A probability estimation method for reliability analysis using mapped Gegenbauer polynomials

Based on the theory of Gegenbauer polynomial approximation, this article presents a probability estimation method for reliability analysis of mechanical systems. Analytical evaluation or statistical analysis is used to compute statistical moments. Then, the well-known Gegenbauer series approach is applied to approximate the probability distribution functions and cumulative distribution functions. The proposed method is completely distribution-free due to the fact that no classical theoretical distributions are assumed in the whole process, and the approximate analytic description provides a universal form of probability distribution function and cumulative distribution function. Five numerical examples involving mathematic functions and mechanical engineering problems are applied to illustrate the application of the proposed method. The results show that the probability distribution functions and cumulative distribution functions of random variables and responses can be accurately and efficiently derived from Gegenbauer polynomial approximation method with high-order statistical moments. © 2013 IMechE.