Algebraic and modal methods for computing high-order sensitivities in asymmetrical undamped system

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.