An enhanced argument principle algorithm for exact complex transcendental eigenvalue analysis of damped structures

Abstract

An efficient and reliable complex transcendental eigenvalue algorithm is proposed for exact modal analysis of built-up structures with a generalized damping model. First, the exact damped dynamic stiffness (DDS) formulations for structural elements with frequency-dependent generalized damping models are developed, which can be assembled directly to model complex built-up structures exactly in the frequency domain. An enhanced argument principle algorithm (EAPA) is then proposed to calculate all complex eigenvalues (either distinct or repeated, up to the arbitrarily required precision) within an interested region in the complex plane based on the DDS matrix. The EAPA is essentially enhanced by five newly proposed techniques to improve efficiency, reliability and robustness. In particular, the two-dimensional bisection method, adaptive step-size techniques, data reuse technique, and parallel computation technique improve the efficiency significantly, whereas the pole calculation technique, two-dimensional bisection method, and data reuse technique enhance the reliability greatly. The efficiency, reliability and robustness of the proposed method is demonstrated by finite element method and other algorithms. Moreover, the proposed enhanced argument principle algorithm can also be used as a reliable and efficient zero-finding tool for transcendental functions in general.