Abstract The exact joint response transition probability density function (PDF) of linear multi-degree-of-freedom oscillators under Gaussian white noise is derived in closed-form based on the Wiener path integral (WPI) technique. Specifically, in the majority of practical implementations of the WPI technique, only the first couple of terms are retained in the functional expansion of the path integral related stochastic action. The remaining terms are typically omitted since their evaluation exhibits considerable analytical and computational challenges. Obviously, this approximation affects, unavoidably, the accuracy degree of the technique. However, it is shown herein that, for the special case of linear systems, higher than second order variations in the path integral functional expansion vanish, and thus, retaining only the first term (most probable path approximation) yields the exact joint response transition Gaussian PDF. Both single- and multi-degree-of-freedom linear systems are considered as illustrative examples for demonstrating the exact nature of the derived solutions. In this regard, the herein derived analytical results are also compared with readily available in the literature closed-form exact solutions obtained by alternative stochastic dynamics techniques. In addition to the mathematical merit of the derived exact solution, the closed-form joint response transition PDF can also serve as a benchmark for assessing the performance of alternative numerical solution methodologies.

中文翻译：

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基于维纳路径积分技术的高斯白噪声下线性多自由度系统的精确闭式解

摘要 基于维纳路径积分(WPI)技术，以封闭形式推导出高斯白噪声下线性多自由度振荡器的精确联合响应转移概率密度函数(PDF)。具体来说，在 WPI 技术的大多数实际实现中，只有前几项保留在与路径积分相关的随机动作的函数扩展中。其余项通常被省略，因为它们的评估表现出相当大的分析和计算挑战。显然，这种近似不可避免地会影响该技术的准确度。然而，这里表明，对于线性系统的特殊情况，路径积分泛函扩展中高于二阶的变化消失，因此，仅保留第一项（最可能的路径近似）会产生精确的联合响应转换高斯 PDF。单自由度和多自由度线性系统都被认为是演示导出解的确切性质的示例。在这方面，本文导出的分析结果也与文献中容易获得的通过替代随机动力学技术获得的封闭形式精确解进行了比较。除了导出精确解的数学优点外，封闭形式的联合响应转换 PDF 还可以作为评估替代数值解法性能的基准。单自由度和多自由度线性系统都被认为是演示导出解的确切性质的示例。在这方面，本文导出的分析结果也与文献中容易获得的通过替代随机动力学技术获得的封闭形式精确解进行了比较。除了导出精确解的数学优点外，封闭形式的联合响应转换 PDF 还可以作为评估替代数值解法性能的基准。单自由度和多自由度线性系统都被认为是演示导出解的确切性质的示例。在这方面，本文导出的分析结果也与文献中容易获得的通过替代随机动力学技术获得的封闭形式精确解进行了比较。除了导出精确解的数学优点外，封闭形式的联合响应转换 PDF 还可以作为评估替代数值解法性能的基准。