Keywords: waveguides, beams, spectral elements, wave propagation, structural dynamics.

Hybrid waveguide-finite element methods have been developed in recent years. The first approach, proposed a decade ago independently by Gavric [1] and Finnveden [2], was called the spectral finite element method (SFEM). More recently, a new method was developed where a standard finite element (FE) code can be used to model a slice of the waveguide and, from this FE model, a waveguide model can be derived and used to compute the spectral relations, the group and energy velocities, and the forced response [3]. Mace et al. [4] have also developed, a few years ago, a method with a similar approach, and have investigated numerical issues based on a previous work by Zhong and Williams [5]. These wave approaches that use a finite element model of a slice of the waveguide are based on the periodic structures theory developed by Mead [6] in the early seventies.

In this paper, after briefly reviewing the methods mentioned previously, the idea of deriving a spectral element in the sense proposed by Doyle [7] in his textbook on wave propagation methods is built from a slice of a rod structure modelled with solid finite elements. This method, called here the wave spectral element method (WSEM) is presented and applied to a straight rod problem. The method has the potential to be applied to more complex structures such as car tires [8,9] and ducts. In the case of ducts, the method can be applied simulate fault detection methods in a straightforward way. The spectral dynamic matrix is obtained, instead of the standard wave propagation solution (propagation modes and wave numbers). Thus, the proposed method can be easily combined with standard finite elements using a mobility approach.

1

A.L. Gavric, "Computation of propagative waves in free rail using a finite element technique", Journal of Sound and Vibration, 185, 531-543, 1995. doi:10.1006/jsvi.1995.0398

2

S. Finnveden, "Exact spectral finite element analysis of stationary vibrations in a railway car structure", Acta Acustica, 2, 433-449, 1994.

3

J.M. Mencik, M.N. Ichchou, "Multi-mode propagation and diffusion in structures through finite elements", European Journal of Mechanics A/Solids, 24, 877-898, 2005. doi:10.1016/j.euromechsol.2005.05.004

4

B.R. Mace, D. Duhamel, M.J. Brennan, L. Hinke, "Finite element prediction of wave motion in structural waveguides", J. Acoust. Soc. Am., 117(5), 2835-2843, 2005. doi:10.1121/1.1887126

5

W.X. Zhong, F.W. Williams, "On the direct solution of wave propagation for repetitive structures", Journal of Sound and Vibration, 181, 485-501, 1995. doi:10.1006/jsvi.1995.0153

6

D.J. Mead, "A general theory of harmonic wave propagation in linear periodic systems with multiple coupling", Journal of Sound and Vibration, 27(2), 235-260, 1973. doi:10.1016/0022-460X(73)90064-3

7

J.F. Doyle, "Wave Propagation in Structures", 2nd. Edition, Springer, New York, 1997.

8

J.C. Delamotte, R.F. Nascimento, J.R.F. Arruda, "Simple models for the dynamic modeling of rotating tires", Shock and Vibration, 15, 2008.

9

J.S. Bolton, H.J. Song, Y.K. Kim, D.E. Newland, "The wave number decomposition approach to the analysis of tire vibration", Proceedings of NOISE-CON, 97-102.

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