Proc IMechE Part F: Identification of a wheel–rail adhesion J Rail … · 2017. 2. 12. · Original Article Identification of a wheel–rail adhesion coefficient from experimental

Original Article

Identification of a wheel–rail adhesioncoefficient from experimental data duringbraking tests

Monica Malvezzi1, Luca Pugi2, Susanna Papini2, Andrea Rindi2

and Paolo Toni2

Abstract

The forces that occur in the wheel–rail interface significantly affect vehicle dynamics, especially in the longitudinal

direction. Conventionally, the tangential component of the force exchanged between the rail and the wheel is expressed

as the product of the normal component of the force, and the so-called adhesion coefficient. This ratio depends on

several parameters that are usually summarized in the term ‘adhesion conditions’. When the adhesion conditions are

degraded (for example, in cases of rain, fog, ice, dead leaves, etc.), and the vehicle is accelerating or braking, pure rolling

conditions between the wheels and the rails do not hold any more, and macroscopic sliding occurs on one or more of

the wheels. The aim of this work is to identify a relationship between adhesion coefficient and some parameters, namely

wheel sliding and train speed, starting from a set of experimental measurements, obtained from test runs conducted with

artificially degraded adhesion conditions.

Keywords

Wheel–rail adhesion, train braking, neural network models, longitudinal dynamics

Date received: 31 January 2012; accepted: 27 June 2012

Introduction

The contact between wheels and rails is very import-ant in the dynamic behaviour of railways vehicles, andhas been studied since the very beginning of railways.The resultant of the pressure distribution that arises inthe contact patch can be represented by a force and aresultant moment (spin). The resulting force can beexpressed as the vectorial sum of one component tan-gent to the wheel in the contact point and anothercomponent normal to the wheel profile in the samepoint. By increasing the tangential force starting fromzero, and fixing the remaining relevant parameters,two different phases can be distinguished.

The first one, named pseudo-sliding, mainly dependson elastic deformation of the two contact bodies. Inthis case, the contact area can be divided into twozones, in the first zone, relative slip between thebodies occurs, while the second zone is characterizedby the adhesion between contact surfaces.1,2 The over-all size of the contact patch depends substantially onthe normal component of the contact force, while therelative dimensions between sliding and adhesion areasdepend on the ratio between the tangential and normalcomponents of the contact forces. The tangential com-ponent of the force exchanged between the wheel andthe rail is given by the integral of the tangential stresses

that arise in the slip zone. With growing longitudinalforces the adhesion zone decreases up to a limit situ-ation where sliding occurs on the whole contact area.This is the second phase, that may occur during brak-ing and traction operations and when adhesionbetween the wheel and the rail is poor. In such situ-ations, the adhesion coefficient is a function of severalparameters (normal load, sliding speed, temperature ofthe two bodies, contact geometry, weather conditions,presence of rain, snow, dead leaves, etc.), and thedependency on some of them may not be easilyexpressed analytically.

The adhesion coefficient �, conventionally definedas the ratio between the tangential and the normalcomponent of the force exchanged between the wheeland the rail (respectively T and N), is expressed as

� ¼ TN

ð1Þ

1Department of Information Engineering, University of Siena, Italy2MDM Lab., Energy Department, University of Florence, Firenze, Italy

Corresponding author:

Monica Malvezzi, Department of Information Engineering, University of

Siena, via Roma 56, Siena, 53100, Italy.

Email: [email protected]

Proc IMechE Part F:

J Rail and Rapid Transit

0(0) 1–12

! IMechE 2012

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DOI: 10.1177/0954409712458490

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The coefficient � depends on several uncontrollableand difficult to quantify factors, and the research ofa law that relates it to other directly measurable vari-ables (train speed, wheel velocity, etc.) is a difficulttask. Some experimental identification procedures ofwheel–rail contact forces are presented in the litera-ture.3–10 On the other hand, the definition of a wheel–rail interaction model is needed for the realization of a‘real-time’ realistic simulator of the dynamic of a rail-way vehicle, which will be used to test in laboratory‘runs’ (by means of properly designed test rigs) thebehaviour of on-board devices that are part ofadvanced train control and monitoring systems, thusreducing the cost of line tests, such as automatic trainprotection and automatic train control (ATP/ATC)on-board elements, wheel slide protection (WSP) sys-tems, anti-slip devices and so on.11 More complex testrigs are necessary to test the interaction between dif-ferent on-board subsystems also in critical conditions:in particular virtual test runs with degraded adhesionconditions may be used for hardware-in-the-loop test-ing of the complex interactions arising between differ-ent on-board subsystems on scaled12 or full-scaleroller rigs.13

A quantitative and reliable model of the wheel–railinteraction forces is needed also to design and developinnovative devices (for example, WSP and anti-skiddevices), in particular, to test, by means of softwaresimulations, the behaviour of innovative control stra-tegies. In this paper, the identification of the wheel–rail adhesion coefficient starting from experimentalmeasurements is described. In particular, researchhas been performed with the goal of obtaining a reli-able and computationally efficient model of thewheel–rail contact to be used in a hardware-in-the-loop simulator of the dynamics of a railway vehicle,devoted to test WSP and odometry systems.

The paper is organized as follows: in the next sec-tion , a brief introduction to wheel–rail interactionsduring degraded adhesion tests is presented. Then, theexperimental data and corresponding vehicle modelare described; a simplified dynamic model adoptedto identify adhesion coefficient from experimentaldata is presented; the proposed identification modelbased on feed-forward neural networks is presented;the results of the training and validation on availableexperimental data are shown; and finally the resultsobtained are discussed.

Problem description

The adhesion coefficient, introduced in the precedingsection, depends on a large number of variables:environmental conditions (e.g. presence of water,snow, ice on the rail surface), materials, temperature,etc. Due to the complexity of the phenomenon, withconstant environmental conditions, it is commonlyaccepted to express adhesion as a function of slidingspeed, �v, defined as the difference between the

vehicle speed and the tangential velocity of thewheel at the point of contact,6,14,15 defined as

�v ¼ v� r! ð2Þ

in a braking phase, and

�v ¼ r!� v ð3Þ

in a traction phase. where v is the vehicle speed, ! isthe angular wheel velocity and r is the wheel contactradius.

As will be explained in the following subsection, inthe so-called pseudo-slip or creepage zone, i.e. the partof the adhesion curve where the sliding between thewheel and the rail is small, and the elastic deform-ations of the bodies determine the entity of the tan-gential force exchanged, the adhesion coefficientdepends on the relative slip,1,2 given by the followingexpression

�v ¼ v� r!v

ð4Þ

in a braking phase, and

�v ¼ r!� vr!

ð5Þ

in a traction phase.

Wheel–rail contact forces in rolling conditions

A complete theoretical study of the adhesion in themicro-slip zone is described in by Kalker1 andJohnson,2 but its results cannot be used directly toobtain the wheel–rail model to be used for hardware inthe loop (HIL) simulation of safety relevant subsystemslike WSP systems, mainly for the following reasons:

. the accurate models described in previously citedstudies are not adequate to simulate macroscopicsliding occurring with degraded adhesionconditions;

. considering real-time requirements for hardware-in-the-loop testing and the necessity of the simula-tion of many axles/contact points (at least four)involve the availability of limited computationalresources and the preference for algorithms thatcan be solved in a known deterministic time;

. finally, for the application we are interested in, themodel of the vehicle is planar, so tri-dimensionalmodels considering multi-contact patches are toosophisticated in this case.

Given the magnitude of the tangential and thenormal component of the contact force, numericalprocedures are available1 that evaluate the shapeand the dimensions of the contact area and subdivide

2 Proc IMechE Part F: J Rail and Rapid Transit 0(0)



it into the sliding and the adhesion zone.16 This pro-cedure is used for the analysis of the phenomena thatoccur in the contact area between two elastic bodies,which are rolling with respect to the other.

For small values of creepage, the relationshipbetween the creepage and the creep force can be con-sidered linear, so longitudinal Fx and lateral Fy forcesand the moment Mz can be calculated according toKalker theory

Fx ¼ �f11�

Fy ¼ �f22�� f23�

Mz ¼ f23�� f33�

ð6Þ

where f11, f22, f33, f23 are the linear creep coefficients,depending on the contact ellipse semi-axis and on thematerial properties, their values are tabulated and canbe found, for example, in Kalker.1 The coefficients �, �and � represent, respectively, the longitudinal,lateral and spin creepage components, and aredefined as17

� ¼ vrc � irvow

� ¼ vrc � trvow

� ¼ :rw � nrvow

ð7Þ

In the above expressions, vrc is the wheel speed atthe contact point, ir is the unit vector that identifiesthe rail longitudinal direction, tr is the unit vector tan-gent to the contact surface and orthogonal to ir (itidentifies the lateral direction), nr is the unit vectornormal to the contact surfaces in the contact point,vow is the magnitude of the velocity of the wheelsetcentre of mass.

The tangential forces evaluated with this linearmodel have to be saturated, in order to approximatea limited known friction factor �. The saturation ofcreep forces is performed according to the modifiedJonshson–Vermeulen formulation,17 by defining acoefficient ks as

ks¼

�N

FR

FR

�N

� ��19

FR

�N

� �2� 127

FR

�N

� �3" #for FR43�N

�N

FRfor FR43�N

8>>>><>>>>:

ð8Þ

where � is the wheel–rail friction factor, N is the

normal component of the force, FR ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2x þ F2y

qis

the total creep force and Fx and Fy are the creep

forces in the x and y directions, respectively.

Wheel–rail contact forces when sliding is present

When the full contact zone is sliding, pure rollingconditions do not hold any more, and macroscopicsliding between the wheel and the rail occurs. In thisphase, the adhesion coefficient depends on the abso-lute slip, defined in equation (4). The relationshipbetween the adhesion coefficient � and the absolutesliding �w was investigated by many researchersbecause its behaviour is fundamental for the perform-ance anti-skid and WSP devices.6 This relationship isdefined on the basis of experimental results, but, sincemacro-sliding occurs, for example, in the presence ofcontaminants in the contact area, the variability ofadhesion coefficient due to non-homogeneous contactconditions is very high, and the identification of itsdependence on slip is difficult.

To obtain a unique model, valid either for themicro-slip and the macro-slip ranges, it is necessaryto express the adhesion factor � as a function of therelative slip, given by the ratio between the absoluteslip and the speed.

Since in this work, the longitudinal dynamics isanalysed, the relative slip in this case is evaluated asthe component of the above defined creepage in thelongitudinal direction.

An example of relative slip/adhesion curve is pre-sented in Boiteux,6 and is shown in Figure 1. As canbe seen, two curves are present: the continuous onerefers to a loss of adhesion, i.e. to an increment of theslip, while the dashed curve is relative to a recovery ofadhesion, i.e. to a decrement of the slip, and their typ-ical shapes are clearly different.

The behaviour of the adhesion factor withdegraded conditions seems to be affected also by vehi-cle speed, as confirmed by experimental tests and theliterature.6,7 The dependency on velocity has beenoften explained with the so-called energetic or polish-ing effect: degraded adhesion conditions are mainlydue to contaminants present in the wheel–rail inter-face, when high sliding occurs, the energy dissipatedby the friction produces localized heating that maypartially destroy or remove contaminants; materialfrom sliding surfaces is removed and the tribologicalfeatures of the rolling surfaces are altered. As a con-sequence, the equivalent friction adhesion factor ismodified. This is also called the polishing effect sincethe sliding of a wheel clears the rail from the contam-inants, improving the adhesion conditions found bythe following ones. This effect is often claimed to beresponsible for the hysteresis behaviour of the adhe-sion factor during repeated cycles of contact losses.The cleaning effect mainly depends on the specificenergy dissipated during the sliding per unit oflength, which depends on vehicle speed; as a conse-quence, the effect due to energy dissipation producesan alteration of the adhesion factor that depends onspeed and from the wear number associated with slid-ing. Relative sliding is a good index to evaluate this

Malvezzi et al. 3



phenomenon since it is approximately proportionalto the specific energy dissipated over a unitary arclength. Other parameters, e.g. unmodelled contactforce fluctuations due to the flexible behaviour ofboth wheel and rail, may contribute to modifyingthe behaviour of the adhesion factor with speed.

Data from test runs

The theoretical models and qualitative considerationsdescribed in the preceding section are useful for thecomprehension of the wheel–rail contact phenomena,but do not give quantitative information on the adhe-sion curve.

In order to verify these qualitative behaviours andto obtain quantitative information, a set of experi-mental data, from a series of test runs, is analysed,to estimate, by means of a simplified dynamicalmodel, the adhesion coefficient between the wheeland the rail. The experimental data used to identifythe wheel–rail adhesion coefficient are obtained froma set composed of 27 braking tests,11 conducted with asingle vehicle, where the adhesion between the rail andthe wheels is degraded by spraying a water-basedsolution of soap (cleonsol) on the wheel or on the rail.

The tests were originally devoted to verify thebehaviour of WSP systems and were conductedaccording to Union Internationale des Chemins des

Fer (UIC) regulations on WSP homologation proced-ures.18 In these tests, braking performance is evaluatedin critical adhesion conditions, artificially reproduced.Braking performance is usually evaluated by means ofbraking stopping distance and deceleration; however,other parameters are often measured to obtain moredetailed information on the system behaviour.The tests described in this paper were conducted witha single vehicle, a passenger coach named UIC Z1. InFigure 2(a), the device used to spray the solution of con-taminants is shown. Alternatively, on some tests pre-scribed by UIC regulations, degraded adhesionconditions are realized by applying directly on the railsthe solution of contaminants as shown in Figure 2(b).

The following measured values were available forthis study:

. absolute vehicle speed v;

. vehicle wheel angular velocities !i, i¼ 1 , . . . , 4;

. pressure in the brake cylinder pi, i¼ 1 , . . . , 4.

Each measurement is sampled with a time step of2.4ms, corresponding to 417Hz. Train speed is mea-sured by a radar Doppler sensor, whose signal is typ-ically affected by high-frequency noise and spikescorresponding to track irregularities,19 then it is fil-tered using a digital linear low-pass fourth-orderfilter with a cut-off frequency of 1.5Hz. Wheel

Figure 1. Typical qualitative behaviour of the slip/adhesion curve, adapted from Boiteux et al.6




velocities are measured with angular speed sensors,whose signals are affected by misalignment and quant-ization errors. Typically, their noise level is lower thanthe radar Doppler one, on the other hand, the dynam-ics of the wheel angular speed is faster than the vehicleone. In order to preserve the information content ofaxle speed measurements, angular speed sensors(tachometers) are filtered with a low-pass filter witha cut-off frequency of 4Hz. Experimental data areelaborated using the MATLAB standard tools forsignal processing.

Figure 3 shows one experimental braking test:Figure 3(a) shows the vehicle speed and the velocityof the first wheel during the braking, as a function oftime, while Figure 3(b) shows the pressure in the cor-responding brake cylinder, as a function of time aswell. It is worth noting that the low adhesion and theintervention of the WSP system lead to an oscillatingbehaviour, both in the wheel tangential velocity and inthe brake cylinder pressure. The amplitude and fre-quency of such oscillations depend on many factors,including the feature of the WSP system, the dynamicalparameters of the vehicle, etc. The WSP system object-ive is to maintain the slip value in a range in which theadhesion coefficient is high: when the value of the brak-ing force is too high, with respect to the availablewheel–rail adhesion, the wheel begins to slide, andthe sliding value increases rapidly (as can be seen, inthese tests, it reaches values up to 5m/s). The WSPdevice then modulates braking force, in order toreduce and control the slip value. Figure 4 shows, fora similar test, the vehicle speed and the wheel periph-eral velocities relative to four wheels. In this figure, it isworth noting the different behaviour of the four wheelvelocities, due to the different adhesion conditions metby the wheels, which determine different interventionsof the WSP.

In these tests, the rail profile was the UIC 60, andthe wheel was ORES1002; the profiles did not presentsignificant wear, the track was straight and withoutturnouts.20

Simplified dynamic model

The dynamics of each wheelset can be described bythe following differential equation, which is referredto as the simplified model of the wheel sketchedin Figure 5,

Tir ¼ J _!i þ Cf, i ð9Þ

where Ti is the tangential component of the contactforce acting on the ith axle, _!i is the axle angularacceleration, Cf,i is the braking torque, r is the wheelradius and J is the axle moment of inertia. The brak-ing torque Cf,i has been assumed proportional to thebrake cylinder pressure pi

Cf, i ¼ �pi ð10Þ

The proportionality coefficient � depends on the cylin-der area, on the brake rigging ratio, on the brake effi-ciency and on the friction coefficient of brakingmaterials, which in this study was assumed to be0.35.21 The friction factor of thebrake pads isknown from technical documentation of the padmaker, and it is also information available in thebrake plant calculation, which is usually producedas part of the vehicle technical documentation. Thebrake pad friction factor can be easily verified orextrapolated from the braking test with full adhesioncondition conditions exploiting the proportionalitybetween vehicle deceleration and applied brakingforces. From expression (9), Ti can be calculated as

Ti ¼1

rJ _!i þ

1

r�pi ð11Þ

Moreover, vehicle longitudinal deceleration is esti-mated with a simple derivative filter applied to trainspeed measurements. The vertical forces between vehi-cle body and bogies are estimated by means of a

Figure 2. Degraded adhesion tests: (a) device used to produce degraded adhesion condition on the railway vehicle and (b) device

used to produce degraded adhesion condition acting directly on the rails.11

Malvezzi et al. 5



simplified two-dimensional vehicle dynamic model,and then the normal load acting on each axle is eval-uated. For example, the vertical force between thebody and the front boogie can be evaluated as

Nb1 ¼mc � g2þmc � d � h1 � hbð Þ

l1ð12Þ

where mc is the car body mass, g is acceleration due togravity, d is vehicle longitudinal deceleration and h1,hb, h2, l1 and l2 are dimensions correspondingto the geometrical layout of the vehicle shownin Figure 6.

The vertical (normal) load acting on the frontwheel of the front bogie can be evaluated as

N1 ¼Nb1 þmb1 � g

2þmb1 � dec � h2

l2ð13Þ

Vertical loads acting on the other wheels, namely N2,N3 and N4, are calculated in a similar way.

Once the vertical and the longitudinal componentsof the contact force are estimated, the adhesion coef-ficient �i can be evaluated for each wheel and for eachtime sample. In Figure 7, the typical behaviour of theadhesion factor � is shown, as a function of the slip

Figure 3. Example of data measured during a braking test: (a) comparison between train speed and wheel speed and (b) pressure on

first brake axle.




during single adhesion losses. The profile has beencalculated using the experimental data correspondingto the test shown in Figure 4. In particular, curves�i(�v) (index i represents the corresponding axlenumber) are evaluated using the followingrelationship

�i ¼�pi þ Ji _!i

Nirið14Þ

For example, in Figure 8 the adhesion curve obtainedfor the same axle with different braking speeds isshown: it is worth observing that the adhesion curve

depends on speed, as discussed, for example, inPolach.7

Definition of a model based on neuralnetworks

As discussed in the preceding sections, the wheel–railphenomenon is quite complex and depends on severaluncontrollable and even unobservable parameters. Onthe other hand, simulation of degraded adhesion con-ditions is necessary to reproduce complex interactionsthat often arise among different mechatronic on-board systems and railway vehicle dynamics. Typicalapplications are the study and the simulation of WSPsystems, anti-skid traction controls or odometry on-board subsystems.

The goal of the analysis described in this paper is todefine a mathematical model able to describe, in arealistic but computationally efficient way, the behav-iour of wheel–rail adhesion in critical environmentalconditions, as those present in the experimental testsdescribed in the preceding section.

The first variable that has to be taken into accountin this study is the sliding �v, furthermore, as outlinedin the preceding sections, during cyclical sequences inwhich the wheel sliding is increased and decreased, theadhesion factor shows hysteresis. The time derivativeof slip �_v is then useful to recognize the curve relativeto loss of adhesion, i.e. to an increment of the slip,from the one relative to a recovery of adhesion, i.e.to a decrement of the slip.

In this work, a mathematical relationship betweentwo inputs, which in our case, are the wheel slip

Figure 4. Example of data measured during a braking test: vehicle speed and wheel velocities.

T

Cr

f

i

ω,θ

Figure 5. Forces and torques acting on a wheel during

braking.

Malvezzi et al. 7



�v and its time derivative �_v, and the adhesion factor�, which is the output.

Mathematical models used to represent physicalphenomena can be substantially divided into twomain groups: deterministic or mechanistic modelsand data-driven models. The former are usuallyadopted when the phenomenon dynamics is quitesimple and a limited set of experimental data areavailable, while, on the contrary, the latter ones areused when the system physics is uncertain or complexbut several experimental measurements are available.Pugi et al.22 presented a deterministic three-dimen-sional model that allowed the adhesion to be modelledalso in critical conditions. They adopted differenttypes of identification techniques to solve the prob-lem. Data-driven models are also referred to as black

box models: a mathematical structure of the model isusually pre-defined, which depends on a series of par-ameters that are tuned, during an iterative processcalled training in order to replicate as precisely as pos-sible the experimental data. Neural networks belongto this type of system identification methods.

In this paper, a simple and widely known type ofneural network, the multi-layer perceptron, isadopted. A feed-forward neural network23–25 consistsof one or more layers of neurons. The latter layer is adefined output layer, while the others, when they arepresent, are referred to as hidden layers. Each layer iscomposed of a series of neurons, and each neuron is asimple dynamical system that applies a specific trans-fer function to a weighted sum of the inputs. Thestructure of the neural network is defined once the

m1m2m3m4

mb1mb2

mc

h1

h2

l1

l2 l2

Figure 6. Simplified vehicle dynamic model of a railway vehicle in the longitudinal plane.

Figure 7. Wheel–rail adhesion factor as a function of slip calculated from degraded adhesion data of Figure 4, the curves were

obtained elaborating data from the first loss of adhesion at the beginning of the braking phase.




number of layers, the number of neurons for eachlayer, and the neuron transfer function are defined.The identification of a function using data-driventechniques in general requires three steps. The firststep is the collection of reference data; each datumis a vector that contains an input vector and the cor-responding measured output vector. The availabledata are usually divided in two subsets: the trainingdata set and the validation data set. The first one isused for the iterative optimization of network param-eters (training), while the second one is used to evalu-ate system performance. In our application, for thetraining phase, the second axle data were used,while for the validation, we used the third axle ones.This solution was adopted for convenience, even if itcould seem better to choose, for the data sets, meas-urements from all the wheels. However, since the vehi-cle used to perform the tests is symmetric, and thetests are conducted in two different directions, wecould assume that the data sets from the second andthird wheels are quite similar.

Then the network’s architecture (organization ofneurons in the network and definition of the activa-tion function for each neuron) is defined. Finally, thenetwork is trained: reference data are submitted to thenetwork and the values of parameters are updated inan iterative process in order to minimize the errorbetween the network output and the desired output.The error can be defined in several ways; usuallya measurement of the distance is used, in thispaper, we adopted the mean square error. Thechosen architecture is a multi-layer perceptron, usingin the training process, the Levenberg–Marquardtalgorithm.23–25

The performance of a neural network depends onits architecture; for the examined problem, double-layer networks have been chosen, with hyperbolic tan-gent activation functions in the hidden layer andlinear activation functions in the output layer. Thistype of architecture has been chosen since it issimple and widely adopted in the identification ofnon-linear systems starting from noisy measurements.

Some tests were performed to find the optimalnumber of neurons in the hidden layer: if thenumber of neurons is too low, the model is too simpleand is not able to reproduce the complexity of thephysical phenomenon. On the other hand, increasingthe number of neurons, the system complexity obvi-ously increases, also from the computational point ofview. Furthermore, if the network is too complex, it isno more able to capture the physics of the system, butrather tends to copy the input/output experimentalpairs, which usually are affected by measurementnoise, and the system outputs may be not realistic.Some tests showed that in the presented case, the opti-mal size of the hidden neuron layer is 10.

At the end of the training process, the neural net-work defines substantially the function between the

Figure 8. Slip curve from experimental tests of the second axle at 120 km/h and at 160 km/h the curves were obtained elaborating

data from the first loss of adhesion at the beginning of the braking phase.

Figure 9. Error in the training and validation processes.

Malvezzi et al. 9



sliding (and its time derivative) and the adhesion coef-ficient. Results are then analysed and verified using, asinputs, experimental data not included in the trainingprocess.

Figure 9 shows the behaviour of the relative errorfor the training and validation data sets as a functionof the number of training epochs (i.e. number of iter-ations in the training process). The diagram shows agood mutual accordance and a quite similar depend-ency from the number of epochs between the differentdata sets, this result confirms that the training andvalidation data sets have similar informationcontents.

Results

Figure 10 shows, for a test included in the validationdata set, a comparison between the experimental and

the model predicted adhesion values. As can beobserved from the figure, the neural network basedmodel is able to reproduce the experimental datawith a good approximation. This test was obtainedby giving to the neural network identification modelthe inputs from an experiment and observing the cor-responding output. Since the input data are thosefrom experimental measurements, the output of theidentification model presents some oscillations andsome sharp points.

The behaviour in the first part of the diagram, cor-responding to small sliding values, is approximatelylinear, but presents some irregularities due to themeasurement errors. Actually, in this part of the dia-gram, the identification model reliability is not veryhigh, because of the unreliability and noise in theexperimental measurements. In an implementationof the model devoted to reproduce the dynamics ofa railway vehicle, this part could be substituted by a

Figure 10. Slip curve of second axle: comparison between the neural network (N.N.) output and experimental data.

Figure 11. Complete slip curve (simulated), obtained at a speed of about 120 km/h.




linear model as described in the subsection on wheel–rail contact forces in rolling conditions.

The obtained mathematical model based on neuralnetworks is then verified by giving regular patternsand smooth functions as inputs. For example,Figure 11 shows the results obtained by linearlyincreasing the sliding from zero to a maximum valueof about 9m/s and then decreasing it linearly up tozero. The time derivative of the sliding in this exampleis then composed of two constant parts, the first posi-tive and the second negative. This example shows howthe obtained model can reproduce the typical shape ofa simulated cycle composed of a loss of adhesion anda following recovery of rolling conditions, the typicalhysteresis behaviour of the adhesion factor is evident.In the diagram, the part of the curves relative to smallsliding values (around zero) is not displayed, since theidentification procedure is not very reliable in thispart, as previously discussed.

Conclusions

Identification of adhesion behaviour in the wheel–railcontact is a multi-faceted phenomenon that is hard toinvestigate. This paper analyses the problem of theadhesion identification in braking, analysing a reach-able models that correlates slip and adhesion coeffi-cient. The numerical estimation of the adhesioncoefficient from a set of data tests allowed a seriesof absolute slip/adhesion curves to be obtained(foreach axle relative to different train speeds). Then, anidentification of the adhesion coefficient using a stand-ard neural network procedure was implemented: itpermitted the definition of a mathematical model forthe evaluation of the adhesion coefficient as a functionof the wheel sliding. In addition, the obtained modelproduces results compliant with the main propertiesof the observed phenomenon, e.g. the different behav-iour corresponding to loss of adhesion and recoveryof adhesion. Since the simulated phenomena are alsodominated by stochastic disturbances, clearly, a per-fect fit of experimentally measured time histories isnot possible.

The proposed adhesion model is limited to a bi-dimensional approach that constrains possible appli-cations to planar vehicle models which are often usedfor real-time simulation on HIL test rigs such as thosedescribed in Pugi et al.11 or to full-scale roller rigssuch as those described in Allota et al.12 andMalvezzi et al.13 Further work will be devoted to pro-duce degraded adhesion models able to reproduce anear to realistic behaviour also on multi-patch, three-dimensional contact models usually used for multi-body simulations.

Funding

This research received no specific grant from anyfunding agency in the public, commercial, or not-for-profitsectors.

Acknowledgements

The authors wish to thank Trenitalia SPA and in particular

the manager and the technicians of the experimental struc-tures and laboratories of Florence, which provided theexperimental data, and whose help and experience were fun-

damental for the development of this work. The authors arealso grateful to Francesco Maraschiello, who collaboratedwith them to elaborate experimental measurements during

the preparation of his Master’s Thesis.

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Appendix 1

Notation

dec (m/s2) train decelerationf11, f22, f33, f23 linear creep coefficientsir unit vector that identifies longitu-

dinal directiong (m/s2) gravity accelerationh1, h2, hb, l2 (m) vehicle dimensional features,

shown in Figure 6ks coefficient for the tangential con-

tact force saturationmb1, mb2 (kg) bogie masses

mc (kg) car body massmi (kg) mass of each wheelnr unit vector normal to the contact

surfaces in the contact pointpi (Pa) pressure at the ith brake cylinderr (m) wheel radiusrf (m) wheel braking radiustr unit vector tangent to the lateral

directionv (m/s) train speedvow (m/s) magnitude of the velocity of the

wheelset centre of massvrc (m/s) wheel speed at the contact pointCf (Nm) braking moment at the wheelCf,i (Nm) braking moment at the ith wheelFx (N) contact force component in the

longitudinal directionFy (N) contact force component in the

lateral directionFR (N) resultant of longitudinal and lat-

eral contact force componentsJi (kg m

2) wheelset moment of inertiaN (N) normal component of the contact

forceNi (N) normal contact force at the wheel

iNbi (N) vertical force between the car

body and the ith bogieT (N) tangential component of the con-

tact forceTi (N) tangential contact force at the

wheel i

� proportionality coefficientbetween brake pressure and braketorque

� adhesion coefficient�, � and � longitudinal, lateral and spin

creepage�v (m/s) relative slip! (rad/s) angular wheel velocity!i (rad/s) angular velocity of the ith wheel!�(rad/s2) angular wheel acceleration

_!i (rad/s2) angular acceleration of the ith

wheel�v (m/s) absolute slip




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Proc IMechE Part F: Identification of a wheel–rail adhesion J Rail … · 2017. 2. 12. · Original Article Identification of a wheel–rail adhesion coefficient from experimental