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

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.

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