Keywords: Timoshenko's flexural waves, second spectra bifurcation, numerical simulations, wave number dispersion.

The presence of the second spectrum flexural wave propagation as described by Timoshenko's beam theory was first clamed by Traill-Nash and Collar [1]. The exact theory for wave propagation in circular rods of infinite length (in the context of the mathematical theory of elasticity) have shown [2] Timoshenko's beam theory to be very accurate for the prediction of flexural natural frequencies and wave propagation characteristics. It is important to point out that the exact theory predicts the existence of an infinite number of modes of wave propagation and these are usually categorized according to the nature of the lowest mode as flexural, torsional or longitudinal. The lowest wave mode propagation predicts by Timoshenko's beam theory is in accordance with the exact theory (the maximum discrepancy is less than one-tenth of one percent for the practical range). The second mode, which corresponds with the second spectrum agrees with the second flexural mode of the exact theory at long wavelength, but wavelength shortens diverge considerably [2]. However, these two modes of wave motions have to be considered to attend mathematical consistency.

In previous paper of the author [3] the attention was devoted to the problem of the additional velocity dispersion and spurious reflections of Timoshenko's flexural waves produced by numerical simulation for propagating second spectrum wave, that is, before bifurcation condition [4]. This paper complements the first one considering the second spectrum mode after bifurcation condition, when evanescent waves (or ringing motions) take place. Predominantly spatially damped vibrations are presented in this case.

The spectral analysis of wave motion is adopted. Spectral analysis or frequency domain synthesis is found to be very efficient approach in analysing waves. Furthermore, spectral analysis is not much a solution technique as it is a different insight into the wave mechanics [4]. The main advantage of the spectral approach is that the close connection between waves and vibration becomes apparent.

The wave propagation problems have been solved using lumped mass models [5] as they considerably decrease the amount of operations if an explicit time integration algorithm is used. Recently a new motivation has taken place. The modern third order implicit integration algorithms have been recently proposed [6] involving more than one set of implicit equations to be solved at each step. In these cases, if the lumped mass matrix is considered the amount of operations may become similar to those usual second order methods.

A new lumped mass matrix to solve Timoshenko's flexural wave propagation problems in which the dynamic equilibrium of the moments is also considered. The best attributes of this lumped mass matrix may be its low numeric velocity dispersion in comparison with classical lumped mass matrix. The numerical integration of the flexural wave equation by using the finite element method via semi-discretized technique introduces additional velocity dispersion and spurious wave motions as it has been established previously [5].

1

R.W. Trail-Nash, A.R. Collar, "The effects of shear flexibility and rotary inertia on the bending vibration of beams", Quart Journ Mech Appl Math, VI, 186-222, 1953. doi:10.1093/qjmam/6.2.186

2

N.G. Stephen, "Second frequency spectrum of Timoshenko beams", Journal of Sound and Vibration, 80(4), 578-582, 1982. doi:10.1016/0022-460X(82)90501-6

3

J.E. Laier, "Dispersive and spurious reflections of Timoshenko's flexural waves in numerical simulations. Adv Engng Software, 33. 605-610, 2002. doi:10.1016/S0965-9978(02)00078-9

4

J.F. Doyle, "Wave Propagation in Structures" Springer-Verlag New York Inc., Second Edition, 1997.

5

J.E. Laier, "Hermitian lumped mass matrix formulations for flexural wave propagation", Commun. Numer. Meth. Eng., 14, (1998), 43-49. doi:10.1002/(SICI)1099-0887(199801)14:1<43::AID-CNM132>3.0.CO;2-A

6

J. Kujawski, R.H. Gallagher, "A generalized least-squares family ofalgorithms for transient dynamic analysis", Earth. Eng. Struct. Dyn., 18, (1989), 539-550. doi:10.1002/eqe.4290180408¬

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