Nonlinear Systems with Singular Diffusion Matrices: A Broad Perspective Including Hysteresis Modeling

In Chap. 6, the Wiener path integral technique is extended using a constrained variational formulation to treat a broader class of systems whose governing stochastic differential equations are characterized by singular diffusion matrices. In this regard, the governing equations of motion can be cast as a set of underdetermined stochastic differential equations coupled with a set of excitation-free equations. The latter, which can be of arbitrary (nonlinear) form, are construed as constraints. In this manner, a constrained variational problem is derived that is solved by employing various numerical schemes for determining the most probable path and for obtaining the system response joint probability density function. The chapter concludes with diverse numerical examples referring to structures with only a subset of their degrees-of-freedom excited; nonlinear systems exhibiting hysteresis modeled via additional auxiliary state equations, e.g., Bouc-Wen formalism; and energy harvesters with coupled electromechanical equations.