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DUALITY SYSTEM in APPLIED MECHANICS and OPTIMAL CONTROL Advances in Mechanics and Mathematics DUALITY SYSTEM IN APPLIED MECHANICS AND OPTIMAL CONTROL Advances in Mechanics and Mathematics Volume 5 Series Editors: David Y. Gao Virginia Polytechnic Institute and State University, U.S.A. Ray W. Ogden University of Glasgow, U.K. Advisory Editors: I. Ekeland University of British Columbia, Canada K.R. Rajagopal Texas A&M University, U.S.A. W. Yang Tsinghua University, P.R. China DUALITY SYSTEM IN APPLIED MECHANICS AND OPTIMAL CONTROL by Wan-Xie Zhong Dalian University of Technology KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 1-4020-7881-1 Print ISBN: 1-4020-7880-3 ©2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2004 Kluwer Academic Publishers Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstore at: http://ebooks.kluweronline.com Contents v Preface xi Introduction 1 0.1, Introduction to Precise Integration method 4 0.1.1, Homogeneous equation, algorithm for exponential matrix. 4 0.1.2, Solution of inhomogeneous equation 6 0.1.3, Precision analysis 7 0.1.4, Discussions on time-variant system or non-linear system 9 Chapter 1, Introduction to analytical dynamics 11 1.1, Holonomic and nonholonomic constraints 11 1.2, Generalized displacement, degrees of freedom and virtual displacement 14 1.3, Principle of virtual displacement and the D’Alembert principle 16 1.4, Lagrange equation 18 1.5, Hamilton variational principle 19 1.6, Hamiltonian canonical equations 24 1.6.1, Legendre transformation and Hamiltonian canonical equations 24 1.6.2, Cyclic coordinate and conservation 28 1.7, Canonical transformation 29 1.8,Symplectic description of the canonical transformation 32 1.9, Poisson bracket 34 1.9.1, Algebra of Poisson bracket 35 1.10, Action 37 1.11, Hamilton-Jacobi equation 38 1.11.1,A simple harmonic oscillator 39 1.11.2, Time invariant system 40 1.11.3, Separation of variables 41 1.11.4, Separation of variables for linear systems 42 Chapter 2, Vibration theory 47 2.1, Single degree of freedom vibration 47 2.1.1, Linear vibrations 47 2.1.2, Parametric resonance 49 2.1.3, Introduction to non-linear vibration 52 2.1.3.1, Limit cycle 52 2.1.3.2, Duffing equation 54 vi 2.1.3.3, Simplified solution method for Duffing equation 57 2.2, Vibration of multi-degrees of freedom system 61 2.2.1. Free vibration with no damping, eigen-solutions 62 2.2.2, Constraints, count of eigenvalues 66 2.2.2.1, Inclusion theorem 67 2.2.2.2, Dynamic stiffness matrix and eigenvalue count 68 2.2.3, Eigenvalue sign count and substructure analysis 70 2.2.3.1, Mixed energy, the eigenvalue count for dual variables 72 2.2.3.2, Substructure combination and eigenvalue count of mixed energy 75 2.2.3.3, Essence of modal synthesis method 77 2.2.4, Subspace iteration for the eigen-solution of symmetric matrices 79 2.2.4.1, Ortho-normalization algorithm 80 2.2.4.2, Subspace projection and its eigen-solutions 80 2.2.4.3, Subspace iteration 80 2.2.4.4, Transition of the subspace 82 2.2.5, Eigen-solutions of asymmetric real matrix 82 2.2.5.1, Dual subspace iteration for asymmetric matrix 83 2.2.6, Singular value decomposition 85 2.2.6.1, QR decomposition 86 2.2.6.2, Singular value decomposition 86 2.3, Small vibration of gyroscopic systems 87 2.3.1, Method of separation of variables, eigen-problem 89 2.3.2, Positive definite Hamilton function 92 2.3.2.1, Variational principle of eigenvalues 93 2.3.2.2, Eigenvalue count and the inclusion theorem 96 2.3.3, Indefinite Hamilton function 98 2.3.3.1, Effect of gyroscopic force to the stability of vibration 100 2.3.3.2, Symplectic eigenvalue problem and its algorithm 102 2.3.3.3, Symplectic eigen-solution of skew-symmetric matrix 106 2.3.3.4, Numerical example 113 2.4, Non-linear vibration of multi-degrees of freedom system 115 2.4.1, Non-linear internal parametric resonance 120 2.4.2, Non-linear internal sub-harmonic resonance 124 2.5, Discussion on the stability of gyroscopic system 127 2.5.1, Gyroscopic system with positive definite Hamilton function 127 2.5.2, Case of indefinite Hamilton function 129 Chapter 3, Probability and stochastic process 135 3.1, Preliminary of probability theory 135 3.1.1, Probability distribution function and probability density function 136 3.1.2, Mathematical expectation, variance and covariance 137 3.1.3, Expectation of a random vector and its co-variance matrix 138 3.1.4, Conditional expectation and covariance of random vector 139 3.1.5, Characteristic function of random variable 140 3.1.6, Normal distribution 140 3.1.7, Linear transformation and combination of Gauss random vectors 142 3.1.8, Least square method 143 3.2, Preliminary of stochastic process 147 vii 3.2.1, Stationary and non-stationary stochastic process 149 3.2.2, Ergodic stationary process 149 3.3, Quadratic moment stochastic process (regular process) 150 3.3.1, Continuity and differentiability of a regular stochastic process 151 3.3.2, Mean square integration 152 3.4, Normal stochastic process 153 3.5, Markoff process 154 3.6, Spectral density of stationary stochastic process 155 3.6.1, Wiener-Khintchin relation 155 3.6.2, Direct spectral analysis of stationary stochastic process 156 3.6.3, White noise 157 3.6.4, Wiener process 158 Chapter 4, Random vibration of structures 161 4.1, Models of random excitation 163 4.1.1, Stationary random excitations 163 4.1.2, Non-stationary random excitations 165 4.2, Response of structures under stationary excitations 167 4.2.1, Random response of single degree of freedom system 167 4.2.2, Multi-degrees of freedom system under single source excitation 170 4.2.2.1, Case of inconsistent damping 176 4.2.2.2, Single source multi-point excitation with different phases 177 4.2.3, Stationary response of structure to multi-source excitations 178 4.2.3.1, Spectral expansion 179 4.2.3.2, Response analysis 179 4.3, Response under excitation of non-stationary stochastic process 180 4.3.1, Response under uniformly modulated non-stationary excitation 180 4.3.2, Response under evolutionary modulated non-stationary excitation 181 Chapter 5, Elastic system with single continuous coordinate 183 5.1, Fundamental equations of Timoshenco beam theory 183 5.2, Potential energy density and mixed energy density 185 5.3, Separation of variables, Adjoint symplectic ortho-normality 188 5.3.1, Adjoint symplectic orthogonality 189 5.3.2, Expansion theorem 191 5.4, Multiple eigenvalues and the Jordan normal form 192 5.4.1. Wave propagation for Timoshenco beam and its extension 194 5.4.2, Physical meaning of symplectic orthogonality—work reciprocity 197 5.5, Expansion solution of the inhomogeneous equation 201 5.6, Two end boundary conditions 202 5.7, Interval mixed energy and precise integration method 206 5.7.1, Displacement method analysis 207 5.7.2, Mixed energy, the dual variables 210 5.7.3, Riccati differential equation and its precise integration 213 5.7.4, Power series expansion 215 5.7.5, Interval combination 216 5.7.6, Precise integration for the fundamental interval 218 5.7.7, Precise integration for asymmetric Riccati equations 222 viii 5.8, Eigenvector based solution of Riccati equations 229 5.8.1, Analytical solution applied to the symmetric Riccati equations 235 5.8.2, Algorithm for the eigen-solutions of a Hamilton matrix 237 5.8.3, Transform to real value computation 240 5.8.4, Transformation for purely imaginary eigenvalues 242 5.9, Stepwise integration method by means of sub-structural combination 245 5.10, Influence function of single continuous coordinate system 249 5.10.1, Reciprocal theorems of the impulse influence matrix functions 254 5.10.2, Precise integration of impulse influence matrix function 257 5.10.3, Numerical example of impulse influence matrix function 259 5.10.4, Application 260 5.11, Power flow 262 5.11.1, Algebraic Riccati equation (ARE) 263 5.11.2, Transmission waves 265 5.11.3, On power flow orthogonality 266 5.12, Wave scattering analysis 268 5.13, Wave induced resonance 270 5.14, Wave propagation along periodical structures 272 5.14.1, Dynamic stiffness matrix of a fundamental substructure 273 5.14.2, Energy band and eigen-solutions of a symplectic matrix 273 5.14.3, Pass-band analysis for periodical wave-guides 275 5.14.4, Dynamic stiffness of a fundamental period and eigenvalue count 277 5.14.5, Computation of the pass-band eigenvalue 280 Chapter 6, Linear optimal control, theory and computation 283 6.1, State space of linear system 284 6.1.1, Input-output description and state space description 284 6.1.1.1, Continuous-time and discrete-time systems 287 6.1.2, State space description of single-input single-output system 289 6.1.3, Integration of linear time-invariant systems 291 6.1.4, Frequency domain analysis, transfer function 294 6.1.5, Controllability and observability of a linear system 294 6.1.5.1, Controllability of a steady system 295 6.1.5.2, Observability of a steady system 295 6.1.6, Linear transformation 296 6.1.7, Realization of a transfer function in state space 297 6.1.8, Duality principle for controllability and observability 298 6.1.9, Discrete-time control 300 6.2, Theory of stability 301 6.2.1, Stability analysis under Lyapunov meaning 302 6.2.2, Lyapunov method of stability analysis 303 6.3, Prediction, filtering and smoothing 305 6.4, Prediction and its computation 308 6.4.1, Mathematical model for prediction 309 6.4.2,
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