On kinematic and selfexcited rail vehicle–track interaction
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Abstract
The paper contains description of dynamical problems connected with the selfexcitation and kinematic excitation of rail vehicle and railway track. Some phenomena are presented which may create high loads resulting in track degradation and fatigue of wheelset axles. An alternative approach to rail vehicle hunting is proposed. Examples of experimental investigations are given which show that the dynamical load acting on the track can be much higher than the static load.
Keywords
Railway vehicle Track Kinematic excitation Hunting Dynamic overload1 Introduction
The dynamic interaction of a rail vehicle with its track is a very complex phenomenon. Some effects are still not well understood even by specialists in this field. A wide palette of models is required to cover the variety of different aspects of the topic. The majority of experimental observations made on real vehicle–track systems can be explained using theoretical analyses and numerical investigations. Their results may help the designers of railway vehicles to increase the safety and comfort of travel and to reduce the maintenance costs of tracks and trains. A study of some selected problems concerning the influence of system parameters on the stability of a track–train system in relative motion is presented in the following study.
2 Hunting as a classic train–track instability of two subsystems in relative motion
In the following, another reason and model of train sinusoidal motion (hunting) are presented. It is more general than the usual models and can be applied also for magnetic levitated trains (MAGLEV).
In the real case, we can observe phenomena showing train instability like hunting, which has a limit cycle. In reality, the hunting amplitude is limited by nonlinearities occurring in real systems. To the problem of a nonlinear analysis of hunting as a selfexcited vibration, we will devote a separate paper.
3 Wheel–rail selfexcited vibration
The above question was explained during investigation of influence of the vibration external excitation on friction selfexcited mechanical system [6]. It was pointed out using relatively complicated friction model dependent on the process history that it is possible to limit the selfexcitation by means of additional vibrational excitation. Comparing the results, one can conclude that vibration of the bridge excited by the train can reduce the sleep waves in the rails.

\(x=KX/F_0\)—dimensionless displacement,

K—spring stiffness,

\(F_0\)—constant normal reaction,

\(d_1 ,d_2\)—internal and external viscous damping, respectively,

\(w=s\dot{x},\)—relative velocity,

\(t_s\)—time of sticking,

\(\delta =\mathrm{d}f_s /\mathrm{d}t\)—force rate.
The crucial points are connected with initiation of sliding which depends on the force rate and time of sticking (experimentally measured in [6]). Than we observe increase in the force associated with damping, after reaching maximal force the system is selfexcited till the speed reach maximum value. The change of acceleration sign is connected with further decreasing force and positive damping. On the end of this part of characteristic, we obtain sticking or rapidly increasing frictional force and the process is further continued.
In case of higher slip velocities, the sticking does not occur and we are faced with motion called semiharmonic vibration. The corresponding friction characteristic is shown in Fig. 13.
4 Selfexcited system subjected to additional vibration
It is visible in Fig. 14 that the solution consists of vibration of two frequencies. The first one is connected with selfexcitation and the second one with additional excitation. Increasing slightly the frequency of the additional excitation \(\eta \) (<0.5%), the response after a few cycles changes qualitatively to that of amplitude about three times smaller than the previous one. The change is visible in the righthand side graph in Fig. 14. Such a change (after a few cycles) is possible only in case of a historydependent model of friction. The displacement–velocity phase plane portraits for the abovementioned two kinds of solutions are shown in Fig. 15. The graphs are qualitatively different although the frequencies are very close to each other.
Talking about additional vibrational excitation on the bridge, we can supplement the above study and mention a few kinds of kinematic excitation of rail vehicles. As the source of such excitation, we can mention: periodically spaced corrugation, periodically spaced sleepers, supports of estacade and the outofround railway wheels (wheel polygonalization) creating also a kind of periodic kinematic excitation.
As a source of similar excitation, we can mention: periodically spaced corrugation, periodically spaced sleepers, supports of estacades and wheels poligonalization (the outofround railway wheels). The above phenomena, similarly as the earlier mentioned sleep waves on the Brixental bridge, can generate a periodic kinematic excitation.
The kinematically excited vibration can be solved using Floquet’s theory and looking for a traveling wave form of the solution. Contrary to a wavy form of interaction between two continuous systems (Fig. 2), in the case of train modeled as lumped system the problem of traveling and oscillating concentrated load (force) acting on a beam must be solved. Such a problem was formulated by Mathews [8] who proposed the solution in the form of standing waves. In that paper, the boundary conditions were not properly formulated, particularly the condition of radiation. The traveling wave solution with correct formulation of the boundary conditions allows to solve the problem taking into account the group velocity of generated waves. The proper solution was given in 1986 by Bogacz and Krzyżyński [9]. Such a solution allows us to solve various problems of discrete dynamical interaction of the train as a lumped system with a track modeled as continuous systems. As the practical example of the theory application, we can call the train–track dynamical interaction and its stability.
5 Concluding remarks
A study of selected problems of vehicle–track interaction concerning the influence of hybrid and continuous modeling on the stability and selfexcitation of systems in relative motion is given. An alternative approach to the classic rail vehicles hunting [10], based on instability of two continuous, onedimensional systems in relative motion is proposed. The presented results extend our previous considerations presented in [3, 11] and in [12]. The case devoted to the frictioninduced vibration which is the reason of many damages of trains and tracks was considered. It was shown that an additional vibrational excitation of proper frequency can reduce the frictioninduced vibration and its negative consequences for the train and track structure.
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