IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE University of London 6758&785$/'<1$0,&$1$/<6,6$1' 7(67,1*2)&283/('6758&785(6 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 Department of Mechanical Engineering Imperial College of Science, Technology and Medicine London SW7 October, 2000 $%675$&7 Engineering structures normally exist in the form of assemblies of several components or substructures, and the modelling, prediction and optimisation of the assembled structures are presently achieving far from the required accuracy and reliability, and from that which is attained for the individual components. This is the essential problem addressed by this research. The difficulties in achieving the required accuracy are almost certainly due to the variety and complexity of joint types and to the lack of an accurate estimation of the interactions between substructures. This thesis provides two approaches for joint parameter identification, a least-squares method based and a neural network based one. Their mathematical backgrounds are thoroughly presented and their validity is examined by numerical case studies. The coupling analysis method has the equal importance to the joint parameter identification. Two branches of coupling methods, CMS and FRF-based, are systematically investigated. Two new methods, one in each branch, are developed to take joint effects into account in the analysis. Numerical studies show that these methods are accurate and efficient. The significance of modal incompleteness and measurement noise to the coupling analysis is also estimated. Two relevant issues in joint modelling and substructure coupling are also discussed in this thesis. They are (i) non-linearity considerations in joint modelling and substructure coupling and (ii) impact of rotational DOF information. The former reviews the progress of nonlinear joint modelling as well as the analysis methods dealing with nonlinear coupling problem, indicating that the nonlinear behaviour of fastening joints is not significant. The latter demonstrates the importance of the RDOF-related information in both joint modelling and FRF coupling analysis. i $&.12:/('*(0(176 I am deeply indebted to Professor D J Ewins for his supervision throughout this research work. It is his initiatives and instructions that enabled me to achieve developments in this area. I am also grateful to the other members of the staff who helped me a lot during these years, Dr. Imregun, Mr. Robb, Mrs. Savage and Mr. Woodward. I wish to express my appreciation to my colleagues in the Dynamics Section of Imperial College for their friendly cooperation and useful discussions. I would to express my gratitude to Bosch GmbH and CVCP. It would have been impossible for me to study here without the financial support from them. Special thanks to my parents for all their love, encouragement and understanding. Finally, I wish to thank my wife and son for their love and sacrifice. They shared my happiness and disappointments. ii 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 coordinates iii 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 2 eigen-value β proportional coefficient of damping matrix σ singular value or standard deviation of added noise γ percentage of noise iv Symbols A substructure A B substructure B C coupling coordinate of assembled structure, subscript C coupling coordinate on substructure A of assembled structure, subscript ~ C 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 c~ 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 v 7DEOHRI &RQWHQWV &+$37(5 *(1(5$/,1752'8&7,21 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 &+$37(5 /,1($5-2,1702'(//,1*²/($67648$5(60(7+2' 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 &+$37(5 /,1($5-2,1702'(//,1*²1(85$/1(7:25.0(7+2' 3.1. GENERAL IDEAS ........................................................................................................................46 vi 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