| Preface |
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xi | |
| Introduction |
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1 | (3) |
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Introduction to Precise Integration method |
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4 | (7) |
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Homogeneous equation, algorithm for exponential matrix. |
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4 | (2) |
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Solution of inhomogeneous equation |
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6 | (1) |
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7 | (2) |
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Discussions on time-variant system or non-linear system |
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9 | (2) |
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Introduction to analytical dynamics |
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11 | (36) |
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Holonomic and nonholonomic constraints |
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11 | (3) |
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Generalized displacement, degrees of freedom and virtual displacement |
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14 | (2) |
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Principle of virtual displacement and the D'Alembert principle |
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16 | (2) |
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18 | (1) |
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Hamilton variational principle |
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19 | (5) |
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Hamiltonian canonical equations |
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24 | (5) |
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Legendre transformation and Hamiltonian canonical equations |
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24 | (4) |
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Cyclic coordinate and conservation |
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28 | (1) |
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29 | (3) |
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Symplectic description of the canonical transformation |
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32 | (2) |
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34 | (3) |
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Algebra of Poisson bracket |
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35 | (2) |
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37 | (1) |
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38 | (9) |
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A simple harmonic oscillator |
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39 | (1) |
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40 | (1) |
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41 | (1) |
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Separation of variables for linear systems |
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42 | (5) |
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47 | (88) |
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Single degree of freedom vibration |
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47 | (14) |
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47 | (2) |
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49 | (3) |
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Introduction to non-linear vibration |
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52 | (1) |
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52 | (2) |
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54 | (3) |
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Simplified solution method for Duffing equation |
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57 | (4) |
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Vibration of multi-degrees of freedom system |
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61 | (26) |
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Free vibration with no damping, eigen-solutions |
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62 | (4) |
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Constraints, count of eigenvalues |
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66 | (1) |
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67 | (1) |
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Dynamic stiffness matrix and eigenvalue count |
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68 | (2) |
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Eigenvalue sign count and substructure analysis |
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70 | (2) |
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Mixed energy, the eigenvalue count for dual variables |
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72 | (3) |
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Substructure combination and eigenvalue count of mixed energy |
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75 | (2) |
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Essence of modal synthesis method |
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77 | (2) |
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Subspace iteration for the eigen-solution of symmetric matrices |
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79 | (1) |
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Ortho-normalization algorithm |
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80 | (1) |
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Subspace projection and its eigen-solutions |
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80 | (1) |
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80 | (2) |
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Transition of the subspace |
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82 | (1) |
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Eigen-solutions of asymmetric real matrix |
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82 | (1) |
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Dual subspace iteration for asymmetric matrix |
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83 | (2) |
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Singular value decomposition |
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85 | (1) |
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86 | (1) |
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Singular value decomposition |
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86 | (1) |
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Small vibration of gyroscopic systems |
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87 | (28) |
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Method of separation of variables, eigen-problem |
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89 | (3) |
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Positive definite Hamilton function |
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92 | (1) |
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Variational principle of eigenvalues |
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93 | (3) |
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Eigenvalue count and the inclusion theorem |
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96 | (2) |
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Indefinite Hamilton function |
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98 | (2) |
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Effect of gyroscopic force to the stability of vibration |
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100 | (2) |
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Symplectic eigenvalue problem and its algorithm |
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102 | (4) |
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Symplectic eigen-solution of skew-symmetric matrix |
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106 | (7) |
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113 | (2) |
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Non-linear vibration of multi-degrees of freedom system |
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115 | (12) |
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Non-linear internal parametric resonance |
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120 | (4) |
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Non-linear internal sub-harmonic resonance |
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124 | (3) |
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Discussion on the stability of gyroscopic system |
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127 | (8) |
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Gyroscopic system with positive definite Hamilton function |
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127 | (2) |
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Case of indefinite Hamilton function |
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129 | (6) |
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Probability and stochastic process |
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135 | (26) |
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Preliminary of probability theory |
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135 | (12) |
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Probability distribution function and probability density function |
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136 | (1) |
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Mathematical expectation, variance and covariance |
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137 | (1) |
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Expectation of a random vector and its co-variance matrix |
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138 | (1) |
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Conditional expectation and covariance of random vector |
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139 | (1) |
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Characteristic function of random variable |
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140 | (1) |
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140 | (2) |
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Linear transformation and combination of Gauss random vectors |
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142 | (1) |
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143 | (4) |
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Preliminary of stochastic process |
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147 | (3) |
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Stationary and non-stationary stochastic process |
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149 | (1) |
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Ergodic stationary process |
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149 | (1) |
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Quadratic moment stochastic process (regular process) |
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150 | (3) |
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Continuity and differentiability of a regular stochastic process |
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151 | (1) |
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152 | (1) |
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Normal stochastic process |
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153 | (1) |
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154 | (1) |
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Spectral density of stationary stochastic process |
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155 | (6) |
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Wiener-Khintchin relation |
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155 | (1) |
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Direct spectral analysis of stationary stochastic process |
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156 | (1) |
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157 | (1) |
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158 | (3) |
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Random vibration of structures |
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161 | (22) |
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Models of random excitation |
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163 | (4) |
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Stationary random excitations |
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163 | (2) |
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Non-stationary random excitations |
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165 | (2) |
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Response of structures under stationary excitations |
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167 | (13) |
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Random response of single degree of freedom system |
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167 | (3) |
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Multi-degrees of freedom system under single source excitation |
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170 | (6) |
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Case of inconsistent damping |
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176 | (1) |
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Single source multi-point excitation with different phases |
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177 | (1) |
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Stationary response of structure to multi-source excitations |
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178 | (1) |
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179 | (1) |
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179 | (1) |
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Response under excitation of non-stationary stochastic process |
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180 | (3) |
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Response under uniformly modulated non-stationary excitation |
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180 | (1) |
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Response under evolutionary modulated non-stationary excitation |
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181 | (2) |
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Elastic system with single continuous coordinate |
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183 | (100) |
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Fundamental equations of Timoshenco beam theory |
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183 | (2) |
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Potential energy density and mixed energy density |
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185 | (3) |
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Separation of variables, Adjoint symplectic ortho-normality |
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188 | (4) |
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Adjoint symplectic orthogonality |
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189 | (2) |
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191 | (1) |
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Multiple eigenvalues and the Jordan normal form |
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192 | (9) |
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Wave propagation for Timoshenco beam and its extension |
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194 | (3) |
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Physical meaning of symplectic orthogonality---work reciprocity |
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197 | (4) |
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Expansion solution of the inhomogeneous equation |
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201 | (1) |
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Two end boundary conditions |
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202 | (4) |
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Interval mixed energy and precise integration method |
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206 | (23) |
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Displacement method analysis |
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207 | (3) |
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Mixed energy, the dual variables |
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210 | (3) |
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Riccati differential equation and its precise integration |
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213 | (2) |
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215 | (1) |
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216 | (2) |
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Precise integration for the fundamental interval |
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218 | (4) |
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Precise integration for asymmetric Riccati equations |
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222 | (7) |
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Eigenvector based solution of Riccati equations |
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229 | (16) |
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Analytical solution applied to the symmetric Riccati equations |
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235 | (2) |
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Algorithm for the eigen-solutions of a Hamilton matrix |
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237 | (3) |
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Transform to real value computation |
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240 | (2) |
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Transformation for purely imaginary eigenvalues |
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242 | (3) |
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Stepwise integration method by means of sub-structural combination |
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245 | (4) |
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Influence function of single continuous coordinate system |
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249 | (13) |
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Reciprocal theorems of the impulse influence matrix functions |
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254 | (3) |
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Precise integration of impulse influence matrix function |
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257 | (2) |
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Numerical example of impulse influence matrix function |
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259 | (1) |
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260 | (2) |
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262 | (6) |
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Algebraic Riccati equation (ARE) |
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263 | (2) |
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265 | (1) |
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On power flow orthogonality |
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266 | (2) |
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268 | (2) |
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270 | (2) |
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Wave propagation along periodical structures |
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272 | (11) |
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Dynamic stiffness matrix of a fundamental substructure |
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273 | (1) |
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Energy band and eigen-solutions of a symplectic matrix |
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273 | (2) |
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Pass-band analysis for periodical wave-guides |
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275 | (2) |
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Dynamic stiffness of a fundamental period and eigenvalue count |
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277 | (3) |
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Computation of the pass-band eigenvalue |
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280 | (3) |
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Linear optimal control, theory and computation |
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283 | (160) |
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State space of linear system |
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284 | (17) |
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Input-output description and state space description |
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284 | (3) |
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Continuous-time and discrete-time systems |
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287 | (2) |
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State space description of single-input single-output system |
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289 | (2) |
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Integration of linear time-invariant systems |
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291 | (3) |
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Frequency domain analysis, transfer function |
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294 | (1) |
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Controllability and observability of a linear system |
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294 | (1) |
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Controllability of a steady system |
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295 | (1) |
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Observability of a steady system |
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295 | (1) |
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296 | (1) |
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Realization of a transfer function in state space |
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297 | (1) |
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Duality principle for controllability and observability |
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298 | (2) |
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300 | (1) |
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301 | (4) |
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Stability analysis under Lyapunov meaning |
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302 | (1) |
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Lyapunov method of stability analysis |
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303 | (2) |
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Prediction, filtering and smoothing |
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305 | (3) |
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Prediction and its computation |
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308 | (20) |
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Mathematical model for prediction |
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309 | (1) |
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Prediction of one dimensional system |
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310 | (2) |
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Prediction of multi-degrees of freedom system |
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312 | (2) |
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314 | (1) |
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Precise integration of inhomogeneous equations |
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315 | (2) |
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Precise integration of the Lyapunov differential equation |
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317 | (2) |
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Precise integration of the algebraic Lyapunov equation |
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319 | (2) |
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Integration of asymmetric Lyapunov equation |
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321 | (3) |
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Solution with modulated input |
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324 | (2) |
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Disturbance of colored noise |
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326 | (1) |
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Some comments for precise integration of the extended system |
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327 | (1) |
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328 | (49) |
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Model of linear estimation |
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329 | (1) |
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Model of discrete-time system |
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329 | (1) |
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Model of continuous-time system |
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330 | (1) |
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Kalman filter analysis for discrete-time linear system |
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330 | (5) |
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Correlated dynamic and measurement noises |
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335 | (1) |
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Continuous-time Kalman-Bucy filtering analysis |
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336 | (4) |
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Correlated dynamic and measurement noises |
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340 | (2) |
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Continuous-time Kalman-Bucy filter under colored noises |
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342 | (1) |
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343 | (2) |
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345 | (2) |
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Differential equations for the interval matrices and vectors |
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347 | (3) |
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Physical interpretation of the solution of Riccati equation |
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350 | (1) |
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Precise integration of the Riccati differential equation |
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351 | (5) |
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Analytical solution of Riccati equation based on eigen-solutions |
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356 | (1) |
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Solution of single step filter equation |
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357 | (3) |
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Analytical single step integration for the filter equation |
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360 | (4) |
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Analytical equation for single step integration |
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364 | (2) |
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Taylor expansion of precise stepwise filtering |
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366 | (3) |
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Interval combination within the η interval |
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369 | (3) |
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Integration of filter equation for the whole interval |
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372 | (3) |
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375 | (2) |
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Optimal smoothing and computations |
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377 | (14) |
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Optimal smoothing of continuous-time linear system |
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378 | (3) |
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Interval mixed energy and differential equation for smoothing |
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381 | (4) |
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Mean value and variance of smoothing |
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385 | (3) |
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Precise integration for single time step |
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388 | (1) |
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Three kinds of smoothing algorithms |
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389 | (1) |
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389 | (1) |
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390 | (1) |
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390 | (1) |
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391 | (16) |
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Theory of LQ optimal control for the future time interval |
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392 | (3) |
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395 | (1) |
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Gram matrices of controllability and observability |
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396 | (1) |
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Positive definiteness of the Riccati matrix |
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397 | (4) |
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Stability analysis based on Lyapunov second method |
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401 | (1) |
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Precise computation of LQ control |
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402 | (1) |
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Precise solution of Riccati differential equation |
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403 | (1) |
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Integration of state differential equation |
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404 | (2) |
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Measurement feedback optimal control |
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406 | (1) |
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407 | (36) |
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Analysis of H∞ state feedback robust control |
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410 | (3) |
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Extended Rayleigh quotient |
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413 | (2) |
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415 | (3) |
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418 | (1) |
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419 | (4) |
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423 | (4) |
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Solution of dual equations |
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427 | (1) |
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Extended Rayleigh quotient |
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428 | (2) |
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430 | (2) |
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Precise integration method |
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432 | (1) |
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433 | (2) |
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Synthesis variational principle for measurement feedback control |
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435 | (2) |
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437 | (1) |
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Variance matrix of the state estimation vector |
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438 | (1) |
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439 | (4) |
| Concluding remarks |
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443 | |
Other versions by this Author