Dynamics of Large Structures and Inverse Problems

Abdelkhalak El Hami, INSA-Rouen, France Bouchaib Radi, Setta Hassan First University, Morocco.

• Preface xiChapter 1 Introduction to Inverse Methods 11.1 Introduction 11.2 Identification methods 31.3 Identification of the strain hardening law 61.3.1 Example of an application 81.3.2 Validation test 91.3.3 Hydroforming a welded tube 11Chapter 2 Linear Differential Equation Systems of the First Order with Constant Coefficients: Application in Mechanical Engineering 152.1 Introduction 152.2 Modeling dissipative systems 152.2.1 Intrinsic solutions of autonomous systems 172.2.2 Intrinsic solutions 172.2.3 Intrinsic solutions of the adjoining system 192.2.4 Relation between the intrinsic solutions of s and s* 192.2.5 Relation between modal matrices X and X* 202.3 Autonomous system general solution 212.3.1 Direct solution by using the exponential matrix 212.3.2 Indirect solution by modal transformation 232.4 General solution of the complete equation 242.4.1 Direct solution by the exponential matrix 242.4.2 Indirect solution by modal transformation 242.4.3 General solution in the particular case of harmonic excitation 262.5 Applications to mechanical structures 272.5.1 Discrete mechanical structure at n degrees of freedom, linear, regular and non-dissipative 272.5.2 Discrete mechanical structure at n DOF, linear, regular and dissipative 292.5.3 Intrinsic vector norm 322.5.4 Particular solution of the system with a harmonic force 342.6 Inverse problems: expressions of the M, B, K matrices according to the intrinsic solutions 36Chapter 3 Introduction to Linear Structure Dynamics 413.1 Introduction 413.2 Problems in structure dynamics 413.2.1 Finite elements method 433.2.2 Modal superposition method 443.2.3 Direct integration 463.2.4 Newmark method 463.2.5 The θ Wilson method 473.2.6 Modal analysis of the sandwich beam 49Chapter 4 Introduction to Nonlinear Dynamic Analysis 534.1 Introduction 534.2 Linear systems 544.2.1 Generalities 544.2.2 Simple examples of large displacements 564.2.3 Simple example of a variable 584.2.4 Simple example of dry friction 584.2.5 Material nonlinearities 594.3 The nonlinear 1 DOF system 604.3.1 Generalities 604.3.2 Movement without non-dampened excitation 614.3.3 Case of a stiffness in the form � (1 + �� 2) 624.3.4 Movement with non-dampened excitation 654.3.5 Movement with dampened excitation 684.4 Nonlinear N DOF systems 714.4.1 Generalities 714.4.2 Nonlinear connection with periodic movement 724.4.3 Direct integration of the equations 74Chapter 5 Condensation Methods Applied to Eigen Value Problems 775.1 Introduction 77Contents vii5.2 Mathematical generality: matrix transformation 785.3 Dynamic condensation methods 805.4 Guyan condensation 845.5 Rayleigh–Ritz method 875.6 Case of a temporary problem 905.6.1 Simplification with a full modal basis 91Chapter 6 Linear Substructure Approach for Dynamic Analysis 1056.1 Generalities 1056.2 Different types of Ritz vectors 1076.2.1 Stress vectors of the j st substructure �� (j) 1076.2.2 Attachment vectors of the j st substructure �� (j) 1086.2.3 Displacement field type vectors in dynamic regimes 1086.3 Synthesis of eigen solutions of the assembled structure: formulation by an energetic method (Lagrange with multiplicators) 1116.3.1 Equilibrium equation of the k st isolated substructure �� (k) 1116.3.2 Ritz basis for the k the substructure �� (k) 1126.3.3 Compatibilities between substructure �� (1) and �� (2) 1136.3.4 Lagrangian L of the assembled structure 1136.4 Craig and Bampton substructuration method 1166.4.1 Formulation of base relations in the case of two substructures 1176.4.2 Assembly of two substructures 1196.4.3 Restoring physical DOF 1206.4.4 Comments 1216.5 Mixed method 1216.5.1 Formation in the case of a single secondary SS 1226.5.2 Reconstructing the assembled structure 1226.5.3 Comments 1236.6 Methods with eigen vectors with free common contours 1246.6.1 Stiffness method of coupling 1246.6.2 Solution to [6.39] Ritz transformation 1276.6.3 Formulation based on the dynamic flexibility matrices: search for the assembled structure’s eigen solutions 1296.6.4 Formulation in the case of two �� (k) , k = 1,2, etc 1306.7 Method systematically introducing an intermediary connection structure 1336.7.1 Formation 1336.7.2 Introducing Ritz vectors 1366.7.3 Introducing fitting conditions 1376.7.4 Equilibrium equations of the assembled structure 1396.7.5 Normalization of the assembled structure’s eigen vectors 1406.7.6 Critique of the method 141Chapter 7 Nonlinear Substructure Approach for Dynamic Analysis 1457.1 Introduction 1457.2 Dynamic substructuration approaches 1477.2.1 Linear case 1487.2.2 Nonlinear case 1497.3 Nonlinear substructure approach 1517.3.1 Vibration equations of a substructure 1527.3.2 Fixed interface problem 1537.3.3 Static raising problem 1557.3.4 Representation of the system in Craig-Bampton’s linear base 1557.3.5 Model reduction with the Shaw and Pierre approach 1577.3.6 Assembling substructures 1597.4 Proper orthogonal decomposition for flows 1607.4.1 Properties of the POD modes 1617.4.2 POD snapshot 1627.4.3 Script of low-order dynamic systems 1637.5 Numerical results 1687.5.1 Modal analysis 1717.5.2 Decomposition of the circular acoustic cavity 1737.5.3 Decomposition of the elastic ring 174Chapter 8 Direct and Inverse Sensitivity 1778.1 Introduction 1778.2 Direct sensitivity 1808.2.1 Definition of the state’s sensitivity matrix x(t) 1808.2.2 Sensitivity equations 1808.2.3 Simple direct applications 1828.3 Sensitivity of eigen solutions 1838.3.1 Direct numerical method 1838.3.2 Derivatives of the eigen vectors according to the modal bases 1848.3.3 Derivatives of eigen vectors based on the exact expressions 1878.4 First derivative of a particular solution 1908.4.1 Scalar case (primarily didactic) 1908.4.2 General case 1908.5 Grouping the sensitivity relations together 1918.5.1 Variations 1918.5.2 Grouping the eigen values and eigen vectors together 1928.6 Inverse sensitivity 1958.6.1 Overdetermined case: 2a > m 1968.6.2 Unique solution: 2a = m 1978.6.3 Underdetermined case: 2a < m 198Chapter 9 Parametric Identification and Model Adjustment in Linear Elastic Dynamics 2059.1 Introduction 2059.2 Study in the elastic dynamics of mechanical structures 2069.2.1 Provisional calculations of behavior based on mathematical models 2079.2.2 Identification 2079.3 Parametric identification – use of a test for constructing weaker calculation models 2089.3.1 Introduction 2089.3.2 Error minimization in the behavioral equation 2099.3.3 Error minimization on the outputs 2109.3.4 Combined estimation of the state and the parameters 2119.4 Some basic methods in parametric identification 2119.4.1 Linear dependency with respect to the parameters and estimation in the sense of the least squares 2119.4.2 Estimation of parameters in the sense of maximum likelihood 2129.4.3 Estimation of the vector p by the Gauss–Newton method Bayes formulation Vector z(p) nonlinear function of p 2149.4.4 Non-random least squares method 2189.4.5 Quasi-linearization method 2209.5 Parametric correction of finite elements models in linear elastic dynamics based on the test results 2219.5.1 Highlighting a few difficulties 2229.6 M model adjustment: k∈ � c, c by minimizing the matrix norms by the correction matrices δm, δk 2239.6.1 Principle of Baruch and Bar-Itzhack method 2249.6.2 Kabe, Smith and Beattie methods 2269.7 M model adjustment: k∈ � c, c by minimizing residue vectors made up based on local correction matrices ΔM I , ΔK I 2279.7.1 Minimization of formed residue based on the behavior equation 2289.7.2 Minimization of formed reside based on outputs 228Chapter 10 Inverse Problems in Dynamics: Robustness Function 23510.1 Introduction 23510.2 Convex models 23610.2.1 Definitions 23610.2.2 Direct problem 23710.2.3 Inverse problem 23710.3 Robustness function 23810.3.1 Monocriterion response 23810.3.2 Multicriteria response 23810.4 Solution methods 23910.4.1 Interval arithmetic 23910.4.2 Optimization method 24010.5 Numerical calculations 24410.6 Applications 24510.6.1 Dual-recessed beam 24510.6.2 Square 25110.7 Conclusion 256Chapter 11 Modal Synthesis and Reliability Optimization Methods 25911.1 Introduction 25911.2 Design reliability optimization in structural dynamics 26011.2.1 Frequential hybrid method 26011.2.2 Optimization condition of the hybrid problem 26611.3 The SP method 27011.3.1 Formulation of the problem 27111.3.2 Implementation of the SP approach 27311.4 Modal synthesis and RBDO coupling methods 28111.5 Discussion 286Appendix 289Bibliography 299Index 307