Abstract The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc-Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.

中文翻译：

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具有奇异扩散矩阵的非线性结构系统的随机响应确定：具有约束的维纳路径积分变分公式

摘要 本文推广了用于确定随机激励非线性振荡器的联合响应概率密度函数 (PDF) 的维纳路径积分 (WPI) 近似半解析技术，以说明具有奇异扩散矩阵的系统。指示性示例包括（但不限于）仅激发部分自由度的系统、通过附加辅助状态方程进行滞后建模，以及具有耦合机电方程的能量收集器。一般而言，上述系统的运动控制方程可以转化为一组欠定随机微分方程和一组确定性常微分方程。后者可以是任意形式，在这里被解释为对由随机激励驱动的系统运动的约束。接下来，对 WPI 使用半经典近似处理会产生一个确定性约束变分问题，以通过数值方式解决以确定最可能的路径；因此，为了以计算有效的方式评估系统联合响应 PDF。这是结合 Rayleigh-Ritz 方法和适当的优化算法完成的。考虑了与线性和非线性约束方程有关的几个数值例子，包括各种多自由度系统、地震激励下的线性振荡器和遵循 Bouc-Wen 形式主义表现出滞后的非线性振荡器。