Multi-parameter sensitivity algorithms can be used to construct a Hessian matrix and second-degree Taylor expansion. In terms of an asymmetric dynamic system, two multi-parameter sensitivity algorithms are proposed in this paper. The modal method with its consistence proof is firstly derived to compute the first- and second-order sensitivities of the eigenpair, and the algebraic method with its stability proof is also proposed. One significant difference between the algebraic method and the modal method is that the algebraic method uses the derivative of the normalization condition as the bordered equation to remove the singularity of the coefficient matrix in the sensitivity dominant equation, whereas the modal method uses the derivative of the normalization condition as the supplementary equation to determine the special coefficient in the modal superposition for a normalized undamped mode shape. As both the proposed methods adopt the same normalization condition, the resulting sensitivities are consistent with each other. Three numerical example are used to determine the correctness, accuracy and validity of the proposed methods in three cases: a single-parameter system, a two-parameter system, and a special system which has complex eigenpairs caused by the asymmetric property of the dynamic system.

中文翻译：

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计算非对称无阻尼系统高阶灵敏度的代数和模态方法

多参数灵敏度算法可用于构建 Hessian 矩阵和二阶泰勒展开。针对非对称动态系统，本文提出了两种多参数灵敏度算法。首先推导出具有一致性证明的模态方法来计算特征对的一阶和二阶灵敏度，并提出具有稳定性证明的代数方法。代数法与模态法的一个显着区别是，代数法以归一化条件的导数为边界方程，去除敏感度主导方程中系数矩阵的奇异性，而模态法使用归一化条件的导数作为补充方程来确定归一化无阻尼振型的模态叠加中的特殊系数。由于所提出的两种方法都采用相同的归一化条件，因此产生的灵敏度彼此一致。通过三个数值算例来确定所提方法在三种情况下的正确性、准确性和有效性：单参数系统、双参数系统和由于动态系统的不对称性质导致的具有复杂特征对的特殊系统.