Problems Of Repeated Eigenvalues Of Random Structures

The history and designing research for stochastic finite element method are reviewed firstly. Then the merits and defects of conventional methods are proposed. The conventional perturbation stochastic finite element method can consider uncertainty of mass, damp, hardness, stimulus etc. as variables or uncertainties, but these variables are usually supposed smallness. One Recursive stochastic finite element method will be introduced in this paper, the new stochastic finite element method that can solve problems including random variables of large fluctuation levels uses discrete random fields to express elastic module. A series of certain recursive equations like perturbation method will be developed. Solving these equations, we can get expectation and mean square covariance of random structural eigenvalues. The stochastic structure with repeated eigenvalues is taken as research objects in this paper, and structural free vibration characters and solutions are investigated by the stochastic finite element method. Hence exploring research works on 5 aspects have been done around the purpose.1. Summarizing random field discretization, the paper compares the merits and defects of three conventional discrete methods, then introduce theories of spectrum decomposition and Monte-Carlo simulating method about stochastic process.2. Introducing the development of perturbation method, especially the principle of matrix perturbation method, this paper researches changes of structural eigenvalues and eigenvectors with structural parameters' variables, and analyzes matrix perturbation method of one order and two order in structures with isolated eigenvalues, then extend this method to structures with repeated eigenvalues.3. Considering structural materials' randomness and using perturbation stochastic finite element method to research it, isolated and repeated eigenvalues in structures will be analyzed. Then the computational expressions of numerical characters about structural modes are developed.4. Utilizing the new recursive stochastic finite element method to analyze free vibration of random structures with isolated and repeated eigenvalues, thencombining theories about random fields, a series of certain recursive equations will be developed. By solving these equations, statistical expressions of expectation and mean square covariance about eigenvalues will be gotten.5. Through two typical numerical examples about repeated eigenvalues, perturbation and recurisive stochastic finite element methods are compared with Monte-Carlo simulation method. We will find it that: perturbation stochastic finite element method can solve merely problems of isolated and repeated eigenvalues in the condition of random variables of small fluctuation levels; Monte-Carlo simulation method can solve these problems in the all fluctuation levels, but it costs too much time; the results of recursive stochastic finite element method can approach better those of Monte-Carlo simulation method than that of perturbation stochastic element method, even in the condition of large fluctuation levels, and this news stochastic finite element method can save much of much calculational time.In the end, the content of this paper has been summarized and some problems, which need further study, have been proposed.