Derivatives of repeated eigenvalues and corresponding eigenvectors of damped systems
18 Jun 2007-Applied Mathematics and Mechanics-english Edition (Editorial Committee of Applied Mathematics)-Vol. 28, Iss: 6, pp 837-845
TL;DR: In this article, a procedure for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems is presented, where derivatives are calculated in terms of the eigen values and eigen vectors of the second-order system, and the use of rather undesirable state space representation is avoided.
Abstract: A procedure is presented for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the second-order system, and the use of rather undesirable state space representation is avoided. Hence the cost of computation is greatly reduced. The efficiency of the proposed procedure is illustrated by considering a 5-DOF non-proportionally damped system.
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25 Nov 2013
TL;DR: In the majority of practical cases this approach is adopted more for mathematical convenience than for modeling the physics of vibration damping as mentioned in this paper, and it has become standard practice to use this approach in dynamics, covering a wide range of applications from aerospace to civil engineering.
Abstract: 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.
81 citations
Cites methods from "Derivatives of repeated eigenvalues..."
...Derivatives of repeated complex eigenvalues and corresponding eigenvectors of nonproportionally damped systems were considered in [HUI 07]....
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Book•
25 Nov 2013
43 citations
TL;DR: 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.
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TL;DR: In this article, the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given to calculate the particular solution by solving a linear system with nonsingular coefficient matrix.
Abstract: In this paper, we consider computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem. In the proposed method, the eigenvector derivatives of the symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given to calculate the particular solution by solving a linear system with nonsingular coefficient matrix, the method is numerically stable and efficient. Two numerical examples are included to illustrate the validity of the proposed method.
15 citations
TL;DR: In this article, the problem of computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of a symmetric quadratic eigenvalue problem was considered.
Abstract: We consider computation of the derivatives of the semisimple eigenvalues and corresponding eigenvectors of a symmetric quadratic eigenvalue problem. Using the normalization condition, we can compute the derivatives of the differentiable eigenvalues of the quadratic eigenvalue problem. Using the constrained generalized inverse, we present an efficient algorithm to compute the particular solutions to the governing equation of the derivatives of eigenvectors. The proposed method is suitable for the computation of the eigenpair derivatives of a symmetric quadratic eigenvalue problem when the first-order derivatives of eigenvalues are distinct or repeated. A numerical example is included to illustrate the validity of the proposed method.
10 citations
References
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TL;DR: In this paper, the authors extend the Multiple Damage Location Assurance Criterion (MDLAC) by introducing two methods of estimating the size of defects in a structure, which are illustrated using numerical data for two truss structures and validated experimentally using a three-beam test structure.
376 citations
"Derivatives of repeated eigenvalues..." refers background in this paper
...The derivatives of eigenvalues and eigenvectors with respect to system parameters are extremely important in a variety of problems, such as structural optimization [1] , model updating [2] and damage detection [ 3 ] . Andrew and Tan [4] , Xie and Dai [5,6] , Zhang and Zerva [7] ,e t al. dis-...
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TL;DR: Three fundamentally different approaches to design sensitivity analysis are presented and it is indicated that the state space and design space methods are more general than the virtual load method.
Abstract: Design sensitivity analysis plays a central role in structural optimization, since virtually all optimization methods require computation of derivatives of structural response quantities with respect to design variables. Three fundamentally different approaches to design sensitivity analysis are presented. These have been used extensively in the structural optimization literature. They are the virtual load method, the state space method, and the design space method. An analysis of these methods indicates that the state space and design space methods are more general than the virtual load method. Moreover, the virtual load method, when applicable, is a special case of the state space method. Any one of these procedures may be incorporated into an optimality criterion or a mathematical programming method for structural optimization.
282 citations
"Derivatives of repeated eigenvalues..." refers background in this paper
...The derivatives of eigenvalues and eigenvectors with respect to system parameters are extremely important in a variety of problems, such as structural optimization [ 1 ] , model updating [2] and damage detection [3] . Andrew and Tan [4] , Xie and Dai [5,6] , Zhang and Zerva [7] ,e t al. dis-...
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TL;DR: In this article, an efficient algorithm with proven numerical stability is derived for computation of eigenvalue and eigenvector derivatives of damped vibratory systems with multiple eigenvalues. But the algorithm is not applicable to non-proportionally damped systems.
100 citations
TL;DR: In this paper, a method to calculate derivatives of eigenvectors of damped discrete linear dynamic system with respect to the system parameter is presented, which avoids using the state-space approach.
62 citations
TL;DR: New sufficient conditions for analytic dependence of eigenvalue functions, $\lambda(\alpha)$, on $\alpha$ in a neighborhood of $\alpha_0$ are obtained and connections with known results on self-adjoint problems are made.
Abstract: The eigenvalue problem for non-self-adjoint, analytic matrix functions of two variables, $L(\lambda,\alpha)$, is examined with emphasis on the case when, at a fixed $\al_0$, $L(\lambda,\alpha_0)$ has a multiple, semisimple eigenvalue $\lambda_0$. New sufficient conditions for analytic dependence of eigenvalue functions, $\lambda(\alpha)$, on $\alpha$ in a neighborhood of $\alpha_0$ are obtained. An algorithm for generating Taylor coefficients of perturbed eigenvalues and eigenvectors is studied and the existence of positive radii of convergence is established. Connections with known results on self-adjoint problems are made.
51 citations