This paper investigates a non-intrusive hybrid uncertainty propagation framework in mechanical dynamic systems, utilizing polynomial chaos expansion (PCE) and Legendre inclusion function. Uncertainties with substantial knowledge and information are conceptualized as stochastic parameters and described using PCE, while Legendre polynomials are employed to represent uncertain, bounded parameters, specifically interval uncertainties. During the computation, the PCE model for each time step is developed through Galerkin projection and sparse grid numerical integration. Consequently, the statistical moments of the dynamic responses relative to the stochastic parameters are derived from the orthogonality of the PCE and transformed into functions reflecting the interval parameters. The interval bounds for these statistical moments are further determined using the Legendre inclusion function, which is obtained through optimal Latin hypercube sampling and collocation methods. Three dynamics examples, respectively, modeled by differential–algebraic equations, finite element method, and commercial software proves its applicability and superiority. Detailed assessment demonstrates that this method offers high computational efficiency and accuracy. As a non-intrusive approach, it poses no particular limitations on numerical methods for solving differential equations, making it universally applicable.

中文翻译：

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通过多项式混沌展开和勒让德区间包含函数实现机械动力学问题的混合不确定性传播


本文利用多项式混沌展开（PCE）和勒让德包含函数，研究了机械动态系统中的非侵入式混合不确定性传播框架。具有大量知识和信息的不确定性被概念化为随机参数并使用 PCE 进行描述，而勒让德多项式则用于表示不确定的有界参数，特别是区间不确定性。在计算过程中，通过伽辽金投影和稀疏网格数值积分建立了每个时间步长的PCE模型。因此，动态响应相对于随机参数的统计矩是从 PCE 的正交性导出的，并转换为反映区间参数的函数。使用勒让德包含函数进一步确定这些统计矩的区间界限，该函数是通过最佳拉丁超立方采样和配置方法获得的。分别用微分代数方程、有限元法和商业软件建模的三个动力学实例证明了其适用性和优越性。详细的评估表明该方法具有较高的计算效率和准确性。作为一种非侵入式方法，它对求解微分方程的数值方法没有特别的限制，使其具有普遍适用性。