Keywords: railway, dynamics, frequency domain, rail, peak force.

This paper is concerned with calculating the peak wheel-rail contact forces associated with flatbed rail two-axle freight wagons. The research described in the paper is part of ongoing research being undertaken in the Departments of Civil Engineering at Trinity College Dublin and N.U.I. Galway in conjunction with Iarnród Éireann.

The contact forces between rail and wheel are a very important factor in the design of railway track and rolling stock suspension systems. From the track viewpoint, the peak loading determines the rail section required and places limits on the levels of track irregularity that are acceptable. From a rolling stock viewpoint, the suspension characteristics are a major influence on the magnitude of the peak loading. Knowledge of both peak forces and the forces' power spectral density functions is desirable from a design viewpoint. The wheel-rail contact forces can be measured [1]. However, modelling the system facilitates the design of vehicle suspension systems such that peak forces, both high and low, are minimised. It is desirable to minimise unloading of the wheels because unloading leaves the wheels liable to derail.

The paper details the dynamic analysis of flatbed rail freight wagons in both the time and frequency domains. Time domain analyses have the advantage of simplicity. The simple nature of this approach allows for complexity in the model where required. This allows the Hertzian contact to be modelled as a non-linear spring and facilitates modelling loss of contact. However, time domain analyses are dependent on the details of the vertical railhead mis-alignment and thus varying the initial conditions will affect the peak forces calculate. Analysis in the frequency domain is very useful for predicting peak events and identifying the fundamental dynamic characteristics of a system [2]. A frequency domain analysis is eminently suited to problems where the input data, railhead mis-alignment in this case, is best described statistically.

This particular study is concerned with the effect of the vehicle suspension system on the peak vertical wheel-rail contact forces. Thus for the purpose of this particular analysis a very simple 2 DOF model is sufficient. The model does not include the dynamic response of the track system and thus is a simplification. However, this simplification facilitates the development of closed form solutions describing the response, including undamped natural frequencies, of the rolling stock's suspension in terms of the basic suspension characteristics.

The frequency domain analysis of the flatbed wagon's response yields transfer functions describing the relative displacements of the wheel and rail and the axle and body in terms of the vertical rail profile irregularities. Formulating the problem in terms of the relative displacements facilitates calculating the peak wheel-rail contact force and the peak force in the suspension system.

Calculation of the peak forces is achieved by combining the transfer functions with a power spectral density function describing the vertical irregularities of the rail profile. This generates an output power spectral density function. By integrating the output power spectral density functions it is possible to calculate the RMS values for the relative displacements.

The peak contact forces are obtained from the RMS values. Much work has been done on relating the peak coordinate values to the RMS values so that it is possible to infer extreme values directly from root means squared values [3]. For processes that have a zero mean the RMS corresponds to the standard deviation . In many engineering applications it has been traditional to consider extreme events in terms of the number of standard deviations from the mean. However, there are many more rigorous techniques, such as methods that are based on the first passage principle, that model the distribution of peak events explicitly. One approach is to apply a peak factor based on the first passage principal [3]. These approaches require that the length of track for which the peak event is being sought is specified [4].

1

Esveld, Coenraad, "Modern Railway Track", MRT Productions, Duisburg, West Germany, 1989

2

L. Sun and T.W. Kennedy, "Spectral Analysis and Parametric Study of Stochastic Pavement Loads", ASCE Journal of Engineering Mechanics, 128(3), 318-327, 2002. doi:10.1061/(ASCE)0733-9399(2002)128:3(318)

3

B. Basu and V.K. Gupta, "Stochastic seismic response of single degree of freedom systems through wavelets", Engineering Structures, 22, 1714-1722, 2000. doi:10.1016/S0141-0296(99)00109-1

4

Frýba, Ladislav, "Dynamics of Railway Bridges", Thomas Telford, London, 1996

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