“does god play dice with corrugations?”: environmental effects on growth

“Does god play dice with corrugations?”: Environmental effectson growth

P.A. Meehan n, P.A. Bellette, R.J. HorwoodThe University of Queensland, Australia

a r t i c l e i n f o

Article history:Received 31 October 2013Accepted 24 November 2013

Keywords:Rolling contact mechanicsWear modellingTractionCorrugationsEnvironmental effects

a b s t r a c t

Corrugation growth has perplexed many researchers for several decades with remaining challengesincluding its reliable prediction in the field. In the present paper, the effect of environmental variationson corrugation growth is investigated using field measurements and mechanics-based modelling.Statistically significant relationships between average daily rainfall, humidity and the growth of railcorrugation were investigated using meterological and railway site field monitoring of a metropolitannetwork test site with a recurring rail corrugation of about 95 mm wavelength. Corrugation growth rate(Gr) was determined by systematically measuring the longitudinal rail profile with a Corrugation AnalysisTrolley (CAT) on the 240 m radius, narrow gauge, concrete-sleepered curve. Both roughness generatedand weld initiated profile growths were investigated. The weather data was obtained from records heldby the Bureau of Meteorology. The modelling is developed to provide insight and mechanics basedanalysis of the field corrugation growth under changes in environmental conditions and vehicle speedvariability. Results show a strong correlation between variations in rainfall and corrugation growth that isconsistent with changes in steady state wear. Changes in contact patch geometry and the damagemechanism as the corrugation amplitude grows are shown to have substantial but slower effects ongrowth. The results are used to explain observed changes in corrugation growth rate under variablespeed control at the same site.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

The growth of corrugations has intrigued and perplexed manyengineers and researchers for several decades. This has motivatedongoing research into understanding, simulating and possiblycontrolling the phenomenon (see reviews of [1,2,14]). Althoughmuch is known now, one of the remaining challenges is thereliable prediction of corrugation growth in the field usingmechanics-based modelling. In particular, although many corruga-tion models and laboratory testrigs have been developed andutilised to provide important insight into the phenomena, reliableprediction of the growth rate of corrugations in the field remainsan enigma. This is perhaps not surprising considering that the fieldoffers conditions that are substantially uncontrolled as comparedto simulation and laboratory environments. Despite these difficul-ties, useful field measurements and observations have beenperformed such as described in [2–6]. For instance field resultspertaining to the steady wear parameter indicate that (steady)wear resistant rail is also more resistant to unsteady wear orcorrugations. Field results pertaining to the application of friction

modifiers [2,4,5] in reducing the occurrence of rail corrugations,verify the importance of changes in friction coefficient and stick-slip behaviour on corrugation growth.

Although these previous studies have provided very useful obser-vations, the environmental effects of varying rainfall and humidity aswell as contact profile have not been carefully monitored andquantified under field conditions. Indeed, perhaps the lack of reliable,repeatable, field corrugation growth results is due to these randomenvironmental effects; a question that prompted the light-hearted titleof this paper. In response, the importance of the environmental effectsof rainfall, humidity as well as variations in vehicle speed, contactgeometry and the damage mechanism on corrugation growth wasdetermined using field measurements and mechanics modelling ofthe corrugation growth process on a metropolitan field site.

2. Background theory – corrugation growth under fieldvariations

An analytical model for corrugation growth prediction devel-oped previously by the authors [15,16,24] was utilised for insightand mechanics based analysis of the field data taking into accountthe effects of environmental conditions, contact patch widening,deformation variations and vehicle pass speed variation/

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Please cite this article as: P.A. Meehan, et al., “Does god play dice with corrugations?”: Environmental effects on growth, Wear (2013),http://dx.doi.org/10.1016/j.wear.2013.11.027i

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distribution. The conceptual block diagram for the corrugationformation model is depicted in Fig. 1 (based on [2,3,4,7,8,11],etc.),where a feedback process occurs over multiple wheelset passesdue to the interaction of vehicle/track vibration dynamics I,contact mechanics II and a damage/wear mechanism III, initiatedby an initial longitudinal rail profile irregularity.

The model has been used for considering either vertical or lateralvibration modes, interacting with longitudinal or lateral contact forces.In such a way, both straight track [24,13] and cornering conditions[9,15] may be modelled. In the present case, cornering conditions,dominated by vertical vibrations interacting with lateral traction, slipand wear variations, are modelled based on evidence from previousmodelling and field investigations at the same site [15,19]. The contactmechanics model is based on the quasistatic microslip and considerssmall linear variations about nominal non-linear operating conditionsof which the sensitivity coefficients may be derived from any quasi-static creep theory (i.e. [12,20,21,25]). Note that typical high slipcornering conditions can be modelled based on the variations abouta nominal full sliding creep curve condition. The wear model is basedon assuming that the rate of wear is proportional to the frictionalpower dissipated. Due to computational efficiency, it allows for aninvestigation into measurable reductions in growth rate that may beachieved by using different pass speed distributions, for a range ofinitial track profiles and changing frictional/wear conditions overmany vehicle passages. The field condition variations in contact patchgeometry, rainfall, humidity would primarily affect parameters asso-ciated with the wheel/rail rolling contact mechanics and wear on therail surfaces (as indicated in Fig. 1).

Essentially, the conceptual closed loop feedback system forcorrugation growth may be represented by a recursive equation ofgrouped railway parameters of the form [15,16,24]:

Znþ1

Zn¼ Kb

11þkcðRrðωÞþRwðωÞÞ

þ1 ð1Þ

where zn is the nth pass longitudinal rail profile, Kb is a groupedparameter representing the sensitivity of wear variations towheel/rail contact deflection variations, kc is the contact stiffness,and Rr and Rw are the vertical rail and wheel receptance functionsof frequency ω. Kb encapsulates the contact mechanics and wearprocess parameters and is defined by

Kb ¼ ΔzoCξPkc=Po; Δzo ¼ koQξ

2bρð2Þ

where Δzo is the steady state wear (uncorrugated rail longitudinalprofile wear per pass), CξP is the sensitivity of lateral creep to normal

force fluctuations, Po is the steady state normal force, Q is the lateraltraction force, ξ is the creep, b is the contact patch width and ρ is therail density and k0 is the wear coefficient. Details of the derivationof these results have been provided in [12,15,16,24]. Further, underthe assumptions that (1) onemode of vibration of damping constant ζidominates the response of the combined wheel/rail system and(2) lateral frictional power variations dominate in cornering, thegrowth of corrugations can be given by [15,24]

Znþ1

Zn

��¼ 1þGr ; Gri � Kb 1þ Kci

4ζi 1þ ζið Þ

� �ð3Þ

The accumulated wear Zn can be summed up over multiple passes asshown subsequently in Eqs. (7) and (8), however the growth rateparameter Gri can be determined without performing this process. Inparticular, the corrugation growth equation is in the form of com-pounding interest where Gri is the approximate exponential growthrate parameter for mode/frequency i, equivalent to an interest rate pervehicle passage. The quotient term involving the parameters Kci and ζiis a measure of the growth contribution from the vehicle/trackcorrugation mode vibration and will be determined by the character-istic properties of the structural dynamics. Kci is a grouped parameter,representing the modal sensitivity of the wheel/rail relative displace-ment to a change in input longitudinal profile. It is proportional to theratio of contact stiffness to the corrugation modal stiffness of mode i.More details are provided in [24]. This simplified but efficient modeldoes not include the effects of contact filtering for short pitchcorrugations [10,11].

For realistic railway parameters, Kb, Kci and ζiare alwayspositive valued. It is noted that an exponential growth rate isexpected only to occur initially when changes in environmental,vehicle passing speed, contact patch geometry and wear/deforma-tion conditions are small. However, like interest rates, it is a usefulmeasure of local/present growth and of slow and fast variationsfrom nominal corrugation growth conditions.

When a randomly distributed sequence of pass speeds isintroduced of distribution p(x) the expected amplitude ratio ofthe transfer function becomes [15]

Znþ1

Zn¼ exp

Z 1

�1ln Kb

11þkcðRrðV=λÞþRwðV=λÞÞ

þ1��

��pðxÞdx ð4Þ

where multiple wheel passes at varying speeds are considered byvarying the frequency as a ratio of speed V to fixed wavelength, λ.Note the contact mechanics and wear parameter Kb and hence thesteady state wear Δzo is unaffected by speed variations.

2.1. Mechanisms of environmental effects on corrugation growthrate

The field condition variations in contact patch geometry, rain-fall and humidity would primarily affect parameters associatedwith the wheel/rail rolling contact mechanics II and wear on therail surfaces III. In particular, Eqs. (1)–(4) highlight that the growthrate parameter Gr is directly proportional to the contact mechanicsand wear parameter, Kb. Therefore mechanisms for the action ofenvironmental effects are expected to act through this parameter.In particular, by inspection of Eq. (4) changes in rainfall andhumidity may be expected to cause changes in corrugation growthrate Gr via four options; (a) the wear coefficient, k0, (b) the steadystate traction, Q, (c) the steady state creep, ξ0, and/or (d) thesensitivity of creep to normal force fluctuations, CξP. These arediscussed sequentially in the following:

(a) The wear coefficient, k0, is likely to change as interfaciallubricants are known to change the wear behaviour of steelin rolling contact. For instance, recent experimental results[26,27] with liquid friction modifiers at fixed frictional power

Fig. 1. Feedback model for corrugation formation (based on [15]).

P.A. Meehan et al. / Wear ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





showed that the wear coefficient was drastically reduced.Under fixed frictional power and accounting for associatedchanges in the creep curve, the experimentally measured wearcoefficient was seen to reduce by up to 5–10 times. Water dueto rainfall or relative humidity condensate will act in a similarfashion, lowering steady and dynamic wear rates. Furtherexperimental results are required to confirm this effect.

(b) The steady state traction, Q, will be expected to change underdifferent environmental conditions via creep law changes. Inparticular, the effect of wet running conditions on the creepcurve is well discussed in the literature; see for example [21].The generic effect of a wet contact can be seen in Fig. 2.

In steady cornering the wheelset will typically take up a fixedangle of attack with respect to the rail, generating large lateralcreep. It can be seen that the traction force given by the contact fora fixed creep (arising from a fixed angle of attack) will be higherfor a dry contact than a wet contact (approximately 1.5 timeshigher for saturated creep). Through Eq. (2) this can be seen togive rise to larger steady state wear in dry conditions than wet.

(c) and (d) Changes to the creep curve will also give differentequilibrium cornering conditions, which will have effects onboth the steady state creep ξ0 and the sensitivity of creep tonormal force fluctuations CξP. The likely effect of these for abogie in cornering is not obvious and requires more detailedfield data augmented with accurate vehicle dynamics simula-tions to quantify. This is beyond the scope of the present paper.

3. Field site and methods

The field site investigated is a section of suburban track prone tocorrugations occurring on a curve with a 242m radius, a recom-mended speed of 50 km/h and a cant of 56 mm. The traffic iscomposed of 3 and 6 carriage Electrical Multiple Unit (IMU/SMU)trains, half of which stop at the previous station and half that runexpress with a recommended speed of 60 km/h. Corrugations weremeasured to have a dominant wavelength in the 90–110mm range, asdetailed in the following. As part of an ongoing study into corrugationgrowth at various sites around Australia (see [23]) measurements ofthe longitudinal rail profile at this site have been recorded over aperiod of 4 years using a Corrugation Analysis Trolley [2].

3.1. Environmental parameters measurement

Rainfall and humidity were the two environmental parameterschosen for investigation as they have been highlighted in previous

research as influencing steady wear and contact mechanics [28].Rainfall data was collected from Bureau of Meteorology records forthe ‘Brisbane’ station which was assumed to be indicative (onaverage) of the amount of rain at the suburban Brisbane site some17 km away. The cumulative rainfall observed between each CATtest was divided by the number of days in that period, to give anaverage rainfall per day for each interval. This was comparedagainst variations in the corrugation growth rate parameter Gr andsteady state wear Δz for comparison.

3.2. Rail profile measurement and analysis

CAT measurements of longitudinal rail profile were taken semi-periodically (averaging approximately every 2 months) betweeninitial grinding in September 2006 and final grinding in May 2009.These CAT measurements were used to obtain the corrugationgrowth over the entire test section as well as that initiated by aweld over the entire period. The 3rd Octave filtering techniqueswere utilised on raw CAT output files of longitudinal profile toidentify corrugation wavelength as shown in Fig. 3.

The corrugation growth was then obtained by plotting themagnitude of the RMS displacement of the dominant corrugation100 mm band for each measurement against the total number ofdriven wheel passes over that interval.

As described in Section 2, exponential corrugation growth istheoretically expected due to the feedback nature of the corruga-tion growth mechanism for static rail conditions, i.e.

Zn ¼ Z0eGrn ð5Þwhere Zn is the nth pass profile amplitude and Gr is the corruga-tion growth rate parameter. A local corrugation growth rate isdefined to capture the variation in Gr due to variations inenvironmental parameters, contact geometry, the damagemechanism, etc. over time. The local corrugation growth ratebetween the mth and nth passes can then be determined bylooking at the ratio of corrugation amplitudes between successivemeasurements, i.e.

Gr �log Zn

Zm

� �n�m

ð6Þ

Corrugation growth was also captured as initiated by a weld forcomparison using an analysis of CAT measurements near athermite weld calibrated to less regular miniprof measurements.In particular, the rate of peak to peak weld growth and steady statewear on the test site was estimated by observing the growth of the‘dip’ with a wavelength of about 80 mm around the heat affectedzone (HAZ) of a thermite weld. The methodology is based on theimpulse response of the corrugation growth rate transfer function:

ZnðωÞZ0ðωÞ

¼ ∏N

n ¼ 1

Δz0CξP

P0

kc1þkcðRrðxnωÞþRwðxnωÞÞ

þ1� �

ð7Þ

where Zn is the Fourier transform of the longitudinal rail profile onthe nth pass, and xn is the ratio of the current pass speed to the

Fig. 2. Wet and dry measured creep curves (reproduced from [21]).

Fig. 3. The 3rd Octave graph of profile from the test section for 1,053,256 passes.

P.A. Meehan et al. / Wear ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





mean speed. If the impulse response is taken, the total response isgiven by [17]

ZnðωÞZ0ðωÞ

¼ ð1þKbÞNð1þhigher order termsÞ ð8Þ

In the time domain the impulse area will grow proportionally toð1þKbÞN as detailed in [17].

The peak to trough size of the dip was determined for each CATmeasurement over the entire grinding cycle. A representativemeasurement is shown in Fig. 4.

Note the filtering range 30–300 mm was used based on theperceived reliability from the moving accelerometer based device(the CAT). The resulting weld growth can then be determined as afunction of driven wheel passes (as shown in Fig. 5). This alsoprovides a measurement from which the contact mechanics andwear parameter Kb and the steady state wear Δzo can be deter-mined directly using Eqs. (2) and (8). According to the theorypresented in Section 2, these parameters should be immune tovariations in the pass speed distribution so this effect can beisolated.

3.3. Isolating environmental variations in corrugation growth

Exponential growth of corrugations is expected only to occurwhen the effect of changes in the quasistatic conditions such as

contact patch geometry and wear/plastic deformation conditionsare small. In the field measurements these effects slowly changethe steady local corrugation growth rate Gr over time as theamplitude increases. In particular, widening of the contact patchoccurs due to conforming wear and/or deformation of wheel andrail surfaces over many wheelset passages. In addition as thecorrugations become larger in amplitude, the dynamic forcesincrease in level to conditions under which plastic deformationoccurs in addition to wear, i.e. the damage mechanism changes.Changes in the damage mechanism may also be due to non-uniform distribution of wear in relation to the corrugation profile.These effects need to be taken into account when analysing thefield data to isolate the effect of environmental variations whichoccur on a much smaller time scale. Experimental evidence [27]suggests that the broadening of the lateral contact patch widthand changes in the damage mechanism on steady state wear, overa number of wheelset passes may be described by

Δz0 ¼k0Q0ξ0

2ρb0ðnþn0Það9Þ

Since corrugation growth rate is proportional to the steady statewear (Eqs (1)–(3)) the reduction in steady state wear is equal tothe reduction in corrugation growth rate suggesting a local growthrate function of the same form. Hence a high pass filter of the formof Eq. (9) has been applied to the local corrugation growth data toisolate the effects of environmental fluctuations occurring on amuch smaller time scale (i.e. order of 103 wheelset passages ascompared to 106).

4. Results and discussions

The methods described in Section 3 were applied to obtain andanalyse the field data to identify correlations between environ-mental, steady state wear and corrugation growth variations asdetailed in the following.

Corrugation growth history was obtained first using the CATmeasured RMS displacement of the 100 mm band at multipleintervals over 2.6 years. For comparison, the growth in steady statewear was also obtained using the methods described in Section 3as shown in Fig. 5.

Fig. 5(a) depicts an initial exponential like growth in corrugationsthat is retarded slowly over time. The similarity in the trend of growthis clear when comparing Fig. 5(a) and (b) confirms that corrugationgrowth initiated from initial roughness is similar that initiated from aweld. On a smaller time scale there is some evidence of local variationsin these trends. To investigate these more directly, the local corruga-tion and steady state wear growth rates were measured as describedin Section 3.2 and are shown in Fig. 6.

The most obvious trend in Fig. 6(a) and (b) is the progressivedecrease in corrugation growth rate throughout the growth cycle dueto the effects of contact patch widening and deformation changeswith time. To take this into account, a trend-line based on contactpatch widening was fitted to the experimentally derived growth rateby assuming the power law form of Eq. (9) with a¼1.64. The similarityof the trends in Fig. 6(a) and (b) highlights a proportional dependenceof corrugation growth rate Gr to the contact mechanics and wear, Kb,and steady state wear, Δzo, parameters as theoretically defined in Eqs.(1)–(3) and in Section 2.

To isolate the environmental effects on corrugation growth,measured variations in the growth rate from this trend-line of Eq.(9) were determined and compared against measured variations inhumidity and rainfall as shown in Fig. 7. In particular, thevariations are highlighted by plotting the local growth rate as aratio of the smooth trend-line. Fig. 7 visually shows an approx-imate inverse (or negative) correlation between variations in the

Fig. 4. CAT profile (30–300 mm filtered) for heat affected zone (HAZ) in thermiteweld at test site.

Fig. 5. Growth in (a) 100 mm band 3rd Octave of measured longitudinal rail profileat test section and (b) weld/HAZ dip size.






environmental factors (rainfall and relative humidity) and corru-gation growth rate, respectively.

The statistical significance of these correlations was quantifiedas shown in Fig. 8.

Fig. 8 confirms a strong correlation (o 0.1% probability of nocorrelation based on p test) between variations in rainfalland corrugation growth rate. The correlation with humidity ismarginal (o10% probability of no correlation based on p test).Also humidity and rainfall were shown to be correlated tothemselves (R¼0.62, p¼0.03 for the interval considered), so thecorrelation from rainfall is considered the primary effect.

Correlations between variations in the environmental factorsand steady state wear calculated from the weld growth werealso determined but no significant correlations were found forthe small time scale variations. This may be due to differences inthe contact and wear mechanics in the heat affected zone near theweld. However, the strong correlations in trends shown betweencorrugation and weld growth in Fig. 5 and corrugation growth rateand contact mechanics and wear parameter (and steady statewear) in Fig. 6 over the long period shown, indicate that theseparameters may encapsulate the effects of environmental growthin the long term.

5. Implications for corrugation control

The field results shown in the previous section were applied toa corrugation case study in which both environmental and passspeed variations occurred. In particular, the same site has been the

focus of an investigation of pass speed variation on corrugationgrowth rate (see [27,,15] for details). In particular, the distributionof passing train speed was controlled such that the average speedstandard deviation over the section was controlled and increasedfrom a nominal 4.4 km/h to 7.6 km/h. This speed-based corruga-tion control system (CCS) was turned on when the corrugationamplitude exceeded approximately 40 μm as shown in Fig. 9.During this control phase (CCS ON) the vehicle passing speedstandard deviation was increased by 1.7 times, while maintainingthe same average speed as the CCS Off condition. The predictedreduction in corrugation growth rate (without consideringenvironmental variations) due to this increase in pass speeddistribution was 38% [15]. However, the field measured reductionobtained was approximately 60% as shown in Fig. 9 and Table 1.

To account for the difference in predicted and measuredperformance, measurements of rainfall during this period wereperformed in order determine if environmental factors had sig-nificantly changed. During the control period (CCS ON) the averagerainfall increased from 2.170.4 mm/day up to 4.171.2 mm/day.According to the inverse correlation identified in Fig. 7 this higherrainfall will lower the corrugation growth rate. Based on this, thelinear trend-line for correlation between corrugation growth andrainfall shown in Fig. 8 was used to predict the influence of theincreased rainfall in this period on corrugation growth. Note thatany negative predictions of growth rate have been set instead tozero in these calculations. These growth rate estimates are out-lined in Table 1 and shown compared to the observed profilegrowth in Fig. 9.

The results in Table 1 highlight a close match between thepredicted (�63%) and field (�60%) results once environmental

Fig. 6. Histories of (a) local corrugation growth rate in the 100 mm band and(b) contact mechanics and wear parameter (proportional to steady state wear)estimated from weld growth; as functions of wheelset passes.

Fig. 7. Field measured (a) rainfall and (b) relative humidity versus local corrugationgrowth rate.






effects have been taken into account. Note that the effect of rainfall orvehicle passing speed variation, individually provide a substantial dropin corrugation growth. These results highlight the importance ofconsidering environmental variations on corrugation growth. It is

noted that in this case study, the effects of contact patch wideningand/or the damage mechanism based on the results in Fig. 6, areassumed to be relatively small and slow after the CCS is activated andrelatively small as compared to the environmental effects over thesubsequent period investigated. However, it is recommended to studythis more closely in future field investigations.

6. Conclusions

Environmental effects have been investigated in the field basedon insight gained using mechanics-based modelling of thegrowth-rate of wear-type corrugations. The relationship betweenaverage daily rainfall, humidity and the growth of rail corrugationwas observed by comparing weather data to results from themonitoring of longitudinal rail profile on a suburban test site inAustralia. Corrugation growth rate (Gr) was determined by system-atically measuring the longitudinal rail profile with a CorrugationAnalysis Trolley (CAT) on a 240 m radius, narrow gauge, concrete-sleepered curve on the metropolitan network that was known tosuffer from significant short-pitch rail corrugation of about 95 mmwavelength. The field results show a strong correlation betweenvariations in rainfall and the growth rate of corrugations inthe field.

Results also showed a strong correlation between corrugationgrowth along the test site and peak to peak weld growth which ispredicted via mechanics based modelling. Contact patch wideningand changes in the the damage mechanism are shown to beimportant effects to consider as the amplitude of corrugationsgrow. The results were then applied to a field case under variationsin environmental conditions and pass speed distribution and showthat changes in growth rate were substantial and able to be(retrospectively) predicted. The results have important implica-tions for quantifying field corrugation growth and baselining theeffect of field corrugation control strategies such as frictionmodifiers and variable speed control.

Acknowledgements

The authors are grateful to the Rail CRC for the funding of thisresearch, and the support from the CRC for Rail Innovation,Queensland Rail, Steering Committee Chair John Powell and allother members of the committee.

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Table 1Corrugation growth rate predictions under different conditions.

Condition Growth rate parameter, Gr

Uncontrolled 1.27�10�6

Uncontrolled, rainfall adjusted (�46%) 0.69�10�6

Controlled (�38%) 0.78�10�6

Controlled and rainfall adjusted (�63%) 0.47�10�6

Measured (�60%) 0.5�10�6



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“does god play dice with corrugations?”: environmental effects on growth

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