A quasi-Gaussian approximation method for the Duffing oscillator with colored additive random excitation

Statistical analysis of stochastic dynamical systems is of considerable importance for engineers as well as scientists. Engineering applications require approximate statistical methods with a trade-off between accuracy and simplicity. Most exact and approximate methods available in the literature to study stochastic differential equations (SDEs) are best suited for linear or lightly nonlinear systems. When a system is highly nonlinear, e.g., a system with multiple equilibria, the accuracy of conventional methods degrades. This problem is addressed in this article, and a novel method is introduced for statistical analysis of special types of essentially nonlinear SDEs. In particular, second-order dynamical systems with nonlinear stiffness and additive random excitations are considered. The proposed approximate method can estimate second-order moments of the state vector (namely position and velocity), not only in the case of white noise excitation, but also when the excitation is a correlated noise. To illustrate the efficiency, a second-order dynamical system with bistable Duffing-type nonlinearity is considered as the case study. Results of the proposed method are compared with the Gaussian moment closure approximation for two types of colored noise excitations, one with first-order dynamics and the other with second-order dynamics. In the absence of exact closed-form solutions, Monte Carlo simulations are considered as the reference ideal solution. Results indicate that the proposed method gives proper approximations for the mean square value of position, for which the Gaussian moment closure method cannot provide good estimations. On the other hand, both methods provide acceptable estimations for the mean square value of velocity in terms of accuracy. Such nonlinear SDEs especially arise in energy-harvesting applications, when the ambient vibration can be modeled as a wideband random excitation. In such conditions, linear energy harvesters are no longer optimal designs, but nonlinear broadband harvesting techniques are hoped to show much better performance.