Stochastic Dynamics of Structures

Abdelkhalak El Hami is Professor at the Institut National des Sciences Appliquées, Rouen, France. He is the author of many articles and books on optimization and uncertainty. Bouchaib Radi is Professor in the Faculty of Sciences and Technology at the University of Hassan Premier, Settat, Morocco. His research interests are in such areas as structural optimization, parallel computation, contact problem and metal forming. He is the author of many scientific articles and books.

• Preface  xi Chapter 1. Introduction to Structural Dynamics  11.1. Composition of problems relating to dynamic structures 21.1.1. Finite element method  41.1.2. Modal superposition method 51.1.3. Direct integration 61.2. Structural optimization  81.2.1. Design optimization 91.2.2. Shape optimization  91.2.3. Topological optimization 101.2.4. Definitions and formulation of an optimization problem  121.3. Structures with uncertain parameters 131.3.1. Monte Carlo simulation 141.3.2. Analytic method  151.3.3. Stochastic finite element method  151.3.4. Fluid logic method  161.3.5. Reliability method  171.3.6. Reliability optimization 211.4. Conclusion 23Chapter 2. Decoupled Systems 252.1. Introduction  252.2. Problems with structural dynamics  252.2.1. Movement equation  252.2.2. Hooke’s law  262.2.3. Variational formulation  272.2.4. Estimation by finite elements  272.2.5. Resolution in the frequency domain 282.2.6. Solution in the temporal domain 292.2.7. Reduction of the model  312.3. Acoustic problems 422.3.1. Wave equation: formulation pressure  422.3.2. Variational formulation  432.3.3. Estimation by finite elements  432.3.4. Solution in the frequency domain  442.3.5. Model fluid reduction 452.4. Conclusion 55Chapter 3. Coupled Systems  573.1. Introduction  573.2. Mathematical formulation 573.2.1. Behavior equations  573.2.2. Conditions for fluid–structure coupling 583.3. Variational formulation  593.4. Estimation by finite elements 593.4.1. Estimation of unknown physical values 593.4.2. Integration of variational forms 603.5. Vibro-acoustic problem  603.6. Hydro-elastic problem 613.6.1. Calculation of the elementary matrix of the fluid–structure interaction  643.6.2. Dynamic study  653.7. Reduction of the model  673.7.1. Modal superposition method for the paired system 673.7.2. Direct calculation 713.7.3. Calculation with modal reduction  723.7.4. Modal synthesis method for paired systems  743.7.5. Direct numerical calculation 813.7.6. Numerical calculation with modal superposition  833.8. Conclusion 84Chapter 4. Reliability and Meshless Methods in Mechanics  854.1. Introduction to non-networking methods  854.2. Moving least squares 884.2.1. Properties of MLS form function  944.2.2. Base functions 954.2.3. Weight functions 964.3. Galerkin mesh-free method  984.4. Imposition of essential limiting conditions 1034.4.1. Variational principle modified with Lagrange multipliers 1034.4.2. Variational principle modified without Lagrange multipliers 1044.4.3. Variational principle with a charge 1054.4.4. Connection with meshing of finite elements  1064.5. Integration in the EFG method  1074.6. Description of EFG method algorithms 109Chapter 5. Mechanical Systems with Uncertain Parameters 1155.1. Introduction  1155.2. Monte Carlo simulation  1165.3. Disturbance methods 1165.3.1. Expansion into a second-order Taylor series  1185.3.2. Muscolino distortion method  1245.3.3. Disturbance methods and modal reduction methods  1275.4. Projection onto polynomial chaos  1315.4.1. Moments of the response function in frequency 1345.4.2. Moments of dynamic response 1355.4.3. Projection onto polynomial chaos with modal reduction  1375.5. Conclusion 149Chapter 6. Modal Synthesis Methods and Stochastic Finite Element Methods  1516.1. Introduction  1516.2. Linear dynamic problems 1526.2.1. Equations of motion 1526.2.2. Solutions in the transient regime 1536.2.3. Solutions in the harmonic regime  1546.3. Modal synthesis methods 1556.3.1. Introduction  1556.3.2. Sub-structure assembly technique  1576.3.3. Fixed interface method  1586.3.4. MacNeal’s free interface method  1616.3.5. Free interface method 1636.3.6. Hybrid method  1666.3.7. Reduction in degrees of freedom of the interface 1666.4. Stochastic finite element methods  1686.4.1. Introduction  1686.4.2. Discretization of random fields 1696.4.3. Methods of moments 1716.5. Conclusion 179Chapter 7. Stochastic Modal Synthesis Methods  1817.1. Introduction  1817.2. Taylor series expansion of the modal equations of a stochastic structure 1817.2.1. Expression of the mean values and covariances  1847.3. Muscolino perturbation method 1847.3.1. Expansion of the modal equations of a stochastic structure 1857.4. Stochastic fixed interface method  1867.4.1. Taylor series expansion  1867.5. Stochastic modal synthesis method 1917.5.1. Introduction  1917.6. Conclusion 236Chapter 8. Dynamic Response of a Structure with Uncertain Variables to a Given Excitation 2378.1. Introduction  2378.2. Perturbation method  2388.2.1. Taylor series expansion of the equations of motion 2388.2.2. Muscolino perturbation method 2418.3. Stochastic modal synthesis method 2428.4. Projection onto homogeneous chaos 2458.5. Coupling modal synthesis methods with projection onto homogeneous chaos 2488.6. Conclusion 264Chapter 9. Stochastic Frequency Response Function  2659.1. Introduction  2659.2. Calculation of the stochastic frequency response function 2669.3. Calculation of the stochastic frequency response function with modal synthesis methods  2709.4. Conclusion 281Chapter 10. Modal Synthesis Methods and Reliability Optimization Methods  28310.1. Introduction  28310.2. Combining modal synthesis and RBDO methods 28310.3. Conclusion  294Chapter 11. Stochastic Model of Transmission in a Wind Turbine  29511.1. Introduction  29511.2. Modeling the dynamic behavior of the gearing system in a wind turbine 29511.3. Dynamic response of a two-step gear system in a wind turbine with uncertain variables  29611.3.1. Dynamic model of a two-step wind turbine transmission  29611.3.2. Study using the polynomial chaos method  29911.3.3. Perturbation method study 30911.3.4. Comparison of the different methods 31511.4. Conclusion  317Bibliography 319Index 327