اسم المؤلف
Jimin He and Zhi-Fang Fu
التاريخ
19 فبراير 2022
المشاهدات
445
التقييم
(لا توجد تقييمات)

Modal Analysis
Jimin He and Zhi-Fang Fu
Contents
Preface xi
1 Overview of modal analysis 1
1.1 Introduction 1
1.2 What is modal analysis? 2
1.3 What is modal testing? 3
1.4 Applications of modal analysis 3
1.4.1 Troubleshooting 4
1.4.2 Correlation of finite element model and
experimental results 4
1.4.3 Structural modification 5
1.4.4 Sensitivity analysis 5
1.4.5 Reduction of mathematical models 5
1.4.6 Forced response prediction 5
1.4.7 Force identification 6
1.4.8 Response prediction 6
1.4.9 Substructure coupling 6
1.4.10 Structural damage detection 6
1.4.11 Active vibration control 7
1.5 Practical applications of modal analysis 7
1.6 Historical development of modal analysis 9
2 Mathematics for modal analysis 12
2.1 Basic matrix concepts 12
2.1.1 Trace of a matrix 13
2.1.2 Determinant of a matrix 13
2.1.3 Norm of a matrix 14
2.1.4 The rank of a matrix 15
2.1.5 Similarity of matrices 15
2.2 Linear simultaneous equations 15
2.3 Matrix inversion 18
2.3.1 Inverse of a non-singular real matrix 18
2.3.2 Inverse of a square complex matrix 19
2.3.3 Pseudo inverse of a matrix 20
2.4 Decomposition of a matrix 212.4.1 LU decomposition 21
2.4.2 QR decomposition 22
2.4.3 Schur decomposition 23
2.4.4 Spectrum decomposition 23
2.4.5 Submatrix decomposition 24
2.4.6 Singular value decomposition (SVD) 24
2.4.7 Eigenvalue decomposition 25
2.4.8 Cholesky decomposition 26
2.5 The matrix eigenvalue problem 26
2.6 Derivatives of matrices 29
2.6.1 Derivatives of a bilinear form 30
2.6.2 Derivatives of matrix traces 31
2.7 Perturbation 31
2.8 The least-squares method 33
2.9 Partial fraction expansion 34
2.10 Laplace transform and transfer function 35
2.11 Fourier series and Fourier transform 37
2.12 Variable separation method for partial differential equations 38
2.13 Poles and zeros of a polynomial function 40
2.14 State–space concept 40
2.15 Time series analysis 43
2.15.1 AR model 45
2.15.2 ARMA model 45
2.16 The z-transform 47
3 Basic vibration theory 49
3.1 Modelling of a vibration problem using mathematical models 49
3.2 Basic concepts of vibration 49
3.3 Free vibration of an SDoF system 50
3.4 Harmonic vibration of an SDoF system 52
3.5 Vibration of an SDoF system due to an arbitrary force 54
3.6 Free and harmonically forced vibration of an MDoF system 55
3.7 Energy approach 57
3.7.1 Virtual work principle 58
3.7.2 D’Alembert’s principle 59
3.7.3 Kinetic energy 59
3.7.4 Potential energy 61
3.7.5 Lagrange’s equation 62
3.8 Vibration of continuous systems 65
3.8.1 The vibrating string 66
3.8.2 Vibrations of membranes 68
3.8.3 Longitudinal vibration of a bar 73
3.8.4 Transverse vibration of a beam 75
4 Modal analysis theory of an SDoF dynamic system 79
4.1 Frequency response functions of an SDoF system 79
4.2 Graphical display of a frequency response function 81
vi Contents4.2.1 Amplitude–phase plot and log–log plot 82
4.2.2 Real and imaginary plots 84
4.2.3 Nyquist plot 85
4.2.4 Dynamic stiffness plot 86
4.3 Properties of the FRF of an SDoF system 88
4.3.1 Asymptoticity of log–log plots 88
4.3.2 Circularity of the Nyquist plot 90
4.3.3 Linearity of dynamic stiffness 93
5 Modal analysis of an undamped MDoF system 94
5.1 Normal modes and orthogonality of an undamped MDoF system 94
5.1.1 Normal modes of an undamped MDoF system 94
5.1.2 Orthogonality properties of an undamped MDoF system 97
5.2 Frequency response functions of an undamped MDoF system 100
5.2.1 Dynamic stiffness matrix and receptance matrix 100
5.2.2 Physical interpretation of a receptance FRF 102
5.2.3 Display of an FRF of an undamped MDoF system 103
5.3 Mass-normalized modes and modal model of an
undamped MDoF system 104
5.4 Frequency response functions and the modal model 106
5.4.1 Decomposition of an FRF using modal data 106
5.4.2 Other forms of an FRF 108
5.5 Asymptote properties of FRFs of an undamped MDoF system 108
5.6 Other forms of orthogonality properties of an
undamped MDoF system 112
5.6.1 Orthogonality between spatial model and response model 112
5.6.2 Orthogonality between measured modes and their
reciprocal modal vectors 113
5.6.3 Orthogonality between submatrices 114
5.6.4 Orthogonality with incomplete data 116
5.7 Harmonic response of an undamped MDoF system using FRFs 116
5.8 Anti-resonances and minima of an FRF 118
5.8.1 Anti-resonances and spatial data 118
5.8.2 Anti-resonances and modal data 119
5.8.3 Minima of an FRF 121
6 Modal analysis of a damped MDoF system 123
6.1 Proportional damping models 123
6.2 Non-proportional viscous damping model 125
6.3 Non-proportional structural damping model 127
6.4 Mass-normalized modes of a damped MDoF system 128
6.5 Frequency response functions of a damped MDoF system 128
6.5.1 Dynamic stiffness matrix and receptance matrix 128
6.5.2 Composition of a receptance FRF using vibration modes 129
6.5.3 Display and properties of an FRF of a damped
MDoF system 130
Contents vii6.6 Time response of a damped MDoF system 135
6.7 Forced normal modes of a damped MDoF system 136
6.8 Remarks on complex modes 138
7 Frequency response function measurement 140
7.1 Introduction 140
7.2 A general measurement set-up 141
7.2.1 Excitation mechanism 141
7.2.2 Accelerometer 142
7.2.3 Force transducer 144
7.3 Preparation of the test structure 145
7.4 Selection of excitation forces 147
7.4.1 Sinusoidal excitation 147
7.4.2 Random excitation 148
7.4.3 Pseudo random excitation 148
7.4.4 Impact excitation 148
7.5 Different estimates of an FRF and effects of noise 149
7.6 Two incompletenesses of measured data 152
7.7 Initial assessment of measured FRF data 153
7.7.1 Repeatability check of the measured FRF data 154
7.7.2 Reciprocity check of the measured FRF data 154
7.7.3 Linearity check of the measured FRF data 154
7.7.4 Special characteristics of an FRF 155
8 Modal analysis methods – frequency domain 159
8.1 Introduction 159
8.2 Detection of vibration modes from measured FRF data 160
8.3 Derivation of modal data from FRF data – SDoF methods 162
8.3.1 Peak-picking method 163
8.3.2 Circle fit method 164
8.3.3 Inverse FRF method 168
8.3.4 Least-squares method 170
8.3.5 Dobson’s method 171
8.4 Derivation of modal data from FRF data – MDoF methods 174
8.4.1 Rational fraction polynomials 174
8.4.2 Lightly damped structures 176
9 Modal analysis methods – time domain 180
9.1 Least-squares time domain method 181
9.2 Ibrahim time domain (ITD) method 182
9.3 Random decrement method 188
9.4 ARMA time series method 191
9.5 Least-squares complex exponential (LSCE) method 193
9.6 Summary of time domain modal analysis 195
viii Contents10 Multi-input multi-output modal analysis methods 198
10.1 Introduction 198
10.2 Estimation of FRFs for MIMO testing 199
10.2.1 No input noise – [ ( )] Hˆ 1 ω model 200
10.2.2 No output noise – [ ( )] Hˆ 2 ω model 204
10.2.3 Both input and output noise – [ ( )], [ ( )], H H ˆ ˆ 3 4 ω ω
[ ( )], [ ( )], H H ˆ ˆ v ω ω s models 206
10.3 Frequency domain poly-reference modal analysis method 209
10.4 Time domain global modal analysis method 212
10.4.1 Establishment of a mathematical model 212
10.4.2 Estimation of coefficient matrix [A] 215
10.4.3 Identification of modal parameters 216
10.5 Eigensystem realization algorithm method 217
10.5.1 State–space equation of an NDoF system 218
10.5.2 Impulse response matrix 219
10.5.3 Construction of a Hankel matrix 219
10.5.4 Impulse response matrix and system matrices 220
10.5.5 Eigensystem realization algorithm 220
11 Local structural modification 224
11.1 The objectives of structural modification 224
11.2 Direct and inverse problems 224
11.3 Modal properties after structural modification – reanalysis 225
11.4 Local structural modification by mass and stiffness changes 227
11.4.1 Unit rank modification 227
11.4.2 Mass modification of a lumped system 228
11.4.3 Stiffness modification of a lumped system 230
11.5 Local structural modification by physical parameter changes 233
11.6 Optimization of structural dynamic characteristics 236
11.6.1 Fixing a resonance and an anti-resonance
during modification 236
11.6.2 Pole–zero cancellation 238
11.6.3 Creation of a frequency range of no resonance 238
12 System identification using neural network 241
12.1 Introduction 241
12.2 Neural network model 242
12.2.1 Neuron 242
12.2.2 Topological structure of neural network 243
12.2.3 Learning algorithm of a neural network 244
12.3 BP network and its algorithm 245
12.4 Application of neural network in nonlinear system identification 248
12.5 Examples of using neural network for system identification 251
Contents ix13 Applications of modal analysis on real structures 257
13.1 Introduction 257
13.2 Modal analysis of a car chassis 257
13.3 Modal analysis of a lathe 260
13.4 Modal analysis of a shaper 262
13.5 Modal analysis of a combustion locomotive structure 263
13.6 Modal analysis of a power generator 265
13.7 Modal analysis of a flat flood gate 266
13.8 Modal analysis used for stability diagnosis 269
13.9 Modal analysis used for stump quality check 272
13.10 Modal analysis of ancient bronze bell 274
13.11 Modal analysis of bus roll cage structure for optimum
rollover design 279
Index 287
Accelerance, 80
Accelerometer, 141, 142
Accelerometer, mounting, 143
Activation function, 243
Active vibration control, 7
Acoustic modal analysis, 9
Ambient excitation, 180
Anti-resonance(s), 118
Apparent mass, 81
Applied forces, 58
AR model, 45
Arbitrary force, 54
Architecture, 243
ARMA model, 45, 191
Armature-bearing, 265
Artificial neural network (ANN), 241, 248
Asymptoticity, 88
Asymptote, 108
Asymptote lines, 90
Auto-correlation, 260
Auto-spectrum, 149, 260
Bar, 73
Backpropagation network, 245
Baseband frequency, 283
Beam, 75
Beating signal, 278
Bending modes, 260
Bending moment, 77
Bilinear form, 31
Black box, 140, 249
Boltzmann machine learning rule, 245
Boundary conditions, 67, 74, 76
BP network, 245
Bronze bell, 274
Built-in feedback, 244
Bus frame, 281
Bus role cage, 279
Cantilevers, 231
Car chassis, 257
Chattering, 261
Cholesky decomposition, 26
Circularity, 90
Circle fit method, 164
Circumferential mode shape, 278
Cofactor, 13
Coherence, 151, 201–204
Combustion locomotive, 263
Complex modes, 138
Compressors, 269
Condition monitoring, 7
Connective weights, 242
Constraint forces, 58
Continuous systems, 65
Controllability, 220
Correlation, 4
Coupling matrix, 42
Crashworthiness, 279
Cross sectional area, 234, 235
Cross spectrum, 149
Crystals, 144, 145
Cutting tool, 260
D’Alembert’s Principle, 59
Damage detection, 6
Damping, 53, 54
non-proportional, 125–128
proportional, 123
structural, 54
viscous, 53
Damage loss factor, 163
Damage ratio, 163
Decomposition of a matrix, 21
Decomposition:
Cholesky decomposition, 26
eigenvalue decomposition, 25
FRF, 106
LU decomposition, 21
QR decomposition, 22
Shur decomposition, 23
Index288 Index
singular value decomposition, 24
spectrum decomposition, 23
submatrix decomposition, 23
Decibel scales
Degrees of freedom
Deflection, 77
Delayed sampling, 183
Derivative, derivatives, of
bilinear form, 30
matrix, 29
matrix trace, 31
Different estimates, 149
Diagonalize, 99
Direct problem, 224
Dobson’s method, 171
Double pendulum, 64
Dual channel spectral analysis, 149
Dynamic equilibrium, 59
Dynamic stiffness, 81
Dynamic stiffness matrix, 100–101, 128
Eigenvalue decomposition, 25
Eigenvalue problem, 26
Elasticity modulus, 234, 273
Elemental matrices, 234, 235
Energy approach, 57
Euclidian norm, 14
Excitation forces, 147
Excitation:
ambient, 188
Hammer, 142
impact, 148
pseudo random, 148
random, 148
shaker, 142
sinusoidal, 147
Excitation frequency, 224
Excitation mechanism, 141
Fast Fourier transform, 10
Feedback, 244
Feedforward, 244
Feeding motion, 261
Finite element model, FE model, 4, 264, 275,
276, 280
Fixing a natural frequency, 237
Fixing a resonance, 236
Fixing an anti-resonance, 236
Flood gate, 266
Force identification, 6
Force response prediction, 5
Force transducer, 141, 144
Forced normal modes, 136
Fourier series, 9, 37
Fourier transform, 37
Free-free boundary condition, 258
Free vibration, 50
Frequency incompleteness, 152
Frequency response analyser, 258, 262
Frequency response function, 79, 80, 100,
(FRF):
accelerance, 80
mobility, 80
receptance, 80
Frobenius norm, 14
FRF:
different estimates of, 149
display of, 81–88, 103–104, 130–136
point, 101
other forms of, 107
special characteristics of, 155
transfer, 101
Gaussian elimination, 16, 19
Generalized coordinates, 58
Generalized damping, 125
Generalized mass, 60, 98
Generalized stiffness, 62, 98
Generalized velocity, 60
Generalized δ-rule, 345
Generator assembly, 265, 266
Half point points, 163
Hammer excitation, 142
Harmonic response, 116
Harmonic vibration, 52
Hebb rule, 244
Hidden layer, 244
Hopfield network, 245
Impulse response, 220, 271
Impulse response function, 180
In situ, 145
Incomplete data, 116
Incompleteness, 152
Infinitesimal, 57
Initial assessment, 153
Initial conditions, 67, 76
Input layer, 244
Input matrix, 42
Input vector, 42
Decomposition (Contd.)Index 289
Interconnection, 242
Inverse:
of a real matrix, 18
of a complex matrix, 19
pseudo, 20
Inverse Laplace transform, 36
Inverse problem, 224
ITD, 182–187
Kinetic energy, 59
Kronecker vector, 113
Lagrangian, 63
Lagrange’s equation, 62
Laplace transform, 35
Lathe, 260
Layer:
hidden, 244
input, 244
multi-, 244
output, 244
Least-squares method, 33, 170
Length, 234
Lightly damped structures, 176
Linear regression model, 45
Linear simultaneous equations, 15
Linearity, 93, 154
Local structural modification, 224, 231
Longitudinal vibration, 73
LR algorithm, 27
LR method, 27
LU decomposition, 21
Lumped system, 228, 230
Mass density, 234, 273
Mass line, 90
Mass normalized modes, 104, 105, 128
Mass normalized mode shapes
Mathematical model, 5, 49, 212
Matrix:
complex, 20
derivatives, 29
determinant, 13
eigenvalue, 26
hermitian, 23
inversion, 18
lower triangular, 21
norm, 14
pertubation, 31
rank, 15
similarity, 15
trace, 13
transformation, 213
upper triangular, 19
MDoF, 94
Measurement set-up, 141, 142
Mechanical impedance, 81
Membranes, 68
Method, methods:
ARMA time series, 191
circle fit, 164
Dobson’s, 171
eigensystem realization algorithm, 217
frequency domain poly-reference, 209
ibrahim time domain (ITD), 182
least-squares, 170
least-squares time domain, 181
least-squares complex exponential, 193
inverse FRF, 168
MDoF, 174
peak picking, 163
random decrement, 188
rational fraction polynomials, 174
SDoF, 162
time domain global, 212
MIMO, 147, 162, 198
Minima, 118
Minimum realization, 218
Mobility, 80
Modal analysis, 2
Modal constant, 108
Modal damping, 124
Modal mass, 98
Modal data, 119
Modal model, 105
Modal stiffness, 98
Modal testing, 3
Mode shape, 56
Modelling, 49
method, methods)
frequency domain, 159
time domain, 180
Modification:
local structural, 224
mass, 228
stiffness, 230
structural, 5, 224
Mounting, 143
Multiple coherence, 203
Natural frequency, 51
Neural network, 241
Neural network model, 242
Neuron, 242, 243
Neutralization, 238290 Index
Nodal lines, 71
Nodal point, 224
Noise and vibration harshness (NVH), 7
Noise reduction, 8
Non-proportional, 125–128
Non-singular, 16
Non-trivial solution, 237
Non-unique solution, 16
Nonlinear changes, 235
Nonlinear system identification, 248
Norm, 14
Normal modes, 94
NVH, 7
Observability, 220
Optimization, 236
Ordinary coherence, 201
Orthogonality, 72, 97, 112–116
Orthogonality properties, 97, 112–116
Output error, 244
Output equation, 42
Output layer, 244
Output matrix, 42
Output vector, 42
Overestimator, 150
Parallel model, 250
Partial coherence, 202
Partial differential equations, 38
Partial fraction expansion, 34
Peak FRF value, 163
Peak picking method, 164
Pendulum, 50
Perturbation, 31
Phase delay, 248
Physical elements, 224
Physical interpretation, 102
Physical parameter changes, 233
Perturbation, 31
Piezoelectric, 144
Plot:
amplitude-phase, 82, 130
dynamic stiffness, 86, 134
linear-log, 83
log-log, 88, 103, 130
Nyquist, 85, 133
real and imaginary, 84, 132
Pole-zero cancellation, 238
Poles, 40
Polynomial function, 40
Poly-reference, 209
Point FRF, 101
Potential energy, 61
Power amplifier, 141
Power generator, 265
Power spectrum, 271
Pressure sensor, 267
Principle modes, 94
Probability density, 268
Proportional damping, 123
Pseudo inverse, 20
Pseudo random, 148
QR decomposition, 22
QR method, 27
Random decrement, 188
Rank deficient, 15
Rational fraction polynomials, 174
Reanalysis, 225
Receptance, 80
Receptance FRF matrix, 101
Receptance matrix, 100–101, 128
Reciprocal modal vector, 113
Reciprocity, 101, 154
Repeatability, 154
Residue, 34
Residues, 81
Response model, 105
Response prediction, 6
Response segments, 188
Rod, 73
Rollover, 280
Rotors, 269
Rotor-bearing, 265
Rubber springs, 257
S form, 243
Scaling factor, 235
Self-regulated learning, 245
Self-excitation, 269
Seismic mass, 144
Sensitivity analysis, 5
Series-parallel model, 250
Shaker, 141
Shaper, 262
Shear force, 77
Shur decomposition, 23
Signal conditioner, 141
Signal generator, 141
Signoid function, 243
Simulated annealing, 245Index 291
Simulation diagram, 42
Singular value decomposition, 24
Single-degree-of-freedom (SDoF) system, 50
Sound spectrum, 275, 279
Spatial data, 118
Spatial incompleteness, 152
Spectral analysis, 43
Spectrum decomposition, 23
Stability, 269, 279
Stability diagnosis, 269
Standard deviation, 268
Static neural network, 250
State equations, 41
State-space, 40
State-space equation, 41, 218
State-space model, 40, 41
State-space transformation, 235
State-space vector, 235
State variables, 41, 218
State vector, 42
Stationary random process, 268
Steepest decent method, 245
Stiffness line, 90
Structural modification, 5, 224, 230, 233
Structure, preparation, 145–147
Stump quality check, 272
Substructure coupling, 6
Subframe, 7
Submatrix, 124
Submatrix decomposition, 23
Sweep sinusoidal excitation, 261
System equation, 42
System identification, 241, 242
Tailstock, 262
Tapped delay line (TDL), 248
Three dimensional plot, 82
Threshold form, 243
Time response, 135
Time series analysis, 43
Tool carriage, 262
Topological structure, 243
Topology, 242
Torsional mode, 260
Transfer FRF, 101
Transfer function, 35, 42, 55
Transformation matrix, 213
Transverse vibration, 75
Triaxial accelerometer, 263
Troubleshooting, 4
Turbine, 265
Underestimator, 150
Unique solution, 16
Unit rank modification, 227
Variable separation method, 38
Vector diagram, 54
Vibrating string, 66
Vibration:
basic concept, 49
free, 50
harmonic, 52
longitudinal, 73
transverse, 68, 75
Viener filter, 218
Virtual displacement, 58
Virtual work, 57
Virtual work principle, 58
Wave propagation, 272
Weighted least squares method, 33
Weightings, 244
Weights, 243
z-transform, 47
Zeros, 40
δ-rule, 245

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