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Perturbation theory and the Rayleigh quotient

Perturbation theory and the Rayleigh quotient
Perturbation theory and the Rayleigh quotient
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
0022-460X
Chan, K.T.
7d33b162-af28-4970-9080-3de063cec990
Stephen, N.G.
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Young, K.
33daac74-a29f-4676-9a99-c86ab703ecd8
Chan, K.T.
7d33b162-af28-4970-9080-3de063cec990
Stephen, N.G.
af39d0e9-b190-421d-86fe-28b793d5bca3
Young, K.
33daac74-a29f-4676-9a99-c86ab703ecd8

Chan, K.T., Stephen, N.G. and Young, K. (2011) Perturbation theory and the Rayleigh quotient. Journal of Sound and Vibration, 330. (doi:10.1016/j.jsv.2010.11.001).

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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

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Published date: 2011

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

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Date deposited: 14 Dec 2010 09:11
Last modified: 19 Jul 2019 23:13

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

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