Novel Modal Method for Efficient Calculation of Complex Eigenvector Derivatives

Abstract

A novel modal superposition method is presented for calculating eigenvector derivatives in self-adjoint or non-self-adjoint systems. The method is especially appropriate for determination of derivatives of many eigenvectors and requires only one eigenvalue and its corresponding right and left eigenvectors. Two parameters for the eigenvalue under consideration are introduced to obtain a generic derivative formula, which contains most of the available modal superposition methods as a branch. Matrices of the modified system are decomposed with available lower modes to avoid repeatedly solving equations for different eigenvalues. The contribution of truncated higher modes is expanded in a continued product of a few matrices, polynomials, and one generalized power series, whereby rapid convergence can be achieved easily, even in the vicinity of the convergent boundary, without additional computation cost. All other modal superposition coefficients can be neglected intrinsically with the determination of introduced parameters, but a simple condition for coefficients associated with lower-order eigenvalues should be satisfied in a special case. Numerical examples show that the method is efficient and can give results comparable to the exact solutions. The method is applicable to various damped systems and closed-mode cases.