Analytical solution of the generalized density evolution equation for stochastic systems: Euler-Bernoulli beam under noisy excitations and nonlinear vibration of Kirchhoff plate

The Generalized Density Evolution Equation (GDEE) describes the evolution of probability densities driven by physical processes. The numerical solution of the GDEE, implemented through a fully developed computational framework, is referred to as the Probability Density Evolution Method (PDEM). However, the absence of analytical solutions presents challenges for error calibration in numerical methods. In this study, analytical solutions of the GDEE are derived, focusing primarily on stochastic dynamic systems. The forced vibration of an Euler-Bernoulli beam subjected to random excitations is first analyzed, yielding analytical solutions for mid-span displacement response. For lower dimensional scenarios, two cases are examined: random harmonic loading and random step loading, both involving uncertainties in structural parameters. Results reveal that the corresponding displacement responses are non-Gaussian and non-stationary random processes. For higher dimensional scenarios, additional noise excitation is considered. By employing the Stochastic Harmonic Function (SHF) representation, noise excitation is effectively approximated as a superposition of finite random harmonic loads. Analytical derivations demonstrate that the SHF representation gradually converges toward the actual noise as the expansion terms increase. Furthermore, to illustrate the versatility of the developed analytical method, a nonlinear free vibration analysis of a Kirchhoff plate without external excitations is presented, showcasing its applicability to broader structural dynamic problems. These analytical solutions provide valuable benchmarks for further in-depth research into the PDEM, especially for the calibration of numerical methods.