It was shown previously that the synergy of the multi-degree-of-freedom (multi-DOF) and the mechanical stopper techniques is an effective way to further broaden the operation bandwidth of vibration energy harvesters. Considering the stochastic factor at the same time, this paper is devoted to developing a theoretical method for analyzing the dynamical characteristics of an impact-engaged 2DOF energy harvesting system under stochastic excitation, in which the impact is modeled as a nonlinear piecewise function. The existence of the impact and multi-DOF brings complexity in analyses, and the main theoretical method utilized in this paper is the stochastic averaging method which is suitable for multi-DOF quasi-nonintegrable systems. After that, the joint and marginal probability density functions of the system are derived, and numerical examples are shown to verify the effectiveness of the theoretical method. It is noteworthy that the theoretical method has the potential to be extended to more complex multi-DOF systems with stoppers. Subsequently, the stochastic response, bifurcations in the stochastic and corresponding deterministic systems, and the relationship between the D-bifurcation and the mean square voltage are discussed. Effects of crucial system coefficients, the impact stiffness and distance on the system are investigated as well.

中文翻译：

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带塞子的二自由度能量收集系统的随机响应和分岔

之前的研究表明，多自由度（multi-DOF）和机械限位器技术的协同作用是进一步拓宽振动能量收集器工作带宽的有效途径。同时考虑随机因素，本文致力于开发一种分析随机激励下的冲击接合2DOF能量收集系统的动力学特性的理论方法，其中冲击被建模为非线性分段函数。冲击和多自由度的存在给分析带来了复杂性，本文采用的主要理论方法是随机平均法，适用于多自由度准不可积系统。之后，导出系统的联合和边际概率密度函数，并通过数值算例验证了理论方法的有效性。值得注意的是，该理论方法有可能扩展到更复杂的带塞子的多自由度系统。随后，讨论了随机响应、随机和相应确定性系统中的分岔以及 D 分岔与均方电压之间的关系。还研究了关键系统系数、冲击刚度和距离对系统的影响。讨论了D分岔与均方电压的关系。还研究了关键系统系数、冲击刚度和距离对系统的影响。讨论了D分岔与均方电压的关系。还研究了关键系统系数、冲击刚度和距离对系统的影响。