• Structural Engineering and Mechanics
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• v.52 no.5
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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.

• East Asian mathematical journal
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• v.35 no.1
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• pp.41-50
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• 2019

This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.267-277
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• 2011

This paper is concerned with eigenvalues of second-order vector equations on time scales with boundary value conditions. Properties of eigenvalues and matrix-valued solutions are studied. Relationships between eigenvalues of different boundary value problems are discussed.

• Proceedings of the KIEE Conference
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• 1997.07c
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• pp.772-774
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• 1997

Small-signal stability analysis of large power systems needs the calculation of eigenvalues. But we use the partial eigenanalysis to calculate rightmost eigenvalues, since system matrix is very large. This paper proposes the Arnoldi-Chebyshev method identifying rightmost eigenvalues. This method constructs an optimal ellipse containing unwanted eigenvalues and use the Chebyshev iteration to get approximate eigenvetors corresponding to wanted eigenvalues, and then applies the modified Arnoldi method to calculate wanted eigenvalues.

• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.243-248
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• 1997

A full multigrid scheme was used in computing some eigenvalues of the Laplace eigenvalues problem with the Dirichlet bound-ary condition. We get a system of algebraic equations with an aid of finite difference method and apply subspace iteration method to the system to compute first some eigenvalues. The result shows that this is very effective in calculating some eigenvalues of this problem.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.151-161
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• 2008

We prove that a countably condensing operator defined on a closed wedge in a Banach space has a fixed point if it is strictly quasibounded, by using an index theory for such operators. From this, the existence of eigenvalues and surjectivity are deduced.

• Structural Engineering and Mechanics
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• v.18 no.3
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• pp.331-342
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• 2004

The vibration control problem of structures with random parameters is discussed, which is approximated by a deterministic one. A method for calculating the standard deviations of eigenvalues of the closed-loop systems is presented by using the random perturbation. The method presented in this paper will not require the distribution function of the random parameters of the systems other than their means and variances. Similarly, the distribution function of the random eigenvalues will not be computed other than their means and variances. The standard deviations of eigenvalues of the uncertain closed-loop systems can be used to estimate the stability robustness. The present method is applied to a vibration control system to illustrate the application. The numerical results show that the present method is effective.

• Structural Engineering and Mechanics
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• v.10 no.5
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• pp.427-440
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• 2000

The formulas for the perturbation analysis with modes of close eigenvalues are derived in this paper. Emphasis is made on the consistency of the straightforward perturbation process, given the complete terms of perturbations in the zeroth-order, which is a form of Rayleigh quotient, and in the higher-orders. By dividing the perturbation of eigenvector into two parts, the first-order perturbation with respect to the modes of close eigenvalues is moved into the zeroth-order perturbation. The normality condition is employed to compute the higher-order perturbations of eigenvector. The algorithm can be condensed to a single mode with a distinct eigenvalue, and this can accelerate the convergence of the perturbation analysis. The example confirms that the perturbation approximation obtained from the suggested procedure is in a good accuracy on the eigenvalues, eigenvectors, and normality.

• Proceedings of the KIEE Conference
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• 1996.07b
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• pp.734-736
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• 1996

This paper presents a method of calculating a selective number of eigenvalues in power systems, which are rightmost, or are largest modulus. The modified Arnoldi method in conjunction with implicit shift OR-algorithm is used to calculate the rightmost eigenvalues. Algorithm requires neither a prior knowledge of the specified shifts nor the calculation of inverse matrix. The key advantage of the algorithm is its ability to converge to the wanted eigenvalues at once. The method is compared with the modified Arnoldi method combined with S-matrix transformation, where the eigenvalues having the largest modulus are to be determined. The two methods are applied to the reduced Kansai system. Convergence characteristics and performances are compared. Results show that both methods are robust and has good convergence properties. However, the implicit shift OR method is seen to be faster than the S-matrix method under the same condition.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1315-1325
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• 2021

In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean n-space ℝⁿ. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the (k + 1)^th eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.