Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.