An Improvement On Solving The Eigenvalues Of Stochastic Structure With Homotopy Analysis Method

The randomness in the response of structure was not take into consideration orbeen ignored during the traditional design and analysis of engineering structures. Theanalysis and calculation results of deterministic models could meet the actualsituation only if the original system’s variability is small enough. But the reality isthat the traditional mechanical model and analysis method are not able to reflect thereal property fully, which is caused by the influence of the material properties,geometry, boundary conditions and structural physical characteristics, etc. Thus it isof great significance to take the randomness of structural into consideration in theprocess of structure dynamic characteristics research. By now, the analysis of thisproblem is mainly implemented by the Monte-Carlo simulation method and theperturbation method.The Monte Carlo simulation method is time-consuming, especially even morefor large systems. The perturbation method, usually involves in the low order ofperturbation, and only works well in the case where the fluctuation of randomquantity is small as well as structural parameters are of Gaussian distribution. Forthese reasons, The Recursive Stochastic Finite Element Method proposed byprofessor Huang can solve the problem of large random variation well, and there is nolimit to the distribution of random parameters, which has been used to study thestochastic structure dynamic analysis and reliability issues successfully. But there is acertain error for special or greater variability of random structure compared with thesolution of Monte-Carlo simulation method. In this paper, the characteristic equationfor the structure is reconstructed with the idea of the homotopy analysis method. Thenthe zero order deformation equation is used to introduce the parameter h into therecursive stochastic finite element method, and expand the convergence domain ofthe recursive stochastic finite element method, which is simple and effective.In this paper, the following aspects were studied:1. Introduce a new random finite element method in detail, which is calledrecursive random finite element method. Using the nonorthogonal polynomials toexpanse the eigenvalues and eigenvectors, then establish a series of deterministicrecursive equation by merging the similar items. After that, calculate the extension coefficient of the nonorthogonal polynomials. Thus the mean of eigenvalue andeigenvector can be reached, so as the variance. The proposed method is effective andpractical through typical examples by comparing the results between the recursivestochastic finite element method and Monte Carlo simulation.2. The homotopy method was proposed by professor Liao of Shanghai JiaotongUniversity, which could be used to solve the problem of strong nonlinear. In thispaper, the multivariable generalized Taylor expression of complex function has beenderivated with this theory, and proved the convergence theorem.3. Appling the multivariable generalized Taylor series expansion to the fourthorder recursive stochastic finite element method. With the idea of the homotopyanalysis method, the characteristic equation for the structure is reconstructed, then，the zero order deformation equation is used to introduce the parameter h into therecursive stochastic finite element method. By comparing the relationship betweenthe plasticity modulus-frequency and the certainty elastic modulus-frequency to findout the most suitable h value, in order to achieved the purpose of circular frequencycorrection. Through concrete examples with two random variables, verified that thisnew method can better approaching with Monte-Carlo simulation results fully.