Structural Dynamics and Vibration in Practice Book

Preface     xiii
Acknowledgements     xv
Basic Concepts     1
Statics, dynamics and structural dynamics     1
Coordinates, displacement, velocity and acceleration     1
Simple harmonic motion     2
Time history representation     3
Complex exponential representation     5
Mass, stiffness and damping     7
Mass and inertia     7
Stiffness     10
Stiffness and flexibility matrices     12
Damping     14
Energy methods in structural dynamics     16
Rayleigh's energy method     17
The principle of virtual work     19
Lagrange's equations     21
Linear and non-linear systems     23
Systems of units     23
Absolute and gravitational systems     24
Conversion between systems     26
The SI system     27
References     28
The Linear Single Degree of Freedom System: Classical Methods     29
Setting up the differential equation of motion     29
Single degree of freedom system with force input     29
Single degree of freedom system with base motion input     33
Free response of single-DOF systems by direct solution of the equation of motion     34
Forced response of the system by direct solution of the equation of motion     38
The Linear Single Degree of Freedom System: Response in the Time Domain     45
Exact analytical methods     46
The Laplace transform method     46
The convolution or Duhamel integral     50
Listings of standard responses     53
'Semi-analytical' methods     55
Impulse response method     56
Straight-line approximation to input function     56
Superposition of standard responses     56
Step-by-step numerical methods using approximate derivatives     59
Euler method     60
Modified Euler method     62
Central difference method     62
The Runge-Kutta method     65
Discussion of the simpler finite difference methods     69
Dynamic factors     70
Dynamic factor for a square step input     70
Response spectra     72
Response spectrum for a rectangular pulse     72
Response spectrum for a sloping step     74
References     76
The Linear Single Degree of Freedom System: Response in the Frequency Domain     77
Response of a single degree of freedom system with applied force     77
Response expressed as amplitude and phase     77
Complex response functions     81
Frequency response functions     83
Single-DOF system excited by base motion     86
Base excitation, relative response     87
Base excitation: absolute response     91
Force transmissibility     93
Excitation by a rotating unbalance     94
Displacement response     95
Force transmitted to supports     96
References     97
Damping     99
Viscous and hysteretic damping models     99
Damping as an energy loss     103
Energy loss per cycle - viscous model     103
Energy loss per cycle - hysteretic model     104
Graphical representation of energy loss     105
Specific damping capacity     106
Tests on damping materials     108
Quantifying linear damping     108
Quality factor, Q     108
Logarithmic decrement     109
Number of cycles to half amplitude     110
Summary table for linear damping     111
Heat dissipated by damping     112
Non-linear damping     112
Coulomb damping     113
Square law damping     113
Equivalent linear dampers     114
Viscous equivalent for coulomb damping     115
Viscous equivalent for square law damping     116
Limit cycle oscillations with square-law damping     117
Variation of damping and natural frequency in structures with amplitude and time     117
Introduction to Multi-degree-of-freedom Systems     119
Setting up the equations of motion for simple, undamped, multi-DOF systems     119
Equations of motion from Newton's second law and d'Alembert's principle     120
Equations of motion from the stiffness matrix     120
Equations of motion from Lagrange's equations     121
Matrix methods for multi-DOF systems     122
Mass and stiffness matrices: global coordinates     122
Modal coordinates     126
Transformation from global to modal coordinates     127
Undamped normal modes     132
Introducing eigenvalues and eigenvectors     132
Damping in multi-DOF systems     142
The damping matrix     142
Damped and undamped modes     143
Damping inserted from measurements     144
Proportional damping     145
Response of multi-DOF systems by normal mode summation     147
Response of multi-DOF systems by direct integration     155
Fourth-order Runge-Kutta method for multi-DOF systems     156
Eigenvalues and Eigenvectors     159
The eigenvalue problem in standard form     159
The modal matrix     161
Some basic methods for calculating real eigenvalues and eigenvectors     162
Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination     162
Matrix iteration     165
Jacobi diagonalization     168
Choleski factorization     177
More advanced methods for extracting real eigenvalues and eigenvectors     178
Complex (damped) eigenvalues and eigenvectors     179
References     180
Vibration of Structures     181
A historical view of structural dynamics methods     181
Continuous systems     182
Vibration of uniform beams in bending     182
The Rayleigh-Ritz method: classical and modern     189
Component mode methods     194
Component mode synthesis     195
The branch mode method      208
The finite element method     213
An overview     213
Equations of motion for individual elements     221
Symmetrical structures     234
References     235
Fourier Transformation and Related Topics     237
The Fourier series and its developments     237
Fourier series     237
Fourier coefficients in magnitude and phase form     243
The Fourier series in complex notation     245
The Fourier integral and Fourier transforms     246
The discrete Fourier transform     247
Derivation of the discrete Fourier transform     248
Proprietary DFT codes     255
The fast Fourier transform     256
Aliasing     256
Response of systems to periodic vibration     260
Response of a single-DOF system to a periodic input force     261
References     265
Random Vibration     267
Stationarity, ergodicity, expected and average values     267
Amplitude probability distribution and density functions     270
The Gaussian or normal distribution     274
The power spectrum     279
Power spectrum of a periodic waveform      279
The power spectrum of a random waveform     281
Response of a system to a single random input     286
The frequency response function     286
Response power spectrum in terms of the input power spectrum     287
Response of a single-DOF system to a broadband random input     288
Response of a multi-DOF system to a single broad-band random input     296
Correlation functions and cross-power spectral density functions     299
Statistical correlation     299
The autocorrelation function     300
The cross-correlation function     302
Relationships between correlation functions and power spectral density functions     303
The response of structures to random inputs     305
The response of a structure to multiple random inputs     305
Measuring the dynamic properties of a structure     307
Computing power spectra and correlation functions using the discrete Fourier transform     310
Computing spectral density functions     312
Computing correlation functions     314
Leakage and data windows     317
Accuracy of spectral estimates from random data     318
Fatigue due to random vibration     320
The Rayleigh distribution      321
The S-N diagram     322
References     324
Vibration Reduction     325
Vibration isolation     326
Isolation from high environmental vibration     326
Reducing the transmission of vibration forces     332
The dynamic absorber     332
The centrifugal pendulum dynamic absorber     336
The damped vibration absorber     338
The springless vibration absorber     342
References     345
Introduction to Self-Excited Systems     347
Friction-induced vibration     347
Small-amplitude behavior     347
Large-amplitude behavior     349
Friction-induced vibration in aircraft landing gear     350
Flutter     353
The bending-torsion flutter of a wing     354
Flutter equations     358
An aircraft flutter clearance program in practice     360
Landing gear shimmy     362
References     366
Vibration testing     367
Modal testing     368
Theoretical basis     368
Modal testing applied to an aircraft     369
Environmental vibration testing     373
Vibration inputs     373
Functional tests and endurance tests     374
Test control strategies     375
Vibration fatigue testing in real time     376
Vibration testing equipment     377
Accelerometers     377
Force transducers     378
Exciters     378
References     385
A Short Table of Laplace Transforms     387
Calculation of Flexibility Influence Coefficients     389
Acoustic Spectra     393
Index     397