Vibration of Discrete and Continuous Systems

Vibration of Discrete and Continuous Systems
Third Edition
Ahmed Shabana
Contents
1 Introduction . 1
1.1 Kinematics of Rigid Bodies 1
1.2 Dynamic Equations 6
1.3 Single Degree of Freedom Systems 11
1.4 Oscillatory and Nonoscillatory Motion 17
1.5 Other Types of Damping 24
1.6 Forced Vibration 28
1.7 Impulse Response . 36
1.8 Response to an Arbitrary Forcing Function . 39
1.9 Linear Theory of Vibration 42
2 Lagrangian Dynamics 55
2.1 Generalized Coordinates 55
2.2 Virtual Work and Generalized Forces . 59
2.3 Lagrange’s Equation . 64
2.4 Generalized Inertia Forces . 71
2.5 Kinetic Energy . 74
2.6 Strain Energy 81
2.7 Hamilton’s Principle . 84
2.8 Conservation Theorems . 88
2.9 Virtual Work and D’Alembert’s Principle 93
2.10 Vibration of Complex Nonlinear Systems 95
3 Multi-degree of Freedom Systems . 107
3.1 Equations of Motion . 108
3.2 Undamped Free Vibration . 118
3.3 Orthogonality of the Mode Shapes . 123
3.4 Rigid-Body Modes 133
3.5 Conservation of Energy . 140
3.6 Forced Vibration of the Undamped Systems 143
xi3.7 Viscously Damped Systems 145
3.8 General Viscous Damping . 151
3.9 Approximation and Numerical Methods . 160
3.10 Matrix-Iteration Methods 165
3.11 Method of Transfer Matrices . 179
3.12 Nonlinear Systems and Modal Decomposition 189
4 Vibration of Continuous Systems . 201
4.1 Free Longitudinal Vibrations . 202
4.2 Free Torsional Vibrations . 214
4.3 Free Transverse Vibrations of Beams . 221
4.4 Orthogonality of the Eigenfunctions 232
4.5 Forced Vibrations . 239
4.6 Inhomogeneous Boundary Conditions 247
4.7 Viscoelastic Materials 250
4.8 Energy Methods 252
4.9 Approximation Methods 255
4.10 Galerkin’s Method 265
4.11 Assumed-Modes Method 269
5 The Finite Element Method 279
5.1 Assumed Displacement Field . 281
5.2 Comments on the Element Shape Functions 287
5.3 Connectivity Between Elements . 290
5.4 Formulation of the Mass Matrix . 296
5.5 Formulation of the Stiffness Matrix 300
5.6 Equations of Motion . 309
5.7 Convergence of the FE Solution 313
5.8 Higher-Order Elements . 317
5.9 Spatial Elements 321
5.10 Large Rotations and Deformations . 329
5.11 Sub-structuring Techniques 333
6 Methods for the Eigenvalue Analysis 343
6.1 Similarity Transformation . 344
6.2 Polynomial Matrices . 346
6.3 Equivalence of the Characteristic Matrices . 350
6.4 Jordan Matrices 354
6.5 Elementary Divisors . 359
6.6 Generalized Eigenvectors . 361
6.7 Jacobi Method . 365
6.8 Householder Transformation . 367
6.9 QR Decomposition 372
xii ContentsAppendix A: Linear Algebra . 379
A.1 Matrices 379
A.2 Matrix Operations 381
A.3 Vectors 387
A.4 Eigenvalue Problem . 395
References . 401
Index . 405
Contents xiii introduction
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