كتاب Solving Vibration Analysis Problems Using Matlab

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Solving Vibration Analysis Problems Using Matlab
Rao V. Dukkipati . Ph D. I*.K.
Fellow of ASM E and CSME
Professor and Chair
Graduate Program Director
Department of Mechanical Engineering
Fairfield University
Fairfield, Connecticut USA

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Contents
PREFACE (iv)
ACKNOWLEDGEMENTS (vi)
1.1 Classification of Vibrations 1
1.2 Elementary Parts of Vibrating Systems . 2
1.3 Periodic Motion . 3
1.4 Discrete and Continuous Systems 4
1.5 Vibration Analysis 4
1.5.1 Components of Vibrating Systems . 6
1.6 Free Vibration of Single Degree of Freedom Systems . 8
1.6.1 Free Vibration of an Undamped Translational System . 8
1.6.2 Free Vibration of an Undamped Torsional System 10
1.6.3 Energy Method 10
1.6.4 Stability of Undamped Linear Systems . 11
1.6.5 Free Vibration with Viscous Damping 11
1.6.6 Logarithmic Decrement 13
1.6.7 Torsional System with Viscous Damping 14
1.6.8 Free Vibration with Coulomb Damping 14
1.6.9 Free Vibration with Hysteretic Damping 15
1.7 Forced Vibration of Single-degree-of-freedom Systems 15
1.7.1 Forced Vibrations of Damped System . 16
1.7.1.1 Resonance 18
1.7.2 Beats . 19
1.7.3 Transmissibility . 19
1.7.4 Quality Factor and Bandwidth 20
1.7.5 Rotating Unbalance . 21
1.7.6 Base Excitation 21
1.7.7 Response Under Coulomb Damping 22
1.7.8 Response Under Hysteresis Damping . 22
1.7.9 General Forcing Conditions And Response . 22
1.7.10 Fourier Series and Harmonic Analysis . 23
(xi)1.8 Harmonic Functions . 23
1.8.1 Even Functions 23
1.8.2 Odd Functions . 23
1.8.3 Response Under a Periodic Force of Irregular Form . 23
1.8.4 Response Under a General Periodic Force 24
1.8.5 Transient Vibration . 24
1.8.6 Unit Impulse 25
1.8.7 Impulsive Response of a System 25
1.8.8 Response to an Arbitrary Input . 26
1.8.9 Laplace Transformation Method 26
1.9 Two Degree of Freedom Systems 26
1.9.1 Equations of Motion 27
1.9.2 Free Vibration Analysis 27
1.9.3 Torsional System . 28
1.9.4 Coordinate Coupling and Principal Coordinates 29
1.9.5 Forced Vibrations 29
1.9.6 Orthogonality Principle 29
1.10 Multi-degree-of-freedom Systems . 30
1.10.1 Equations of Motion 30
1.10.2 Stiffness Influence Coefficients 31
1.10.3 Flexibility Influence Coefficients . 31
1.10.4 Matrix Formulation . 31
1.10.5 Inertia Influence Coefficients . 32
1.10.6 Normal Mode Solution 32
1.10.7 Natural Frequencies and Mode Shapes . 33
1.10.8 Mode Shape Orthogonality . 33
1.10.9 Response of a System to Initial Conditions . 33
1.11 Free Vibration of Damped Systems 34
1.12 Proportional Damping 34
1.13 General Viscous Damping . 35
1.14 Harmonic Excitations . 35
1.15 Modal Analysis for Undamped Systems . 35
1.16 Lagrange’s Equation 36
1.16.1 Generalized Coordinates . 36
1.17 Principle of Virtual Work . 37
1.18 D’Alembert’s Principle . 37
1.19 Lagrange’s Equations of Motion 38
1.20 Variational Principles 38
1.21 Hamilton’s Principle . 38
References . 38
Glossary of Terms . 40
2.1 Introduction 53
2.1.1 Starting and Quitting MATLAB 54
2.1.2 Display Windows 54
2.1.3 Entering Commands . 54
2.1.4 MATLAB Expo 54
2.1.5 Abort 54
2.1.6 The Semicolon (;) 54
2.1.7 Typing % 54
2.1.8 The clc Command . 54
2.1.9 Help 55
2.1.10 Statements and Variables 55
2.2 Arithmetic Operations . 55
2.3 Display Formats . 55
2.4 Elementary Math Built-in Functions . 56
2.5 Variable Names 58
2.6 Predefined Variables 58
2.7 Commands for Managing Variables 59
2.8 General Commands 59
2.9 Arrays 61
2.9.1 Row Vector . 61
2.9.2 Column Vector . 61
2.9.3 Matrix . 61
2.9.4 Addressing Arrays . 61
2.9.4.1 Colon for a Vector 61
2.9.4.2 Colon for a Matrix . 62
2.9.5 Adding Elements to a Vector or a Matrix 62
2.9.6 Deleting Elements . 62
2.9.7 Built-in Functions . 62
2.10 Operations with Arrays . 63
2.10.1 Addition and Subtraction of Matrices 63
2.10.2 Dot Product 64
2.10.3 Array Multiplication . 64
2.10.4 Array Division . 64
2.10.5 Identity Matrix 64
2.10.6 Inverse of a Matrix 64
2.10.7 Transpose 64
2.10.8 Determinant . 65
2.10.9 Array Division . 65
2.10.10 Left Division 65
(xiii)2.10.11 Right Division 65
2.10.12 Eigenvalues and Eigenvectors . 65
2.11 Element-by-element Operations . 66
2.11.1 Built-in Functions for Arrays . 67
2.12 Random Numbers Generation . 68
2.12.1 The Random Command 69
2.13 Polynomials . 69
2.14 System of Linear Equations 71
2.14.1 Matrix Division 71
2.14.2 Matrix Inverse . 71
2.15 Script Files 76
2.15.1 Creating and Saving a Script File 76
2.15.2 Running a Script File 76
2.15.3 Input to a Script File . 76
2.15.4 Output Commands 77
2.16 Programming in Matlab . 77
2.16.1 Relational and Logical Operators 77
2.16.2 Order of Precedence 78
2.16.3 Built-in Logical Functions 78
2.16.4 Conditional Statements 80
2.16.5 NESTED IF Statements . 80
2.16.6 ELSE and ELSEIF Clauses 80
2.16.7 MATLAB while Structures . 81
2.17 Graphics 82
2.17.1 Basic 2-D Plots . 83
2.17.2 Specialized 2-D Plots . 83
2.17.2.1 Overlay Plots . 84
2.17.3 3-D Plots . 84
2.17.4 Saving and Printing Graphs . 90
2.18 Input/Output In Matlab . 91
2.18.1 The FOPEN Statement . 91
2.19 Symbolic Mathematics . 92
2.19.1 Symbolic Expressions 92
2.19.2 Solution to Differential Equations . 94
2.19.3 Calculus 95
2.20 The Laplace Transforms 97
2.20.1 Finding Zeros and Poles of B(s)/A(s) 98
2.21 Control Systems 98
2.21.1 Transfer Functions 98
2.21.2 Model Conversion 98
(xiv)2.22 The Laplace Transforms 101
10.11.1 Finding Zeros and Poles of B(s)/A(s) . 102
Model Problems and Solutions . 102
2.23 Summary . 137
References 138
Problems . 138
3.1 Introduction 150
3.2 Example Problems and Solutions . 150
3.3 Summary . 197
Problems . 198

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