Shape Optimisation of Railway Wheel Profile under Uncertainties. Ivan Y. Shevtsov, Valeri L. Markine, Coenraad Esveld

Shape Optimisation of Railway Wheel Profile under Uncertainties Ivan Y. Shevtsov, Valeri L. Markine, Coenraad Esveld Delft University of Technology, Delft, THE NETHERLANDS Abstract Optimisation of a railway vehicle-track system is a complex process. The paper presents a procedure for optimal design of a wheel profile based on geometrical wheel/rail contact characteristics such as the rolling radii difference (RRD). The procedure uses an optimality criteria based on a RRD function. The criteria accounts for stability of wheelset, cost efficiency, minimum wear of wheels and rails as well as safety requirements. The shape of the wheel profile is varied during the optimisation process in order to satisfy the optimality criteria. The proposed optimum design procedure has been applied to improve the performance of the metro trains in Rotterdam, The Netherlands (RET), which were suffering from severe wheel wear and as a result the hunting of the vehicles. The results of the optimisation have shown that the performance of railway vehicle can be improved by improving the contact properties of the wheel and rail. Using the proposed procedure a new wheel profile has been obtained and applied to the RET metro trains. Due to the application of the optimised wheel profile the instability of the metro trains has been eliminated and the lifetime of the wheels has been increased from 25 km to 12 km. It was found that track gauge variation significantly affecting the wheel lifetime. Currently, research is carried out to incorporate this uncertainty in the wheel profile design procedure. Introduction Modern railways are developing systems. Numerous changes are introduced to increase its productivity, serviceability and profitability. However, in the field of the wheel/rail interface, any change should be carefully investigated before being applied in the field. Moreover, as it will be shown in this paper, even when no difference is expected in the system behaviour as a result of the renewal, unexpected side effects can deteriorate this behaviour. The development of appropriate combinations of wheel/rail profiles improves vehicle-track dynamic interaction and reduces the wear. The design of a wheel profile is an old problem and various approaches were developed to obtain a satisfactory combination of wheel and rail profiles. Usually, it is possible to find an optimal combination when dealing with a closed railway system, i.e. when only one type of rolling stock is running on a track and no influence of other types of railway vehicles is present. Examples of such systems are heavy haul and tram lines. In the present paper such closed system is considered, namely the Rotterdam metro network (RET) in The Netherlands (Figure 1). The kinematical properties of wheel and rail contact, such as rolling radius, contact angles and wheelset roll angle vary as the wheelset moves laterally relative to the rails. The nature of the functional dependence between these geometrically constrained variables and the wheelset lateral position is defined by the cross-sectional shape of wheel and rail. By studying the geometrical characteristics of the contact between wheel and rail it is possible to judge on the dynamic behaviour of the wheelset and dynamic properties (like stability) of the vehicle. The wheel and rail cross-sectional shapes define not only the kinematical and dynamical properties of the wheelset but also the physical properties such as contact stresses, creep and wear. Therefore, an illdesigned combination of wheel and rail profiles can be a source of various railway problems such as high wear rate of the wheels, instability (hunting) of a wheelset as well as rolling contact fatigue defects of the rails. These problems on their turn lead to cost inefficiency and unsafety in exploitation. Typically, a wheel profile was designed using a trial and error approach. A choice of a wheel crosssectional shape was mostly based on a designer intuition and experience as well as using some measurement data. During the last decades a number of efforts have been made to use numerical

methods in wheel design process. Smith and Kalousek [16] developed a numerical procedure for design of a wheel profile described by a series of arcs. The process of designing a new wheel profile for North American railroads is described by Leary et al. [6] wherein alternative profiles were derived through two techniques: an average worn wheel profile and profiles based on expansions of rail shapes. One procedure for design of a wheel profile using a numerical optimisation technique was proposed in Shevtsov et al. [12], [13]. The suggested there optimality criteria for wheel profile design were based on a Rolling Radii Difference (RRD) function. In this procedure the optimisation searches for an optimum wheel profile by minimizing the difference between target (desired) and actual RRD functions. To solve the minimization problem an optimisation procedure based on Multipoint Approximations based on Response Surface fitting (MARS method) was used. In Shevtsov et al. [14], [15] and Markine et al. [8] the procedure was further developed and three approaches for choosing of a target RRD function were suggested. Similar approach was proposed in Shen et al. [11] wherein the contact angle function was used for the design of railway wheel profiles. Persson and Iwnicki [1] use direct optimisation procedure based on a genetic algorithm for design of a wheel profile for railway vehicles. In this paper a procedure described in Shevtsov et al. [14], [15] is applied to improve the performance of metro vehicles that were suffering from severe wheel wear and instability. Figure 1: RET metro, Rotterdam, The Netherlands. Courtesy of www.retmetro.nl. Wheel/rail contact properties An important characteristic of the contact between wheel and rail is the rolling radius of a wheel at the contact point [3]. This radius can be different for the right and the left wheel as a wheelset is moving along a track ( r1 and r2 respectively, as shown in Figure 2). When a wheelset is in a central position with respect to the track the rolling radius of the left and the right wheel are the same, namely r1 = r2 = r. An instantaneous difference between the rolling radius of the right and the left wheel can be defined as a function of the lateral displacement y of a wheelset with respect to its central position (Figure 2), according to: r ( y ) r1( y ) r2 ( y ). (1) The rolling radius difference (RRD) is one of the main characteristics of wheel/rail contact that defines the behaviour of a wheelset on a track. For example to pass a sharp curved track without slippage between wheels and rails an appropriate RRD of the wheelset is needed. Avoiding the slippage is important from the wheel and rail wear point of view.

r 2 =r-.5 r y r 1 =r+.5 r 2γ r O r y w z w Figure 2: Rolling radius ( r 1 and r 2 ) corresponding to wheelset displacement y, the wheels are conical, γ is the conicity of the wheels. Generally, the rolling radius difference is a non-linear function of the lateral displacement y of a wheelset, which from geometrical considerations (Figure 2) can be written as: r = 2 y γ ( y ), (2) where γ is the wheel conicity. The effective or equivalent conicity γ e is determined at a certain lateral displacement y = y. It should be noted that for a purely conical wheel the equivalent conicity is equal to the cone angle of the wheel, i.e. γ = γ e. Due to wear a wheel profile changes (Figure 3) and consequently its RRD function. Examples of rolling radius difference functions (also known as a y r curve) for purely conical and worn profiles are given in Figure 4. Wheel flange Flange root Wheel tread Field side Wheel hollowness Worn wheel Unworn wheel Back of flange Flange root gage Flange corner face gage face Wheel flange wear Tread wear Wheel tread rail ball Field side contact False flange Figure 3: Worn and unworn wheels. Determination of geometrical contact characteristics for given wheel and rail profiles, wheel and rail gauge, and railhead cant angle is a well-known problem already solved for many years. These nonlinear characteristics were investigated by Wickens [17], Cooperrider et al. [2] and De Pater [9]. A linear conical wheel profile widely used earlier has bi-linear characteristics of the rolling radius difference, as shown in Figure 4. This results in shocks when the wheel flange and rail come into contact during the wheelset motion. A worn wheel better matches a rail and therefore has a smoother RRD function. However, high conicity of a worn wheel reduces the critical speed of a wheelset and results in severe vehicle oscillations (hunting). An optimum wheel profile should satisfy: low wheel wear rate; wheelset stability; acceptable contact stresses.

Obviously, an optimum profile is a compromise between stability and curving behaviour. Traditionally such a compromise is achieved by manually modifying the wheel shape to find satisfactory contact characteristics in combination with given rail. However, this design approach is quite time consuming and expensive. r worn profile conical profile 2γ e y y y flangeway clearance Figure 4: Rolling radius difference functions ( y r curves). The use of numerical simulations and optimisation methods looks more efficient. The approach discussed in this paper concentrates on finding a wheel profile satisfying an optimised y r curve. When the optimum rolling radius difference function r( y ) for a given rail profile is known, one can try to solve an inverse problem in order to find a wheel profile with such a RRD. This approach is formulated as an optimisation problem. Wheel design procedure The wheel profile design procedure is shown schematically in Figure 5. The first step is an analysis of the current wheel/rail profiles. Profile measurements with for instance the MiniProf system are used to analyse the wheelset contact properties to design a target RRD function. The second step is the definition of an optimum (target) RRD function. The problem of finding a wheel profile corresponding to the target RRD function was formulated as an optimisation problem. The problem has been solved using the MARS optimisation method, which was successfully used for various real-life applications [7]. Since the dynamic properties of a vehicle are not directly controlled during the optimisation process, which in fact reduces the computational efforts of the optimisation, they must be verified afterwards. The optimised profile is tested for stability, wear and dynamic contact stresses with the ADAMS/Rail computer package [1]. If the dynamic performance of a vehicle with the obtained wheel profile does not satisfy the imposed requirements, the RRD function should be adjusted and the optimisation should be performed again in an iterative process. Design variables To describe the geometry of a wheel profile, a number of points on its flange, flange root and tread are chosen. Connected by a piecewise cubic Hermite interpolating polynom, these points define the shape of the wheel profile, as shown in Figure 6. The positions of these points can be varied in order to modify the profile. For tangent track only the points ambient to the wheel tread are considered. In this case stability is the dominant factor. In curves points ambient to the flange root and the field side are selected. Here wear and contact stresses are dominant.

Wheel/rail profiles measurements Choice of target RRD Solution of inverse problem no Requirements satisfied? yes Optimised wheel profile New wheel profile Analysis of dynamic behavior using ADAMS/Rail TM Figure 5: Flowchart of wheel profile design procedure. z y Wheel profile Constrained points Moving points Figure 6: Wheel profile, moving and constrained points. Objective function and constraints Since the optimum wheel profile is defined by the target RRD the difference between the resulting RRD and the target RRD should be as small as possible. The target RRD function is divided on three parts which are responsible for tangent track, curved track and sharp curves. For tangent track conicity at y = should be maintained in the range.25-.2 for respectively TGV trains and metro trains to provide a sufficiently high critical speed of a vehicle. On a curved track with a large radius the corresponding rolling radius difference must allow the wheelset to find the radial position in a curve to prevent wheel creepage. In sharp curves wheels are expected heavy flange contact. In this case the RRD must be as high as possible. The complete procedure of selection of the target RRD function is described in [15]. Two safety requirements are included, i.e. wheel flange thickness and minimum flange angle to prevent vehicle derailment. Dynamic analysis The vehicle is modelled using ADAMS/Rail [1], see Figure 7. The dynamic simulations are performed on specially designed tracks. On straight track the wheel profile is tested for vehicle stability and wear. On curved track the wheel profile is tested for curving behaviour, wear and RCF. Wheel wear is estimated using the wear index W calculated according to [5]: W = F1 ξ + F 2 η, (3) where F 1 is the longitudinal creep force; ξ is the longitudinal creepage; F 2 is the lateral creep force; η is the lateral creepage.

Figure 7: ADAMS/RAIL vehicle model. Wheel profile design for RET metro History The wheel profile design procedure described in the previous section has been applied to improve the performance of some RET metro trains. In 1999 the existing NP46 rails were replaced by S49 rails. The profiles looked similar and therefore no difference in the train performance was expected. However, after the introduction of the new rail profile the metro trains started to experience severe lateral vibrations. At the same time a high level of wheel wear was observed after a relatively small mileage. In fact these vibrations were caused by the worn wheels. Immediately after these vibrations had been observed the wheels were re-profiled to prevent derailment. Thus, due to the replacement of the rails the lifetime of the wheels was reduced from 12 km (with the rails NP46) to 25 km (with the rails S49). The Railway Engineering Group of TU Delft was commissioned a research work to improve vehicle stability and wheel life. Analysis of the problem The problem has been solved in several steps. First of all a comparison of the old and new rails has been performed. It should be noted that the S49 rail is installed with an inclination of 1:4 whereas NP46 rail is put normal. Even though the rail profiles NP46 and S49 look similar (Figure 8a) there is a small difference between these two profiles which only becomes visible while zooming in the top of the rails as shown in Figure 8b. However, such small differences, in the order of tenths of a millimetre, are responsible for substantial changes in vehicle behaviour. In the next step the wheel/rail contact characteristics of the old and new situation have been investigated. The wheel/rail contact points for new (unworn) wheel profile UIC51 with the unworn NP46 and S49 rails respectively are shown in Figure 9. In this figure the lines between the wheel and rail profiles connect the corresponding contact points, which were calculated per.5 mm of the lateral wheelset displacement. The lateral wheelset displacements are shown above the wheel profile. The coordinate system in this figure is the wheelset coordinate system ywoz w (Figure 2) with the origin in the centre of the wheelset in neutral position. It should be noted that in this figure the wheel is shifted 1 mm vertically. By comparing Figure 9a and Figure 9b one can observe a discontinuity (a big jump) in the position of the contact point on the S49 rail around the neutral wheelset position. Since the displacements in the range +/- 2 mm typically correspond to the motion of a wheelset on a straight track a lot of such jumps will occur during the vehicle motion. Due to these contact point jumps the wheel profile wears very rapidly. This ultimately results in very big jumps of the contact point as shown in Figure 1. This figure shows the contact points of a worn (measured) UIC51 wheel profile and a S49 rail. These big jumps of the contact point were also the source of the vibrations observed in the metro trains. On the other hand the wheel/rail combination UIC51-NP46 has more uniformly distributed contact points on a straight track (corresponding to wheelset displacements of +/- 2 mm) as shown in Figure 9a. As a result the wear rate of the UIC51 wheels on the NP46 rails was much lower and therefore the wheel lifetime was relatively long (12 km).

As it was mentioned earlier, the rolling radius difference plays an important role in the vehicle dynamics and has therefore been investigated next. The RRD functions for the UIC51 wheel profile on respectively the NP46 and S49 rails are shown in Figure 11. This figure reveals that the UIC51/S49 wheel/rail combination has a much higher inclination compared to the UIC51/NP46 combination. This also means that the corresponding equivalent conicity (2) is higher for S49 than for NP46. The high conicity was the reason for large vehicle vibrations, high tread wear and ultimately the relatively short lifetime of wheels. 1-4 -2 2 4-1 -2 1-3 -2-1 1 2 3-1 -3-4 -2 S49 (1:4) NP46-3 a. b. S49 (1:4) NP46 Figure 8: Comparison of rails NP46 and S49 (a. not zoomed; b. zoomed in). -37-38 6 4 2-2 -4-6 -8-1 -37-38 6 4 2-2-4-6 -8-1 -39-39 -4-41 -42 8 1-4 -41-42 8 1-43 -43-44 -44-45 7 72 74 76 78 8-45 a. b. 7 72 74 76 78 8 Figure 9: Contact points of unworn wheel UIC51 and rails - NP46 (a.), S49 (b.). -37-38 -39 2 4 6-2-4-6 -8-1 -4-41 -42 8 1-43 -44-45 7 72 74 76 78 8 Figure 1: Contact points of worn (measured) UIC51 wheel and S49 rail.

6 5 r, mm 4 3 2 1-6 -4-2 -1 2 4 6-2 -3-4 -5 UIC51_NP46 (target) Optimised_S49 UIC51_S49-6 Figure 11: RRD function: UIC51 wheel on NP46 rail; UIC51 wheel on S49 rail, Optimised wheel on S49 rail. Design of a wheel profile After the source of the vehicle instability problem was found, the next step was to improve (optimise) the wheel profile. The UIC51/NP46 wheel/rail combination (see Figure 11) has a conicity of.2. The metro vehicle on wheels with such conicity is stable. Therefore, the UIC51/NP46 RRD function has been chosen as the target RRD function in the optimum wheel profile design procedure described earlier. The shape of the optimum wheel profile is shown in Figure 12. Even though the changes in the shape are not significant, they result in quite substantial changes of the RRD function as shown in Figure 11. The RRD function of optimised profile on S49 rail is very close to the target RRD, which means good results of optimisation. The wheel/rail contact points are shown in Figure 13. It can be observed that the contact points are very well distributed in the range from -4 mm to 4 mm of the lateral wheelset displacement. Such contact will result in much lower wheel wear. Dynamic simulations on a straight track have been performed in order to analyse vehicle stability and wheel wear. The track has the length of 7 m whereas at the distance of 5 m from the beginning of the track a horizontal ramp (height - 5 mm, width -.1 m) has been introduced. The ramp induces initial disturbance to a wheelset leading to oscillations of the bogie. As a result of the optimisation the critical velocity of a metro train has been increased from 5 m/s for unworn UIC51 to 6 m/s for the optimised wheel profile. The critical velocity for the worn UIC51 profile was 3 m/s, which explained the observed stability problems. The lateral displacements of its wheelset are shown in Figure 14. The metro train was travelling on a straight track with the speed of 2 m/s. From this figure it can be seen that the oscillations of the wheelset due to the lateral ramp are damping out relatively fast and the amplitude of these oscillations is not large. Therefore, it can be concluded that the motion of the metro vehicle with the optimised wheel profile is stable at the operational speed of 2 m/s. From Figure 14 one can also see that the unworn UIC51 wheel profile on the S49 rail delivers a stable behaviour of a wheelset at 2 m/s as well, which was observed in practice. The wear indexes of a wheelset with the optimised and UIC51 wheel profiles are shown in Figure 15. From this figure it can be seen that the wear index for the unworn UIC51/S49 wheel/rail combination is significantly higher than for the optimised wheel profile (lower wear). The explanation can be found in Figure 16. The optimised wheel on the S49 rail has always single-point contact, while the unworn UIC51/S49 wheel/rail combination always remain in double-point contact, causing high wear rates. So the optimised profile prevents double point contact and thus reduces wheel wear.

High wheel tread wear leads to a hollow worn profile, causing vehicle instability due to high or negative conicity. This is prevented with the optimised wheel profile. 1-1 3 4 5 6 7 8 9 1-3 Initial profile (UIC51) -5 Constrained points Moving points -7 Optimised profile -9-11 Figure 12: Initial and optimised wheel profiles. -37-38 6 4 2-2 -4-6 -8-1 -39-4 -41-42 8 1-43 -44-45 7 72 74 76 78 8 Figure 13: Contact points of optimised wheel profile and S49 rail. Lateral displ., mm 1..8.6.4.2 Time, s. -.2 1 2 3 4 5 6 7 8 9 1 -.4 -.6 -.8 Optimised profile UIC51-1. Figure 14: Lateral displacements of wheelset with optimized and UIC51 wheels on S49 rail, velocity 2 m/s.

Wear index, N 5. 4. Optimised profile UIC51 3. 2. 1.. Time, s 1 2 3 4 5 6 7 8 9 1 Figure 15: Wear index of the left front wheel. Optimised and UIC51 wheel profiles on S49 rail. Contact point, mm 15 1 5-5 -1-15 -2-25 Time, s 1 2 3 4 5 6 7 8 9 1 Optimised_CP1 UIC51_CP1 UIC51_CP2 Figure 16: Position of contact point(s) on the left front wheel. Optimized (single-point contact) and UIC51 (double-point contact) wheel profiles on S49 rail. Current work The optimised wheel profile has been applied to metro trains in Rotterdam (the Netherlands). Unexpectedly low lifetime of the designed wheel profile was observed during the first tests. The wheels were running only 55 km before hunting occurs. Of course it was an improvement comparing with the initial 25 km. However, the aim of 12 km was not achieved. An additional investigation was done. It was found that the track gauge, which was assumed as the standard 1435 mm during the optimisation of the wheel profile, was in reality 2 mm narrower. That is why the optimised wheel profile was adapted to the narrower gauge by reducing the flange width. After the adaptation the life of the wheels has reached 12 km and no lateral vibrations of the metro trains with the optimised wheel profile have been observed. Despite that the optimisation of the wheel was successful an open question remains. The influence of the track gauge variation was not taken into account during the optimisation, but appeared to be an important factor affecting the wheel lifetime. At present time work is carried out to incorporate this factor in the optimisation procedure. The simplest solution is to use conical wheel profiles which are not so affected by the track gauge variation. However, such profiles have bad performance in curves, which will lead to

increase of the wear. Therefore the challenge is to design a wheel profile which preserves its curving performance while being less affected by the track gauge. Results on this research will be presented during the congress. Conclusions In this article a numerical method for wheel profile design was discussed, based on optimisation of the Rolling Radius Difference function. A new wheel profile was obtained by minimising the difference between target and current RRD. MARS method was chosen as a general optimisation technique. The design procedure allows developing wheel profiles with a priory defined contact properties. Using this procedure the wheel profile design for the RET metro trains has been improved. The results of the dynamical simulations have shown that the performance of railway vehicle is improved by improving the contact properties of wheel and rail. Double point contact between wheel and rail produces high wear and leads to hollow wear of wheels. The new wheel profile was implemented on the RET metro trains. As a result of this the instability of the metro trains was eliminated and the lifetime of the wheels was increased from 25 km to 12 km as per February 26. TU Delft is focusing research work to incorporate the uncertainties (such as track gauge, wheel profile tolerances) in the wheel profile design procedure. Acknowledgements The authors would like to thank RET for providing measurement data of wheel and rail profiles and metro vehicle data. References [1] ADAMS/Rail, MSC.Software Corporation MSC.ADAMS, (25), http://www.mscsoftware.com. [2] N.K. Cooperrider, E.H. Law et al. Analytical and Experimental Determination of Nonlinear Wheel/Rail Constraints, Proceedings ASME Symposium on Railroad Equipment Dynamics, (1976). [3] R.V. Dukkipati. Vehicle Dynamics, Boca Raton: CRC Press, (2), ISBN -8493-976-X. [4] C. Esveld. Modern Railway Track, (Second Edition), Zaltbommel: MRT-Productions, (21), ISBN 9-84-324-3-3. (www.esveld.com). [5] J.J. Kalker. Three-Dimensional Elastic Bodies in Rolling Contact, Dordrecht: Kluwer Academic Publishers, (199), ISBN -7923-712-7. [6] J.F. Leary, S.N. Handal, and B. Rajkumar. Development of freight car wheel profiles a case study, Proceedings of the Third International Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Cambridge, U.K., July 22-26, (199), ISBN 444-88774-1 also in Wear, Volume 144, pp. 353-362, (1991). [7] V.L. Markine. Optimisation of the Dynamic Behaviour of Mechanical Systems, PhD Thesis, TU Delft: Shaker Publishing BV, (1999), ISBN 9-423-69-8. [8] V.L. Markine, I.Y. Shevtsov, C. Esveld. Shape Optimisation of Railway Wheel Profile. Proceedings of 21st International Congress of Theoretical and Applied Mechanics. August 15-21, 24, Warsaw, Poland. (On CD-ROM) (24) ISBN 83-89697-1-1. [9] A.D. de Pater. The Geometrical Contact Between Track and Wheelset, Vehicle System Dynamics, Volume 17, pp. 127-14, (1988). [1] I. Persson, S.D. Iwnicki. Optimisation of railway profiles using a genetic algorithm, Vehicle System Dynamics, Supplement to Vol. 41, pp. 517-527, (24). ISBN 9-265-1972-9. [11] G. Shen, J.B. Ayasse, H. Choller, I. Pratt. A unique design method for wheel profiles by considering the contact angle function, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, Volume 217, pp. 25-3, (23).

[12] I.Y. Shevtsov, V.L. Markine, C. Esveld. One procedure for optimal design of wheel profile, Proceedings of the IQPC conference on Achieving Best Practice in Wheel/Rail Interface Management, Amsterdam, The Netherlands, January 31 - February 1, (22). [13] I.Y. Shevtsov, V.L. Markine, C. Esveld. Optimisation of Railway Wheel Profile Using Mars Method, Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC/ Structures, Structural Dynamics, and Material Conference, Denver, Colorado, USA, 22-25 April, 22. Paper AIAA 22-732. ISBN 1-56347- 56-X (On CD-ROM) (22). [14] I.Y. Shevtsov, V.L. Markine, C. Esveld. Optimal design of wheel profile for railway vehicles, Proceedings 6th International Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Gothenburg, Sweden, June 1-13, 23, pp. 231-236, ISBN 91-631-3928-6 (23). [15] I.Y. Shevtsov, V.L. Markine, C. Esveld. Optimal design of wheel profile for railway vehicles, Wear, Volume 258, Issues 7-8, pp. 122 13, (25). ISSN 43-1648. [16] R.E. Smith, J. Kalousek. A design methodology for wheel and rail profiles for use on steered railway vehicles, Proceedings of the Third International Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Cambridge, U.K., July 22-26, 199, ISBN 444-88774-1 also in Wear, Volume 144, pp. 329-342, (1991). [17] A.H. Wickens. The Dynamic Stability of Railway Vehicle Wheelsets and Bogies Having Profiled Wheels, International Journal of Solids Structures, Volume 1, pp. 319-341, (1965).