• Structural Engineering and Mechanics
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• 제52권5호
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• pp.903-918
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• 2014

Large space structures may have resonant low eigenvalues and often these appear with closely-spaced natural frequencies. Owing to the coupling among modes with closely-spaced natural frequencies, each eigenvector corresponding to closely-spaced eigenvalues is ill-conditioned that may cause structural instability. The subspace to an invariant subspace corresponding to closely-spaced eigenvalues is well-conditioned, so a method is presented to design the feedback control law of intelligent structures with closely-spaced eigenvalues in this paper. The main steps are as follows: firstly, the system with closely-spaced eigenvalues is transformed into that with repeated eigenvalues by the spectral decomposition method; secondly, the computation for the linear combination of eigenvectors corresponding to repeated eigenvalues is obtained; thirdly, the feedback control law is designed on the basis of the system with repeated eigenvalues; fourthly, the system with closely-spaced eigenvalues is regarded as perturbed system on the basis of the system with repeated eigenvalues; finally, the feedback control law is applied to the original system, the first order perturbations of eigenvalues are discussed when the parameter modifications of the system are introduced. Numerical examples are given to demonstrate the application of the present method.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2004년도 추계학술대회논문집
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• pp.721-726
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• 2004

A simplified method for the computation of first second and higher order derivatives of eigenvalues and eigenvectors derivatives associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m is the number of multiplicity of the repeated eigenvalues. The algebraic equation developed can be used to compute derivatives of both eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space it is numerically stable and very efficient compared to previous methods. This method can be consistently applied to structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam is considered.

• Structural Engineering and Mechanics
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• 제42권4호
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• pp.449-469
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• 2012

For generally damped linear systems with repeated eigenvalues and defective eigenvectors, this study provides a decomposition method based on residue matrix, which is suitable for engineering applications. Based on this method, a hybrid approach is presented, incorporating the merits of the modal superposition method and the residue matrix decomposition method, which does not need to consider the defective characteristics of the eigenvectors corresponding to repeated eigenvalues. The method derived in this study has clear physical concepts and is easily to be understood and mastered by engineering designers. Furthermore, this study analyzes the applicability of step-by-step methods, including the Newmark beta and Runge-Kutta methods for dynamic response calculation of defective systems. Finally, the implementation procedure of the proposed hybrid approach is illustrated by analyzing numerical examples, and the correctness and the effectiveness of the formula are judged by comparing the results obtained from the different methods.

• 한국강구조학회 논문집
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• 제8권3호통권28호
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• pp.21-34
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• 1996

A numerical method is presented for computation of eigenvector derivatives used an iterative procedure with guaranteed convergence. An approach for treating the singularity in calculating the eigenvector derivatives is presented, in which a shift in each eigenvalue is introduced to avoid the singularity. If the shift is selected properly, the proposed method can give very satisfactory results after only one iteration. A criterion for choosing an adequate shift, dependent on computer hardware is suggested ; it is directly dependent on the eigenvalue magnitudes and the number of bits per numeral of the computer. Another merit of this method is that eigenvector derivatives with repeated eigenvalues can be easily obtained if the new eigenvectors are calculated. These new eigenvectors lie "adjacent" to the m (number of repeated eigenvalues) distinct eigenvectors, which appear when the design parameter varies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The results are compared with those of Nelson's method which can find the exact eigenvector derivatives. For the case of repeated eigenvalues, a cantilever beam is considered. The results are compared with those of Dailey's method which also can find the exact eigenvector derivatives. The design parameter of the cantilever plate is its thickness, and that of the cantilever beam its height.

• Structural Engineering and Mechanics
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• 제27권5호
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• pp.527-540
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• 2007

The modal controller of single-input system cannot stabilize the defective system with positive real part of repeated eigenvalues, because some of the generalized modes are uncontrollable. In order to stabilize the uncontrollable modes with positive real part of eigenvalues, the multi-input system should be introduced. This paper presents a recursive procedure for designing the feedback controller of the multi-input system with defective repeated eigenvalues. For a nearly defective system, we first transform it into a defective one, and apply the same method to manage. The proposed methods are based on the modal coordinate equations, to avoid the tedious mathematic manipulation. As an application of the presented procedure, two numerical examples are given at end of the paper.

• 대한전기학회:학술대회논문집
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• 대한전기학회 1993년도 하계학술대회 논문집 A
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• pp.76-78
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• 1993

This paper describes an efficient method of computing any desired number of the most unstable eigenvalues and eigenvectors of a large scale multi-machine power system. Approximate eigenvalues obtained by Hessenberg process are refined using Rayleigh quotient iteration with cubic convergence property. If further eigenvalues and eigenvectors are needed, the procedure described above are repeated with deflation. The proposed algorithm can cover all the model types of synchronous machines, exciters, speed governing system and PSS defined in AESOPS. The proposed algorithm applied to New England test system with 10 machines and 39 buses produced the results same with AESOPS in faster computation time. Also eigenvectors computed in Rayleigh quotient iteration makes it possible to make eigen-analysis for improving unstable modes.

• Structural Engineering and Mechanics
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• 제15권5호
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• pp.551-562
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• 2003

This paper presents a procedure for designing feedback controllers for defective systems with repeated eigenvalues, and also for a nearly defective system with close eigenvalues. For the nearly defective system, we first transform it into a defective one, and then apply the same method to deal with the nearly defective system. A method for computing the gain matrices is discussed here. The methodologies proposed are based on the modal coordinate equation to avoid the tedious mathematical manipulation. As an application of the present procedure, a numerical example is given.

• 대한기계학회논문집A
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• 제40권4호
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• pp.363-372
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• 2016

강체와 유연체가 혼합된 다종 구조 시스템의 동특성을 개선을 위한 최적화를 수행하는 경우, 일반적으로 그 시스템의 고유 진동수를 높이게 된다. 강체와 유연체의 시스템을 동시에 다루는 위상 최적화 정식화가 있으나, 그 시스템의 고유 진동수를 다룬 연구는 드물며, 특히 목적하는 진동수가 중복 고유 진동수 의 하나로 되는 경우를 다룬 연구는 보고된 바 없다. 본 연구에서는 중복 고유 진동수를 다루어야 하는 경우에 나타나는 수치적 문제를 해결하였으며 그 방법을 활용한 위상최적설계 정식화와 민감도 해석을 제시하였다. 그 다음, 몇 가지 수치 예제를 통해 제안된 정식화의 타당성을 검증해 보았다.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2011년도 춘계학술대회 논문집
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• pp.443-444
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• 2011

This paper discusses a model order reduction for large order rotor dynamics systems results from the finite element discretization. Typical rotor systems consist of a rotor, built-on parts, and a support system, and require prudent consideration in their dynamic analysis models because they include unsymmetric stiffness, localized nonproportional damping and frequency dependent gyroscopic effects. When the finite element model has a very large number of degrees of freedom because of complex geometry, repeated dynamic analyses to investigate the critical speeds, stability, and unbalanced response are computationally very expensive to finish within a practical design cycle. In this paper, the Krylov-based model order reduction via moment matching significantly speeds up the dynamic analyses necessary to check eigenvalues and critical speeds of a Nelson-Vaugh rotor system. With this approach the dynamic simulation is efficiently repeated via a reduced system by changing a running rotational speed because it can be preserved as a parameter in the process of model reduction. The Campbell diagram by the reduced system shows very good agreement with that of the original system. A 3-D finite element model of the Nelson-Vaugh rotor system is taken as a numerical example to demonstrate the advantages of this model reduction for rotor dynamic simulation.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2001년도 가을 학술발표회 논문집
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• pp.515-522
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• 2001

A simplified method is presented for the computation of eigenvalue and eigenvector derivatives associated with repeated eigenvalues. In the proposed method, adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m is the number of multiplicity of the repeated eigenvalue. One algebraic equation developed can be computed eigenvalue and eigenvector derivatives simultaneously. Since the coefficient matrix of the proposed equation is symmetric and based on N-space, this method is very efficient compared to previous methods. Moreover the numerical stability of the method is guaranteed because the coefficient matrix of the proposed equation is non-singular, This method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam and a 5-DOF mechanical system in the case of a non-proportionally damped system are considered as numerical examples. The design parameter of the cantilever beam is its width, and that of the 5-DOF mechanical system is a spring.