A Novel Method for the Probabilistic Solutions of Dynamical Systems with Both Random Excitation and Random Parameters

In this paper, a novel method, named improved Sobol integration method, is proposed to obtain the probability density function (PDF) of the responses of the dynamical systems containing random parameters and subjected to random excitation. First, the system containing random parameters is converted into an ensemble of deterministic systems by sampling the random parameters from their respective PDFs. After that, the conditional PDFs of the response of deterministic systems under random excitation are obtained analytically. By employing conditional PDFs and the PDFs of the random system variables, the overall PDF of the system responses is obtained by integrating the product of the conditional PDFs of the responses and the PDFs of random system parameters. Then, the proposed method is employed to complete the integration. It offers advantages in terms of efficiency over conventional Sobol integration method and Monte Carlo simulation. Notably, it provides accurate solutions in the tails of the PDF of system responses, which is crucial for the reliability analysis of systems. The obtained PDF solutions are compared with those from the conventional Sobol integration method and the Monte Carlo simulation to test the effectiveness and efficiency of the proposed method. Particularly for the systems containing multiple random variables, which can be a challenge problem for estimating the tail behavior of the PDF solutions of system responses.