The first order and second order bistable Duffing oscillator are selected as typical models for analyzing the stochastic characteristics of double-well nonlinear oscillators excited by sinusoidal force and Gaussian white noise. By employing cyclostationary Gaussian analysis, two solution modes of the almost low-energy and high-energy oscillations are identified. A criterion is proposed to classify two energy modes in oscillation. Cyclostationary numerical Gaussian linearization, analytical Gaussian linearization, and non-Gaussian linearization are developed for analyzing the stochastic response of the bistable oscillator. The time-averaged cyclostationary responses of probability density, mean, and second moment by employing the Gaussian and non-Gaussian linearization are compared and validated with Monte Carlo method. The present study contributes a significant and novel engineering method for analyzing double-well oscillators under white noise and sinusoidal excitations. In addition, a classification criterion, which is important in operation, to ensure high energy orbit is proposed and validated.
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