The University of Southampton
University of Southampton Institutional Repository

On the application of finite element analysis to wave motion in one-dimensional waveguides

On the application of finite element analysis to wave motion in one-dimensional waveguides
On the application of finite element analysis to wave motion in one-dimensional waveguides
This thesis considers issues concerning the application of the wave finite element (WFE) method to the free and forced vibrations of one-dimensional waveguides. A short section of the waveguide is modelled using conventional finite element (FE) methods. A periodicity condition is applied and the resulting mass and stiffness matrices are post-processed to yield the dispersion relations and so on. First, numerical issues are discussed and methods to reduce the errors are proposed. FE discretisation errors and errors due to round-off of inertia terms are described. A method using concatenated elements is proposed to reduce those round-off errors. Conditioning of the eigenvalue problem is discussed. An application of singular value decomposition is proposed to reduce errors in numerically determining eigenvectors together with Zhong’s formulation of the eigenvalue problem. Effects of the modelling of the cross-section on conditioning are shown. Three methods for numerically determining the group velocity are compared and the power and energy relationship is seen to be reliable. The WFE method is then applied to complicated structures and its accuracy evaluated. Dispersion curves are shown including purely real, purely imaginary and complex wavenumbers. Free wave propagation in a plate strip with free edges, a ring and a cylindrical strip is predicted and the results compared with analytical or numerical solutions to the analytical dispersion equations. In particular, dispersion curves for freely propagating flexural waves, including attenuating waves, are presented. Complicated phenomena such as curve veering, non-zero cut-on phenomena and bifurcations are observed as results of wave coupling in the wave domain. A method of decomposition of the power is proposed to reduce the size of the system matrices and to investigate the wave characteristics of each wave mode. The wave approach is then used to predict the forced response. A well-conditioned formulation for determining the amplitudes of directly excited waves is proposed. The forced response is determined by considering wave propagation and subsequent reflection at boundaries. Numerical examples of a beam, a plate and a cylinder are shown. Inclusion of rapidly decaying waves is discussed. As a practical application, free and forced vibrations of a tyre are analysed. The complicated cross-section of a tyre is modelled using a commercial FE package. Frequency dependent material properties of rubber are included. Free wave propagation is shown including attenuating waves and predicted responses are compared with experiment. Effects of the size of the excited region are discussed.
Waki, Yoshiyuki
ec106335-5a20-4cb2-9aa0-a3c869dd80a8
Waki, Yoshiyuki
ec106335-5a20-4cb2-9aa0-a3c869dd80a8

Waki, Yoshiyuki (2007) On the application of finite element analysis to wave motion in one-dimensional waveguides. University of Southampton, Institute of Sound and Vibration Research, Doctoral Thesis, 226pp.

Record type: Thesis (Doctoral)

Abstract

This thesis considers issues concerning the application of the wave finite element (WFE) method to the free and forced vibrations of one-dimensional waveguides. A short section of the waveguide is modelled using conventional finite element (FE) methods. A periodicity condition is applied and the resulting mass and stiffness matrices are post-processed to yield the dispersion relations and so on. First, numerical issues are discussed and methods to reduce the errors are proposed. FE discretisation errors and errors due to round-off of inertia terms are described. A method using concatenated elements is proposed to reduce those round-off errors. Conditioning of the eigenvalue problem is discussed. An application of singular value decomposition is proposed to reduce errors in numerically determining eigenvectors together with Zhong’s formulation of the eigenvalue problem. Effects of the modelling of the cross-section on conditioning are shown. Three methods for numerically determining the group velocity are compared and the power and energy relationship is seen to be reliable. The WFE method is then applied to complicated structures and its accuracy evaluated. Dispersion curves are shown including purely real, purely imaginary and complex wavenumbers. Free wave propagation in a plate strip with free edges, a ring and a cylindrical strip is predicted and the results compared with analytical or numerical solutions to the analytical dispersion equations. In particular, dispersion curves for freely propagating flexural waves, including attenuating waves, are presented. Complicated phenomena such as curve veering, non-zero cut-on phenomena and bifurcations are observed as results of wave coupling in the wave domain. A method of decomposition of the power is proposed to reduce the size of the system matrices and to investigate the wave characteristics of each wave mode. The wave approach is then used to predict the forced response. A well-conditioned formulation for determining the amplitudes of directly excited waves is proposed. The forced response is determined by considering wave propagation and subsequent reflection at boundaries. Numerical examples of a beam, a plate and a cylinder are shown. Inclusion of rapidly decaying waves is discussed. As a practical application, free and forced vibrations of a tyre are analysed. The complicated cross-section of a tyre is modelled using a commercial FE package. Frequency dependent material properties of rubber are included. Free wave propagation is shown including attenuating waves and predicted responses are compared with experiment. Effects of the size of the excited region are discussed.

Text
P2415.pdf - Other
Download (2MB)

More information

Published date: December 2007
Organisations: University of Southampton

Identifiers

Local EPrints ID: 52061
URI: http://eprints.soton.ac.uk/id/eprint/52061
PURE UUID: cbea31e7-1e80-4170-8f3b-1b710b595b4f

Catalogue record

Date deposited: 09 Jun 2008
Last modified: 13 Mar 2019 20:46

Export record

Contributors

Author: Yoshiyuki Waki

University divisions

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×