scholarly journals An Improved Method for Computing Eigenpair Derivatives of Damped System

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Pingxin Wang ◽  
Jieer Wu ◽  
Xibei Yang
Keyword(s):  
Homogeneous Equation ◽  
Condition Numbers ◽  
Improved Method ◽  
Numerical Examples ◽  
Repeated Eigenvalues ◽  
Damped System ◽  
Particular Solution ◽  
Relative Errors ◽  
Damped Systems ◽  
Derivatives Of

The calculation of eigenpair derivatives plays an important role in vibroengineering. This paper presents an improved algorithm for the eigenvector derivative of the damped systems by dividing it into a particular solution and general solution of the corresponding homogeneous equation. Compared with the existing methods, the proposed algorithm can significantly reduce the condition number of the equation for particular solution. Therefore, the relative errors of the calculated solutions are notably cut down. The results on two numerical examples show that such strategy is effective in reducing the condition numbers for both distinct and repeated eigenvalues.

Eigenvector Derivatives of Generalized Nondefective Eigenproblems With Repeated Eigenvalues

1995 ◽  
Vol 117 (1) ◽  
pp. 207-212 ◽  
Author(s):  
Y.-Q. Zhang ◽  
W.-L. Wang
Keyword(s):  
Recent Work ◽  
New Method ◽  
First Eigenvalue ◽  
Numerical Examples ◽  
Repeated Eigenvalues ◽  
Generalized Eigenproblem ◽  
Eigenvalue Derivatives ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector ◽  
Derivatives Of

A new method is presented for computation of eigenvalue and eigenvector derivatives associated with repeated eigenvalues of the generalized nondefective eigenproblem. This approach is an extension of recent work by Daily and by Juang et al. and is applicable to symmetric or nonsymmetric systems. The extended phases read as follows. The differentiable eigenvectors and their derivatives associated with repeated eigenvalues are determined for a generalized eigenproblem, requiring the knowledge of only those eigenvectors to be differentiated. Moreover, formulations for computing eigenvector derivatives have been presented covering the case where multigroups of repeated first eigenvalue derivatives occur. Numerical examples are given to demonstrate the effectiveness of the proposed method.

Derivatives of repeated eigenvalues and corresponding eigenvectors of damped systems

2007 ◽  
Vol 28 (6) ◽  
pp. 837-845 ◽  
Author(s):  
Hui-qing Xie ◽  
Hua Dai
Keyword(s):  
Repeated Eigenvalues ◽  
Damped Systems ◽  
Derivatives Of

Computation of Eigenvalues and Eigenvectors of a Mistuned Bladed Disk

2006 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Eigenvalues And Eigenvectors ◽  
Numerical Examples ◽  
Bladed Disk ◽  
Repeated Eigenvalues ◽  
Derivatives Of ◽  
Computation Of Eigenvalues

This paper deals with fundamental aspects of variations in eigenvalues and eigenvectors of a bladed disk due to mistuning. First, the existence of derivatives of repeated eigenvalues and corresponding eigenvectors is thoroughly examined. Next, an algorithm is developed to compute these derivatives. It is shown how a Taylor series expansion can be used to efficiently compute eigenvalues and eigenvectors of a mistuned system. This methodology is developed for perturbations in both repeated and unrepeated eigenvalues of the tuned system. Lastly, numerical examples are presented.

Computation of Eigenvalues and Eigenvectors of a Mistuned Bladed Disk Via Unidirectional Taylor Series Expansions

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Taylor Series ◽  
Taylor Series Expansion ◽  
Series Expansions ◽  
Eigenvalues And Eigenvectors ◽  
Numerical Examples ◽  
Bladed Disk ◽  
Repeated Eigenvalues ◽  
Taylor Series Expansions ◽  
Derivatives Of ◽  
Computation Of Eigenvalues

This paper deals with the computation of eigenvalues and eigenvectors of a mistuned bladed disk. First, the existence of derivatives of repeated eigenvalues and corresponding eigenvectors is discussed. Next, an algorithm is developed to compute these derivatives. It is shown how a Taylor series expansion can be used to efficiently compute eigenvalues and eigenvectors of a mistuned system. Numerical examples are presented to corroborate the validity of theoretical analysis.

Eigenvector Derivatives of Generalized Nondefective Eigenproblems With Repeated Eigenvalues

1993 ◽  
Author(s):  
Yong-Qiang Zhang ◽  
Wen-Liang Wang
Keyword(s):  
Recent Work ◽  
New Method ◽  
First Eigenvalue ◽  
Numerical Examples ◽  
Repeated Eigenvalues ◽  
Generalized Eigenproblem ◽  
Eigenvalue Derivatives ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector ◽  
Derivatives Of

A new method is presented for computation of eigenvalue and eigenvector derivatives associated with repeated eigenvalues of the generalized nondefective eigenproblem. This approach is an extension of recent work by Dailey and by Juang et al. and is applicable to symmetric or nonsymmetric systems. The extended phases read as follows. The differentiable eigenvectors and their derivatives associated with repeated eigenvalues are determined for generalized eigenproblem, requiring the knowledge of only those eigenvectors to be differentiated. Moreover, formulations for computing eigenvector derivatives have been presented covering the case where multi-groups of repeated first eigenvalue derivatives occur. Numerical Examples are given to demonstrate the effectiveness of the proposed method.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

2016 ◽  
Vol 6 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Lei Zhu ◽  
Wei-Wei Xu ◽  
Xing-Dong Yang
Keyword(s):  
Algebraic System ◽  
Condition Numbers ◽  
Linear Algebraic System ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Interest In Science ◽  
Perturbation Bounds

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse

2015 ◽  
Vol 96 (1) ◽  
pp. 201-210 ◽  
Author(s):  
Pingxin Wang ◽  
Hua Dai
Keyword(s):  
Generalized Inverse ◽  
Repeated Eigenvalues ◽  
Damped Systems

Diagonal Dominance and the Decoupling Approximation in Damped Discrete Linear Systems

2007 ◽  
Author(s):  
Matthias Morzfeld ◽  
Nopdanai Ajavakom ◽  
Fai Ma
Keyword(s):  
Diagonal Element ◽  
Diagonal Dominance ◽  
Damping Matrix ◽  
Damped System ◽  
Modal Damping ◽  
Principal Coordinates ◽  
Discrete Linear Systems ◽  
Modal Coupling ◽  
Decoupling Approximation ◽  
Damped Systems

The principal coordinates of a non-classically damped linear system are coupled by nonzero off-diagonal element of the modal damping matrix. In the analysis of non-classically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is widely accepted that if the modal damping matrix is diagonally dominant, then errors due to the decoupling approximation must be small. In addition, it is intuitively believed that the more diagonal the modal damping matrix, the less will be the errors in the decoupling approximation. Two quantitative measures are proposed in this paper to measure the degree of being diagonal dominant in modal damping matrices. It is demonstrated that, over a finite range, errors in the decoupling approximation can continuously increase while the modal damping matrix becomes more and more diagonal with its off-diagonal elements decreasing in magnitude continuously. An explanation for this unexpected behavior is presented. Within a practical range of engineering applications, diagonal dominance of the modal damping matrix may not be sufficient for neglecting modal coupling in a damped system.

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