A Study of the Bounds on Eigenvalues of a Transcendental Dynamic Stiffness Matrix Provided by a Simply Derived Linear Matrix Pencil

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Abstract
The approximate representation of a transcendental dynamic stiffness matrix by a simple matrix pencil is studied herein. For cases in which linearization is performed below the first pole of the dynamic stiffness matrix, interesting and important bounding properties on exact eigenvalues are established. If the linearization is performed above the first pole, it is not possible to make general assertions regarding bounds on any of the exact eigenvalues. The accuracy of estimates, provided by the eigenvalues of a simple matrix pencil representation of a transcendental stiffness matrix, of the eigenvalues of that matrix, is studied numerically in relation to plane frame vibration problems in a latter section of the paper.

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