Keywords: railway mechanics, contact problems, dynamics, beam theory, wave propagation.

In railway mechanics, the Euler-Bernoulli beam is frequently used to model rails. For some problems it is sufficient to substitute the contact forces exacted by a single wheel by a Dirac force applied at the point of contact. For a harmonically variable force with a constant speed, analytical solutions are available [1,2]. The problem formulated in [3,4] was first solved without taking into account the Sommerfeld condition. The solution obtained in [3] is not correct. It can be taken as an approximate solution, only in the sub-critical case, where all wave amplitudes decrease with distance from the source of the disturbances. In [4,5], it was shown that imperfections in the geometries of rails and wheels may lead to strong speed fluctuations of the contact point. The contact point may jump and the value and direction of the load may vary. For regular patterns changes are periodic, but they may be far from harmonic [5]. This may influence the stability of the rolling motion of the wheel-set and long-term changes in the geometry arising from wear [6].

Classical solutions to the moving load problem for the Euler-Bernoulli beam were obtained in [1] by means of Fourier techniques in the form of a superposition of certain dispersive waves. A discussion of the dispersion relation leads to characteristic regions in the speed or velocity plane. On the boundaries of those regions resonance occurs, no stationary solutions exist. Transient solutions to initial-value problems do not exhibit the classical resonance. Numerical solutions obtained using the method proposed in the paper coincide with the analytical solution for the constant speed [2]. The method can be applied in the transient and the periodic speed cases. Quick transgressions of the critical curves do not lead automatically to dangerous concentrations of energy. In the long run, pseudo-stationary wave forms develop and are preserved.

1

R. Bogacz, T. Krzyzynski, K. Popp, "On the Generalization of Mathews' Problem of the Vibrations of a Beam on Elastic Foundation", ZAMM, 69(8), 243-252, 1989. doi:10.1002/zamm.19890690804

2

W. Gerbig, "Dynamische Euler-Bernoulli-Balkengleichung unter bewegter Last", Diploma thesis, University of Rostock, 2011.

3

P.M. Mathews, "Vibrations of a Beam on Elastic Foundation", ZAMM, 38(3-4), 105-115, 1958. doi:10.1002/zamm.19580380305

4

R. Bogacz, Z. Kowalska, "Computer simulation of the interaction between a wheel and a corrugated rail", Eur. J. Mech. A/Solids, 20, 673-684, 2001. doi:10.1016/S0997-7538(01)01150-0

5

R. Bogacz, K. Frischmuth, "Abrazion and percussion effects in rail-wheel contact", Journal of Theoretical and Applied Mechanics, 50(1), 119-130, 2012.

6

R. Bogacz, T. Krzyzynski, K. Popp, "Wave propagation in two dynamically coupled periodic systems", Proc. International Symposium on Dynamics of Continua, Bad Honnef, Shaker Verlag, 55-64, 1998.

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