Algorithms for the Nonlinear Eigenvalue Problem

Ruhe, Axel

doi:10.1137/0710059

Cited by 260 publications

(181 citation statements)

References 14 publications

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“…Under certain conditions, Equation (14) reduces to a polynomial eigenvalue problem [15], or to a generalized eigenvalue problem [16]. For example, in the absence of controls equations, viscous and structural damping and gyroscopics, the eigenvalue problem reduces to the real, generalized eigenvalue problem:…”

Section: Point 0: Free-vibrationmentioning

confidence: 99%

Continuation Methods for Nonlinear Flutter

Meyer¹

2016

Aerospace

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Continuation methods are presented that are capable of treating frequency domain flutter equations, including multiple nonlinearities represented by describing functions. A small problem demonstrates how a series of continuation processes can find all limit-cycle oscillations within a specified region with a reasonable degree of confidence. Curves of the limit-cycle amplitude variation with velocity, indicating regions of stability and instability with colors, give a compact view of the nonlinear behavior throughout the flight regime. A continuation technique for reducing limit-cycle amplitudes by adjusting various system parameters is presented. These processes are economical enough to be a routine part of aircraft design and certification.

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Section: Point 0: Free-vibrationmentioning

confidence: 99%

Continuation Methods for Nonlinear Flutter

Meyer¹

2016

Aerospace

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“…The disadvantages of this approach are that the eigenvalue problem (6.4) is non-linear and the components of the A k matrix are defined in terms of integrals and hence may be costly to evaluate. The solution of non-linear eigenvalue problems are considered in references [55], [73] and [84]. Unfortunately, standard algorithms for solving non-linear eigenvalue problems are not generally available.…”

Section: Chapter 6 Interior Modal Analysismentioning

confidence: 99%

The Boundary Element Method in Acoustics

Kirkup¹

2000

Journal of Computational Acoustics

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“…Nonlinear eigenvalue problems arise in various applications [1][2][3]. Numerical methods for solving matrix eigenvalue problems with nonlinear dependence on the spectral parameter were constructed and investigated in the papers [4][5][6][7][8][9][10][11][12][13]. Mesh methods for solving differential nonlinear eigenvalue problems were studied in the papers [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Eigenvibrations of a bar with load

Samsonov

́ev

2017

MATEC Web Conf.

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Abstract. The differential eigenvalue problem describing eigenvibrations of an elastic bar with load is investigated. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

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Algorithms for the Nonlinear Eigenvalue Problem

Cited by 260 publications

References 14 publications

Continuation Methods for Nonlinear Flutter

Continuation Methods for Nonlinear Flutter

The Boundary Element Method in Acoustics

Eigenvibrations of a bar with load

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