1 Concepts from Vibrations 1
1.1 Newton’s Laws 2
1.2 Moment of a Force and Angular Momentum 5
1.3 Work and Energy 6
1.4 Dynamics of Systems of Particles 10
1.5 Dynamics of Rigid Bodies 14
1.5.1 Pure translation relative to the inertial space 15
1.5.2 Pure rotation about a fixed point 16
1.5.3 General planar motion referred to the mass center 18
1.6 Kinetic Energy of Rigid Bodies in Planar Motion 21
1.6.1 Pure translation relative to the inertial space 21
1.6.2 Pure rotation about a fixed point 22
1.6.3 General planar motion referred to the mass center 22
1.7 Characteristics of Discrete System Components 23
1.8 Equivalent Springs，Dampers and Masses 27
1.9 Modeling of Mechanical Systems 39
1.10 System Differential Equations of Motion 44
1.11 Nature of Excitations 48
1.12 System and Response Characteristics.The Superposition Principle 53
1.13 Vibration about Equilibrium Points 57
1.14 Summary 66
Problems 67
2 Response of Single-Degree-of-Freedom Systems to Initial Excitations 80
2.1 Undamped Single-Degree-of-Freedom Systems.Harmonic Oscillator 81
2.2 Viscously Damped Single-Degree-of-Freedom Systems 87
2.3 Measurement of Damping 94
2.4 Coulomb Damping.Dry Friction 98
2.5 Plotting the Response to Initial Excitations by MATLAB 101
2.6 Summary 102
Problems 103
3 Response of Single-Degree-of-Freedom Systems to Harmonic and Periodic Excitations 109
3.1 Response of Single-Degree-of-Freedom Systems to Harmonic Excitations 110
3.2 Frequency Response Plots 114
3.3 Systems with Rotating Unbalanced Masses 120
3.4 Whirling of Rotating Shafts 122
3.5 Harmonic Motion of the Base 128
3.6 Vibration Isolation 131
3.7 Vibration Measuring Instruments 132
3.7.1 Accelerometers——high frequency instruments 133
3.7.2 Seismometers——low frequency instruments 135
3.8 Energy Dissipation.Structural Damping 137
3.9 Response to Periodic Excitations.Fourier Series 141
3.10 Frequency Response Plots by MATLAB 149
3.11 Summary 150
Problems 151
4 Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations 157
4.1 The Unit Impulse.Impulse Response 158
4.2 The Unit Step Function.Step Response 162
4.3 The Unit Ramp Function.Ramp Response 165
4.4 Response to Arbitrary Excitations.The Convolution Integral 168
4.5 Shock Spectrum 174
4.6 System Response by the Laplace Transformation Method.Transfer Function 177
4.7 General System Response 184
4.8 Response by the State Transition Matrix 186
4.9 Discrete-Time Systems.The Convolution Sum 189
4.10 Discrete-Time Response Using the Transition Matrix 198
4.11 Response by the Convolution Sum Using MATLAB 201
4.12 Response by the Discrete-Time Transition Matrix Using MATLAB 202
4.13 Summary 203
Problems 204
5 Two-Degree-of-Freedom Systems 208
5.1 System Configuration 209
5.2 The Equations of Motion of Two-Degree-of-Freedom Systems 211
5.3 Free Vibration of Undamped Systems.Natural Modes 215
5.4 Response to Initial Excitations 224
5.5 Coordinate Transformations.Coupling 225
5.6 Orthogonality of Modes.Natural Coordinates 229
5.7 Beat Phenomenon 233
5.8 Response of Two-Degree-of-Freedom Systems to Harmonic Excitations 238
5.9 Undamped Vibration Absorbers 240
5.10 Response of Two-Degree-of-Freedom Systems to Nonperiodic Excitations 243
5.11 Response to Nonperiodic Excitations by the Convolution Sum 248
5.12 Response to Initial Excitations by MATLAB 250
5.13 Frequency Response Plots for Two-Degree-of-Freedom Systems by MATLAB 252
5.14 Response to a Rectangular Pulse by the Convolution Sum Using MATLAB 252
5.15 Summary 254
Problems 255
6 Elements of Analytical Dynamics 262
6.1 Degrees of Freedom and Generalized Coordinates 263
6.2 The Principle of Virtual Work 265
6.3 The Principle of D’Alembert 267
6.4 The Extended Hamilton’s Principle 268
6.5 Lagrange’s Equations 273
6.6 Summary 276
Problems 277
7 Multi-Degree-of-Freedom Systems 280
7.1 Equations of Motion for Linear Systems 281
7.2 Flexibility and Stiffness Influence Coefficients 285
7.3 Properties of the Stiffness and Mass Coefficients 290
7.4 Lagrange’s Equations Linearized about Equilibrium 294
7.5 Linear Transformations.Coupling 297
7.6 Undamped Free Vibration.The Eigenvalue Problem 301
7.7 Orthogonality of Modal Vectors 309
7.8 Systems Admitting Rigid-Body Motions 310
7.9 Decomposition of the Response in Terms of Modal Vectors 316
7.10 Response to Initial Excitations by Modal Analysis 320
7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix 323
7.12 Geometric Interpretation of the Eigenvalue Problem 325
7.13 Rayleigh’s Quotient and Its Properties 331
7.14 Response to Harmonic External Excitations 336
7.15 Response to External Excitations by Modal Analysis 337
7.15.1 Undamped systems 337
7.15.2 Systems with proportional damping 340
7.16 Systems with Arbitrary Viscous Damping 345
7.17 Discrete-Time Systems 355
7.18 Solution of the Eigenvalue Problem.MATLAB Programs 359
7.19 Response to Initial Excitations by Modal Analysis Using MATLAB 361
7.20 Response by the Discrete-Time Transition Matrix Using MATLAB 363
7.21 Summary 363
Problems 365
8 Distributed-Parameter Systems：Exact Solutions 374
8.1 Relation between Discrete and Distributed Systems.Transverse Vibration of Strings 375
8.2 Derivation of the String Vibration Problem by the Extended Hamilton Principle 380
8.3 Bending Vibration of Beams 383
8.4 Free Vibration.The Differential Eigenvalue Problem 389
8.5 Orthogonality of Modes.Expansion Theorem 403
8.6 Systems with Lumped Masses at the Boundaries 408
8.7 Eigenvalue Problem and Expansion Theorem for Problems with Lumped Masses at the Boundaries 414
8.8 Rayleigh’s Quotient.The Variational Approach to the Differential Eigenvalue Problem 423
8.9 Response to Initial Excitations 431
8.10 Response to External Excitations 439
8.11 Systems with External Forces at Boundaries 443
8.12 The Wave Equation 447
8.13 Traveling Waves in Rods of Finite Length 449
8.14 Summary 457
Problems 458
9 Distributed-Parameter Systems：Approximate Methods 464
9.1 Discretization of Distributed-Parameter Systems by Lumping 465
9.2 Lumped-Parameter Method Using Influence Coefficients 468
9.3 Holzer’s Method for Torsional Vibration 473
9.4 Myklestad’s Method for Bending Vibration 484
9.5 Rayleigh’s Principle 493
9.6 The Rayleigh-Ritz Method 499
9.7 An Enhanced Rayleigh-Ritz Method 516
9.8 The Assumed-Modes Method.System Response 523
9.9 The Galerkin Method 529
9.10 The Collocation Method 533
9.11 MATLAB Program for the Solution of the Eigenvalue Problem by the Rayleigh-Ritz Method 539
9.12 Summary 541
Problems 543
10 The Finite Element Method 549
10.1 The Finite Element Method as a Rayleigh-Ritz Method 550
10.2 Strings，Rods and Shafts 554
10.3 Higher-Degree Interpolation Functions 563
10.4 Beams in Bending Vibration 574
10.5 Errors in the Eigenvalues 581
10.6 Finite Element Modeling of Trusses 583
10.7 Finite Element Modeling of Frames 597
10.8 System Response by the Finite Element Method 604
10.9 MATLAB Program for the Solution of the Eigenvalue Problem by the Finite Element Method 608
10.10 Summary 611
Problems 612
11 Nonlinear Oscillations 616
11.1 Fundamental Concepts in Stability.Equilibrium Points 617
11.2 Small Motions of Single-Degree-of-Freedom Systems from Equilibrium 628
11.3 Conservative Systems.Motions in the Large 639
11.4 Limit Cycles.The van der Pol Oscillator 644
11.5 The Fundamental Perturbation Technique 646
11.6 Secular Terms 649
11.7 Lindstedt’s Method 652
11.8 Forced Oscillation of Quasi-Harmonic Systems.Jump Phenomenon 656
11.9 Subharmonics and Combination Harmonics 663
11.10 Systems with Time-Dependent Coefficients.Mathieu’s Equation 667
11.11 Numerical Integration of the Equations of Motion.The Runge-Kutta Methods 672
11.12 Trajectories for the van der Pol Oscillator by MATLAB 679
11.13 Summary 680
Problems 682
12 Random Vibrations 685
12.1 Ensemble Averages.Stationary Random Processes 686
12.2 Time Averages.Ergodic Random Processes 689
12.3 Mean Square Values and Standard Deviation 691
12.4 Probability Density Functions 692
12.5 Description of Random Data in Terms of Probability Density Functions 699
12.6 Properties of Autocorrelation Functions 702
12.7 Response to Arbitrary Excitations by Fourier Transforms 703
12.8 Power Spectral Density Functions 708
12.9 Narrowband and Wideband Random Processes 710
12.10 Response of Linear Systems to Stationary Random Excitations 718
12.11 Response of Single-Degree-of-Freedom Systems to Random Excitations 722
12.12 Joint Probability Distribution of Two Random Variables 727
12.13 Joint Properties of Stationary Random Processes 730
12.14 Joint Properties of Ergodic Random Processes 733
12.15 Response Cross-Correlation Functions for Linear Systems 735
12.16 Response of Multi-Degree-of-Freedom Systems to Random Excitations 738
12.17 Response of Distributed-Parameter Systems to Random Excitations 743
12.18 Summary 746
Problems 748
Appendix A.Fourier Series 752
A.1 Orthogonal Sets of Functions 752
A.2 Trigonometric Series 754
A.3 Complex Form of Fourier Series 757
Appendix B.Laplace Transformation 759
B.1 Definition of the Laplace Transformation 759
B.2 Transformation of Derivatives 760
B.3 Transformation of Ordinary Differential Equations 760
B.4 The Inverse Laplace Transformation 761
B.5 Shifting Theorems 761
B.6 Method of Partial Fractions 762
B.7 The Convolution Integral.Borel’s Theorem 765
B.8 Table of Laplace Transform Pairs 767
Appendix C.Linear Algebra 768
C.1 Matrices 768
C.1.1 Definitions 768
C.1.2 Matrix algebra 770
C.1.3 Determinant of a square matrix 772
C.1.4 Inverse of a matrix 774
C.1.5 Transpose，inverse and determinant of a product of matrices 775
C.1.6 Partitioned matrices 776
C.2 Vector Spaces 777
C.2.1 Definitions 777
C.2.2 Linear dependence 778
C.2.3 Bases and dimension of vector spaces 778
C.3 Linear Transformations 779
C.3.1 The concept of linear transformations 779
C.3.2 Solution of algebraic equations.Matrix inversion 781
Bibliography 787
Index 789