Abstract 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.

中文翻译：

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分析双井非线性振荡器的循环平稳高斯和非高斯线性化

摘要 选取一阶和二阶双稳态Duffing振子作为典型模型，分析正弦力和高斯白噪声激发的双井非线性振子的随机特性。通过循环平稳高斯分析，确定了几乎低能和高能振荡的两种求解模式。提出了一种标准来对振荡中的两种能量模式进行分类。循环平稳数值高斯线性化、解析高斯线性化和非高斯线性化被开发用于分析双稳态振荡器的随机响应。采用高斯和非高斯线性化对概率密度、均值和二阶矩的时间平均循环平稳响应进行了比较和验证，并用蒙特卡罗方法进行了验证。本研究为分析白噪声和正弦激励下的双井振荡器提供了一种重要且新颖的工程方法。此外，还提出并验证了一个在运行中很重要的分类标准，以确保高能轨道。