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# Perturbation theory and the Rayleigh quotient

Chan, K.T., Stephen, N.G. and Young, K. (2011) Perturbation theory and the Rayleigh quotient. Journal of Sound and Vibration, 330.

Record type: Article

## Abstract

The characteristic frequencies ? of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ?, it is shown that when an approximate eigenfunction is known with an error O(?N), the Rayleigh quotient gives the frequency with an error O(?2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theory

Text
Chan_11.pdf - Other

Published date: 2011

## Identifiers

Local EPrints ID: 169273
URI: http://eprints.soton.ac.uk/id/eprint/169273
ISSN: 0022-460X
PURE UUID: b75b586c-e6a8-4346-aa1e-622bcb7aa0ea

## Catalogue record

Date deposited: 14 Dec 2010 09:11

## Contributors

Author: K.T. Chan
Author: N.G. Stephen
Author: K. Young