كتاب Advanced Mechanical Vibrations - Physics, Mathematics and Applications
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 كتاب Advanced Mechanical Vibrations - Physics, Mathematics and Applications

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Advanced Mechanical Vibrations
Physics, Mathematics and Applications
Paolo Luciano Gatti  

كتاب Advanced Mechanical Vibrations - Physics, Mathematics and Applications  A_m_v_10
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Contents
Preface xi
Acknowledgements xiii
Frequently used acronyms xv
1 A few preliminary fundamentals 1
1.1 Introduction 1
1.2 Modelling vibrations and vibrating systems 1
1.3 Some basic concepts 3
1.3.1 The phenomenon of beats 5
1.3.2 Displacement, velocity, acceleration and decibels 6
1.4 Springs, dampers and masses 8
2 Formulating the equations of motion 13
2.1 Introduction 13
2.2 Systems of material particles 14
2.2.1 Generalised co-ordinates, constraints
and degrees of freedom 15
2.3 Virtual work and d’Alembert’s principles –
Lagrange and Hamilton equations 16
2.3.1 Hamilton’s equations (HEs) 20
2.4 On the properties and structure of Lagrange’s equations 24
2.4.1 Invariance in the form of LEs
and monogenic forces 24
2.4.2 The structure of the kinetic energy
and of Lagrange equations 24
2.4.3 The energy function and the conservation of energy 28
2.4.4 Elastic forces, viscous forces and
Rayleigh dissipation function 29viii Contents
2.4.5 More co-ordinates than DOFs:
Lagrange’s multipliers 32
2.5 Hamilton’s principle 34
2.5.1 More than one independent variable:
continuous systems and boundary conditions 38
2.6 Small-amplitude oscillations 44
2.7 A few complements 48
2.7.1 Motion in a non-inertial frame of reference 48
2.7.2 Uniformly rotating frame 51
2.7.3 Ignorable co-ordinates and the Routh function 53
2.7.4 The Simple pendulum again: a
note on non-small oscillations 56
3 Finite DOFs systems: Free vibration 59
3.1 Introduction 59
3.2 Free vibration of 1-DOF systems 59
3.2.1 Logarithmic decrement 65
3.3 Free vibration of MDOF systems: the undamped case 67
3.3.1 Orthogonality of eigenvectors and normalisation 68
3.3.2 The general solution of the undamped
free-vibration problem 70
3.3.3 Normal co-ordinates 72
3.3.4 Eigenvalues and eigenvectors sensitivities 78
3.3.5 Light damping as a perturbation
of an undamped system 80
3.3.6 More orthogonality conditions 82
3.3.7 Eigenvalue degeneracy 83
3.3.8 Unrestrained systems: rigid-body modes 84
3.4 Damped systems: classical and non-classical damping 87
3.4.1 Rayleigh damping 88
3.4.2 Non-classical damping 90
3.5 GEPs and QEPs: reduction to standard form 92
3.5.1 Undamped Systems 93
3.5.2 Viscously damped systems 94
3.6 Eigenvalues sensitivity of viscously damped systems 96
4 Finite-DOFs systems: Response to external excitation 99
4.1 Introduction 99
4.2 Response in the time-, frequency- and s-domains:
IRF, Duhamel’s integral, FRF and TF 100Contents ix
4.2.1 Excitation due to base displacement,
velocity or acceleration 105
4.3 Harmonic and periodic excitation 107
4.3.1 A few notes on vibration isolation 110
4.3.2 Eccentric excitation 112
4.3.3 Other forms of FRFs 114
4.3.4 Damping evaluation 116
4.3.5 Response spectrum 117
4.4 MDOF systems: classical damping 120
4.4.1 Mode ‘truncation’ and the
mode-acceleration solution 122
4.4.2 The presence of rigid-body modes 125
4.5 MDOF systems: non-classical viscous
damping, a state-space approach 126
4.5.1 Another state-space formulation 129
4.6 Frequency response functions of a 2-DOF system 133
4.7 A few further remarks on FRFs 137
5 Vibrations of continuous systems 139
5.1 Introduction 139
5.2 The Flexible String 140
5.2.1 Sinusoidal waveforms and standing waves 142
5.2.2 Finite strings: the presence of
boundaries and the free vibration 143
5.3 Free longitudinal and torsional vibration of bars 148
5.4 A short mathematical interlude: Sturm–Liouville problems 150
5.5 A two-dimensional system: free
vibration of a flexible membrane 156
5.5.1 Circular membrane with fixed edge 158
5.6 Flexural (bending) vibrations of beams 162
5.7 Finite beams with classical BCs 163
5.7.1 On the orthogonality of beam eigenfunctions 167
5.7.2 Axial force effects 168
5.7.3 Shear deformation and rotary
inertia (Timoshenko beam) 170
5.8 Bending vibrations of thin plates 174
5.8.1 Rectangular plates 176
5.8.2 Circular plates 180
5.8.3 On the orthogonality of plate eigenfunctions 181
5.9 A few additional remarks 182x Contents
5.9.1 Self-adjointness and positive-definiteness
of the beam and plate operators 182
5.9.2 Analogy with finite-DOFs systems 185
5.9.3 The free vibration solution 188
5.10 Forced vibrations: the modal approach 190
5.10.1 Alternative closed-form for FRFs 199
5.10.2 A note on Green’s functions 201
6 Random vibrations 207
6.1 Introduction 207
6.2 The concept of random process, correlation
and covariance functions 207
6.2.1 Stationary processes 212
6.2.2 Main properties of correlation
and covariance functions 214
6.2.3 Ergodic processes 216
6.3 Some calculus for random processes 219
6.4 Spectral representation of stationary random processes 223
6.4.1 Main properties of spectral densities 227
6.4.2 Narrowband and broadband processes 229
6.5 Response of linear systems to
stationary random excitation 232
6.5.1 SISO (single input–single output) systems 233
6.5.2 SDOF-system response to broadband excitation 236
6.5.3 SDOF systems: transient response 237
6.5.4 A note on Gaussian (normal) processes 239
6.5.5 MIMO (multiple inputs–multiple outputs) systems 241
6.5.6 Response of MDOF systems 243
6.5.7 Response of a continuous system to distributed
random excitation: a modal approach 245
6.6 Threshold crossing rates and peaks distribution
of stationary narrowband processes 249
Appendix A: On matrices and linear spaces 255
Appendix B: Fourier series, Fourier and
Laplace transforms 289
References and further reading 311
Index 31
Index
accelerance 114, 138
acceleration 6
accelerometer 7
action integral 34
amplitude 3
complex 4
peak 3
angular frequency 3
asymptotically stable system 64
base excitation 105–106
basis 261
orthonormal 264–267
beams 162–167
axial-force effects 168–170
Euler-Bernoulli 163, 187, 197
Timoshenko 163, 170–174
Rayleigh 173
shear 172
beats 5–6, 76
biorthogonality 275
bode diagram 108
boundary condition (BC) 40, 143
geometric (or imposed) 40
natural (or force) 40
boundary-value problems (BVP) 150
broadband process 230–232
BVP see boundary-value problem
Caughey series 89
Cholesky factorisation 83
Christoffel symbol 27
Clapeyron’s law 30
complete orthonormal system 154
condition number 287
configuration space 93
constraint
bilateral16
equations 15
forces 16, 33
holonomic 15
non-holonomic 15
rheonomic 15
scleronomic 15
unilateral 16
continuous system 2
convolution 101, 225, 298
co-ordinates
Cartesian 15
generalised 15
ignorable (or cyclic) 53
correlation function 209, 214–216
correlation matrix 241
covariance function 209, 214–216
critically damped system 62
dashpot 9
d’Alembert solution 141, 310
damping
classical 87–90, 120–122
coefficient 9
critical 61
evaluation 116–117
matrix 47
non-classical 90–92, 126–133
light damping perturbation 80–81
ratio 61
decibel 7–8
degree of freedom (DOF) 2, 16
deterministic vibration 2
differential eigenvalue problem 186
dirac delta ‘function’ 300–304
discrete system 2318 Index
displacement 6
Duhamel integral 101
DOF see degree of freedom
dynamic coupling 77
dynamic magnification factor 107
dynamic potential 27
eccentric excitation 112–114
eigenfunction 145, 151
eigenvalue 67, 145, 151, 274
degeneracy 68, 83, 278
sensitivity 78–81, 96–97
eigenvector 67, 274
left 130, 275
right 130, 275
sensitivity 78–81
energy
function 28
kinetic 18
potential 19
strain 30
ensemble 211
equation
Bessel 159, 180
Helmholtz 157
ergodic random process 216–219
Euler’s buckling load 169
Euler relations 3
extended Hamilton’s principle 36
flexibility (or compliance) 9
force
conservative 18
elastic 29–30
fictitious 50, 53
generalised 18
gyroscopic 27
inertia 19
monogenic 24
nonconservative 19
viscous 30–32
forced vibration 48
SDOF systems 100–103, 107–110
continuous systems 190–199
Fourier
coefficients 289
series 289–294
transform 294–300
free vibration 48
SDOF systems 59–65
MDOF systems 67–68, 70–72
continuous systems 148–150,
156–167, 174–181, 188–190
frequency 3
equation 67, 144
fundamental 67, 289
natural 60
of damped oscillation 62
ratio 102
frequency response function (FRF)
102–103, 114–116, 137–138,
199–201
estimate 235
estimate 235
modal 121, 192, 244
FRF see frequency response function
function
Bessel 159, 180
coherence 236
complementary 99
Green 201–205
harmonic 3
Heaviside 302
periodic 109, 289
weight 151
Gaussian (or normal) processes
239–241
generalised momenta 20
generalised potential 24
generalised eigenvalue problem (GEP)
68, 92–96, 274
Gibbs’ phenomenon 294
Gram-Schmidt orthonormalisation
process 267–268
Green’s formula 151
half-power bandwidth 116
Hamilton function (or
Hamiltonian) 20
Hamilton equations 21
non-holonomic form 34
Hamilton’s principle 34–37
extended 36
harmonic excitation 107–108
Hermitian form 282
independent random variables 210
impulse response function (IRF) 100
modal 120, 191, 244
initial value problem (IVP) 14, 141Index 319
inner product 265, 268–269
IRF see impulse response function
isomorphism 264
Lagrange’s equations 18
non-holonomic form 33
standard form 19
Lagrange function (or Lagrangian) 19
Lagrange identity 151
Lagrange’s multipliers 32–34
Lagrangian density 38, 157, 175
Laplace transform 304–310
linear system 1
logarithmic decrement 65–66
mass 9
apparent 116
coefficients 46
matrix 46
modal 69
matrix 255
defective 278
determinant 258
diagonal 257
diagonalisable 277
dynamic 93
exponential 282–283
Hermitian (or self-adjoint) 256,
279–281
Hermitian adjoint 256
inverse 259
negative-definite 282
negative-semidefinite 282
nondefective 278
nonsingular 259
normal 257
orthogonal 257
positive-definite 282
positive-semidefinite 282
rank 259
similar 270, 277
singular 259
skew-hermitian 256
skew-symmetric 256
spectral 70
symmetric 256, 279–281
trace 258
transpose 256
unitary 257
MDOF see multiple degree of freedom
mean 210
mean square value 210, 291
matrix 245
mechanical impedance 116
membrane free vibration 156–157
circular 158–162
rectangular 157–158
multiple inputs-multiple outputs
(MIMO) systems 241–243
mobility 114, 117
modal
damping ratio 86
force vector 120
matrix 70, 281
participation factor 120
mode
acceleration method 122–124
complex 81, 91–92
shape 67, 145
truncation 122
moments (in probability) 209
central 210
non-central 210
multiple degrees of freedom (MDOF)
systems
multiplicity
algebraic 68, 275, 278
geometric 83, 278
narrowband process 229–230
peak distribution 251–254
threshold crossing rates 249–251
natural system 25, 28
Newton’s laws 13
non-inertial frame of reference 48–53
nonlinear system 1
non-natural system 25
norm
matrix 285–288
vector 265–266
normal (or modal) co-ordinates 72–73,
120, 244
normalisation 68–70, 147, 274
mass 69
Nyquist plot 108, 117
operator
beam 182–183
biharmonic 175
Laplace (also Laplacian) 157320 Index
operator (Cont.)
linear 263–267
mass 186
plate 183–185
stiffness 186
Sturm-Liouville 150
self-adjoint 152
orthogonality 68–70, 82, 92
mass 69
of beam eigenfunctions 167–168
of plate eigenfunctions 181–182
stiffness 69
overdamped system 62
Parseval’s relation 291, 296–297
pendulum
compound (or physical) 44
double 23, 44
simple 21–23, 56–57
period 3
periodic excitation 109–110
Phase angle 3
phase space 93
phasor 4
plates 174–176
circular 180–181
flexural stiffness 175
Kirchhoff theory 174
Mindlin theory 174
rectangular 176–180
power spectral density (PSD) 223,
227–228
matrix 241
principle
d’Alembert 17
of least action 35
of virtual work 17
superposition 1
uncertainty 299
probability
density function (pdf) 208
distribution function (PDF) 208
progressive wave 141
PSD see power spectral density
quadratic eigenvalue problem (QEP)
80, 92–96
quadratic form 282
random
process 207, 219–223
variable (r.v.) 207
vibration 2, 207
Rayleigh
damping 88
dissipation function 30–32
distribution 251, 253
quotient 154
receptance 114
reciprocity 122, 205
resonance 108, 110
response
indicial 104
pseudo-static 123
resonant 109
spectrum 117–119
transient237–239
Riemann-Lebesgue lemma 292
rigid-body modes 83–87, 125, 149
rod vibration 148–150
root mean square 210
rotary (or rotatory) inertia 170, 173,
174
Routh function (or Routhian) 53–55
sample function 211
SDOF see single degree of freedom
shaft vibration 148–150
shear deformation 163, 170, 172, 174,
187
single degree of freedom (SDOF)
systems 92, 233, 236–239,
244, 251
singular value decomposition 283–285
single input-single output (SISO)
systems 233–236
spectral density see power spectral
density
spectral matrix 70
spring 8
equivalent 9–11
stable system 64
standard deviation 210
standard eigenvalue problem (SEP) 68,
273–276
standing wave 142
stationary random process 212–214,
223–227
weakly stationary (WS) 214
state space 93–96, 126–133
stiffness 8
coefficients 45Index 321
coupling 77
dynamic 116
matrix 46
modal 69
stochastic process see random process
string vibrations 140–148
Sturm-Liouville problem (SLp) 146,
150–156
regular 150
singular 161
theorem
bandwidth 299–300
Bauer-Ficke 288
Betti’s 182
Dirichlet 292
Euler’s 29
Murnaghan-Wintner 284
Schur’s decomposition 283–284
spectral for normal matrices 281
transmissibility
force 111
motion 110
transfer function (TF) 102
uncorrelated random variables 211
underdamped system 62
unstable system 64
variance 210
varied path 35
vector space 260–263
dimension 261–262
subspace 261
velocity 6
group 162, 173
phase 162, 173
vibration isolation 110–112
virtual displacement 16
viscous damper 8
wave equation 140, 148
wavelength 142
wavenumber 142, 309
Weibull distribution 253
white noise 231
band-limited 231
Wiener-Khintchine relations 224


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