Computation of Eigensolution Derivatives for Nonviscously Damped Systems Using the Algebraic Method

Abstract

T HE computation of the eigensolution derivatives plays a significant role in dynamic model updating, structural design optimization, structural dynamic modification, damage detection andmany other applications. Themethods to calculate eigensolution derivatives are well established for undamped and viscous damped systems. These common methods can be divided into the modal method, Nelson’s method, and the algebraic method. Fox and Kapoor [1] first proposed the modal method for symmetric undamped systems by approximating the derivative of each eigenvector as a linear combination of all undamped eigenvectors. Later, Adhikari and Friswell [2] and Adhikari [3] extended the modal method to the more general asymmetric systems with viscous and nonviscous damping, respectively. To simplify the computation of eigensolution derivatives, Nelson [4] proposed a method, which requires only the eigenvector of interest by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution for symmetric undamped systems. Later, Friswell and Adhikari [5] extended Nelson’s method to symmetric and asymmetric systemswith viscous damping. Recently, Adhikari and Friswell [6] extended Nelson’s method to symmetric and asymmetric nonviscously damped systems. However, Nelson’s method is lengthy and clumsy for programming. Lee et al. [7] derived an efficient algebraic method, which has a compact form to compute the eigensolution derivatives by solving a nonsingular linear system of algebraic equations for symmetric systems with viscous damping. Later, Guedria et al. [8] extended the algebraic method to general asymmetric systems with viscous damping. Recently, Chouchane et al. [9] wrote an excellent review of the algebraic method for symmetric and asymmetric systems with viscous damping and extended their method to the second-order and high-order derivatives of eigensolutions. In this note, the algebraic method will be extended to symmetric and asymmetric systems with nonviscous damping. The equations of motion describing free vibration of anN-degreeof-freedom (DOF) linear system with nonviscous (viscoelastic) damping can be expressed by [3,6]: