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

Abstract

SUMMARYIn 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.

Derivatives of complex eigenvectors with distinct and repeated eigenvalues

“…This means that the V R in Equation (27) is a particular solution to Equation (23). Denoting the (n +m)×(n +m) coefficient matrix in Equation (27) as E, we may prove that the matrix E is non-singular.…”

“…from which Equation (28a) is simplified to Equation (23). This means that the V R in Equation (27) is a particular solution to Equation (23).…”

Approximate method for eigensensitivity analysis of a defective matrix

“…Substituting (17) into (11) and the ith equation in (15) we obtain (19) is an eigenvalue problem with λ(1) ν as its eigenvalue and d(i) as its corresponding eigenvector. Note that if λ(1) = 0 then the ith equation in (15) and all of the equations in (16) constitute a system of linear homogeneous algebraic equations with Q(i) as its coefficient matrix.…”

Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues

“…These procedures will complicate the derivations as well as the practical computations. Adhikari (1999), , Adhikari and Friswell (2001), Choi et al (2004), and Guedria et al (2006) presented some numerical methods for computing derivatives of simple eigenvalues and corresponding eigenvectors of the quadratic eigenvalue problem without use of linearization. Friswell and Adhikari (2000) extended Nelson's method to the quadratic eigenvalue problem.…”