This paper leverages recent progress on orthonormal splines for solving uncertainty quantification (UQ) problems from linear structural dynamics. The resulting methods, premised on spline chaos expansion (SCE) and spline dimensional decomposition (SDD), both construe Fourier-like expansion of a dynamic system response of interest with respect to measure-consistent orthonormalized basis splines in input random variables and standard least-squares regression for estimating the expansion coefficients. The SCE and SDD methods are capable of capturing high nonlinearity and non-smoothness, if they exist, in a stochastic dynamic response markedly better than the polynomial chaos expansion (PCE) method. However, due to the tensor-product structure, SCE, like PCE, also suffers from the curse of dimensionality. In contrast, SDD, equipped with a desirable dimensional hierarchy of input variables, deflates the curse of dimensionality to a great extent. Numerical results from frequency response analysis of a two-degree-of-freedom dynamic system indicate that a low-order SCE with fewer basis functions removes or markedly reduces the spurious oscillations generated by high-order PCE in estimating the response statistics. Finally, a high-dimensional modal analysis of a fighter jet comprising 110 random variables was conducted, demonstrating the ability of SDD in solving large-scale UQ problems.

本文利用正交样条的最新进展从线性结构动力学解决不确定性量化 (UQ) 问题。由此产生的方法，以样条混沌展开 (SCE) 和样条维数分解 (SDD) 为前提，都解释了感兴趣的动态系统响应的傅立叶式展开，涉及输入随机变量和标准最小样条中的测量一致正交化基样条 -用于估计膨胀系数的平方回归。SCE 和 SDD 方法能够在随机动态响应中捕获高度非线性和非平滑性（如果存在），明显优于多项式混沌展开 (PCE) 方法。然而，由于张量积结构，SCE 和 PCE 一样，也受到维数灾难的影响。相比之下，SDD，配备了理想的输入变量维度层次结构，在很大程度上消除了维度灾难。二自由度动态系统的频率响应分析的数值结果表明，具有较少基函数的低阶 SCE 消除或显着减少了估计响应统计量时由高阶 PCE 产生的虚假振荡。最后，对包含 110 个随机变量的战斗机进行了高维模态分析，证明了 SDD 解决大规模 UQ 问题的能力。二自由度动态系统的频率响应分析的数值结果表明，具有较少基函数的低阶 SCE 消除或显着减少了估计响应统计量时由高阶 PCE 产生的虚假振荡。最后，对包含 110 个随机变量的战斗机进行了高维模态分析，证明了 SDD 解决大规模 UQ 问题的能力。二自由度动态系统的频率响应分析的数值结果表明，具有较少基函数的低阶 SCE 消除或显着减少了估计响应统计量时由高阶 PCE 产生的虚假振荡。最后，对包含 110 个随机变量的战斗机进行了高维模态分析，证明了 SDD 解决大规模 UQ 问题的能力。