Chapter 1 Introduction and Vibration of Single-Degree-of-
Freedom Systems ...................................... 1
1.1 Introduction ............................................... 1
1.1.1 Degrees of Freedom and Generalized Coordinates ...... 1
1.1.2 Scope of Study ...................................... 7
1.2 Newton's Second Law, Angular Momentum, and Kinetic
Energy ..................................................... 8
1.2.1 Particles ........................................... 8
1.2.2 Systems of Particles ................................ 9
1.2.3 Rigid Bodies ....................................... 13
1.3 Components of Vibrating Systems ........................... 17
1.3.1 Inertia Elements ................................... 17
1.3.2 Stiffness Elements ................................. 22
1.3.3 Energy Dissipation ................................. 30
1.3.4 External Energy Sources ............................ 34
1.4 Modeling of One-Degree-of-Freedom Systems ................. 38
1.4.1 Introduction and Assumptions ....................... 38
1.4.2 Static Spring Forces ............................... 39
1.4.3 Derivation of Differential Equations ............... 42
1.4.4 Model Systems ...................................... 48
1.4.5 One-Degree-of-Freedom Models of Continuous
Systems ............................................ 49
1.5 Qualitative Aspects of One-Degree-of-Freedom Systems ...... 56
1.6 Free Vibrations of Linear Single-Degree-of-Freedom
Systems ................................................... 63
1.7 Response of a Single-Degree-of-Freedom System Due to
Harmonic Excitation ....................................... 70
1.7.1 General Theory ..................................... 70
1.7.2 Frequency-Squared Excitation ....................... 73
1.7.3 Motion Input ....................................... 75
1.7.4 General Periodic Input ............................. 80
1.8 Transient Response of a Single-Degree-of-Freedom System ... 82
Chapter 2 Derivation of Differential Equations Using
Variational Methods ................................. 87
2.1 Functionals ............................................... 87
2.2 Variations ................................................ 91
2.3 Euler-Lagrange Equation ................................... 93
2.4 Hamilton's Principle ..................................... 100
2.5 Lagrange's Equations for Conservative Discrete Systems ... 104
2.6 Lagrange's Equations for Non-Conservative Discrete
Systems .................................................. 112
2.7 Linear Discrete Systems .................................. 122
2.7.1 Quadratic Forms ................................... 122
2.7.2 Differential Equations for Linear Systems ......... 125
2.7.3 Linearization of Differential Equations ........... 127
2.8 Gyroscopic Systems ....................................... 130
2.9 Continuous Systems ....................................... 136
2.10 Bars, Strings, and Shafts ................................ 138
2.11 Euler-Bernoulli Beams .................................... 150
2.12 Timoshenko Beams ......................................... 166
2.13 Membranes ................................................ 170
Chapter 3 Linear Algebra ..................................... 173
3.1 Introduction ............................................. 173
3.2 Three-Dimensional Space .................................. 174
3.3 Vector Spaces ............................................ 177
3.4 Linear Independence ...................................... 182
3.5 Basis and Dimension ...................................... 185
3.6 Inner Products ........................................... 189
3.7 Norms .................................................... 193
3.8 Gram-Schmidt Orthonormalization Method ................... 197
3.9 Orthogonal Expansions .................................... 202
3.10 Linear Operators ......................................... 206
3.11 Adjoint Operators ........................................ 212
3.12 Positive Definite Operators .............................. 219
3.13 Energy Inner Products .................................... 222
Chapter 4 Operators Used in Vibration Problems ............... 225
4.1 Summary of Basic Theory .................................. 225
4.2 Differential Equations for Discrete Systems .............. 227
4.3 Stiffness Matrix ......................................... 227
4.4 Mass Matrix .............................................. 233
4.5 Flexibility Matrix ....................................... 234
4.6 М-1К and AM .............................................. 240
4.7 Formulation of Partial Differential Equations for
Continuous Systems ....................................... 242
4.8 Second-Order Problems .................................... 245
4.9 Euler-Bernoulli Beam ..................................... 253
4.10 Timoshenko Beams ......................................... 262
4.11 Systems with Multiple Deformable Bodies .................. 266
4.12 Continuous Systems with Attached Inertia Elements ........ 272
4.13 Combined Continuous and Discrete Systems ................. 278
4.14 Membranes ................................................ 283
Chapter 5 Free Vibrations of Conservative Systems ............ 287
5.1 Normal Mode Solution ..................................... 287
5.2 Properties of Eigenvalues and Eigenvectors ............... 292
5.2.1 Eigenvalues of Self-Adjoint Operators ............. 292
5.2.2 Positive Definite Operators ....................... 297
5.2.3 Expansion Theorem ................................. 298
5.2.4 Summary ........................................... 302
5.3 Rayleigh's Quotient ...................................... 303
5.4 Solvability Conditions ................................... 306
5.5 Free Response Using the Normal Mode Solution ............. 309
5.5.1 General Free Response ............................. 309
5.5.2 Principal Coordinates ............................. 314
5.6 Discrete Systems ......................................... 316
5.6.1 The Matrix Eigenvalue Problem ..................... 317
5.7 Natural Frequency Calculations Using Flexibility
Matrix ................................................... 326
5.8 Matrix Iteration ......................................... 330
5.9 Continuous Systems ....................................... 341
5.10 Second-Order Problems (Wave Equation) .................... 342
5.11 Euler-Bernoulli Beams .................................... 360
5.12 Repeated Structures ...................................... 375
5.13 Timoshenko Beams ......................................... 398
5.14 Combined Continuous and Discrete Systems ................. 409
5.15 Membranes ................................................ 414
5.16 Green's Functions ........................................ 430
Chapter 6 Non-Self-Adjoint Systems ........................... 437
6.1 Non-Self-Adjoint Operators ............................... 437
6.2 Discrete Systems with Proportional Damping ............... 441
6.3 Discrete Systems with General Damping .................... 446
6.4 Discrete Gyroscopic Systems .............................. 452
6.5 Continuous Systems with Viscous Damping .................. 458
Chapter 7 Forced Response .................................... 465
7.1 Response of Discrete Systems for Harmonic Excitations ..... 465
7.1.1 General Theory .................................... 465
7.1.2 Vibration Absorbers ............................... 470
7.2 Harmonic Excitation of Continuous Systems ................ 480
7.3 Laplace Transform Solutions .............................. 490
7.3.1 Discrete Systems .................................. 491
7.3.2 Continuous Systems ................................ 497
7.4 Modal Analysis for Undamped Discrete Systems ............. 501
7.5 Modal Analysis for Undamped Continuous Systems ........... 504
7.6 Discrete Systems with Damping ............................ 516
7.6.1 Proportional Damping .............................. 516
7.6.2 General Viscous Damping ........................... 517
Chapter 8 Rayleigh-Ritz and Finite-Element Methods ........... 525
8.1 Fourier Best Approximation Theorem ....................... 525
8.2 Rayleigh-Ritz Method ..................................... 528
8.3 Galerkin Method .......................................... 531
8.4 Rayleigh-Ritz Method for Natural Frequencies and Mode
Shapes ................................................... 532
8.5 Rayleigh-Ritz Methods for Forced Response ................ 551
8.6 Admissible Functions ..................................... 556
8.7 Assumed Modes Method ..................................... 560
8.8 Finite-Element Method .................................... 570
8.9 Assumed Modes Development of Finite-Element Method ....... 575
8.10 Bar Element .............................................. 577
8.11 Beam Element ............................................. 584
Chapter 9 Exercises .......................................... 595
9.1 Chapter 1 ................................................ 595
9.2 Chapter 2 ................................................ 602
9.3 Chapter 3 ................................................ 611
9.4 Chapter 4 ................................................ 614
9.5 Chapter 5 ................................................ 617
9.6 Chapter 6 ................................................ 620
9.7 Chapter 7 ................................................ 622
9.8 Chapter 8 ................................................ 625
References .................................................... 627
Index ......................................................... 629
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