Oscillations and chaotic behaviour of unstable railway wagons over large distances

Chaor.

Sohrons & Fracrals

Vol. 5. No

9, pp 1725-1753. 199 Elsevier Science Ltd Printed in Great Britain. OY60-0779/95S9.?0 + 00

0960-0779(94)00173-l

Oscillations and Chaotic Behaviour of Unstable Railway Wagons over Large Distances J. P. PASCAL INRETS, 2 Avenue du General Malleret Joinville, 94114 Arcveil, France

Abstract-Considering the physico-mathematical problem set at the title of this paper, knowing also that it covers actual circumstances of dynamical derailments of vehicles with large safety consequences, one has to face unusual modelling difficulties. Mainly two results are awaited: (1) the description of the mechanisms allowing derailment and (2) an evaluation of the risk. Obviously (and fortunately), dynamical derailments resulting of random association of track geometry with chaotic oscillations of the wagons are rare, almost impossible to produce experimentally on purpose and consequently difficult to describe. However, the available experimental field is not empty but presents lots of time histories of chaotic signals (forces) which cannot be studied on a deterministic basis. As these vehicles are quite non-linear mechanical systems, the only possibility of their modelling is numerical. Thus, there is only one chance to be effective, it is to develop numerical models using simplifying hypothesis in order to yield the shortest CPU times so as to be able to test and adjust the effects of modelling assumptions and to cover the largest possible field, including influence of wear modifications of the rolling profiles. This task has been done (the development lasted 10 years) and our method, which we do not pretend to be the only one, is explained in this paper focusing on physical assumptions rather than on already published mathematical developments. Among numerous difficulties, we had to face chaotic results of the computations. They appeared when vs/time numerical results of long computations were found depending, for instance, on the type of the micro-processor or on the way of factorising the Fortran statements. We are now convinced that there is no solution to this difficulty in the direction of increasing numerical precision because it comes from the nature of the physical problem itself. We could not prove it, but, as all experimental signals were different from each other, one has to admit that the physical reality that is to be modelled cannot be observed deterministically and, consequently, that, even if it would be possible to develop a deterministic model, it would be impossible to validate it.

1. INTRODUCTION

Railway engineering has been developed in all parts of the world for about one and a half centuries and the associated knowledge forms a large technical whole, covering most scientific domains. Considering the specificity of railways, attention is immediately paid to the rolling of rigid steel wheelsets on tracks equipped of steel rails, and, to reduce again the field, there remains, as the heart of the system, the mechanical contact between the wheels and the rails on which relies all of it. Indeed, the forces that support and guide the vehicles are created there and the stability of the rolling surfaces with respect to wear depends upon it. Thus, the functionality of the railway system, its safety, its maintainability depend primarily upon what happens in the contact patch. Any observer of railway development could state that the technical matter that has just been described: wheel-rail contact, seems, at first, very simple but has not been simply described in books in which engineers could learn how to apply the principles of these mechanisms in order to assessfunctionality, safety, maintainability. In fact, railway science 1725

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has a strong and long experimental basis, but seems to have resisted to mathematics more than is usual in other mechanical fields. In our opinion, the reason is not to be looked for in the dynamical complexity of the track-vehicle system, but at first in the physical stationary phenomenon itself owing to which are created both normal and tangential forces inside the contact area formed between a loaded wheelset rolling over a steel rail. The usual scientific approach to this problem of rolling contact, starts from the theory of elasticity in connection with the theory of dry friction of Coulomb (tangential force is limited by normal load times a “constant” friction coefficient). Owing to the fact that both materials in contact are of the same modulus, the problem can be separated into two independent parts: the first one consists of finding the normal pressure distribution over the contact patch and the second to find the tangential pressure distribution associated to the normal pressure and to local creepages [ 11. It need not be said that, even if the profiles of the bodies in contact were very simple, which is not the case, the introduction of such a non-linear law as Coulomb’s would discourage comprehensive analytical attempts of solutions. In addition, it must be emphasized that actual wheel and rail profiles are sensitive to wear and, consequently, are not stable, not analytically definable and even not precisely known at any specific time when one would like to perform a calculation. Because of wear, any observable behaviour of a vehicle on an actual track of a railway network,* will depend upon the entire history of the network from its beginning, during which wheels and rails have adapted and modified each other’s profiles. Due to this actual complexity, difficult to solve with usual mathematical tools, and even without being able to rely on clearly defined input data, most of the scientific efforts made to analyse the contact problem are being turned to the dynamical dimension of it, which means not to it but on both sides of it. In addition, the main evolution of the railway track has consisted, instead of short jointed rails, to use long welded ones which are correctly modelled by a continuous beam taking into account its natural modes, longitudinal (propagation in the beam-rail) and vertical (vibration modes of the rails and sleepers). Thus, different ways are in front of the researcher willing to investigate the railway mechanisms and the question arises of knowing whether they are depending upon each other and if he must explore the first before studying the second: 0 modal analysis of the dynamical modes excited by the contact in order to calculate the normal forces; l numerical analysis applied to the study of contact forces (normal and tangential) as depending upon geometrical forms of the profiles and upon creepages and their coupling to the dynamics into step by step time integrations. There are no clear answers to this question except that the level of complexity of the entire problem is too high to be solved with existing tools. The main reason is that complex numerical models have to be validated by comparisons to experiments, which means that where there are no reliable measurements there will be no reliable numerical solutions. As the frequency ranges of the first eigen modes studied by the first method, on one hand, are in the acoustic domain without (up to now) available contact forces measurements, and as there are many available data for the ‘classical’ rolling stock dynamics in the sub-acoustic *Excepted newly built tracks with specific vehicles such as French TGV tracks, the usual situation kind of vehicle runs over any track of one network.

is that any

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domain, on the other hand, the way to researchers of the contact problem seems clearly indicated. Fortunately, as often in applied physics, there exists here practical trade-offs that aIlow us to explore simplifying hypotheses with existing tools in order to be able to give a limited, but physically correct, description of the reality which, in physics, remains the experimental field. Consequently, the solutions that we propose in this paper do not pretend to be ‘the solution of the problem of railway contact’ but only a way through which, using numerical methods to integrate non-linear equations, in the frame of proven theories and of simplifying hypothesis, it has been possible to simulate the behaviour of railway vehicles with the same level of approximation as is given by measuring methods of the forces in use in the railway companies, so as to compare and validate. Considering the field: rolling contact as main component of functionality, safety, maintainability of the railway system, our method, has been to build a corpus of numerical codes in order to describe a railway reality which was accessible to forces and wear measurements. More clearly, if one believes for instance that the models to develop should and could be used as tools for predicting wear modifications of the wheels and rails profiles, one knows that the problem deals of tangential forces and creepages and has to be simplified. All possible simplifications of the models, provided that they are compatible with physical laws: are intended to be tested with respect to measured results. Consequently, all particularities of the models, unless they are free from complexity, that would give results beyond possibilities of measurements should be considered as refinements. Consequently also, all simplifications as reducing the dynamics only to its vertical component for instance, that would cancel the origin of the forces IO be modelled. would be considered as objectionable. Fortunately, many parts of the study, especially the first steps of it, as validating stationary derailments in twisted tracks [Zj, have given rise to deterministic results that could be confronted to deterministic tests and led to assessingthe codes step by step. However. while the complexity of the models was increasing, their deterministic behaviour was vanishing in such a way that it became difficult to assess the models by comparing calculation results to experimental ones uniquely on the basis of usual comparisons of time histories. But it appeared, looking at experimental results, that they were also chaotic because it was impossible to get twice the same time histories. Thus: to assessthe models, stochastical analysis had also to be used. With complex models of unstable vehicles, every numerical modification (such as a change in the micro-processor type) led to different solutions after a certain time and it has been a matter oE discussion with railway engineers whether it is acceptable or not that numerical models lead to indeterminant results? This is the reason why we are now practically interested in the theories of chaotic oscillations and their numerical approach. Attempts of solutions have been searched into refining the algorithms used to evaluate the equations and into increasing the numerical precision of the operations on floating point reals, they apparentiy succeeded. But, on one hand, it is quite sure that, for longer computing times: the same problems would arise again, and, on the other hand, there is no consistency between Fortran quadruple precision and the experimental precision of all data and particularly of initial conditions of specific problems. Keeping in mind that the aim of the method is to compute numerically and as realistically as possible the dynamical contact forces developed between wheels and rails of railway vehicles running over actual tracks in order to be able, for instance, to compute the risk of derailment and/or the evolution of profiles with wear, the different hypothesis that

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have been chosen and that are used in the codes Voco of INRETS will be presented in parallel with an example. At first the modelling of the contact itself is described. 2. FIRST HYPOTHESIS:

MODELLING A NON-HERTZIAN HERTZIAN ONES

CONTACT

AS A SUM OF

Let us recall that, in this section, a wheelset is supposed to be a rigid assembly of two rigid wheels connected by a rigid axle (Fig. 1). The track is also a rigid assembly of two rails connected by rigid sleepers. Wheelset and track cooperate together by at least two contact ‘points’ that can be considered rigid or locally elastic. The mechanical principle allowing the axle to roll ‘inside’ the track, i.e. to follow its layout without derailing by climbing over the rail relies on the conicity of the wheel profile owing to which any lateral displacement of the wheelset with respect to the track centreline will produce, in the presence of frictional properties of the materials, through a difference between left and right rolling radii, differential longitudinal creep contact forces that will

Right Fig. 1t Principle

of steering of rigid railway wheelsets

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steer the wheelset so as it will tend to return to the centre: Fig. 1. The damping of this mechanical automatic control is made by the same forces. The running of a wheelset alone is never stable but if at least two wheelsets are connected together to a vehicle frame by means of suspension devices, there usually exist a ‘critical’ speed under which lateral oscillations are damped. In some cases, our wagon is one of them, there is one speed at which the unstability is maximum with stable behaviour at higher speeds. According to this principle, the wheels could be conical with some lateral bumpstop to avoid derailments, but, due to wear, this situation would not last long. As a matter of fact, their profiles are continuous in practice and an approximation of their conicity, the factor 13, known as the “Equivalent Conicity” of the wheelset, has for a long time been considered by railway engineers as a very important parameter characterizing their behaviour both on straight lines and in curves. It has been used in particular to estimate the critical speed of a vehicle by computing the eigenvalues of linearized equations of motion. For this purpose, it has been necessary to determine a constant value of conicity, representing as precisely as possible the actual behaviour of the wheelset. But this method is unable to compute realistic contact forces during unstabilities. Numerical computations can help to approximate better the forces. The simplest method to take the actual shapes of the profiles into account is to suppose that they are rigid. But, in all practical cases, wheel profiles are not conical and the rolling radii of rigid profiles, that are in contact by a mathematical point, present discontinuities as the lateral displacement changes. For numerical computations to proceed, it is necessary, in order to avoid unlimited accelerations and numerical instabilities, to linearize these discontinuities and many methods have been proposed [3]. However, none of these methods is able to represent correctly the physics of the contact which is not rigid but elastic. A correct method should take into account the elasticity of the contacts so as to be able to compute dynamical forces. Here we use measuring techniques of the unloaded profiles of the laser type without contact, records are made in polar coordinates and the smoothing techniques, adapted to the size of contact patches, are cubic splines also in polar coordinates. In spite of the smoothing of the profiles, the assumption of rigid profiles still leads to prediction of instability at very low speeds well below commercial operating speeds. For this reason, attention has been paid to methods accounting for elastic profiles in a way that allows several contact points at the same time on the same wheel [4, 51. Wheelset and rails keep being rigid bodies excepted in their contact areas where they are supposed to be elastic having the modulus, E of steel. Unfortunately, introducing contact elasticity will, with respect to the rigid joint case, increase the problem complexity and consequently, will yield enormous CPU times incompatible with our objectives. There are two reasons for this: l

l

the usual railway contact patches, due to their relatively large dimensions (high loads), compared to the gradients of the curvatures along the profiles, often make these contacts of the non-Hertzian type, which means that they cannot be solved analytically but by finite elements methods; the very high stiffness of these elastic contacts, due to the high elastic modulus E of steel, introduce high frequency modes in the dynamics of the system and oblige us to use integration time steps as small as 1 micro-second, which means a multiplying factor of 100 with respect to the other modes.

Thus we have proposed two simplifying hypothesis in order to solve these difficulties [6, 71: l

it is possible to give a good model of a non-Hertzian contact by a sum of a limited

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l

amount of Hertzian ones; as the normal Hertzian problem is immediately solved by Hertz’s form&e and the tangential can be solved in this case by Kalker’s quick code ‘Fastsim’ [8], a factor of reduction of CPU times greater than 1000 is obtained; it is possible, in the case of a rigid wheelset, to give an acceptable model of its elastic contacts by an identification method consisting, for every relative wheelset-track positions, to calculate first, and only once, the elastic forces, as proposed at previous alinea, and secondly, for each of both wheels, the parameters of one equivalent rigid contact that will produce, using ‘Fastsim’, the same tangential forces. Later on, during dynamical computations, normal forces would be calculated, in the frame of a rigid joint, using the kinematic functions (relative altitude and roll angle with their derivatives with respect to lateral displacement) calculated during the elastic identification and stored in a file. Due to this simplification, another reduction factor of 10 to 100 can be obtained for the CPU times.

2.1.

Normal problem of elastic contacts

The first problem is to find, for both wheels and for any relative positions, where the contacts are located, what is their shape and the pressure distribution on it. The method consists, starting from the knowledge of relative positions, i.e. knowing the penetration of the undeformed profiles, to calculate the pressure using the theory of elasticity. There are generally no analytical solutions except in the rare cases when the main curvatures of the profiles are constant in the patch. In this case, it is possible to use Hertz’s theory: the contact is said to be elliptic and the load N, associated with ellipse El is calculated by considering the maximum penetration 6, of undeformed profiles with the help of Hertz’s formula: where 6,(l) is the penetration for a normal load of 1 N. There are numerical codes able to compute the solutions in the general case. One of them, that we have used, is the code “Contact” of Kalker [9]. This software is devoted to three dimensional elastic bodies in rolling contact and its assumptions are consistent with the cases being considered here. It is able to solve the normal pressure problem as well as the tangential problem. As a matter of fact, it would be nice to be able to use such codes for dynamical computations, but CPU times are much too high (several seconds for one case) and there would be no possibility of applying the codes for practical purposes. We have used it only as a basis of assessmentof simplifying solutions [6]. Calculating dynamics with elastic profiles, relative positions between wheels and rails are obtained at given times with undeformed profiles interpenetrating each other. From undeformed functions it is possible to get normal forces and thus to go on the integration process. Figure 2 shows a practical example of the undeformed distance between a wheel and a rail when the same load is produced either by the contact force calculated by “Contact” or by two equivalent ellipses.* It should be stated that the solution consisting, to sum up, of several elliptical forces, each of them being calculated as if this contact was alone, is in principle wrong because they are depending upon each other. The degree of uncertainty is measured by the difference between the two solutions presented: this difference is practically acceptable: to get the same force, it has been necessary to move the relative distance between the bodies *This example has been discussed in detail in the author’s paper of reference [ll].

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-I5

.,1

-,,-I

I

.,I

-10

4

.I

-I

4

-5

4

.I

-1

.I

0

I

1

3

I

5

6

,

I

0

10

II

LI

13

II

13

Fig, 2. Example of undeformed distance between profiles assuming a load uf 88 kN and comparison with two elliptical contacts calculated automatically.

by less than 10 micrometers; the penetrations are very close: 0.085 mm for “Contact” and Cl.075mm using the ellipses which means nearly the same stiffness. Assuming a sum of Hertzian contacts (eIIipticaI contact points) to represent the elastic contact, there is no difficulty in locating the centre of the ellipses and their eccentricity if the undeformed distance curve presents maxima (higher ellipse of Fig. 2), but it is more difficult if not (2nd ellipse of Fig. 2). These parameters have to be found according to different criteria, one of them being continuity. Figure 3 shows a comparison of the patch shapes for the same case as Fig. 2, and Fig. 4 presents, calculated by “Contact”, the lateral distribution of normal forces across the patch together with their resultant of which the location can be compared with that of the two elliptic contacts.

Fig. 3. Compariswn between the patch shapes at 88 kN and Y = 1 mm.

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Fig. 4. Lateral distribution of normal forces across the mid-line of the patch and resultants

2.2.

Tangential problem of elastic contacts

The validation of the assumption according to which a sum of ellipses can be equivalent to a non-Hertzian contact must be done for the complete system of contact forces including tangential forces. Given creep velocities, the software “Contact” allows us to calculate the tangential forces by a finite element discretization of the patch (rectangles). The inequations of Coulomb are being solved for the total discretized area, once the normal pressure is known, by a linearization method where equations are solved with Gauss elimination and coupled with Newton-Raphson iterations [9]. For the Hertzian cases, ‘Fastsim’ uses a discretization of the same rectangular type but finds an approximated solution by simple additions without iterations [8]. This is the reason why it is very fast. Figure 5 presents, still in the case of Fig. 2, a comparison of the two results allowing us to see the orientations of the creepages. The overall comparison seems good. The presentation of the comparison of amplitudes can be proposed in 3D diagrams such as Fig, 6, where normal and tangential stresses are plotted vertically on top of their own rectangle. From our point of view, the results assessthat this hypothesis is permissable. 3. SECOND HYPOTHESIS: IDENTIFICATION OF A SUM OF HERTZIAN CONTACTS BY ONLY ONE

The principles of this identification method have been explained previously in this paper and in more detail in the reference [4]. Let us say that, for every case where the contact forces system is known by the multi-Hertzian elastic calculation, an equivalent ellipse is searched on each wheel so as to find, using it alone, the same system of forces: The location of this equivalent ellipse (position and contact angle) is found through geometric considerations, according to the weights (loads) and locations of each of primary ellipses. Figure 7 explains the reduction process and Fig. 8 shows an example of reduction. Corresponding to each equivalent ellipse, are attached space parameters of which the function is to recall where the primary ellipses are located in the wheel and rail reference frames (in practice, to limit the size of numerical files, the location in the wheel reference frame is known by a whole number, from 1 to 100, indicating one out of 100 bands of equal surface discretized on the wheel profile; same system for the rails; low numbers are on the active sides) and what is their relative importance (no-dimensional factors in % of

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Fig. 5. Comparison of the directions of creepages found by finite elements (C) or by the sum of two elliptic cases (F).

which the sum is 100). These parameters are used during dynamical computations to distribute the contact forces and the frictional energy at correct places so as to be able to compute surface fatigue and wear of wheels and rails (see chapter 7). From this equivalent location, and using ‘Fastsim’ for calculating tangential forces of this equivalent contact, Kalker’s coefficients and the eccentricity of the equivalent contact can be evaluated so as to fight back the elastic forces of the identification case and so as to assume that, later on, the correct forces of new cases could be found by a simple and quick calculation. Let us say that it would be difficult, perhaps impossible, to come back to a geometrically equivalent profile from the parameters of this equivalent ellipse (we never attempted to do it). But, and this is the important fact, not only are these parameters continuous with respect to the lateral translation of the wheelset across the totality of the track, up to the derailment, but, provided that the axle load and the friction coefficient of the identification case not be too far from the application cases, the forces calculated with the help of the equivalent ellipse have been demonstrated to be close to those found by the most sophisticated available methods [6]. These equivalent parameters are stored in a file of which the entry is the relative lateral wheels&track position Y, (the usual amplitude of Y,, is of f20 mm in 1000 steps).

J. P. PASCAL

NORMALPRESSURE- Y=Omm "voco" “CONTACT”

1~

TANGENTIALFORCES - Y=Omm

j

‘VOCO’

‘CONTACT’

FX

FY

Fig. 6. Example of the approximation of a non-hertzian contact patch (right) by 2 hertzian ones (left).

Special attention must be paid here to the influence of the unavoidable wheelset-track roll angle that usually increases so much, when one wheel tends to climb over one rail, as to change the sign of the lateral velocity of the centre of gravity of the wheelset (situated about 0.5 m above the rails} while the flange of the wheel is continuing its lateral way. For this reason, the entry of the file cannot be the relative lateral wheelset-track position at the level of the centre of gravity of the wheelset. It must be taken at the rails level or even at any lower level. This mechanical property of railway wheelsets, that any attempt to derail needs a complete kinetic energy shifting from translation to roll angle rotation is quite specific and governs the dynamics of derailments.

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Fig. 7. Reduction process: the systems of forces (Nl, Tl) and (N2, T2) are equivalent to one system (N. T)where N has the same direction as the vector NI + N2 and is applied at a point located between the other two.

E

5.0

6.1

87.5

0.052

Fig. 8. Example of reduction of two ellipses to one.

4. THIEkD HYPOTHESIS: THE SPECIFIC WHEELSET-TRACK ROLL ANGLE @ CAN ALSO BE IDENTIFIED AS A FUNCTION

The above identification process is done during one dynamical computation of the elastic forces existing between the loaded wheelset and the track. Initial conditions are taken with the flange of left wheel on top of left rail. An outside lateral force is applied to the wheelset so as to force it on the right and is large enough so as to be able to get the derailing situation on the right rail after crossing the track dynamically. As said above, the stiffness of the elastic contacts is very high and high frequency modes can be excited during these computations with very small time steps (1 ~1. In order not to excite them, the system has been heavily damped and the displacement of the wheelset is

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made with a very small velocity (< 10 mm/s) in such a way that nearly no oscillations are created. Thus we get, with respect to the parameter Yb a continuous function for the relative roll angle and altitude (which are linked) and their first and second derivatives. These functions are important because they are used later on for the calculation of inertial forces during the contacts of the flange of the wheels with the rails. Indeed, the continuity of these derivatives will depend on the level chosen for Yb. If Yb were chosen very low, it would depend very much on the roll angle of which the derivatives would thus be quite continuous. But, in this case, the roll angle would be described with respect to itself and would not mean much. The optimal choice of Yb level will be discussed. Figure 9 presents an example of the roll angle function where a large second derivative is foreseen for (0 = 3 mrd (at Yb = 26 mm, curve A: at rail level, or 40 mm, curve B: if the displacement is measured 5 m under the rail level). This relative wheelset-track location is also the one from which the wheel begins to climb over the rail. # is the right parameter to evaluate the derailment; a limit of 8 mrd is thought significant. These functions describing the roll angle are also numerically recorded in the identification file. It is now assumed that, paying no attention to high frequency modes, these functions do not depend upon low frequency dynamical conditions and can be used to compute the inertial forces of this rigid wheelset in contact with this rigid track according us to usual algebro-differential methods of solving rigid joints. This assumption allows us to avoid the high elastic stiffness of the contact during dynamical computations and consequently to use large nominal time steps (around 0.1 ms). Small time steps will however be necessary, but only at times, mainly when the contacts will cross the locations where roll angles second derivatives are large. Let us try to summarize the identification process that we have described: it allows us to conduct fast dynamic computations of rigid wheelsets, solving dynamically a rigid wheelsettrack continuous joint, taking into account not only the elasticity of the bodies in contact, but also their lack of continuity and the non-Hertzian situations together with a realistic computation of non-linear tangential (creep) forces. As we said previously, and as will be assessedlater, this method was subjected to doubt until its results could be compared to the physical reality which is only described by measurements. Having described the contact itself, we shall now deal with solid mechanics.

5. DYNAMICAL

HYPOTHESIS:

TRACK DEFINITION AND INTEGRATIONS REFERENCE FRAME

INSIDE A FIXED

Testing railway vehicles along actual tracks is also, in some ways, testing the tracks, or, more precisely, measuring the answer of a given vehicle to its excitation by a combination of its own speed and of the “track geometry” on which it is running. Thus, answering questions about the vehicle itself, necessitates being able to distinguish in its calculated/ recorded behaviour, what is due to the vehicle, to the track and to their combinations. Without ignoring the complexity of the vehicle itself, most of the time located in non linear suspensions (dry frictions with clearances, bumpstops . . .) and in the wheel/rail contact (profiles, adhesion . . .), that is to say that the largest part of uncertainty comes from the track of which the parameters are actually the input of the whole system. If everybody agrees that it is necessary to take the track geometry into consideration, there are discussions as to the modelling of track dynamics and its necessary degree of complexity. It is impossible to clarify totally this matter here but, on the basis of our aim,

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Fig. 9. Example of the roll angle vs wheelset/track displacement (A = at rail level, I? - 5 m under the rails) in the case of a large gauge (1460 mm),

excluding such problems as noise or interactions with moving substructures as bridges, it is possible to assume, also on the basis of the limited frequency range of aboard instruments (< 50 Hz), that the longitudinal movements of the track (high frequency-small amplitude of rail-beam and sleepers modes. . .) do not really interfere with the vehicle dynamics (displacements at least 10 times larger) and, consequently that it is not necessary to take it into consideration. Special attention must be paid to the measuring technique of the vertical raising of the rails. Running loaded axles over the track, voids of some amplitude that often exist between sleepers and ballast, and that would not be compressed with unloaded measurements, are cancelled in the signals of raising so that this specific (quite damped by the ballast) track unstability is not seen by the vehicles otherwise than through the corresponding variation of raising. According to this, the track may be characterized vertically mainly by its geometry and secondarily by its dynamical behaviour. However, a degree of freedom of the track must be kept laterally to take into account the elasticity of the rails and of the pads (rubber cushions between rail and sleeper). This is necessary because, in some cases, with new wheel profiles cooperating with worn rails for instance, the contact with the flange can be very brutal and must be calculated as a percussion (lateral forces reaching 3 times the vertical load). Thus, at each wheelset of a train, is associated, in the model, a solid “piece of track” having a mass* and being visco-elastically laterally connected to the ground. In most practical calculations, this system is over damped and does not introduce high frequencies in the force answer. At least three different types of datas are necessary to characterize continuously the track geometry from the vehicle point of view: l Layouts (horizontal and vertical) I Irregularities (lateral, vertical, cant) l Rail profiles and Gauge

5.1.

Calculation

of jbrces and their integration

in a fixed reference frame

One of the most difficult problems to be solved when trying to integrate the forces applied to a railway vehicle, is the problem of, in order to deal with relatively small displacements, having to write the equations in moving reference frames generally attached *Normally, this mass represents 1 or 2 sleepers I- 2 meters of rails and ran es from 300 to 500 kg. The stiffness of its link to the ground is from I to SE8 N/ m and the damping equals SE5 N f m/s; rhe associated frequency ranges from 80 to 200 Ha.

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to the centre of the vehicle. As the line definition is given numerically with sharp variations, these reference frames are highly accelerated and the integration process often becomes difficult. To avoid this difficulty we have generalized the following considerations: l

l

the curvature of a circle is a constant and its double integration with respect to the arc length yields a conical (parabola); it is known that the inertial forces that are necessary to force a mass to follow this parabola with a velocity of which the projection over the abscissa axis is constant, have the constant direction of the ordinate axis and just the same amplitude as the centric forces that are necessary to force the same mass to follow the circle at the same velocity.

This property is general: there is always a transformation of the actual space where actual centrifugal forces acting on a mass following a curve at one speed can be calculated according to Newton’s principles in a new space where they have a constant direction because the curve is transformed. This space, of which the x axis measures the arc length, has other very interesting properties [lo]. This new space is obtained by plotting, in a fixed Cartesian reference, as ordinate Y* the results of a double integration of the curvatures C of the actual layout with respect to the arc length s which is plotted on the abscissa axis: --

Y* =

Cds, c = 2*f/(a*a), 2. a = chord length. f = sag, JJ This space has other very interesting properties according to which all relative lateral displacements in the moving references are now calculated along the ordinate axis and their amplitudes are in direct agreement with the measurements made in the moving reference of the vehicle (first order approximation). The transformation can be done for both horizontal and vertical layouts and is used by the Voco software to perform all its integrations in a fixed reference frame. The advantages are that all equations are written in the fixed reference frame as if the track would always be straight, and afterwards, integrations are made without having to consider trigonometric functions nor transfer matrixes between reference frames nor accelerated frames.? Furthermore, usual recording means for track geometry give the sags f of the rails in the middle of straight chords (around 10 meters long). These sags can be directly interpreted as a measurement of the curvatures of the tracks so that the method can be applied without requiring other specific information. The double integration of these curvatures yields the ‘layouts’ of the track but, due to the length of the measuring chords, does not contain short wave irregularities (l-20 m) that must be added. To this purpose, the same sags are now added as geometric irregularities perpendicular to the moving chord.* 5.2.

Test track definition

Three kilometers of an SNCF line were isolated for the tests. It consists, after a short straight line of 200 m, of 4 curves of 500 m radii, alternatively left and right handed, with an average cant of 150 mm (0.1 radian) and large variations of the gauge (up to 1465 mm). It ends with a straight line of 500 m. Rail profiles have been measured and found very tNumerica1 algorithms are used in order to improve the precision, by directly using the integrated increments to enter the identified file without having to make differences of large reals, *There is quite a lot of work being done at present, in Germany and in France, on the best way to make these additions either with Fourier techniques and filters or with purely numerical calculations. Let us say that we’re using here a mixture of the two solutions.

Modelling of unstable railway wagons

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different for high and low rails. Consequently, the code had to take into account gauge and rail profiles variations. Figure 10 gives the ‘irregularities’ of that section. Figure 11 shows the “layouts” which are made of the results of the double integration of sag-curvatures with respect to the arc length both in horizontal and vertical planes. Attention should be paid to the fixed reference of these signals which can be of large amplitudes (for instance 1200 m for the lateral signal).

Lining

hl,

’

Gauge Variations

Raising

Cant (quick changes)

Fig. 10. Geometric irregularities of the test track.

.I. P. PASCAL

Horizontal

,;= n

Integration

orientation

of the track

of the horizontal orientation

Vertical orientation of the track

Integration

of the vertical orientation I

_...__..._. . ..-...--. ,, I ,, -.----_._-.--.--.. ..-.....- --.-

I.._..-.--

Fig. 11. Horiwnta~ and vertical layours of the test track

6. EVALUATION

OF THE WHEELSET-TRACK

JOINT

It can help to present the shape of the equations that allow us to solve the problem of the rigid joint in order to understand the specificity of the railway wheelset and its difficulty of modelling. The constraint yields algebraic equations which are non linear due to the links between normal and tangential forces. At each time step, these equations must be solved in order to find the contact forces before integration. At times, the algebraic equations give a negative normal force for one wheel, which means that the contact is no more possible on this wheel and that the system of equations must be changed because a new degree of freedom is introduced and has to be integrated. Indeed, the initial conditions at the very beginning of the lift off (wheelset roll angle velocity for instance), are taken from the resuits of the previous time step and shall determine strongly the following results. Consequently, as most of the time, the occasion when a wheel lifts off coincides with a difficult numerical transition, numerical approximations arc not averaged

Modelling of unstable railway wagons

1741

in this case and should be minimized because there lies a source of numerical chaos. In order to improve the evaluation, we have tested different changes of the variable to be integrated and we found that a mixture of the wheelset CoG lateral displacement plus a part of its roll angle (equations (20)-(23)) gives good results. lateral relative displacement wheelset-track at rail or lower level sign of yb yb derivatives vs time lateral relative displacement wheelset-track at COG level y, derivatives vs time lateral acceleration of wheelset CdG, Galilean frame lateral acceleration of track, Galilean frame vertical relative displacement wheelset-track above wheelset COG i,, derivatives vs time relative roll angle of jointed wheelset in the track frame d#e/dy h d’&/dyb” wheelset roll angle in the Galilean frame d0,/d t d”@,/dt2 vertical wheelset displacement, Galilean frame vs time derivatives roll angle and derivatives of track roll angle (cant) vertical displacement and derivatives of track centre contact angles in track frame normal forces calculated at previous time step tangential forces calculated at previous time step unknown normal forces unknown tangential forces component of contact forces normal to track as applied to wheels component of contact forces tangent to track as applied to wheels lateral forces applied to wheelset by suspensions vertical forces applied to wheelset by suspensions lateral forces aplied to track by the ground torque (around ox) applied to wheelset by its suspensions wheelset mass and inertia (around ox) track equivalent mass nominal rolling radius (in central position) Rolling radius on the High wheel Rolling radius on the Low wheel semi gauge lateral distance from wheelset COG to equivalent contact points. Note: all these parameters are algebraic reals: radii are positive.

6.1.

Analytical

culculutiun

of contact

forces and of wheelset accelerations

Roll angles are assumed to remain small so as to validate first order approximations. Reference frame orientation is as presented in Fig. 1.

J. P. PASCAL

1742

Deriving

(I?

yieIds

ca

with:

(3) (4)

also

(5) (6)

and. still first order

(7)

deriving

@I (9) (10)

Equations (9) and (lo), associated to (3) and (4): are the expressions of the joint. To understand them, it is necessary to give expressions with respect to PCwhich is to be integrated. At first we replace 3,:

(11) I.

#l

‘e = (1 - R. * #;)

* CL’,

To simplify the equations we assume: Da =

4:

4i

and Db =

K

(1 - Ro - #;j + (1 - R, * #;)3 and from (4) and (6) we derive simple expressions of the joint:

-(j,

+ RO+J’

(13)

(1 - Ro - #2

6e = Da-J,

+ Db

z:, = Da-YRB-J,

(14) + Db*YRB

+ i’, - 3,.

(13

We observe that (1 - R. * $i) at the denominator of (13j becomes null for a value of #i close to +2? and, as this parameter reaches 2.5 or 3 when the contact point is on the flange of the wheel, the undetermination will arise twice during the climb of the wheel. This clearly shows why the forces will sometimes be difficult to compute.

6.2.

Projection of contact forces on the track frame

To get first approach to the solution and spare iterations, we linearize the tangential forces known from previous time step CH = TH NH which allows us to write:

CB = TB NB

Modelling of unstable railway wagons

1743

with: A = (COS(Y~) + CN -sin),

B = (cos (ye) - CB * sin(yB))

(18)

C = (CH *cos(yff) - sin(

D = (-(CB*cosy,)

(19)

+ sin(ye)).

We have developed three codes A, B, C with three different systems of equations according to the level at which the unknown relative distance wheelset-track is measured: at COG level, unknown is ji,; at rail level, unknown is jjI; at a distance h under rail level, unknown is j;,. This third case, more general, which has given the best results for h = 5 m, is detailed below:

6.3.

of equations with jj, as unknown

Code C-system

we define:

Y, = y, + h.Qc

= y, + (R, + h).@,

with :

= y, + b.Qc

(20)

b=R,+h

(21)

we define also:

yb

b.A

(22)

and then:

y, = y, + b * @‘e- h * &.

(23)

=

yb

-

Mathematically, h is a positive constant which has the dimension of length and must be chosen so as to minimize the numerical difficulties associated with shocks. There is now an additional coupling, which did not exist with y, at the rail level, between the rolling of the track & and the new variable yb. Three modes are possible: the Mode 0 accounts for the joint, wheelset and track being in contact on both sides (yb and @‘eare linked), the Mode 1 accounts for a lift of the high wheel (a), becomes a dof) and the Mode 2 for a lift of the low wheel (ap, becomes also a dof). Each time we have 3 unknowns: NH, NB, j;b, or &e, NB, jjb, or NH, &;,, j;b and 3 Newton’s equations for the 3 degrees of freedom y, z and @. Mode 0 OC-N, (H/zEx) (A/ME). OC=

+ PC.NB

* NH + (K/k!%)

NH + (B/ME)*

(NH and N, > 0) - j,

PC=

- NB - c#&* L’,, = #; - jb2 + 3, - CXSUS/ZEX

F+b*K/ZEX

Mode 1 -&;,

+ F.Ns + (K/ZEX).

- YRB. 6t;, + (B/ME)

- jib =-G

+ b-6,

(H/ZEX) (A/ME).

(25) ((26)

+ i’, + FZSES/ME

QC = G + be CXSUSjZEX

- he I$,,

(NH = 0) h+,

(27)

NB = -CXSUS/ZEX

(28)

-+

* Ns = -4,.

Mode 2 E-N,

(24)

Ns - 4;. YRB* j;, = &jbzYRB

E + b*H/ZEX

b-6,,

= -QC

Sy * E0 + i’,, f FZSESIME

(29)

(NB = 0)

- L’b =-G

+ h-4,

(30)

. NH - &;, = -CXSlJS/ZEX

NI, - YR, .6;, - Mm ji, = FZSESIME RC =2*Sy.

(31) + i’, + 6,. Sy . E, + RC

E0.@;‘;,.~h2.

(32)

J. P. PASCAL

I744

6.4.

Evaluarion

of the solutions of thhissystem

One of the above algebraic systems, chosen according to the level of normal forces previously found, is solved at each time step to find new normal forces and accelerations after a few iterations, using Fastsim, to calculate tangential forces, then the next integration step can be done. Algebra-differential problematic couplings are mainly due to (& * jh2). Where @; is found in the identification file as a function of yb. This displacement and Jib are found by integration of ji,. Taking into account the nature of the profiles, this numerical process often becomes unstable and has to be stabilized by numerical adjustments of the time steps and of the integration algorithm. These adjustments, based on monitoring numerically the wheelset-track relative accelerations and jerks have taken many hours before being able to compute long distances without having to stop because of numerical unstabilities. At this stage? hundreds of different wheelset and track cases have been identified and corresponding dynamical computations have been done without any more difficulties. But, whatever were the numerical improvements, it has never been possible, using Fortran double precision norm, to find identical solutions with two different computers for an unstable wagon for durations longer than 40 seconds, i.e. after about 400,000 time steps. After this time, the solutions are similar but not quite the same. Looking for the origin of these differences, we found, between computers, very small differences (last digit) in the results of transcendental functions which initiate large differences in the solutions. Our conclusion is that this quite non-linear mechanism is so sensitive to initial conditions that it must be said to be chaotic. 7. EXAMPLE OF APPLICATION: RESEARCH OF DERAILING CONDITIONS OF A SHORT 2 AXLES WAGON

There are tens of thousands of these ‘UIC’ wagons running on European tracks with a large range of different behaviours. Some of them, especially when empty, are not stable in particuIar conditions which should be known as precisely as possible in order to decrease the probability of derailments. These conditions deal of both vehicle and track characteristics. After a succession of derailments of these wagons at medium speed, it was decided to undertake tests in order to investigate the influence of parameters. After the derailments, no characteristics of the wagons could be found out of norm and the attention was at first brought to the track and mainly to the gauge. With some of these vehicles extracted from the commercial rolling stock, tests on different tracks did not demonstrate dangerous behaviours and the research of new conditions looked hazardous and nearly impossible. Wagons were of the G69 type. 5.7 m between axles, with the normalized UK dry friction suspension system, of which the damping is due to the friction of the cylindrical parts of suspension rings over steel saddles of same diameter. Modelling their dynamics with Voco opened the range of conditions that could be “tested” and led to the assumption that dangerous behaviours of empty wagons could occur if two conditions would be met simultaneously: an over gauge of the track (leading, with thin flanges of worn wheels, to a lateral gap of the wheelset greater than 30 mm from flange contact to flange contact) and a special shape of the rings/saddles assembly allowing some transversal deflexion of the car body before friction would occur (this is possible mainly with new pieces when the ring is not quite cylindrical and can roll over a saddle of larger diameter without friction, i.e. without damping).

Modelling

1745

of unstable railway wagons

Under these conditions, the code predicted that the wagon was unstable at 80 km/h with a specific mode according to which it was possible to observe large variations of the vertical load of the wheels and, at times, the climbing of a wheel over the rail. Figure 12 shows one of these calculated quasi-derailments characterized by a sudden increase of the wheelsettrack roll angle (here 7 mrd at time 51.08 s) that happens 0.2 s after a complete unloading of that wheel normally loaded at 30 kN. Of course, it has been necessary to verify these predictions by operating actual tests at the SNCF so that a wagon was prepared with new suspension rings and a test site was chosen (see definition at chapter 5) [Ill. The wheelsets of this wagon, of which the profiles were machined according to the profiles found on one derailed wagon,* were equipped with strain gauges on the wheels according to the SNCF method for measuring two components, Y and Q, of Ihe contact forces: Y is their projection over the transverse axis parallel to the wheelset axle and Q is their projection over an upwards perpendicular axis. Longitudinal components are not measured. Front wheels are numbered 11 and 12 (left and right) and rear wheels 21 and 22: Q11 is the vertical contact force of the left wheel of the front axle. Other sensors (displacement across suspensions and accelerometers in the car body) were installed in orde,r to be able to analyse completely the dynamics. Many runs were performed and it appeared immediately, as predicted, that this wagon was unstable in the curves at usual speeds (70-180 km/h) and that the unstable mode (around 1.3 Hz) was the same as described by the computations. Figure 13 shows that these unstable movements (as predicted by the theories of chaotic non linear systems) are not deterministic which means that it is not possible to get exactly twice the same time histories of the signals. Computation results, if signals are considered at one place and one time, are also very sensitive to initial conditions and even to integration algorithms so that, as for the tests, they cannot be said deterministic [12]. Due to this ‘chaotic’ behaviour, it is not possible to predict and produce an accident like a derailment that can occur at the nth passing at the same place, at same speed, same adhesion. But it is possible to observe in the computation results. various cases when the

so.00 Ibf

m.20

50.40

50.61) m.m

Jl 00

5iza

51.W

sl.m

Fig. 12. Example of climbing of left front wheel next to derailment (7 mrd). *This

profile,

is well inside normc

and

accepted as ‘usual’ on SNCF wagons.

T2.w

1746

J. P. PASCAL

Fig. 13. Comparison

of recorded measured signals for 2 successive tests that should be identical.

wheels did not derail but were not so far and, by this way, to approach an estimation of the risk. One test run (M14: 75 km/h, dry rails) has been chosen as a basis for comparisons with one computation (C200). Taking the line into account, this speed of 75 km/h gives practically no cant deficiency (centrifugal forces) and was chosen because the wagon was particularly unstable. Measured and computed signals have been post-processed on the same basis (discretized 200 times/second and filtered) and stochastical values (such as standard deviations or maximums at 99.85%) have been at first compared and found inside each other probability margins. Figure 14 shows side by side (same scales) measured and computed signals of the four ‘vertical’ contact forces Q over the 3 km of the section. Figure 15 gives the same comparison for Y forces. This global comparison allows to assess that the model is able to describe the main mechanisms of this vehicle. A clear correlation can be observed between the amplitude and the frequency of the forces and the location of the vehicle with respect to the gauge variations and to the curves, this proves that the model must take these parameters into

Fig. 14. Comparison

of vertical contact forces measured (M14) and calculated (code A).

1747

Modelling of unstable railway wagons

Ml4 I

1,

2.

-

=.

.:a

;;:

.:

21

I

:..

:I<

.a:

:I

3,

:&.

=.

:Ij

C200 :~

,.I

/.,

Fig. 15. Comparison of lateral contact forces measured (M14) and calculated (C200).

account. Notwithstanding the fact that it is usually thought difficult to predict the level of contact forces of such an unstable wagon, especially for lateral forces, there is a good agreement between the measured and calculated amplitudes. This comes to support the main hypothesis and simplifications of the modelling as well for the treatment of the contact as for the dynamics of the track. Figure 16 shows the comparison of contact forces in more details with an expansion of the time axis. Computed signals have been filtered (low-pass) at 10 and 40 Hz to give a better idea of the frequency content. Measured signals are filtered at 20 Hz. The good agreement between the model and the actual vehicle forces shows that the model can be used to study the influence of different parameters. But, as the frequency bandwidth of the measuring system is limited, it cannot be proven only by their similarity that both calculations and measurements are describing completely the actual phenomenon (considering, for instance, the obvious connections between noise emission and rolling contact, it must be stated that their mechanisms are not included here). But it can be observed that nearly alI the information (energy content) of the vertical force is kept by the 10 Hz signal; this seems to demonstrate that high frequency vertical components (attached to rails and wheels natural modes), that are not taken into account in the model nor in the measurements, have not enough energy to significantly influence the contact; if they did, the strongly nonlinear mechanisms of rolling friction would be modified and the low frequency behaviour would also be affected. 7.1. Influence of the wear of the profiles In front of such chaotic oscillations, we tried to investigate if the model could give some information about wear sensitivity to oscillations and, in turn, about the sensitivity of the wagon stability to the modifications of the profiles due to wear. Two simulations are possible: either the evolution of the rail profiles and their surface fatigue on one specific line ran on by different known vehicles having known wheel

J. I’. PASCAL

Vertical Force Q2 I

sum Y21+ Y22

Vertical Force Q22

Lateral Force Y22

Fig. 16. Comparison of measured (M14) and calculated (CZnO)contact forces.

profiles, or a kind of equivalent numerical test line can be designed and operated with a selection of different known rail profiles in order to represent an equivalence for the life of one vehicle at different speeds on the SNCF network. This second possibility is now being developed and some results are presented. Starting from wheels having the standard ‘SNCF profile’, assuming that wear is mainly produced in curves and that 4 rail profiles could be sufficient, different measured rail profiles are tested, the results are assessedcomparing the calculated wheel profiles to actual ones. At this stage, the results tend to show that the rail profiles of this equivalent line should use no WC60 definition but profiles of rails having between 4 and 10 years on duty in medium curves (500 < R < 800 m, found on UIC lines No. 2 or 3). The cant deficiency should not be maximum but slightly over zero. To this

Modelling of unstablerailway wagons

1749

effect, the frictional energy spent at contacts and its locations (distribution according to the identification factors, see Chapter 3) on the profiles is calculated during dynamical computations at each time step, summed up on each of the discretized zones of the profiles and stored at the end of a run over the track section of 3 km. Assuming a constant factor between energy and a destroyed volume of steel, the profiles of the wheels are reconstructed and an identification process is done again before a new dynamical computation on the same track section. The complete process is said to be one iteration (CPU time -5 hours). According to the published numerical values of the wear factor (1.6 lo-l6 m3/J) [13j, the depth of wear corresponding to these 3 km would be in the micrometer range and the total wear process would last years. In order to increase the efficiency of the process, it is accepted to withdraw at each iteration a maximum of 0.1 mm on the most solicited band and to calculate accordingly the distance covered during this iteration. The complete wear of a profile. as shown on Fig. 201 necessitates 100 iterations and corresponds from 200, to 800,000 km according to the damping of the wagon, the speed. Presenting the results of this study is not easy because of the large amount of information available (dynamics of the vehicle, wears), etc. which needs to be summarized. Figure 17 presents a 3D view of the wheel profiles deformations during the first 15 iterations. In order to improve the understanding of the process, the deformations have been amplified 10 times and plotted over the new profile. Figure 19 shows the results of 100 iterations (amplifying factor: 5). Figure 20 is a 2D view of all 100 profiles (factor 5)

Fig. L7. 3D Presentation

of calculated deformations of a new profile due to frictional (deformations are amplified 10 times).

wear-(15

iterations)

.I. P. PASCAL

1750

F3.g.18. Three-dimensional presentation of calculated deformations of a new profile due to frictional wear-,

(100 iterations) (deformations are amplified 5 times).

Fig.

10. Two-dimensional presentation of profiles deformations same cast: as Fig. 18.

Modelling of unstable railway wagons

(mm) 20

T

(mm)

( (mm)

Ex,

EX

j

: (mm)

Ex

It. n” 75

! (mm)

Fig. 20. Two-dimensional presentation of calculated wheel profiles at different stages of wear. Ex = experimental profile.

that allows us to see oscillating functions. Figure 21 shows that, at the end, the calculated profile is close to the experimental one. Figure 21 presents, in kJ, the average frictionnal energy per wheel versus iteration number. Curve A is obtained for a well damped wagon. At iteration 84, curve B, the damping is decreased. It is interesting to notice that the awaited trend to instability, that is measured by the amount of energy, does not appear immediately, but after two to five iterations. This (rather quick and sensitive) coupling proposed by the model between wear and stability has been noticed several times. This advocates to be very careful before drawing conclusions from a unique set of profiles.

J. P. PASCAL

80

Fig. 21. Calculated

frictional

8,

82

83

84

85

86

87

68

energy depending on iteration number and damping damping, B = poor damping.

89

90

of the wagon.

A = normal

8. CONCLUSIONS

We have presented in this paper an overall view of physical and numerical assumptions, based on the sentence “where there are no reliable measurements there will be no reliable numerical solutions”, in order to build dynamical models allowing, with acceptable CPU times, a good representation of the contact forces of unstable railway vehicles running over actual tracks with curves and irregularities. We have presented results and compared them satisfactorily to experimental measurements made by the SNCF, either on a stochastical basis or on a time history basis. So, we believe that the set of hypotheses presented are consistent with themselves and with the facts, which means that none of the approximations made is too far from physical reality. The chaotic behaviour of these vehicles has been proven experimentally (impossible to get twice the same signals) and numerically. This “chaos” is thought to be mainly linked with the specificity of the railway wheelset according to which it cannot derail, in principle, without transferring its lateral kinetic energy into a rotational one, and this is also numerically difficult to deal with. Thus, we believe that chaos and numerical difficulties are Iinked and for this reason it can be impossible to predict a chaotic behaviour only from computations when the physical phenomenon is not known. Acknowledgement-The author wishes to thank the SNCF for the help that its engineers from the rolling-stock department and from the track department gave to this work which would not have been possible without their cooperation and the great quality of the information they put at his disposal.

RJWE.RENCES I. J. J. Kalker, ‘Three-Dimensional Elastic Bodies in Rolling Contact’, 1st Edn. Kluwer Academic Publishers, Dordrecht (1990). 2. P. Aknin, J. l3. Ayasse, H. Chollet, J. L. Maupu, A. Gautier and M. Paradinas, ‘Quasi-static derailment of a railway vehicle-Comparison between experimental and simulation results’ XIII LAVSD Symposium. Chengdu, China (August 1993).

Modelling of unstable railway wagons 3. Groupe de l&wail Roue-Rail LIB-SNCF, Compatahilitk Roue-Rail. DB, Frankfurt, SNCF, Paris (Nov. 1987).

en Trafic International

1753 ri Grands Vifesse dcs ParamPtres

4. J. P. Pascal and G. Sauvage, ,‘New Method for reducing the Multicontact Wheel/rail Prohlem to one equivalent Rigid Contact Patch’, Proceedings 12th IAVSD-Symposium, Lyon (August 26-10, 1991). S. U’. Kik and G. MGller, ‘Comparison of the Behaviour of different Wheelset-Track Models,’ Proceeding 12th IAVSD-Symposium, Lyon (August X-30, 1991). 6. J. P. Pascal ‘The multi-Hertzian-Contact Hypothesis and equivalent conicity in fhc case of 51(x12 and UIC‘hU analytical wheel/rail profiles’. Vehicle System Dynamics 22(Z), 57-78. Swets 8 Zeitlingrer, Lisse (1993). 7. J. P. Pascal and Cr. Sauvage, ‘The available methods to calculate the Wheel/Rail Forces in Non-&z&an Patches and Rail Damaging’, State of the Art paper of the 13th IAVSD Symposium, in VSD. pp. 263. 275 Swcts & Zcitlmgrer, Lisse (May/July 1X)3). X. J J. Kalker. ‘A fast Algorithm for the Simplified Theory of Rolling Contact’, Vrhicle Jj~stern &w~naic.~ II, pp, l-13. Swets & Zeitlingrer, Lisse (1982). 9. J. J. Kaikcr, ‘CDN:PU+@User5 .titinual I990. TU, Delft (lQ!XI). 10. J B. Ay;isnc and 1. P. Pascal, ‘Dynamique ferroviaire en cnurbe-Codes de calcul ‘VOCO ‘Rapparr /;~&575-L.7’1%“,Arcueil, (?rfai 1990). 11. J Courtin. Wagon G69 & essieux Validation du code de calcul ‘Vcxo’ Rappurr d’essais MEP2 RN %W27/9.3 PZ-I, SNCF Direction du Mattriel Departetnent des Essais et des Laboratoires, (Octobre 1993). 12. J P. Pascal er ul. ‘Calcul dynamique par Voco des forces du COnFaCt roue/rail-Validation par les essais en ligne d’un wagon a essicux tcstt par la SNCF entre Hirson at Charleville’ Hopport INRETS No. 16Y Arcueil (19Y3). 13. .I J. Kalker. ‘Simulation of the development of a railway wheel profile through wear‘ Wear, 150. 35-365 (IUSlj. References 131,ill] and 181.[lOI are available from the following addrescs: 3. 11. SNCFME 15 rue Traver&ire F-75012 Paris 8. 10 [?IRETS/LTN BP 34 F-94114 Arcueil Cedex