We develop a moment equation closure minimization method for the inexpensive
approximation of the steady state statistical structure of nonlinear systems
whose potential functions have bimodal shapes and which are subjected to
correlated excitations. Our approach relies on the derivation of moment
equations that describe the dynamics governing the two-time statistics. These
are combined with a non-Gaussian pdf representation for the joint
response-excitation statistics that has i) single time statistical structure
consistent with the analytical solutions of the Fokker-Planck equation, and ii)
two-time statistical structure with Gaussian characteristics. Through the
adopted pdf representation, we derive a closure scheme which we formulate in
terms of a consistency condition involving the second order statistics of the
response, the closure constraint. A similar condition, the dynamics constraint,
is also derived directly through the moment equations. These two constraints
are formulated as a low-dimensional minimization problem with respect to
unknown parameters of the representation, the minimization of which imposes an
interplay between the dynamics and the adopted closure. The new method allows
for the semi-analytical representation of the two-time, non-Gaussian structure
of the solution as well as the joint statistical structure of the
response-excitation over different time instants. We demonstrate its
effectiveness through the application on bistable nonlinear
single-degree-of-freedom energy harvesters with mechanical and electromagnetic
damping, and we show that the results compare favorably with direct Monte-Carlo
Simulations
