Quantification of dynamic wheel–rail contact forces at short rail irregularities and application to measured rail welds

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Abstract

An analytical model for dynamic wheel–rail interaction at rail surface irregularities of arbitrary shape is presented. The model is formulated in the frequency domain and leads to relatively simple closed-form solutions for the variables which describe the dynamic interaction. With the help of the model, basic features of the dynamic wheel–rail interaction at short rail irregularities are identified. The focus is placed on short irregularities with a broad-band spectrum. The model is applied to calculate dynamic wheel–rail contact forces for a sample population of measured rail welds. General quantitative relationships between the rail geometry, the train speed and the level of the dynamic contact force are derived. Statistical distributions of geometrical properties of rail welds and the corresponding contact forces are derived. This allows for an easy estimation of the level of dynamic forces occurring over a railway network, given the availability of geometrical measurements.

Introduction

Continuously welded rail (CWR) nowadays is the standard in modern railway tracks, and the traditional bolted rail connections are often only still present in the form of insulated rail joints (IRJ) for detection and signalling purposes. The deflection of a bolted rail joint under a moving train axle loading leads to a high dynamic impact component, which is a source of rapid track deterioration and high noise levels, as pointed out in Refs. [1], [2], [3], and therefore the welded continuous connection was a significant improvement.

The most common types of rail welding are flash butt welding and aluminothermic or thermite welding. The application of the first type is almost fully automated, including the positioning of the rail ends to be joined, and therefore often applied in the construction of new tracks, whereas the second type requires handicraft and is therefore often applied in repair and maintenance work. However, the final grinding of the rail surface is in most cases done manually for both types of welding. Only in some cases the welds are being treated by a grinding train, on newly built tracks, yielding a high quality of the surface. In the literature, several studies have been published on rail welds, most of them dealing with metallurgical or related aspects. These aspects include the mechanical and micro-structural properties of welds [4], damage types occurring at welds [5], testing and inspection methods [6], the effects of heat transfer during welding on defect formation [7], [8], the effects of post-welding heat treatment [9], the measurement of residual stresses after welding [10], [11] and their alleviation [12], fatigue crack growth in a weld under residual and temperature stresses [13] and the influence of loose sleepers on the bending fatigue of rail welds [14]. In Ref. [15] attention was paid to the relationship between train loading, weld geometry and weld failure modes. From measurements an approximately linear relationship was found between the train speed and the dynamic impact factor (DAF) occurring at welds.

Rail welds are very important in the context of dynamic wheel–rail interaction and track deterioration. In the majority of the cases, after welding a railhead irregularity along the rail results at the position of the weld. This can be due to inaccurate rail end positioning, the manual grinding of the rail surface, and in many cases the influence of inhomogeneous material shrinkage after cooling down in combination with early grinding. The irregularity has a typical length between 0.5 and 1 m. Therefore, in general the rail welds yield a significant contribution to the short-wave part of the track irregularity spectrum. This aspect the rail weld has in common with or other short track defects, such as short-pitch corrugations [16], [17], [18], squats [19], [20], rail surface damage induced by rolling contact fatigue (RCF) [18], [21], [22], [23], [24], [25], rail end unstraightness [26] and other geometrical imperfections of the wheel–rail interface. An overview is given in Ref. [27]. However, especially on new tracks the short-wave contribution is almost entirely determined by the rail weld geometry. This is very important due to its direct relation with the rate of track deterioration: the different frequency components of the power of the dynamic wheel–rail interaction force are dissipated in the vibrations of different components of the track–vehicle system, and consequently determine their rate of deterioration. Detection of short-wave track defects was addressed in, e.g. Refs. [28], [29].

The relationship between the geometry of rail welds and the dynamic wheel–rail interaction has been dealt with extensively by Steenbergen and Esveld [30], [31], on the basis of finite-element (FE) simulations for a number (239) of measured welds. However, these simulations have a number of disadvantages. These include the sensitivity of quantitative results in the high-frequency regime (500–2000 Hz) to filtering and damping (see also Refs. [32], [33]), the validation of artificial damping and the simulation time needed for large measurement samples.

The investigation of the relationship between track geometry and interaction forces is relevant from another viewpoint. Railway track assessment is widely based on geometrical requirements without direct relationship to dynamic train–track interaction forces. This may be acceptable from the viewpoint of train passenger and vehicle ride comfort, but inconsistent from the viewpoint of track deterioration and the resulting necessity of maintenance, such as pointed out in, e.g. Ref. [34]. Therefore, in Refs. [30], [31] also a more adequate assessment method for rail weld geometry was elaborated, relative to the conventional method which is based of the principle of vertical tolerances. The new method, which determines a Quality Index (QI) for the weld, dependent on the train speed for the track section, aims at a reduction of dynamic wheel–rail forces at welds and has been standardised in the Netherlands.

This paper investigates dynamic wheel–rail interaction at rail welds using an analytical approach. The method will be based on a simplified wheel–rail interaction model, and will be formulated in the frequency domain. The problems connected to FE modelling are avoided, but the method also has a disadvantage: because of the fact that the model is simplified and its linearity, it is not suitable for quantitative predictions with a high level of accuracy. The aim of the approach in this paper is therefore not to calculate quantitative results with a high level of accuracy, but to clarify the basic mechanisms in dynamic wheel–rail interaction due to short disturbances in the interface, to establish theoretical relationships, and to investigate trends. The results then can be compared to the FE results in Ref. [31]. Furthermore, the general formulations allow for an application of the results of the investigation on any wheel–rail interfacial irregularity with a broad-band spectrum.

The structure of this paper is as follows. In Section 2 a model for dynamic wheel–rail interaction is presented, together with the underlying assumptions. This model is described mathematically in Section 3, where also solutions for the main variables are derived. In Section 4 a model is presented to describe rail surface irregularities of arbitrary shape, representing the system excitation. Combination of the results of 3 Mathematical problem formulation and solution, 4 Elementary model for description of rail surface irregularities yields theoretical expressions for the wheel–rail contact force, which are analysed in Section 5. In Section 6, basic features of the dynamic wheel–rail interaction are discussed using model simulations for an elementary excitation. Simulation results for a set of measured rail welds are presented and discussed in Section 7. These results are compared to the FE results obtained in Ref. [31]. Section 8 considers the dynamic wheel–rail interaction at rail welds on a railway network from a statistical viewpoint. In Section 9 the relation between the presence of rail welds and the occurrence of corrugation is discussed, and conclusions are presented.

Section snippets

Model for dynamic wheel–rail interaction at short rail irregularities

The following assumptions are adopted in the model being used to analyse the dynamic wheel–rail contact:

  • 1.

    The irregularity of the rail surface is short compared to the ratio of the train speed to the eigenfrequency of the bogie motion. Under these circumstances (f>20 Hz), the wheelset motion may be considered as isolated from the bogie and car body motion. Generally, this holds for welding irregularities, which have a length-scale of about 1 m [30]. In this case, only the unsprung vehicle mass of

Mathematical problem formulation and solution

The mathematical statement for the model as shown in Fig. 1b is given in the following. The equation of motion for the rail is given byEI4w(x,t)x4+ρA2w(x,t)t2+kfw(x,t)=-F(t)δ(x).

The equation of motion for the wheel system is given bymwd2u(t)dt2+k1u(t)=F(t).

The contact force in the wheel–rail contact point is given asF(t)=kH(-u(t)+w(0,t)+z(t)).

Assuming an infinitely small but non-zero damping in the system, the rail is undisturbed at x=±∞ at any time moment (i.e., Sommerfeld's radiation

Elementary model for description of rail surface irregularities

As has been stated in the Introduction, the present study focuses mainly on rail welds as a field of application. In Refs. [30], [31] a model for the vertical geometry of rail welds was proposed and developed; the aim was to describe real measurements, and therefore the model was discrete. According to Ref. [30], the surface of a rail weld may be described in the longitudinal direction by a sequence of discrete slopes which are Gaussian distributed with zero mean value. Such a spatial or

Parametrical analysis of the wheel–rail contact force

Substitution of Eq. (32) into Eq. (27) for the contact force yields an explicit integral expression for this force. For a single ramp (with a single discrete slope, as depicted in Fig. 2a) the expression becomesFcontact(t)=θVπ0Re((1-e-sΔx/V)/s21/kH+1/(mws2+k1)+1/(8EIk03)eσteiωt)dω.

For the sequence of N discrete slopes, the expression for the contact force may be written as follows:F(t)=1πVn=1Nθn0Re(fn(s))dωin whichs=σ+iω,fn(s)=(1-e-sΔx/V)e-(n-1)sΔx/V1/kH+1/(mws2+k1)+1/(8EIk03)ests2.

It

Basic features of the dynamic wheel–rail interaction

In. Eq. (18) a linearisation of the nonlinear Hertzian contact stiffness has been applied to enable an approach in the frequency domain. The linearisation is applied at the level of the static wheelload (which is chosen as 112.5 kN), yielding a tangent stiffness. Application of the geometrical parameters according to Table 1 then results into the following expression (according to Ref. [39]):kH=8×1022QRwheelRrailhead3=3×1022Q3.

The resulting linear stiffness will lead to reliable results for

Simulation results for measured welds

As has been explained in Ref. [30], the height of rail welds along the rail is measured electronically using digital straightedges with a finite sampling interval. The measurements used in this paper are based on the eddy current principle. The sampling interval equals 5 mm and the measurement basis is 1 m. The measurement signal cannot be used directly as an input in the simulations because the wheel–rail contact patch acts as a ‘filter’ for micro-asperities in the rail surface. The roughness on

Statistical properties of wheel–rail interaction at rail welds

In Ref. [30] it has been demonstrated that the discrete first derivatives of rail welds are Gaussian distributed with zero mean value (except for the tails). The maximum discrete first derivatives of a large population of welds proved to be very well approximated by a lognormal distribution.

In Fig. 15, the maximum absolute inclination and its logarithm (for the 5-point-smoothed signals) are shown as a function of the standard normal variable for the measured weld sample which was used in the

Discussion and conclusions

In Section 7, the time-scale of the P1 peak was found to range from about 0.5 to 0.75 ms, which corresponds to a length-scale (half-wavelength) of about 20–30 mm at a regular line-speed of 38.9 m/s (140 km/h). At 69.4 m/s (250 km/h; high-speed lines) the length-scale corresponding to the P1 force is about 35–48 mm (the time-scale does not vary significantly). Short-pitch rail corrugations have a wavelength which typically ranges from 25 to 80 mm. This means that non-smooth irregularities causing P1

Acknowledgements

The presented research is part of a research programme on the relationship between track geometry in the short-wave regime, dynamic train–track interaction and track deterioration, funded by the Dutch Railway Administrator ProRail (programme of Rolf Dollevoet). ProRail is also acknowledged for the availability of measurement data. The presented research will make part of a Ph.D. under the supervision of emeritus Prof. Coenraad Esveld.

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