A complementary note on Gegenbauer polynomial approximation for random response problem of stochastic structure

Recently, Gegenbauer polynomial approximation was proposed for solving the evolutionary random response problem of a random structure with bounded random parameters under evolutionary random excitations. The bounded random parameters used there are supposed to be proportional to a random variable with λλ-PDF (probability density function). For this kind of random parameter, the Gegenbauer polynomial approximation is the unique correct choice for transforming a random structure into its deterministic equivalent system, which plays a central role in solving the response problem. Actually, the Gegenbauer polynomial approximation bridges the gap between the random structural response problem and the conventional methods. Just through its deterministic equivalent system, the random response problem of a random structure can be solved by any available, analytical and numerical method developed for deterministic systems. But the simple assumption on proportionality to λλ-PDF may bring some unnecessary limitation on symmetry. Since λλ-PDF is symmetrical about its center axis, so are these random parameters. However, not all random parameters have this kind of symmetry. Then, what can we do to lessen the limitation, if random parameters are non-symmetric per se? Besides, the graph of λλ-PDF (see Fig. 1 in the text) shows its features’ dependence on λλ, that is, the smaller the λλ, the more dispersive the PDF. Then, what is the influence of different values of λλ on the evolutionary random responses of the stochastic system? Moreover, is there any qualitative information about the dispersion in response with respect to every individual random parameter? This note is devoted to answering the above two questions and giving a supplementary note on Gegenbauer polynomial approximation. Our study shows that the symmetric limitation can be partly lessened by putting the random parameter into a polynomial function, even a quadratic one, of a random variable with λλ-PDF. On the other hand, the second-order moment of the random responses of the stochastic structure relative to that of a nominal one can be used to describe qualitatively their relative dispersion with respect to each individual random parameter.