Taylor Approximation of Second Degree for Complex Mode in Asymmetrical Damped System

Abstract

Algorithms for the first- and second-order sensitivities of the complex mode for an asymmetrical damped linear discrete dynamic system with respect to design parameters are presented. The first- and second-order sensitivities are used to construct the gradient matrix and the Hessian matrix of the complex mode and to establish its Taylor approximation of the second degree. The symmetrical dynamic system possesses a set of biorthonormal right modes so that the two-sided normalization method is engaged. In contrast, in an asymmetrical dynamic system, the two-sided normalization technique is not effective. In our presentation, the results are derived in terms of a complex mode so that the uniqueness of the complex mode and its sensitivities are kept. The algorithms make the calculations more precise and easier to perform. The usefulness and stability of the derived expressions are demonstrated by considering a half-car model of 7 degrees and a whirling beam of 10 degrees.