Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient met...

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Veröffentlicht in:International journal for numerical methods in engineering 2003-03, Vol.56 (12), p.1795-1814
Hauptverfasser: Yuan, Si, Ye, Kangsheng, Williams, Fred W., Kennedy, David
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container_title International journal for numerical methods in engineering
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creator Yuan, Si
Ye, Kangsheng
Williams, Fred W.
Kennedy, David
description When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick–Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large‐scale structures. Copyright © 2003 John Wiley & Sons, Ltd.
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subjects Algorithms
dynamic stiffness matrix
Dynamics
Eigenvalues
Mathematical analysis
Mathematical models
natural frequencies
Newton's method
Resonant frequency
Rigid-body dynamics
second order convergence
Vibration mode
vibration modes
Wittrick-Williams algorithm
title Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices
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