under
Constant and Purely Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . .
2.1 Responses of Undamped and Damped SDOF Systems
to a Constant Force
Case I: Undamped (ξ = 0) and Underdamped
(0 < ξ < 1)
Case II: Critically Damped (ξ = 1 or ceq = cc )
Case III: Overdamped (ξ > 1 or ceq > cc )
2.2 Response of an Undamped SDOF System
to a Harmonic Excitation
Case I: ω = ωn
Case II: ω = ωn (Resonance)
Case I: ω = ωn
Case II: ω = ωn
2.3 Response of a Damped SDOF System to a Harmonic
Excitation
Particular Solution
Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc )
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Alok Sinha
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Contents
2.4
2.5
2.6
2.7
Case II: Critically Damped (ξ = 1 or ceq = cc )
Case III: Overdamped (ξ > 1 or ceq > cc )
2.3.1 Steady State Response
2.3.2 Force Transmissibility
2.3.3 Quality Factor and Bandwidth
Quality Factor
Bandwidth
Rotating Unbalance
Base Excitation
Vibration Measuring Instruments
2.6.1 Vibrometer
2.6.2 Accelerometer
Equivalent Viscous Damping for Nonviscous Energy
Dissipation
Exercise Problems
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3 Responses of an SDOF Spring–Mass–Damper System
to Periodic and Arbitrary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Response of an SDOF System to a Periodic Force
3.1.1 Periodic Function and its Fourier Series Expansion
3.1.2 Even and Odd Periodic Functions
Fourier Coefficients for Even Periodic Functions
Fourier Coefficients for Odd Periodic Functions
3.1.3 Fourier Series Expansion of a Function
with a Finite Duration
3.1.4 Particular Integral (Steady-State Response
with Damping) Under Periodic Excitation
3.2 Response to an Excitation with Arbitrary Nature
3.2.1 Unit Impulse Function δ(t − a)
3.2.2 Unit Impulse Response of an SDOF System
with Zero Initial Conditions
Case I: Undamped and Underdamped System
(0 ≤ ξ < 1)
Case II: Critically Damped (ξ = 1 or c eq = cc )
Case III: Overdamped (ξ > 1 or c eq > cc )
3.2.3 Convolution Integral: Response to an Arbitrary
Excitation with Zero Initial Conditions
3.2.4 Convolution Integral: Response to an Arbitrary
Excitation with Nonzero Initial Conditions
Case I: Undamped and Underdamped
(0 ≤ ξ < 1 or 0 ≤ c eq < cc )
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Case II: Critically Damped (ξ = 1 or c eq = cc )
Case III: Overdamped (ξ > 1 or c eq > cc )
3.3 Laplace Transformation
3.3.1 Properties of Laplace Transformation
3.3.2 Response of an SDOF System via Laplace
Transformation
3.3.3 Transfer Function and Frequency Response
Function
Significance of Transfer Function
Poles and Zeros of Transfer Function
Frequency Response Function
Exercise Problems
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4 Vibration of Two-Degree-of-Freedom-Systems . . . . . . . . . . . . . . . . .
4.1 Mass, Stiffness, and Damping Matrices
4.2 Natural Frequencies and Mode Shapes
4.2.1 Eigenvalue/Eigenvector Interpretation
4.3 Free Response of an Undamped 2DOF System
Solution
4.4 Forced Response of an Undamped 2DOF System Under
Sinusoidal Excitation
4.5 Free Vibration of a Damped 2DOF System
4.6 Steady-State Response of a Damped 2DOF System
Under Sinusoidal Excitation
4.7 Vibration Absorber
4.7.1 Undamped Vibration Absorber
4.7.2 Damped Vibration Absorber
Case I: Tuned Case ( f = 1 or ω22 = ω11 )
Case II: No restriction on f (Absorber not tuned
to main system)
4.8 Modal Decomposition of Response
Case I: Undamped System (C = 0)
Case II: Damped System (C = 0)
Exercise Problems
5 Finite and Infinite (Continuous) Dimensional Systems . . . . . . . . . .
5.1 Multi-Degree-of-Freedom Systems
5.1.1 Natural Frequencies and Modal Vectors
(Mode Shapes)
5.1.2 Orthogonality of Eigenvectors for Symmetric Mass
and Symmetric Stiffness Matrices
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Alok Sinha
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5.1.3 Modal Decomposition
Case I: Undamped System (C = 0)
Case II: Proportional or Rayleigh Damping
5.2 Continuous Systems Governed by Wave Equations
5.2.1 Transverse Vibration of a String
Natural Frequencies and Mode Shapes
Computation of Response
5.2.2 Longitudinal Vibration of a Bar
5.2.3 Torsional Vibration of a Circular Shaft
5.3 Continuous Systems: Transverse Vibration of a Beam
5.3.1 Governing Partial Differential Equation of Motion
5.3.2 Natural Frequencies and Mode Shapes
Simply Supported Beam
Cantilever Beam
5.3.3 Computation of Response
5.4 Finite Element Analysis
5.4.1 Longitudinal Vibration of a Bar
Total Kinetic and Potential Energies of the Bar
5.4.2 Transverse Vibration of a Beam
Total Kinetic and Potential Energies of the Beam
Exercise Problems
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APPENDIX A: EQUIVALENT STIFFNESSES (SPRING
CONSTANTS) OF BEAMS, TORSIONAL SHAFT, AND
LONGITUDINAL BAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
APPENDIX B: SOME MATHEMATICAL FORMULAE . . . . . . . . . . . . . . . . .
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APPENDIX C: LAPLACE TRANSFORM TABLE . . . . . . . . . . . . . . . . . . . . . . . . 304
References
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Index
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