Variable shifted-pole method for calculating complex mode shape derivatives

Structural dynamical behavior becomes complicated when various dampings act on a real structural system simultaneously, and difficulty arises in calculating complex mode shape derivatives. For this reason, a method for complex mode shape derivative calculation is developed based on modal superposition method with the idea of modal acceleration and shifted-pole. First, a complex shift value is added into governing equations, and a modal iterative formula is obtained with a generalized power series. Then, the combination of the iterative solution and each eigenvector represent mode shape derivative. The generalized dynamical compliance matrix involves available lower-order modes and truncated higher-order modes. The contribution from truncated higher-order modes is approximated by continued product of matrix polynomials and a generalized power series, and then lower-order modes as well as system matrices are used to calculate it. Each shift value is the product of shift coefficient and corresponding complex eigenvalue, and is only related to the modulus of the shift value for the lowest mode and the unit complex eigenvalue of the mode to be calculated. The shift coefficient for the lowest mode is obtained from error analysis. The appropriate iterative number of modal acceleration is also given. Finally, many complex mode shape derivatives of high accuracy can be obtained by decomposing system matrices only once. A numerical example illustrates the validity and efficiency of the present method.