A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Abstract

We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems 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, based on copula functions 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 the 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-offreedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations.

Robustness of Nonlinear Electromagnetic Vibration Energy Harvester Subjected to Random Excitation

“…The most accurate uses Monte Carlo analysis of direct numerical computation of the oscillator response, however, this method is very time consuming since a large amount of simulation must be done on a sufficient time period. For efficiency reasons, we decide to propose a less accurate but faster method for a first analysis: the Moment Equation Copula Closure (MECC) method introduced by Joo et al [28]. This method is pratically described below.…”

“…However, this last method completely fails to capture the behavior of the bi-stable VEH when the equilibrium positions are too far from another; in these conditions, the excitation is not sufficient to enable intra-well are more frequent than inter-well movement and minimization of the MECC cost function doesnt adequately satisfyneither equations (14) nor (18). This weak point of the MECC method has been pointed out by Joo et al [28]. Once again, one note that the MECC method is much faster than Monte Carlo simulations.…”

“…First, the vibration random source is presented. Thereafter, the nonlinear vibration energy harvester model with base excitation is presented together with a semianalytical method [28] to approximate the statistical moments of interest. This method is then compared to direct Monte Carlo simulation while analyzing the behavior of the nonlinear VEH under vibrations iduced by the truck.…”

Heavy-Tailed Response of Structural Systems Subjected to Stochastic Excitation Containing Extreme Forcing Events

“…The statistical response of the background dynamics for the velocity response was obtained in Eq. (17). Noting that from Eq.…”

“…Systems with forcing having these characteristics pose significant challenges for traditional uncertainty quantification schemes. While there is a large class of methods that can accurately resolve the statistics associated with random excitations (e.g., the Fokker-Planck (FP) equation [13,14] for systems excited by white-noise and the joint response-excitation method [15][16][17][18] for arbitrary stochastic excitation), these have important limitations for high dimensional systems. In addition, even for lowdimensional systems determining the part of the probability density function (pdf) associated with extreme events poses important numerical challenges.…”