Eigensensitivity analysis for symmetric nonviscously damped systems with repeated eigenvalues

An efficient algorithm is derived for computation of eigenvalue and eigenvector derivatives of symmetric nonviscously damped systems with repeated eigenvalues. In the proposed method, the mode shape derivatives of the nonviscously damped systems are divided into a particular solution and a homogeneous solution. A simplified method is given to calculate the particular solution by solving a linear equation with non-singularity coefficients, the method is numerically stable and efficient compared to previous methods since the coefficient matrix is non-singularity and numerically stable. The homogeneous solution are computed by the second order derivative of eigenequation. One numerical example is used to illustrate the validity of the proposed method

Computing eigenpair derivatives of asymmetric damped system by generalized inverse

Many existing approaches for asymmetric damped system are based on the assumption that the eigenvalues are simple or semisimple with separated derivatives. This paper presents a new algorithm for computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of asymmetric damped system. Compared with the existing methods, the algorithm can be applicable to problems whether the repeated eigenvalues have well separated derivatives. In the proposed method, the derivatives of eigenvectors are divided into a particular solution and a homogeneous solution, where the particular solution is constructed by using generalized inverse matrix. The effectiveness of the proposed algorithm is illustrated by one numerical example

Eigensensitivity of damped system with defective multiple eigenvalues

This paper considers the sensitivity of defective multiple eigenvalues of reducible matrix pencil, the average of eigenvalues is proved to be analytic, the derivatives of the average eigenvalues and the corresponding eigenvector matrices are obtained when the generalized eigenvalue is reducible. The sensitivity of defective multiple eigenvalues of a quadratic eigenvalue problem dependent on several parameters are also obtained by the result of generalized eigenvalue problem. The results are useful for investigating structural optimal design, model updating and structural damage detection

Computing derivatives of repeated eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems

10.1137/120879841SIAM Journal on Matrix Analysis and Applications3431089-111

A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives

Eigenvalue and eigenvector derivatives with respect to system design variables and their applications have been and continue to be one of the core issues in the design, control and identification of practical engineering systems. Many different numerical methods have been developed to compute accurately and efficiently these required derivatives from which, a wide range of successful applications have been established. This paper reviews and examines these methods of computing eigenderivatives for undamped, viscously damped, nonviscously damped, fractional and nonlinear vibration systems, as well as defective systems, for both distinct and repeated eigenvalues. The underlying mathematical relationships among these methods are discussed, together with new theoretical developments. Major important applications of eigenderivatives to finite element model updating, structural design and modification prediction, performance optimization of structures and systems, optimal control system design, damage detection and fault diagnosis, as well as turbine bladed disk vibrations are examined. Existing difficulties are identified and measures are proposed to rectify them. Various examples are given to demonstrate the key theoretical concepts and major practical applications of concern. Potential further research challenges are identified with the purpose of concentrating future research effort in the most fruitful directions.Ministry of Education (MOE)The first and third authors gratefully acknowledge the financial support from the Singapore Ministry of Education through the award of research project grant AcRF Tier 1 RG183/17