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Structural Dynamic Analysis and Testing of Coupled Structures
IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE
University of London
by
Wenjie Liu
A thesis submitted to the University of London
for the Degree of Doctor of Philosophy and
for the Diploma of Imperial College
120(1&/$785(
Matrices and vectors
C viscous damping matrix of structure
& viscous damping matrix of joint
’ structural damping matrix of joint
E error matrix
F force vector of assembled structure
H receptance matrix
I identity matrix
K stiffness matrix of assembled structure
. stiffness matrix of joint
M mass matrix of assembled structure
0 mass matrix of joint
R residual matrix
T transformation matrix
= joint impedance matrix
c viscous damping matrix of substructure
F viscous damping vector of joint
G structural damping vector of joint
f force vector of substructure
I force vector of joint
k stiffness matrix of substructure
N stiffness vector of joint
m mass matrix of substructure
P mass vector of joint
n noise sequence, a vector
p normal coordinatesiv
q generalised coordinates
x physical coordinates
2 eigen-value matrix, diagonal
mass normalised eigen-vector matrix
mass normalised eigen-vector
eigen-vector matrix
residual attachment mode matrix
Scalars
A cross section area
E Young’s modulus
h(t) response function of unit pulse
H (ω) the simulated noise-free FRF
H~(ω) the simulated noise-contaminated FRF
I bending section modulus
L number of internal DOFs
l length of a beam element
n number of DOFs
nc number of coupling DOFs of an assembly of substructures
n f number of frequency points
ω eigen-value
β proportional coefficient of damping matrix
σ singular value or standard deviation of added noise
γ percentage of noisev
Symbols
A substructure A
B substructure B
C coupling coordinate of assembled structure, subscript
C coupling coordinate on substructure A of assembled structure, subscript
coupling coordinate on substructure B of assembled structure, subscript
I internal coordinates of assembled structure, subscript
c coupling coordinate of a set of substructures, subscript
c coupling coordinate on substructure A in a set of substructures, subscript
coupling coordinate on substructure B in a set of substructures, subscript
h subscript, high frequency range
i internal coordinates of a set of substructures, subscript
l subscript, low frequency range
ℜ real set
Abbreviations
CMS component mode synthesis
CMSJ CMS with joint considered and residual attachment mode compensation
DOF degree of freedom
FE finite element
FRAC frequency response assurance criteria
FRF frequency response function
GJDM general joint description method
K-J Klosterman-Jetmundsen method
LSM least-squares method
PCA principal element analysis
RBF radial basis function
RDOF rotational degree of freedom
TDOF translational degree of freedom vi
1.1. INTRODUCTION TO THE PROBLEM .1
1.2. BRIEF REVIEW OF STATE-OF-THE-ART 2
1.3. PROPOSED DEVELOPMENTS .6
1.4. SUMMARY OF THIS THESIS 7
2.1. INTRODUCTION AND OBJECTIVES .10
2.2. THEORETICAL BACKGROUND .12
2.2.1. Definition of Joint 12
2.2.2. Conditions of Compatibility and Equilibrium 13
2.2.3. Essential Equations 14
2.2.4. Discussion on the Applicability 15
2.3. ALGORITHM FOR SOLVING JOINT PARAMETERS .17
2.3.1. Derivation of the Linear Equations for Joint Parameter Identification .17
2.3.2. Non-partitioned Algorithm .19
2.3.3. Partitioned Algorithm 22
2.4. ROBUSTNESS INVESTIGATION OF THE IDENTIFICATION APPROACHES 24
2.4.1. Numerical Simulation 1: A Crossbeam Structure 24
2.4.2. Numerical Simulation 2: Two Beams Coupled in Line via a Joint 27
2.4.3. Improvement of the Condition of Matrix A 33
2.4.4. Tests with Noise Contaminated Data and Error Analysis 37
2.5. CRITERION OF SELECTING INTERNAL DOFS: THEOREM OF TRANSMISSIBILITY .42
2.6. CONCLUSION 44
3.1. GENERAL IDEAS 46vii
3.2. BRIEF REVIEW OF NEURAL NETWORKS .47
3.2.1. Multi-layer Perceptrons 48
3.2.2. Radial Basis Function Networks .50
3.2.3. Comparison of MLP and RBF Networks .52
3.3. DISCUSSION ON PARAMETER SELECTION 53
3.4. GENERATION OF TRAINING SETS: A PARAMETRIC FAMILY OF FE MODELS 54
3.5. PRINCIPAL COMPONENT ANALYSIS TECHNIQUE 57
3.5.1. Definition 57
3.5.2. PCA and SVD 59
3.6. CASE STUDIES ON NUMERICAL SIMULATIONS .60
3.6.1. Simulation 1 61
3.6.2. Simulation 2 66
3.7. CONCLUSION .70
4.1. INTRODUCTION .72
4.2. REVIEW OF THE ESSENTIAL PRINCIPLES 79
4.2.1. CMS without Residual Compensation .80
4.2.2. Residual Compensation – First and Second Order Approximations .83
4.3. CMS WITH JOINTS CONSIDERED METHOD (CMSJ) 86
4.3.1. Coupling Equations .87
4.3.2. Residual Attachment Modes 89
4.3.3. Treatment of Rigid-Body Modes 93
4.4. CASE STUDIES 93
4.4.1. Clamped-Clamped Beam .94
4.4.2. GARTEUR structure 97
4.5. CONCLUSION 100
5.1. INTRODUCTION .101
5.2. REVIEW OF THE ESSENTIAL PRINCIPLES .103
5.2.1. FRF coupling without joint 103
5.2.2. FRF coupling with joint .104
5.3. GENERAL JOINT DESCRIPTION METHOD – NEW DEVELOPMENT .105viii
5.3.1. Theory background 105
5.3.2. Algorithm .108
5.4. CASE STUDIES 111
5.4.1. Cross beam structure .111
5.4.2. Plate couples with beam 114
5.5. CONCLUSION 125
6.1. MOTIVATIONS .126
6.2. STATE OF THE ART 127
6.2.1. Friction Modelling Based on Coulomb Theory: Lumped Parameter Models .129
6.2.2. Friction Modelling Based on Coulomb Theory: Continuous Contact Models 137
6.2.3. Friction Modelling Based on Other Principles .145
6.3. IMPORTANCE OF NON-LINEAR JOINT MODELLING 150
6.4. STRATEGIES FOR DEALING WITH NON-LINEARITY IN FRF COUPLING 155
6.4.1. The principle of harmonic balance 156
6.4.2. Describing function method 158
6.5. CONCLUSIONS .159
7.1. IMPACT OF RDOF DATA ON JOINT PARAMETER IDENTIFICATION 161
7.1.1. Theoretical Analysis – DOF Incompatibility .162
7.1.2. Numerical Illustrations 163
7.2. IMPACT OF RDOF DATA ON FRF COUPLING ANALYSIS 166
7.2.1. Theoretical Analysis .166
7.2.2. Numerical Illustrations 171
7.3. CONCLUSIONS 177
8.1. JOINT MODELLING .179ix
8.2. COUPLING ANALYSIS 180
8.3. NON-LINEARITY CONSIDERATIONS AND IMPACT OF RDOFS .181
8.4. SUGGESTED FUTURE WORK 182
A.1. INTRODUCTION .184
A.2. THE THEOREM OF TRANSMISSIBILITY .184
A.3. EXTENSION OF THE THEOREM .191
A.3.1. Two systems connected with a single spring: “ – ” connection 191
A.3.2. Systems connected with two springs I: “ < ” connection .193 A.3.3. Two systems connected with two springs II: “ > ” connection .194
A.4. LIMITATION OF THE THEOREM 195
A.5. NUMERICAL DEMONSTRATION 196
A.5. CONCLUSION .199
D.1. NOISE MODELLING – TYPE AND LEVEL 206
D.2. THE THEORY BACKGROUND OF NOISE LEVEL DEFINITION 208
D.3. AN ILLUSTRATION OF NOISE SIMULATION .210
E.1 DERIVATION OF SUBMATRIX ci
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