This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solve...This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.展开更多
Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calcula...Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.展开更多
A new matrix perturbation analysis method is presented for efficient approximate solution of the complex modal quadratic generalized eigenvalue problem of viscously damped linear vibration systems. First, the damping ...A new matrix perturbation analysis method is presented for efficient approximate solution of the complex modal quadratic generalized eigenvalue problem of viscously damped linear vibration systems. First, the damping matrix is decomposed into the sum of a proportional-and a nonproportional-damping parts, and the solutions of the real modal eigenproblem with the proportional dampings are determined, which are a set of initial approximate solutions of the complex modal eigenproblem. Second, by taking the nonproportional-damping part as a small modification to the proportional one and using the matrix perturbation analysis method, a set of approximate solutions of the complex modal eigenvalue problem can be obtained analytically. The result is quite simple. The new method is applicable to the systems with viscous dampings-which do not deviate far away from the proportional-damping case. It is particularly important that the solution technique be also effective to the systems with heavy, but not over, dampings. The solution formulas of complex modal eigenvlaues and eigenvectors are derived up to second-order perturbation terms. The effectiveness of the perturbation algorithm is illustrated by an exemplar numerical problem with heavy dampings. In addition, the practicability of approximately estimating the complex modal eigenvalues, under the proportional-damping hypothesis, of damped vibration systems is discussed by several numerical examples.展开更多
This paper discusses the sensitivity analysis of semisimple eigenvalues and associated eigen-matrix triples of regular quadratic eigenvalue problems analytically dependent on several parameters. The directional deriva...This paper discusses the sensitivity analysis of semisimple eigenvalues and associated eigen-matrix triples of regular quadratic eigenvalue problems analytically dependent on several parameters. The directional derivatives of semisimple eigenvalues are obtained. The average of semisimple eigenvalues and corresponding eigen-matrix triple are proved to be analytic, and their partial derivatives are given. On these grounds, the sensitivities of the semisimple eigenvalues and corresponding eigenvector matrices are defined.展开更多
In this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matrices M,D and K for the quadratic pencil Q(λ)=λ^(2)M+λD+K,so that Q(λ)has a prescribed subset of...In this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matrices M,D and K for the quadratic pencil Q(λ)=λ^(2)M+λD+K,so that Q(λ)has a prescribed subset of eigenvalues and eigenvectors.This method can determine the solvability of the inverse eigenvalue problem automatically.We then consider the least squares model for updating a quadratic pencil Q(λ).More precisely,we update the model coefficient matrices M,C and K so that(i)the updated model reproduces the measured data,(ii)the symmetry of the original model is preserved,and(iii)the difference between the analytical triplet(M,D,K)and the updated triplet(M_(new),D_(new),K_(new))is minimized.In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.展开更多
基金the National Science and Tech-nology Council,Taiwan for their financial support(Grant Number NSTC 111-2221-E-019-048).
文摘This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.
基金he research was supported by the National Natural Science Foundation of China (No. 10271055).
文摘Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.
文摘A new matrix perturbation analysis method is presented for efficient approximate solution of the complex modal quadratic generalized eigenvalue problem of viscously damped linear vibration systems. First, the damping matrix is decomposed into the sum of a proportional-and a nonproportional-damping parts, and the solutions of the real modal eigenproblem with the proportional dampings are determined, which are a set of initial approximate solutions of the complex modal eigenproblem. Second, by taking the nonproportional-damping part as a small modification to the proportional one and using the matrix perturbation analysis method, a set of approximate solutions of the complex modal eigenvalue problem can be obtained analytically. The result is quite simple. The new method is applicable to the systems with viscous dampings-which do not deviate far away from the proportional-damping case. It is particularly important that the solution technique be also effective to the systems with heavy, but not over, dampings. The solution formulas of complex modal eigenvlaues and eigenvectors are derived up to second-order perturbation terms. The effectiveness of the perturbation algorithm is illustrated by an exemplar numerical problem with heavy dampings. In addition, the practicability of approximately estimating the complex modal eigenvalues, under the proportional-damping hypothesis, of damped vibration systems is discussed by several numerical examples.
基金Supported by Shanghai Natural Science Fund(No.15ZR1408400)
文摘This paper discusses the sensitivity analysis of semisimple eigenvalues and associated eigen-matrix triples of regular quadratic eigenvalue problems analytically dependent on several parameters. The directional derivatives of semisimple eigenvalues are obtained. The average of semisimple eigenvalues and corresponding eigen-matrix triple are proved to be analytic, and their partial derivatives are given. On these grounds, the sensitivities of the semisimple eigenvalues and corresponding eigenvector matrices are defined.
基金Research supported by National Natural Science Foundation of China(10571047 and 10861005)Provincial Natural Science Foundation of Guangxi(0991238)。
文摘In this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matrices M,D and K for the quadratic pencil Q(λ)=λ^(2)M+λD+K,so that Q(λ)has a prescribed subset of eigenvalues and eigenvectors.This method can determine the solvability of the inverse eigenvalue problem automatically.We then consider the least squares model for updating a quadratic pencil Q(λ).More precisely,we update the model coefficient matrices M,C and K so that(i)the updated model reproduces the measured data,(ii)the symmetry of the original model is preserved,and(iii)the difference between the analytical triplet(M,D,K)and the updated triplet(M_(new),D_(new),K_(new))is minimized.In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.
