Textbook of Mechanical Vibrations

✍ Scribed by Mahesh Chandra Luintel

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 787
• Category: Library

✦ Table of Contents

Preface
Contents
About the Author
1 Basic Concepts of Vibration
1.1 Introduction
1.1.1 Causes of Vibration
1.1.2 Effects of Vibration
1.2 Simple Harmonic Motion
1.3 Vibration Analysis Procedure
1.3.1 Mathematical Modeling
1.3.2 Mathematical Solution
1.3.3 Physical Interpretation of Mathematical Solution
1.4 Generalized Coordinates
1.5 Degrees of Freedom
1.6 Discrete and Continuous System
1.7 Classification of Vibration
1.8 Review of Dynamics
1.8.1 Kinematics
1.8.2 Kinetics
1.8.3 Principle of Work and Energy
1.8.4 Principle of Impulse and Momentum
2 Modeling of Components of a Vibrating System
2.1 Components of a Vibrating System
2.2 Inertia Elements and Kinetic Energy
2.2.1 Kinetic Energy of a Discrete System Consisting of Particles
2.2.2 Kinetic Energy of a Discrete System Consisting of a Rigid Body
2.2.3 Kinetic Energy of a Continuous System
2.3 Stiffness Elements and Potential Energy
2.3.1 Potential Energy Stored by a Spring
2.3.2 Potential Energy or Strain Energy Stored by a Continuous System
2.3.3 Equivalent System and Equivalent Stiffness for Different Combinations of Springs
2.3.4 Equivalent System and Equivalent Stiffness for Continuous System with Negligible Weight
2.4 Damper and Energy Dissipation
2.4.1 Types of Damping
2.4.2 Energy Dissipation Due to Damping
2.5 External Excitation
3 Derivation of Equation of Motion of a Vibrating System
3.1 Classical Methods for Derivation of Equation of Motion
3.1.1 Newton’s Second Law of Motion
3.1.2 Equivalent System Parameters Method
3.1.3 Principle of Conservation of Energy
3.2 Variational Formulation of Dynamic System
3.2.1 Independent Variable, Function and Functional
3.2.2 Differentiation and Variation
3.2.3 Fundamental Lemma of Variational Calculus
3.3 Euler–Lagrange Equation
3.4 Hamilton’s Principle
3.5 Lagrange’s Equations for Conservative Discrete Systems
3.6 Lagrange’s Equations for Non-Conservative Discrete Systems
4 Response of a Single Degree of Freedom System
4.1 Un-damped Free Response of a SDOF System
4.2 Damped Free Response of a SDOF System
4.2.1 Response of an Over-Damped System
4.2.2 Response of a Critically Damped System
4.2.3 Response of an Under-Damped System
4.3 Forced Harmonic Response of a SDOF System
4.4 Rotating Unbalance
4.5 Vibration Isolation and Transmissibility
4.6 Response of a System to an External Motion
4.7 Vibration Measuring Instruments
4.7.1 Seismometer
4.7.2 Accelerometer
4.8 Response to Multi-Frequency and General Periodic Excitations
4.8.1 Response to Multi-Frequency Excitation
4.8.2 Response to a General Periodic Excitation
4.9 Response to Transient Input Forces
4.9.1 Response Due to a Unit Impulse
4.9.2 Response Due to a General Excitation
4.10 Solution Using the Method of Laplace Transform
4.11 Energy Dissipated in Viscous Damper
4.12 Response of a System with Coulomb Damping
4.12.1 Free Response for a System with Coulomb Damping
4.12.2 Forced Response for a System with Coulomb Damping
4.13 Response of a System with Hysteretic Damping
4.13.1 Free Response for a System with Hysteretic Damping
4.13.2 Forced Response for a System with Hysteretic Damping
5 Response of a Two Degree of Freedom System
5.1 Introduction
5.2 Free Response of an Undamped Two Degree of Freedom System
5.3 Free Response of a Damped Two Degree of Freedom System
5.4 Forced Harmonic Response of a Two Degree of Freedom System
5.4.1 Forced Harmonic Response of an Un-damped Two Degree of Freedom System
5.4.2 Forced Harmonic Response of a Damped Two Degree of Freedom System
5.5 Transfer Functions
5.6 Vibration Absorber
5.7 Semi-definite System
5.8 Coordinate Coupling and Principal Coordinates
5.8.1 Equation of Motion Using x1 and x2 as Generalized Coordinates
5.8.2 Equation of Motion Using x and θ as Generalized Coordinates
6 Response of a Multi-Degree of Freedom System
6.1 Introduction
6.2 Formulation of Equation of Motion in Matrix Form
6.3 Flexibility and Stiffness Matrices
6.3.1 Flexibility Influence Coefficients and Flexibility Matrix
6.3.2 Stiffness Influence Coefficients and Stiffness Matrix
6.3.3 Relationship Between Flexibility and Stiffness Matrix
6.3.4 Reciprocity Theorem
6.4 Natural Frequencies and Mode Shapes of a MDOF System
6.5 Orthogonal Properties of the Eigen-Vectors
6.6 Modal Analysis
6.6.1 Modal Analysis for Un-damped Free Response of a MDOF System
6.6.2 Modal Analysis for Damped Free Response of a MDOF System
6.6.3 Modal Analysis for Forced Response of a MDOF System
6.7 Review Questions
7 Modeling and Response of Continuous System
7.1 Introduction
7.2 Lateral Vibration of a String
7.2.1 Derivation of Equation of Motion Using Newton’s Second Law of Motion
7.2.2 Derivation of Equation of Motion Using Hamilton’s Principle
7.2.3 Free Response for Lateral Vibration of a String
7.2.4 Forced Harmonic Response for Lateral Vibration of a String
7.3 Longitudinal Vibration of a Bar
7.3.1 Derivation of Equation of Motion Using Newton’s Second Law of Motion
7.3.2 Derivation of Equation of Motion Using Hamilton’s Principle
7.4 Torsional Vibration of a Shaft
7.4.1 Derivation of Equation of Motion Using Newton’s Second Law of Motion
7.4.2 Derivation of Equation of Motion Using Hamilton’s Principle
7.5 Transverse Vibration of a Beam
7.5.1 Derivation of Equation of Motion Using Newton’s Second Law of Motion
7.5.2 Derivation of Equation of Motion Using Hamilton’s Principle
7.5.3 Free Response for Transverse Vibration of a Beam
7.5.4 Forced Harmonic Response for Lateral Vibration of a Beam
7.6 Modal Analysis for a Continuous System
7.6.1 Modal Analysis of a Continuous System Governed by Wave Equation
7.6.2 Modal Analysis for Vibration Analysis of a Beam
8 Approximate Methods
8.1 Introduction
8.2 Rayleigh Method
8.2.1 Rayleigh Method for a Single Degree of Freedom System
8.2.2 Rayleigh Method for a Discrete Multi Degree of Freedom System
8.2.3 Rayleigh Method for a Shaft or a Beam Carrying a Number of Lumped Inertia Elements
8.2.4 Rayleigh Method for a Continuous System
8.3 Dunkerley’s Method
8.4 Matrix Iteration Method
8.4.1 Matrix Iteration Using Flexibility Matrix
8.4.2 Determination of Higher Order Modes
8.4.3 Matrix Iteration Using Dynamic Matrix
8.5 Stodola’s Method
8.6 Holzer’s Method
8.6.1 Holzer’s Method for a System Without a Branch
8.6.2 Holzer’s Method for a Branched System
8.7 Myklestad-Prohl Method for Transverse Bending Vibration
8.8 Rayleigh–Ritz Method
8.9 Assumed Mode Method
8.10 Weighted Residual Method