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Wear 265 (2008) 1465–1471

Contents lists available at ScienceDirect

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igh-frequency vertical wheel–rail contact forces—Validationf a prediction model by field testing

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Jens C.O. Nielsen ∗

CHARMEC/Department of Applied Mechanics, Chalmers University of Technology, SE-41

a r t i c l e i n f o

Article history:Accepted 21 February 2008Available online 23 May 2008

Keywords:Vehicle-track interactionInstrumented wheelsetWheel flatRail corrugationVehicle/track modelling

a b s t r a c t

The computer program DIFvertical vehicle–track interesults from two field testest case involves impact lrail on dynamic vertical ware compared. The track mon rail and wheel roughnand measured vertical confor most frequencies beloaccounts for both wheelse

1. Introduction

Small amplitude undulations (roughness, waviness) with abroad spectrum of wavelengths are present on the running surfaces

f wheels and rails. Such roughness leads to a broadband relativeisplacement excitation of wheel and rail, inducing high-frequencyertical wheel–rail contact forces together with vibrations andolling noise. Wheel corrugation due to tread braking with castron brake blocks, and rail corrugation that for example mayevelop in situations with uniform traffic in track sections whereigh levels of traction are present, will increase the excitation

evel [1]. In Sweden, dominating wavelengths for corrugation onangent tracks are usually in the interval 4–8 cm, see Fig. 1. Ampli-udes of the corrugation are in the order of 10 �m. Cast ironread brakes lead to similar wavelengths and amplitudes on theheels.

Wheel flats and insulating rail joints are examples of dis-rete irregularities leading to severe impact loads with significantorce contributions at high frequencies. Such loads exacerbateeterioration of wheelset and track components and lead to

ncreased maintenance costs. The high-frequency contents of ver-ical wheel–rail contact forces generated by these different formsf wheel–rail irregularities are often of significant magnitudes andust not be ignored [2]. Common examples of associated damage

∗ Tel.: +46 31 772 1500; fax: +46 31 772 3827.E-mail address: [email protected].

043-1648/$ – see front matter © 2008 Jens C.O. Nielsen. Published by Elsevier B.V. All rioi:10.1016/j.wear.2008.02.038

othenburg, Sweden

ich is being developed at CHARMEC since the late 1980s, is used to simulateon at high frequencies, from about 20 Hz to at least 2000 Hz. Measuredpaigns are used to validate the vehicle–track interaction model. The firstrom a wheel flat, while the other case studies the influence of a corrugatedrail contact forces. Four vehicle models and two visco-elastic track modelss are calibrated versus test data from laboratory and field tests. Input datare taken from field measurements. Good agreement between calculatedorces is observed, both with respect to magnitude and frequency content,00 Hz. The best agreement is obtained when using a vehicle model thata bogie, instead of using a single wheelset model.

© 2008 Jens C.O. Nielsen. Published by Elsevier B.V. All rights reserved.

mechanisms are irregular wear leading to rail corrugation, rollingcontact fatigue of wheel treads, and plastic deformation of cross-ings and insulating joints. Neglect of contact force contributionsfrom higher frequencies is often a consequence of using a measure-ment technology that is based on static calibration methods thatrequire low-pass filtering of the measured force signals, or using asimulation model where such excitation is not properly accounted

for.

There is a trend towards higher speeds for passenger trains andhigher axle loads for freight traffic. An optimization of the com-pound train–track system is necessary to meet these demands.To achieve this, the use of a tool for simulation of dynamicvehicle–track interaction has several advantages compared tofield testing. Parametric studies are straight-forward and pro-vide controlled testing conditions at low cost, and they lead to asignificantly increased understanding of the complex interactionbetween train and track. However, before such a model can beused with confidence, an extensive validation versus field tests isrequired.

In the present study, measured results from two field test cam-paigns are used to validate the vehicle–track interaction modelin DIFF [3]. The computer program DIFF, which is being devel-oped at CHARMEC since the late 1980s, is used to simulate verticalvehicle–track interaction at high frequencies, from about 20 Hz to atleast 2000 Hz. At lower frequencies, more detailed models of vehi-cle dynamics and ground vibration may be necessary. At higherfrequencies, cross-sectional deformation of the rail may becomeimportant.

ghts reserved.

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of contact forces are discussed in detail in the accompanying paper[4].

3. Vehicle–track interaction model

Fig. 4 illustrates the vehicle–track interaction model in DIFF. Thedynamic vehicle–track interaction is solved in the time domain.The solution is based on an extended state-space vector approach,and a complex-valued modal superposition for the linear trackmodel with non-proportionally distributed damping. The equa-tions of motion for vehicle and track, the constraint equationscoupling wheels and rail, and the solution procedure is presentedin detail in Ref. [3]. In the present study, track and vehicle modelsare taken as linear. By instead performing a direct time integrationof the complete set of equations (not using the modal superposi-tion approach), both vehicle and track may be treated as non-linear.This will however increase simulation times considerably.

A non-linear compressive stiffness of the wheel–rail contact(not shown in Fig. 4) is determined by assuming three-dimensionalcontact according to Hertz. The contact stiffness coefficient CHis calculated using the wheel curvature 1/0.44 m−1, transverse

Fig. 1. Rail corrugation with wavelengths in the interval 4–8 cm. Photo: LennartLundfeldt, Banverket.

2. Instrumented wheelset technology

Wheel–rail contact forces were measured on a high-speed(up to 200 km/h) X2 trailer bogie operating on the line

Stockholm–Goteborg. The extensive test campaign was performedin October 2002. Measurement wheels from Interfleet TechnologySweden, instrumented with strain gauge bridges on each side ofthe wheel disc, were employed, see Fig. 2. The wheels were cali-brated for vertical (Q) and lateral (Y) static wheel–rail contact forces,respectively. At measurements in-field, the signals from the straingauge bridges are processed to give the Q- and Y-forces from therolling wheel. The signals were recorded on data loggers on boardthe train. Also GPS-signals, pulses from reference locations alongthe track and train speed were recorded.

The standard approach for data processing used by InterfleetTechnology includes a low-pass filtering of the measured sig-nals with cut-off frequency 90 Hz. A major challenge has been toextend the frequency range in the force measurements. However,the wheelset has several eigenmodes in the frequency range thatare excited by the short-pitch wheel–rail irregularities, and theinfluence of these modes could lead to inaccurate results whentransforming the measured strains to forces in the wheel–rail con-tact. From an experimental modal analysis, it was concluded thatthe most significant source of error is the radial wheel modewith two nodal diameters at 1712 Hz, see Fig. 3. The error con-

Fig. 2. Strain gauges are glued to the wheel disc. Locations of 16 strain gauges formeasurement of Q-force are shown.

tributions from other wheelset eigenmodes (with or without axlebending) seem to be smaller and restricted to narrow frequencybands. The instrumented wheelset technology and the assessment

Fig. 3. Vehicle model C is based on an FE model of an X2 trailer wheelset includingbrake discs. Shape of radial eigenmode with two nodal diameters is shown.

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In Fig. 6, the calculated receptance of the assembled track modelis compared to the corresponding track receptance that was mea-sured in field. The track was excited by vertical impact excitationon top of the rail head and acceleration was measured under therail head. No static vehicle load was applied. Input data for theballast/subgrade models are kA

b = 100 MN/m, cAb = 82 kN s/m, kB

b1 =300 MN/m, kB

b2 = 150 MN/m, cBb1 = 6 kN s/m and cB

b2 = 190 kN s/m.Note that these data are higher than those corresponding to a trackwithout vehicle load, cf. [7]. This explains the deliberate mismatchof measured and calculated receptances at the resonance below150 Hz.

For track model A, the number of complex-conjugated modepairs accounted for in DIFF was 220 (real part of maximumeigenfrequency was 2900 Hz). Since track model B contains twodegrees-of-freedom per sleeper bay with zero mass (due to visco-elastic models of rail pads and ballast/subgrade), several calculatedeigenmodes are over-critically damped. In this case, the numberof used complex-conjugated mode pairs was 301 (real part ofmaximum eigenfrequency was 2700 Hz) while the number of over-critically damped real-valued modes was 198.

Fig. 4. Principal sketch of dynamic vehicle–track interaction model in DIFF. Vehiclemodel D (two wheelsets) and two different visco-elastic models of the rail pad (Aand B) are shown.

wheel curvature = 0, rail curvature = 0 and transverse rail curva-ture = 1/0.3 m−1 as input data.

3.1. Track model

The discretely supported UIC60 rail is modelled by undampedRayleigh–Timoshenko beam finite elements with bending stiffnessEI = 6.4 MN m2, shear stiffness kGA = 250 MN, mass per unit beamlength m = 60 kg/m and rotational inertia per unit beam lengthmr2 = 0.24 kg m. The (half) sleepers are treated as rigid with massMs = 125 kg. The length of the track model is 100 sleeper bays withsleeper spacing L = 0.65 m and clamped boundaries at the two railends. The track model is here taken as repetitive (all sleeper baysare of the same length, and all discrete rail supports have the samedynamic stiffness), but this is not a requirement of the model. Theloading on the track (including wheel and rail irregularities) is takenas symmetric with respect to a fictive centre line between the tworails. Only vertical interaction is considered.

Collecting input data to the rail and sleeper models is rela-tively straight-forward. Determining input data for rail pads andballast/subgrade is a more complex task because rubber, ballastmaterial, clay, etc. have complicated constitutive relations that aredependent on temperature and the magnitude and frequency ofthe applied load. Two track models that include different models

of rail pads and ballast/subgrade are compared in the present study.In track model A, each rail pad is modelled as a discrete linear elas-tic spring and a viscous damper in parallel (Kelvin model). In trackmodel B, each pad is modelled by a three-parameter visco-elasticmodel (standard solid model), see Fig. 4. In track model A, the sup-port under each sleeper is modelled by a Kelvin model, whereas intrack model B, a four-parameter visco-elastic model is adopted. Thefour-parameter model means two spring–damper sets coupled inseries, with each set containing one elastic spring and one viscousdamper coupled in parallel.

For the rail pads, properties corresponding to a static preload of40 kN were determined based on laboratory measurements on Pan-drol studded, nominally 10 mm thick, rubber pads [5]. The dynamicstiffness calculated by the two visco-elastic models are comparedwith the corresponding measured stiffness (data available up to1000 Hz) in Fig. 5. Good agreement for the stiffness magnitude isobserved for both models. However, the phase of the dynamic stiff-ness is better represented by the three-parameter model. Inputdata for the rail pads are kA

p = 120 MN/m, cAp = 25 kN s/m, kB

p1 =240 MN/m, kB

p2 = 240 MN/m, and cBp = 80 kN s/m.

Fig. 5. Magnitude and phase of dynamic rail pad stiffness. Comparison of rail padmodels A and B with data measured in a laboratory.

Fig. 6. Magnitude and phase of track receptance. Comparison of track models A andB with data measured in the field.

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Fig. 7. Calculated direct receptance of vehicle models A–C (and including a spring

and viscous damper for the primary suspension). The receptances were determinedfor radial excitation in the wheel–rail contact.

3.2. Vehicle model

The influence of the vehicle in the frequency range of interest(20–2000 Hz) is modelled by a discretized mass–spring–dampersystem representing one or two wheelsets in a bogie. The vehiclemodel is moving with speed v from the left end to the right end ofthe track model (so-called moving mass model). Four vehicle mod-els will be compared. Either the wheelset is treated as a rigid mass,or as flexible based on an FE model and calculated Craig–Bamptonmodes (see vehicle model C in Fig. 3) [6]. The load from the partof the train above the primary suspension is modelled as a staticwheel load W = 66.2 kN, which is corresponding to an axle load of13.5 tonnes.

In vehicle model D (see Fig. 4), each wheelset model containstwo degrees-of-freedom: two masses, one spring and one viscousdamper. The large mass Mw corresponds to half the wheelset mass1135/2 kg, whereas the values of the small mass mw (3 kg), spring kw

Fig. 8. Impact load versus train speed for a wheel flat with length 100 mm anddepth 0.9 mm. Axle load 24 tonnes, unsprung half wheelset mass Mw = 1185/2 kg.Measured data: *; calculated data: �. From Ref. [7].

(2008) 1465–1471

(3500 MN/m) and damper cw (155 kN s/m) were selected to matchthe receptance of this wheelset model with the correspondingreceptance that was calculated using the FE model of the wheelset.Most resonances and antiresonances of the wheelset can of coursenot be captured by the simplified model D, but the average of thereceptance at frequencies above 1000 Hz is similar for models Cand D. Vehicle model B is of the same type as vehicle model D,but it contains only one wheelset. Model B (and D) is thus a com-promise between a rigid wheelset model (with only the rigid massMw = vehicle model A) and the FE model (vehicle model C). Thecalculated receptances for vehicle models A–C are compared inFig. 7.

The selection of vehicle model may have a significant influenceon the calculated wheel–rail contact forces at the high frequenciesexcited by e.g. wheel flats or short-pitch rail corrugation. Use of thesimplest rigid model (A) may lead to overestimated contact forces,especially in load cases where the wheel recovers contact with therail after a short period of lost wheel–rail contact. This is becausethe dynamic stiffness of model A is very high at high frequencies,cf. Fig. 7.

4. Validation 1: wheel flat

Before simulation and measurement are compared for the casewith the X2 trailer wheelset, results from a previous validation exer-cise will be repeated in brief [7]. Wheel–rail impact loads calculatedusing the model in Section 3 were compared with loads measuredduring a field test performed in 2000, see [8]. The impact loadswere generated by a wheel flat with length 100 mm and depth0.9 mm. A freight train with axle load 24 tonnes was used and trainspeeds were in the interval 5–100 km/h. Impact loads were mea-sured using a wheel impact load detector that was based on straingauges mounted on the rail web in nine consecutive sleeper bays.The impact load detector was calibrated against the quasi-staticwheel load that was measured when the train passed the test siteat a speed of 5 km/h.

In the calculations, a vehicle model of type B and a track modelof type A were used. Input data were similar to those used in thepresent study [7]. Good agreement between measured and cal-culated impact loads was observed, see Fig. 8. The solid line wasgenerated by a numerical curve fitting of the measured results toa third-degree polynomial in order to provide an indication of theinfluence of train speed on measured impact loads. A local maxi-

mum is observed for train speed 40 km/h.

5. Rail and wheel roughness

In connection with the X2 test campaign in 2002, rail roughnesswas measured at several sites along the line Stockholm–Goteborg.Banverket performed the measurements using the accerelometer-based system Corrugation Analysis Trolley (CAT) [9]. From themeasurement database, the site Vretstorp was selected for thepresent investigation. The roughness level spectrum [10] in Fig. 9shows that the rail is corrugated with dominating wavelengths inthe interval 4–8 cm.

Roughness on five X2 trailer wheels was measured by Johans-son [11]. Three probes in mechanical contact with the wheel treadmeasured the deviation from the mean radius using a techniquedeveloped by the company Ødegaard & Danneskiold-Samsøe [12].The deviation was measured with sampling distance 0.5 mm andamplitude resolution 0.06 �m. The centre probe was positionedat the nominal contact point (70 mm from the flange side of thewheel), while the other two probes were positioned on each sideof the centre probe and separated by a distance of 2 × 10 mm. All

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measured in Vretstorp at two different train speeds.

It should be noted that several of the peaks in the DFT spectra arefrequency-constant. This means that these peaks do not shift withtrain speed. The so-called P2 resonance (wheelset, rail and sleepersmoving in phase on the sleeper support stiffness) is observed as awide peak at around 100 Hz. The other frequency-constant peakscan be distinguished at around 570, 830 and 1100 Hz. It was con-cluded in Ref. [13] that these latter peaks are caused by bendingmodes in the rail that occur between the two wheels in a bogie. Acomparison with a Rayleigh–Timoshenko beam model of the rail,with length corresponding to the bogie wheelbase and pinned-pinned boundary conditions, confirm that these modes correspondto three (570 Hz), four (830 Hz) and five (1100 Hz) half wavelengthsbetween the wheels of the bogie. It has also been shown that suchmodes are important in the generation of rail corrugation [13,14].

Results (power spectral density, PSD) from simulations of theconditions in Vretstorp are compared with the corresponding fieldtests in Figs. 11 and 12. Two samples of rail irregularity profilewere generated based on the (narrowband) rail roughness spec-

Fig. 9. Measured roughness level spectra for corrugated rail in Vretstorp (averageof left and right rails) and X2 trailer wheelset (average of five wheels). Comparisonwith ISO 3095 spectrum (dashed line). Roughness level spectra are evaluated in 1/3octave bands.

wheels had a minimum travelled distance of 100,000 km. The eval-uated wheel roughness level spectrum is shown in Fig. 9. It isobserved that the X2 trailer wheels are very smooth comparedto the rail in Vretstorp (at least in the studied wavelength inter-val 2–250 cm). The influence of wheel roughness on the combinedwheel–rail roughness could therefore be neglected in this valida-tion exercise. A filtering of the roughness profile to account for thesize of the wheel–rail contact patch was performed. This leads toan attenuation of roughness magnitudes for shorter wavelengths.The adopted contact filter is based on the moving average of theirregularity that has been summed over the length of the contactpatch. The irregularity profile is assumed to be constant over thewidth of the contact patch.

6. Validation 2: rail corrugation

Results from the instrumented X2 trailer wheelset when pass-ing test site Vretstorp are used in the present validation exercise,cf. Fig.5. The discrete Fourier transforms (DFT) of the measured ver-tical (Q) wheel–rail contact forces at two different train speeds arecompared in Fig. 10. It is observed that the contributions to thecontact force are significant in the frequency range 500–1350 Hz.These contributions are caused by the rail corrugation with wave-lengths 4–8 cm. The influence of train speed can also be observedby comparing the magnitudes of the DFTs in the frequency inter-val 500–1350 Hz in Fig. 10. The sleeper passing frequency, 68 Hzat 160 km/h and 84 Hz at 197 km/h, are visible as distinct peaksin the spectra. The other distinct peaks, which also are speed-dependent (see peaks at 127 and 255 Hz for 160 km/h, and 157and 314 Hz for 197 km/h), are explained by the procedure that pro-cesses the Q force by alternatively using information from the straingauge bridge on either side of the wheel disc. These latter peaksare thus not caused by the dynamic train–track interaction andshould be ignored when comparing measured and calculated con-tact forces.

Fig. 10. DFT of vertical wheel–rail contact force (average from left and right wheel)

Fig. 11. PSD of vertical wheel–rail contact force in Vretstorp. Comparison of mea-sured and calculated contact forces along a distance of 100 sleeper bays. Train speed160 km/h. Vehicle model D.

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7. Influence of vehicle model

The influence of vehicle model on the calculated vertical con-tact force is illustrated in Fig. 14. Four different vehicle models arestudied, see Section 3. One of the sample rail irregularity profilesdescribed in Section 6 was used again.

All four vehicle models lead to similar results, but there are someimportant differences. The influence of the local rail bending modescan only be captured by model D since the other three vehicle mod-els only contain one wheelset. The influence of several eigenmodesof the wheelset, which are only covered by vehicle model C, leadsto distinct peaks and troughs in the PSD spectrum. Note that theinfluence of the radial mode with two nodal diameters at 1712 Hzleads to a reduction in contact force. Vehicle model A generates thehighest contact forces at frequencies above 1000 Hz.

8. Influence of track model

The influence of track model on the calculated vertical contactforce is illustrated in Fig. 15. Two different track models are studied,see Section 3. One of the sample rail irregularity profiles described

Fig. 12. PSD of vertical wheel–rail contact force in Vretstorp. Comparison of mea-sured and calculated contact forces along a distance of 100 sleeper bays. Train speed197 km/h. Vehicle model D.

trum measured in Vretstorp (covering wavelengths in the interval2–250 cm). The length of each sample profile was 100 sleeper bayscovering the length of the track model. The measured signals werefiltered by applying an inverse transfer function [4] to compen-sate for the erroneous magnification of signals caused by the radialwheel eigenmode with two nodal diameters, see Section 2. Theevaluated distance in both simulation and measurement was 100sleeper bays. However, the simulated response was taken from twoDIFF simulations of vehicle–track interaction in the first 60 sleeperbays. The first 10 sleeper bays in each simulation were eliminatedfrom the assessment since the response in these bays was affectedby the boundary conditions of the left end of the model. The com-putation time for one DIFF simulation (vehicle model D travellingalong 60 sleeper bays of track model B with 100 sleeper bays) wasaround 40 min on a PC with a Pentium 2.0 GHz processor. If insteadtrack model A was used, the computation time was reduced toaround 30 min.

Good agreement is observed for most frequencies below2000 Hz. As discussed above, the spurious peaks in the measuredresults below 500 Hz should be ignored. The P2 resonance peakhas a higher magnitude in the simulated response, but it occurs at

a similar frequency. The local rail bending modes are captured wellby the simulation model (vehicle model D).

Another approach to assess simulation and testing is to com-pare maxima of the low-pass filtered vertical wheel–rail contactforces, see Fig. 13. The measured and simulated contact forceswere low-pass filtered with various cut-off frequencies using afourth-order Butterworth filter. Again, it is observed that DIFFgenerates a higher contact force at low frequencies. However, fora cut-off frequency above 1000 Hz the low-pass filtered forcesfrom simulation and testing are very similar. Note that the con-tact forces at 160 km/h are here somewhat higher than thecontact forces at 197 km/h. The influence of contact force contri-butions from the frequency interval 500–1350 Hz is again seento be significant. Neglect of these contributions would lead toan under-predicted contact force. By observing the relatively lowmagnitudes (cf. static wheel load W = 66.2 kN) of the measuredforces in Fig. 13 that were low-pass filtered with 100 Hz, the influ-ence of vehicle (car body and bogie) resonances on measuredvertical contact forces seems to be small for this loading situa-tion.

Fig. 13. Maximum of vertical wheel–rail contact force in Vretstorp. Influence oflow-pass filtering with different cut-off frequencies. Vehicle model D.

Fig. 14. PSD of vertical wheel–rail contact force in Vretstorp. Train speed 160 km/h.Influence of vehicle model.

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forces—field measurements and influence of corrugation, in: Proceedings

Fig. 15. PSD of vertical wheel–rail contact force in Vretstorp. Train speed 160 km/h.Vehicle model D. Influence of track model.

in Section 6 was used again. It is observed that the choice of trackmodel has a negligible effect on calculated contact forces. This isimportant since the use of track model A leads to a considerablereduction in computation time, cf. Section 6. However, track modelB may be a better choice if sleeper vibrations need to be calculated.

9. Concluding remarks

The computer program DIFF for simulation of dynamicvehicle–track interaction at high frequencies (from 20 Hz to at least

2000 Hz) has been validated versus measured vertical wheel–railcontact forces. Results from two field tests using two differentapproaches to measure contact force, an instrumented wheelsetand a wheel impact load detector, have been used in the assessment.

Both measurement methods were based on strain gauge bridgesthat had been calibrated with respect to static loads. In both cases,the influence of the unsprung mass between the measurementposition and the wheel–rail contact was ignored. For the instru-mented wheelset, wheel eigenmodes will influence the magnitudesof the measured strains that are transformed into wheel–rail con-tact forces. For the wheel impact load detector, the contact force wasdetermined from an equation of equilibrium that included the shearforces in the rail at the two strain gauge positions in one sleeper bay,but the inertia of the rail between these positions was neglected.These are examples of circumstances that may lead to reducedmeasurement accuracy. On the other hand, setting up a simula-tion model that fully takes into account the complex interactionbetween train and track is not an easy task. For example, derivingmodels of rail pads and ballast/subgrade is difficult because rubber,ballast material, clay, etc. have complicated constitutive relationsthat are dependent on temperature and the magnitude and fre-

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(2008) 1465–1471 1471

quency of the applied load. Rail pads and ballast/subgrade have alarge influence on track dynamics at low frequencies (for a mod-ern Swedish track with resilient rail pads up to about 400 Hz, seethe first and second resonance in Fig. 6). Thus, it is probable thatresults from either simulation or measurement are more accuratein different frequency intervals.

Based on the good and consistent agreement between mea-sured and simulated vertical contact forces, both with respect tomagnitude and frequency content, it is argued that the computerprogram DIFF is a useful tool in investigations of vertical dynamicvehicle–track interaction at high frequencies.

Acknowledgements

The work was performed at the Department of AppliedMechanics, Chalmers University of Technology in Goteborg,Sweden. It forms part of the activities within the Centreof Excellence CHARMEC (CHAlmers Railway MEChanics), seewww.charmec.chalmers.se. Measured data were supplied by Inter-fleet Technology Sweden and Banverket.

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[9] S.L. Grassie, M.J. Saxon, J.D. Smith, Measurement of longitudinal rail irregulari-ties and criteria for acceptable grinding, J. Sound Vib. 227 (5) (1999) 949–964.

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