Since Lord Rayleigh introduced the idea of viscous damping in his classic work "The Theory of Sound" in 1877, it has become standard practice to use this approach in dynamics, covering a wide range of applications from aerospace to civil engineering. However, in the majority of practical cases this approach is adopted more for mathematical convenience than for modeling the physics of vibration damping.

Structural Dynamic Analysis with Generalized Damping Models: Analysis

A parallel way for computing eigenvector sensitivity of asymmetric damped systems with distinct and repeated eigenvalues

Design sensitivity and Hessian matrix of generalized eigenproblems

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Structural Dynamic Analysis with Generalized Damping Models: Identification

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]:

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

In general, the derivative of an eigenvector of a vibrating symmetric system is the solution of a singular problem. Further complications are encountered in dealing with asymmetric damped systems for which the left and right eigenvectors and their derivatives become distinct and complex. Several approaches have been proposed to overcome this singularity such as Nelson's method and the modal method. In the present work, a new approach is presented for calculating simultaneously the derivatives of the eigenvalues and their associated derivatives of the left and right eigenvectors for asymmetric damped systems. With the proposed method, the exact eigenderivatives can be obtained by solving a first-order linear algebraic system of equations. The method is applied on a 104 DOF ventilator–rotor system, which is used as an example of an asymmetric damped system with distinct eigenvalues. The diameter of the shaft has been chosen as the design parameter. The comparison of the computational time shows that the proposed method is more efficient than both Nelson's approach and the modal method. Copyright © 2006 John Wiley & Sons, Ltd.

Najeh Ben Guedria

Papers

A direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems

Eigensensitivity computation of asymmetric damped systems using an algebraic approach

Second-order eigensensitivity analysis of asymmetric damped systems using Nelson's method

A Two-Stage Approach to Solve Structural Damage Detection Problem in Plate Structures

An Adaptative Differential Evolution Algorithm for Vibration Level Reduction in Rotordynamics