Random Eigenvalue Analysis For Structures With Random Parameters

In the process of the design and analysis of engineering problems, we always neglect the uncertainty and the error in the realistic structure. That is to say in general the theories of the design and analysis of structures are always established on the basis of the definite mathematics models. When the theories of the certainty models have plentiful progress and obtain very good application in practice, we notice there are always uncertain factors in the structural engineering practices, such as the inaccuracy of the measurement, the complexity of the structures or errors in manufacture, etc. When the structures are large and complex, the combination of the uncertainty can have some effect on the systems, especially in multi-part system. Therefore, it is necessary to design and analyze the structures with uncertain models directly. Nowadays, the research through the certainty models can't satisfy us, we pay more attention to the new method through the uncertainty models that can describe the real process when the engineers work.The uncertainty can be described as following three kinds:1.physical uncertainty. 2.statistical uncertainty. 3.model uncertainty. And for the past decades, these uncertainties have been quantified by some uncertain methods. Generally speaking, according to the mathematical models with uncertainties, there are some models as follows: probability models; fuzzy models; convex models. The most common methods for solving uncertainty problems are to model the structural parameters as a random vector. Stochastic Mathematics and the elementary vibration theories are used in this model. In this model, we often regarded that the design parameter are the random variable and the random field. And the uncertainty of the parameter have caused the uncertainty of eigenvalue and eigenvector. The process as follows: Suppose the uncertainty parameter b is the random variable or the random field, so the eigenvalue and eigenvector are also the random variable or the random field. Its characteristic may act according through the theory of Stochastic Mathematics such as mean value, variance, relativity. In the second chapter of this article, I introduced the method of Stochastic Mathematics and the elementary vibration theories related to stochastic model. The third chapter describes the uncertainty of the system by stochastic model. And the expression of eigenvalue and eigenvector in uncertainty system is given by perturbation theory. The sensitivities of eigenvalue and eigenvector to the design variable and the expectation of the first–order sensitivities is calculated. We can find the relation between the correlated stochastic variables and the non-correlated stochastic variables that have been translated. The expectation of random sensitivity of eigenvalue and eigenvector of the non-correlated stochastic variables that have been translated is calculated.In Chapter 4, we discussed the computation of the expectation of the second-order sensitivities of the eigenvalue. Its expression is ( ) ( )E COV b bin the expectation is the second-order sensitivity matrix of eigenvalues-Hessian matrix. COV ( bi ,bk ) is covariance matrix of the stochastic variable. Only standard deviation and correlation matrix are used In the calculation, so the problem is predigested. second-order perturbation method of eigenvalues of multi-parameter structure is used to calculate Hessian matrix and discuss the method about how to calculate high-order sensitivities of eigenvalues. At first, by using first-order perturbation method of eigenvalues of multi-parameter structure, we obtained the expression of first-order sensitivities of eigenvalue and eigenvector. Secondly, we obtained the expression of second-order sensitivities of eigenvalues and the approximation algorithm of Hessian matrix by the expression of first-order sensitivities of eigenvalue and eigenvector. And an example is showed to sure the method effective.