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Vibration of Mechanical Systems

Book Description

In this textbook all the basic concepts in mechanical vibrations are clearly identified and presented in a concise and simple manner with illustrative and practical examples. Vibration concepts include a review of selected topics in mechanics; a description of single-degree-of-freedom (SDOF) systems in terms of equivalent mass, equivalent stiffness, and equivalent damping; a unified treatment of various forced response problems (base excitation and rotating balance); an introduction to systems thinking, highlighting the fact that SDOF analysis is a building block for multi-degree-of-freedom (MDOF) and continuous system analyses via modal analysis; and a simple introduction to finite element analysis to connect continuous system and MDOF analyses. There are more than 60 exercise problems, and a complete solutions manual. The use of MATLAB® software is emphasised.

Table of Contents

  1. Coverpage
  2. Half title page
  3. Title page
  4. Copyright page
  5. Dedication
  6. Contents
  7. Preface
  8. 1 Equivalent Single-Degree-of-Freedom System and Free Vibration
    1. 1.1 Degrees of Freedom
    2. 1.2 Elements of a Vibratory System
      1. 1.2.1 Mass and/or Mass-Moment of Inertia
        1. Pure Translational Motion
        2. Pure Rotational Motion
        3. Planar Motion (Combined Rotation and Translation) of a Rigid Body
        4. Special Case: Pure Rotation about a Fixed Point
      2. 1.2.2 Spring
        1. Pure Translational Motion
        2. Pure Rotational Motion
      3. 1.2.3 Damper
        1. Pure Translational Motion
        2. Pure Rotational Motion
    3. 1.3 Equivalent Mass, Equivalent Stiffness, and Equivalent Damping Constant for an SDOF System
      1. 1.3.1 A Rotor–Shaft System
      2. 1.3.2 Equivalent Mass of a Spring
      3. 1.3.3 Springs in Series and Parallel
        1. Springs in Series
        2. Springs in Parallel
      4. 1.3.4 An SDOF System with Two Springs and Combined Rotational and Translational Motion
      5. 1.3.5 Viscous Dampers in Series and Parallel
        1. Dampers in Series
        2. Dampers in Parallel
    4. 1.4 Free Vibration of an Undamped SDOF System
      1. 1.4.1 Differential Equation of Motion
        1. Energy Approach
      2. 1.4.2 Solution of the Differential Equation of Motion Governing Free Vibration of an Undamped Spring–Mass System
    5. 1.5 Free Vibration of a Viscously Damped SDOF System
      1. 1.5.1 Differential Equation of Motion
      2. 1.5.2 Solution of the Differential Equation of Motion Governing Free Vibration of a Damped Spring–Mass System
        1. Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc)
        2. Case II: Critically Damped (ξ = 1 or ceq = cc)
        3. Case III: Overdamped (ξ > 1 or ceq > cc)
      3. 1.5.3 Logarithmic Decrement: Identification of Damping Ratio from Free Response of an Underdamped System (0 < ξ < 1)
        1. Solution
    6. 1.6 Stability of an SDOF Spring–Mass–Damper System
    7. Exercise Problems
  9. 2 Vibration of a Single-Degree-of-Freedom System Under Constant and Purely Harmonic Excitation
    1. 2.1 Responses of Undamped and Damped SDOF Systems to a Constant Force
      1. Case I: Undamped (ξ = 0) and Underdamped (0 < ξ < 1)
      2. Case II: Critically Damped (ξ = 1 or ceq = cc)
      3. Case III: Overdamped (ξ > 1 or ceq > cc)
    2. 2.2 Response of an Undamped SDOF System to a Harmonic Excitation
      1. Case I: ω ≠ ωn
      2. Case II: ω = ωn (Resonance)
      3. Case I: ω ≠ ωn
      4. Case II: ω = ωn
    3. 2.3 Response of a Damped SDOF System to a Harmonic Excitation
      1. Particular Solution
      2. Case I: Underdamped (0 < ξ < 1 or 0 < ceq < cc)
      3. Case II: Critically Damped (ξ = 1 or ceq = cc)
      4. Case III: Overdamped (ξ > 1 or ceq > cc)
      5. 2.3.1 Steady State Response
      6. 2.3.2 Force Transmissibility
      7. 2.3.3 Quality Factor and Bandwidth
      8. Quality Factor
      9. Bandwidth
    4. 2.4 Rotating Unbalance
    5. 2.5 Base Excitation
    6. 2.6 Vibration Measuring Instruments
      1. 2.6.1 Vibrometer
      2. 2.6.2 Accelerometer
    7. 2.7 Equivalent Viscous Damping for Nonviscous Energy Dissipation
    8. Exercise Problems
  10. 3 Responses of an SDOF Spring–Mass–Damper System to Periodic and Arbitrary Forces
    1. 3.1 Response of an SDOF System to a Periodic Force
      1. 3.1.1 Periodic Function and its Fourier Series Expansion
      2. 3.1.2 Even and Odd Periodic Functions
        1. Fourier Coefficients for Even Periodic Functions
        2. Fourier Coefficients for Odd Periodic Functions
      3. 3.1.3 Fourier Series Expansion of a Function with a Finite Duration
      4. 3.1.4 Particular Integral (Steady-State Response with Damping) Under Periodic Excitation
    2. 3.2 Response to an Excitation with Arbitrary Nature
      1. 3.2.1 Unit Impulse Function δ(t − a)
      2. 3.2.2 Unit Impulse Response of an SDOF System with Zero Initial Conditions
        1. Case I: Undamped and Underdamped System (0 ≤ ξ < 1)
        2. Case II: Critically Damped (ξ = 1 or ceq = cc)
        3. Case III: Overdamped (ξ > 1 or ceq > cc)
      3. 3.2.3 Convolution Integral: Response to an Arbitrary Excitation with Zero Initial Conditions
      4. 3.2.4 Convolution Integral: Response to an Arbitrary Excitation with Nonzero Initial Conditions
        1. Case I: Undamped and Underdamped (0 ≤ ξ < 1 or 0 ≤ ceq < cc)
        2. Case II: Critically Damped (ξ = 1 or ceq = cc)
        3. Case III: Overdamped (ξ > 1 or ceq > cc)
    3. 3.3 Laplace Transformation
      1. 3.3.1 Properties of Laplace Transformation
      2. 3.3.2 Response of an SDOF System via Laplace Transformation
      3. 3.3.3 Transfer Function and Frequency Response Function
        1. Significance of Transfer Function
        2. Poles and Zeros of Transfer Function
        3. Frequency Response Function
    4. Exercise Problems
  11. 4 Vibration of Two-Degree-of-Freedom-Systems
    1. 4.1 Mass, Stiffness, and Damping Matrices
    2. 4.2 Natural Frequencies and Mode Shapes
      1. 4.2.1 Eigenvalue/Eigenvector Interpretation
    3. 4.3 Free Response of an Undamped 2DOF System
      1. Solution
    4. 4.4 Forced Response of an Undamped 2DOF System Under Sinusoidal Excitation
    5. 4.5 Free Vibration of a Damped 2DOF System
    6. 4.6 Steady-State Response of a Damped 2DOF System Under Sinusoidal Excitation
    7. 4.7 Vibration Absorber
      1. 4.7.1 Undamped Vibration Absorber
      2. 4.7.2 Damped Vibration Absorber
        1. Case I: Tuned Case (f = 1orω22 = ω11)
        2. Case II: No restriction on f (Absorber not tuned to main system)
    8. 4.8 Modal Decomposition of Response
      1. Case I: Undamped System (C = 0)
      2. Case II: Damped System (C ≠ = 0)
    9. Exercise Problems
  12. 5 Finite and Infinite (Continuous) Dimensional Systems
    1. 5.1 Multi-Degree-of-Freedom Systems
      1. 5.1.1 Natural Frequencies and Modal Vectors (Mode Shapes)
      2. 5.1.2 Orthogonality of Eigenvectors for Symmetric Mass and Symmetric Stiffness Matrices
      3. 5.1.3 Modal Decomposition
        1. Case I: Undamped System (C = 0)
        2. Case II: Proportional or Rayleigh Damping
    2. 5.2 Continuous Systems Governed by Wave Equations
      1. 5.2.1 Transverse Vibration of a String
        1. Natural Frequencies and Mode Shapes
        2. Computation of Response
      2. 5.2.2 Longitudinal Vibration of a Bar
      3. 5.2.3 Torsional Vibration of a Circular Shaft
    3. 5.3 Continuous Systems: Transverse Vibration of a Beam
      1. 5.3.1 Governing Partial Differential Equation of Motion
      2. 5.3.2 Natural Frequencies and Mode Shapes
        1. Simply Supported Beam
        2. Cantilever Beam
      3. 5.3.3 Computation of Response
    4. 5.4 Finite Element Analysis
      1. 5.4.1 Longitudinal Vibration of a Bar
        1. Total Kinetic and Potential Energies of the Bar
      2. 5.4.2 Transverse Vibration of a Beam
        1. Total Kinetic and Potential Energies of the Beam
    5. Exercise Problems
  13. Appendix A: Equivalent stiffnesses (spring constants) of beams, torsional shaft, and longitudinal bar
  14. Appendix B: Some mathematical formulae
  15. Appendix C: Laplace transform table
  16. References
  17. Index