Numerical Analysis Of Joint Probability Density Of Structural Natural Frequencies Based On Recursive Stochastic Finite Element Method

The traditional design and analysis of engineering structures are based on deterministic mechanical models, and ignore the uncertain variability of structural system modeling. Actually often a system modeling with mean parameters is built instead of the original structural system. Only as the uncertainty of system modeling is small, the structural analysis can give more reasonable results. However, real-life engineering structural systems are usually associated with some degree of uncertainty. In order to reflect the actual dynamic characteristics of the structures as much as possible, more and more studies aim to use stochastic methods to analyze the dynamic characteristics of the structures with uncertain or random parameters. In the context of structural dynamics, for deeply discussing these uncertainties of the dynamic characteristics, it is necessary to study joint probability density of random natural frequencies. By now, the analysis of the problem is mainly implemented by Monte Carlo simulation method and perturbation method.The Monte Carlo simulation method is time-consuming, especially even more for large systems. The perturbation method, usually involves in the low order of perturbation, and only works well in the case where the fluctuation of random quantities is small as well as structural parameters are of Gaussian distribution. For these reasons, this paper presents a new numerical method, named as recursive stochastic finite element method, this method can solve the problem of the large variability of the random variable, and also simple and effective.In this paper, the following aspects were studied:1. In Gao Hongbo's master's thesis, random parameters are Gaussian distributed random variables, through comparison with the Monte Carlo simulation method and perturbation methods, the effectiveness of the presented method is discussed. In this paper, random parameters are beta distribution or uniformly distributed random variables, through comparison with the Monte Carlo simulation method, explain that this method is not limit to the distribution of random parameters, and also simple and effective.2. Using the non-orthogonal polynomials to express the natural frequencies of the structure establish deterministic recursive equation which similar to perturbation method,and then solve order expansion coefficients of the non-orthogonal polynomials. Then according the distribution types of the random parameters to generate the corresponding sequence of random numbers generated. According to the idea of the Monte-Carlo simulation, using the non-orthogonal polynomials of natural frequency to produce one hundred thousand times samples of the natural frequency. Then compared the results with direct Monte-Carlo simulation method shows the effectiveness of the methods.3. Edgeworth expansion is an asymptotic expansion for solving the non-Gaussian probability density function. This article described the derivation of the Edgeworth asymptotic expansion, The cumulative amount and joint cumulant amount of function in the first-order Edgeworth expansion and second-order Edgeworth expansion were solved. Using this method to solve the probability density function of natural frequencies of the typical structures.4. Using the method in this paper, Monte-Carlo simulation method and Edgeworth asymptotic expansion to solve the probability density function of structure natural frequencies, comparised the results of three methods, to verify the proposed method is better able to fit the results of the Monte-Carlo simulation. By solving the probability density function of structure natural frequencies, mastere the total probability information of basic dynamic characteristics, we are able to analyze the structure much more better.