First-Crossing Problem of Weakly-Coupled Strongly-Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noise

[+] Author and Article Information
Y. J. Wu

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

H. Y. Wang

Shanghai Mitsubishi Elevator Co. Ltd, 649 Changhua Rd., Jingan District, Shanghai, 200041, China

1Corresponding author.

ASME doi:10.1115/1.4039244 History: Received June 14, 2017; Revised January 23, 2018


We study first-crossing problem of two-degrees-of-freedom strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time. A two-DOF Duffing-van der Pol oscillators is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

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