Research ArticleAnalysis of Vibrations Generated by the Presence ofCorrugation in a Modeled Tram Track

Julia I Real Herraacuteiz1 Silvia Morales-Ivorra1 Clara Zamorano Martiacuten2

and Vicente Soler Basauri3

1University Institute for Multidisciplinary Mathematics Polytechnic University of Valencia 46022 Valencia Spain2Foundation for the Research and Engineering in Railways 160 Serrano 28002 Madrid Spain3Department of Applied Mathematics Polytechnic University of Valencia 46022 Valencia Spain

Correspondence should be addressed to Julia I Real Herraiz jureahertraupves

Received 7 August 2014 Revised 2 January 2015 Accepted 20 January 2015

Academic Editor Xiaojun Wang

Copyright copy 2015 Julia I Real Herraiz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In recent years there has been a significant increase in the development of the railway system Despite the huge benefits of railwaysone of the main drawbacks of this mode of transport is vibrations caused by vehicles in service especially in the case of tramscirculating in urban areas Moreover this undesirable phenomenonmay be exacerbated by the presence of irregularities in the rail-wheel contactThus an analytical model able to reproduce the vibrational behavior of a real stretch of tram track was implementedBesides a simulation of different types of corrugation was carried out by calculating in an auxiliary model the dynamic overloadsgenerated by corrugationThese dynamic overloads fed the main model to obtain the vibrations generated and then transmitted tothe track

1 Introduction

The development of new railway networks as well as theimprovement of the already existing tracks has become aworldwide priority At the same time externalities derivedfrom railway operations have been identified as a majordrawback for this mode of transport with vibrations beingone of the externalities which must be highlighted Vibra-tions caused by vehicles in service are propagated fromthe railway superstructure and they may affect people orstructures in the surrounding areas Thus major efforts havebeen focused in minimizing the potential effect of vibratorywaves on elements placed outside of the transport systembeing necessary to study their nature and mechanisms ofgeneration transmission and propagation

Railway vibrations are originated by the rail-wheel con-tact so contact surfaces of both elements have a hugeinfluence on the wave features of generated vibration asstated by [1] Tough different rail irregularities are able tomodify the running surface and therefore cause specificvibrations this study has been focused on the vibrationaleffects caused by corrugation

Corrugation is a widespread pathology which can befound in all types of railroads Reference [2] established thatgiven a particular rail profile corrugation can be consideredin one hand both as a wavelength-fixing mechanism whichdepends on the dynamic features of the railroad and as adamage mechanism induced by dynamic loads This finallyresults in a change in the rail profile determining newwavelengths as far as new damage thus constituting a cyclicand progressive process Besides these authors classified thestructural pathologies depending on the wavelength-fixingmechanism and on the damage mechanisms and proposedsome treatments to reduce them Reference [3] extendedthe research of causes of corrugation and suggested newtreatments to prevent its progress

More recently [4 5] after studying the development ofcorrugation by finite elementmodels (FEM) in different typesof railroads we concluded that curves with small radii favorthe appearance of wear with the damage placed under thosewheels circulating in the inner side of the curve

Furthermore at high speeds the performance againstcorrugation is better in slab tracks than in ballasted tracks Adifferent point of view was contributed by [6] who analyzed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 290164 8 pageshttpdxdoiorg1011552015290164

2 Mathematical Problems in Engineering

the influence of pad stiffness when corrugation appeared inthe low rail of a curve in Bilbaorsquos undergroundThose authorsconsidered a multibody model for the vehicle and applying[7] proposals predicted frequencies at which corrugation ismore likely to occur Besides they demonstrated that elasticpads reduce corrugation by suppressing some wavelengthsthat otherwise will be developed

Different prediction models for corrugation have beendescribed until now acquiring special relevance to know howimperfections in rails surface affect the vibrations inducedto the track Dynamic overloads originated by corrugationwere calculated by [8] and then implemented in a discreteFEM to assess the vibrations derived from itThey concludedthat frequencies associated with those continuous defects canbe determined by the quotient between train speed and thewavelength associated with this imperfection

By using a three-dimensional model of wheel-rail inter-action [9] predicted the evolution of wear as well as its effecton the dynamic behavior of the track According to this studybeing the speed of the vehicle and the wavelength associatedwith corrugation constants the larger the amplitude thehigher the dynamic overloads and wear derived from corru-gation Moreover with the rest of parameters being constantthe greater the corrugation wavelength the lower dynamicoverloads and wear In addition the vertical stiffness of thedifferent elements of the track was noted as a determiningfactor on the corrugation properties (ie amplitude andwavelength)

Reference [10] by using a FEM in a moving referencesystem analyzed the vehicle-track interaction On the surfaceof a rail a sinusoidal corrugation was simulated concludingthat all the dynamic responses (displacement loads andacceleration) induced by this irregularity keep this sinusoidalshape It was also found that the greater the severity of thedefect the higher the dynamic overloads generated As forthe speed its influence on the dynamic response of the trackwas demonstrated When forces in the wheel-rail contactwere measured it was found that both displacements andaccelerations grew with speed

In the present paper an analytical model based on thatused by [11] was implemented to reproduce the vibratorybehavior of a tram track Vibration data obtained in acampaign of measurements were used to calibrate the model

Once proved the ability of the model to reproduce thevibratory phenomenon from the real track corrugation willbe introduced in the model and its effect on the vibrationstransmitted to the ground will be analyzed This will requirethe use of an auxiliary model as that presented by [12]is able to calculate those dynamic overloads generated bycorrugation and transmitted to the track Thus once thedynamic overloads are obtained they will be introducedin the main model to calculate the vibrations generated bythis irregularity Furthermore a sensibility analysis of thedifferent parameters involved in the generation of dynamicoverloadswill be carried out to unravel thosewho are relevantin the vibration phenomenon and thus act on them

This model may be useful to predict vibrations generatedby the track corrugation and after establishing acceptablevibratory levels it may allow determining the optimum

RailElastomer

155 cm Concrete blocks

22 cm Reinforced concrete

25 cm Lean concrete

Ground

Figure 1

moment to carry out the maintenance operations Further-more the present analytical model can also be employedto calculate those track overloads due to corrugation whichgenerate not only an aggravation of the rail irregularities butalso premature deterioration of the rest of the track as statedby [3]

2 Analytical Modelization of the Track

Theaimof this section is to describe themodelization processof the stretch of the tramway track studied as well as itsanalytical resolution In addition the data obtained in ameasurement campaign and used to calibrate the model willbe presented Thus a model able to reproduce accurately thevibratory behavior of the studied track has been obtained andthen discussed

21 Description of the Tramway Track Stretch The cross-section of the track studied is constituted by a Ph37Nrail embedded in elastomeric material which are housedin a 155 cm thick surface layer made of concrete blocksThe surface layer is on a reinforced concrete slab of 22 cmthickness which rests on a 25 cm thick layer of lean concreteas shown in Figure 1

The tram in service on this stretch of track is a Voss-loh 4100 series as shown in Figure 2 Its service speed is35 kmhmdashwith 100 kmh being its maximum speedmdashandtransmits an axle load of 100 kN The vehicle is composed oftwo motorized passenger cars at both extremes and anotherplaced in the middle of them The distance between axles ofa same bogie is 2m and the distances between bogies are inturn 1254m 470m and 1254m

22 Description of the Track Model A bidimensional modelable to calculate stresses and displacements is consideredfollowing [11 13]

A system integrated by three layers representing thedifferent materials that compose the track is developed Asstated in the previous section a Phoenix rail and the elas-tomeric material are placed in the first layer and composed ofconcrete blocksThe second layer represents the concrete slaband the third one the lean concrete Note that an indefinitedepth for ground located under the last of these three layers(Boussinesq half space) is assumed

Mathematical Problems in Engineering 3

Figure 2

The rail has been represented as a Timoshenko beamfollowing [14] with its movement equations being thosedescribed by [3]

1205881198601205972119908

1205971199052=120597

120597119909(119860119896119866(

120597119908

120597119909minus 120579)) + 119902 (119909 119905)

1205881198681205972120579

1205971199052=120597

120597119909(119864119868

120597120579

120597119909) + 119860

119896119866(

120597119908

120597119909minus 120579)

(1)

where 119860 is the cross-sectional area 119860119896is the shear cross-

sectional area 119864 corresponds to Young modulus 119868 is theinertia in 119910-axis direction 120588 is the rail mass density 119908 is thevertical rail displacement 120579 is the angular displacement and119902(119909 119905) is the applied load depending on position and time

Otherwise the features of materials composing the layersare shown in Table 1

By assuming a low viscosity between layers and followingthe researchmade by [15] the displacements can be expressedby means of the following vectorial equation

( + 120583)nabla119909119911(nabla119909119911d) + 120583nabla2

119909119911d = 120588120597

2d1205971199052 (2)

while d = (119906(119909 119911 119905) 0 V(119909 119911 119905)) represents the displacementof each layer with 120588 being the mass density for each layerMeanwhile and 120583 are the parameters employed to describethe viscoelastic behavior of each layer as stated by [15]Thoseparameters will be calibrated by using experimental data

23 Load Modeling When modeling loads two differentsituationsmust be distinguishedOnonehand the quasistaticloads which are due to the weight of the train traveling at aspeed119881- must be taken into account On the other hand thedynamic overloads which are the result of irregularities in therail-wheel contact will be discussedThese irregularities maybe caused by defects in rails or wheels joints between rails orcorrugation

The introduction of the harmonic loads which representthe dynamic overloads in the model is made following [16]These harmonic loads can be written as 119875

119895(119905) = 119875

0119895cos(120603

119895119905)

where 1198750119895and 120603

119895represent respectively the magnitude and

the characteristic angular frequency for each loadThe effect of quasistatic loads can be introduced as a

dynamic overload with a null characteristic angular fre-quency as stated by [13] or following the Zimmermannformulation as described by [17] Because of computationaltime reasons in this paper the second procedure will befollowed

Table 1

119864 (Pa) ] 120588 (kgm3)Paving blocks 225 lowast 10and9 025 2400Reinforced concrete 273 lowast 10and10 025 2400Lean concrete 225 lowast 10and10 02 2300

According to the Zimmermann method described by[17] the displacement of the rail due to a quasistatic load canbe expressed as

1198894119911

1198891199094+119896

119864119868119911 = 0 (3)

Imposing the appropriate boundary conditions (point loadsymmetry of the deformed shape of the rail and disappear-ance of loading effect with distance) leads to the followingexpression for the vertical displacement developed by [17]

119911 =119876

2119887119862

4radic119887119862

4119864119868119890minus119909119871V (cos 119909

119871V+ sin 119909

119871V) (4)

where 119876 is the transmitted load 119887 is the gauge 119862 is anequivalent ballast coefficient and 119871V is an elastic lengthdefined as

119871V =4radic119887119862

4119864119868 (5)

Once rail vertical displacements have been calculated byrenaming 119909 = 119881 sdot 119905 where 119881 is the tram speed and 119905 is timethey are differentiated twice with respect to time to obtain thedesired accelerations

24 Analytical Solution Following [16] in order to solve thehigh complexity that often implies integration of movementequations the Lame potentials and Fourier Transformwill beused Thus the movement of the beam representing the railcan be written as

(1198601198701198661198962(1 +

119860119870119866

1198681205881205962 minus 119860119870119866 minus 1198641198681198962

) minus 1205881198601205962) 119908 (119896 120596)

= 119902 (119896 120596) minus 1198861205901198851198851(119896 0 120596)

(6)

And the expressions for layers motion expressed in terms ofLame potentials are

1198892 120593

1198891199112minus 1198772

119871

120593 = 0

1198892 120595

1198891199112minus 1198772

119879

120595 = 0

(7)

where 120593 and 120595 are the Lame potentials expressed in thefrequency-wave number domain The 119877

119871

2 and 119877119879

2 functions

4 Mathematical Problems in Engineering

represent the longitudinal and shear wave transmissionspeed respectively with their expressions being as follows

1198772

119871= 1198962minus

1205962

1198882119871minus 119894120596 (120582lowast + 2120583lowast) 120588

1198772

119879= 1198962minus

1205962

1198882119879minus 119894120596120583lowast120588

(8)

where 120582lowast and 120583lowast are the parameters which define the

damping of the track and 119888119871and 119888119879represent the velocities

of the longitudinal and shear waves in the layerThen the expressions for the displacements (9) and (10)

and stresses (11) and (12) can be written as

119906 (119896 119911 120596) = 119894119896120593 minus119889120595

119889119911 (9)

V (119896 119911 120596) =119889120593

119889119911+ 119894119896120595 (10)

120590119911119911(119896 119911 120596) =

120582(

1198892 120593

1198891199112minus 1198962 120593) + 2120583(

1198892 120593

1198891199112minus 119894119896

119889120595

119889119911) (11)

120590119911119909(119896 119911 120596) = 120583(2119894119896

119889120593

119889119911+1198892 120595

1198891199112+ 1198962 120595) (12)

Solving the system (7) and replacing 120593 and 120595 in (9)ndash(12)

119906119895(119896 119911 120596) = 119894119896 (119860

1119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 119877119879119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

V119895(119896 119911 120596) = 119877

119871119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

minus 119894119896 (1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119885119895(119896 119911 120596) = 119862

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198622119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119909119895(119896 119911 120596) = 119863

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198632119895

(1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

(13)

where

1198621119895

= (120582119895+ 2120583119895)1198772

119871119895minus120582119895119896

1198622119895

= minus2119894119896120583119895119877119879119895

1198631119895

= 2119894119896120583119895119877119871119895

1198632119895

= 120583119895(1198962+ 1198772

119879119895

)

(14)

Boundary conditions are set so that the displacementsbetween adjacent layers are equal and stresses between themare balanced Furthermore no horizontal rail movement andthe vanishing of radiated vibrations as the soil depth increasesare assumed

Therefore replacing the boundary condition of each layerin (13) follows the expression M sdot A = q where M isa coefficient matrix A is the vector of coefficients to becalculated and q is the vector of loads acting on the system

Following [15] once the motion and stresses expressionshave been obtained in terms of deepness and in frequency-wave number domain they are returned to the space-timedomain by using the inverse Fourier transform In order tooptimize this process the displacements are written in termsof the wave number domain and then the expression ofthe inverse Fourier transform is converted into an additionfollowing [14]

The accelerations in each layer are obtained by dif-ferentiating twice with respect to time the displacementexpressions from the inverse Fourier transform

Total accelerations generated and transmitted to thesurrounding ground are obtained through superposition ofquasistatic and dynamic overloads

In order to provide reliability to the model a measure-ment camping was carried out as aforementioned Dataobtained by triaxial accelerometers FastTracer Sequoia havebeen processed and compared with those calculated in themodel After calibrating the damping parameters of thelayers the result of comparison between model and real datais shown in Figure 3

Figure 3 shows the ability of the model to represent thepeaks of vibrations and clearly reproduce the effect of passingpassenger cars and their bogies through the stretch of thetrack However as shown in Figure 3 measured data (green)is much noisier than the modeled one (red) These noisieraccelerogram recordings may be due to the numerous factorsintervening in the real vibratory phenomenon such as railsand wheels imperfections In contrast only few harmonicloads have been considered in the model to simulate theseirregularities Amplitudes and angular frequencies of theseharmonic loads have been obtained after analyzing theFourier spectrum of the registered data taking their moresignificant values To more accurate results a larger numberof harmonics might be considered with the drawback of anincreased computational time

It must be highlighted that due to the typology of trackstudied it was not possible to set the accelerometers in therail so they were placed next to the elastomeric materialwhere the rail was housed Then the data registered withthe accelerometers were influenced by the presence of thiselastomeric material

3 Analysis of Corrugation

Simulation and analysis of the effect of corrugation in themodel presented in the previous section will be addressedThe procedure will be as follows dynamic overloads causedby the presence of corrugation on the rail will be calculated byusing an auxiliary quarter car model and then the influence

Mathematical Problems in Engineering 5

minus3minus2minus1

01234

minus6 minus4 minus2 0 2 4 6 8 10t (s)

a(m

s2)

Figure 3

in dynamic overloads of parameters which define rail irregu-larities will be discussed

Dynamic overloads generate an increase of vibrations andwear with the first one being the objective of this researchThus overloads will feed the model described in Section 2as harmonic forces in order to obtain the accelerationsgenerated by rail irregularities

31 Quarter Car Model The vehicle can be modeled in dif-ferent ways depending on the number of degrees of freedomgiven to the system it can range from hundreds of degrees offreedom allowed by commercial multibody systems (MBS) tosystems with a single degree of freedom constituted by amassattached to the track by a spring

Between both extreme cases the following can be found(i) the quarter car model which will be presented below (ii)the half car system it takes into account the sprung massesthe semisprung masses and the unsprung masses and (iii)the full car models these allow us to take into account notonly the displacements of masses but also their rotation withrespect to each axis

Nevertheless from the point of view of the main modeland the fact that the loads introduced only have a verticalcomponent using a full bogie model would be an unneces-sary computational expense Moreover [18] experimentallydemonstrated that the influence on the dynamic stresses ofthe sprung and semisprung masses is negligible compared tothe unsprung masses

Therefore to calculate the dynamic overloads generatedby corrugation a quarter car system will be implementedusing MATLAB software In this model the masses of thevehicle are discomposed in sprung and unsprung massesThe compatibility of strengths and displacements betweenboth masses are solved by a spring and a damping elementMeanwhile the strength transmission between the unsprungmasses and the rail is performed by an equivalent springwhich takes into account not only the stiffness of the rail-wheelHertzian contact but also the stiffness of the underlyingtrack

The expressions governing the behavior of the quarter carmodel are as follows

119898211990910158401015840

2+ 1198962(1199092minus 1199091) + 1198882(1199091015840

2minus 1199091015840

1) = 0

119898111990910158401015840

1minus 11988821199091015840

2+ 11988821199091015840

1minus 11989621199092+ (1198962+ 1198961) 1199091minus 1198961119911 = 0

(15)

x2

x1

m1

k1

k2 c2

m2

z

Figure 4

Table 2

Quarter car model values1198981(kg) 500

1198961(Nm) 32000000

1198982(kg) 4000

1198962(Nm) 501745

1198882(Nsdotsm) 875

where 1198981 unsprung mass per wheel 119896

1 contact stiffness

1198982 sprung mass per wheel 119896

2 primary stiffness 119888

2 primary

damping 119909119894 displacement of the 119894th mass 1199091015840

119894 velocity of the

119894th mass and 11990910158401015840119894 acceleration of the 119894th mass

In Figure 4 a quarter car model sketch can be seenIn Table 2 the values for the mass stiffness and damping

are shownProblem resolution is performed by the Laplace transfer-

ence function and once the accelerations of bothmasses havebeen obtained dynamic overloads are calculated as follows

119865dyn = 119898111990910158401015840

1+ 119898211990910158401015840

2 (16)

32 SensitivityAnalysis Real rail profilesmay be compared toa sum of sinusoids of different amplitudes and wavelengthsTherefore in order to analyze the effect of a particularcorrugation phenomenon the profile of the affected track ismodeled as a sinusoidal function which is defined by wearamplitude and wavelength Furthermore the speed of thetrain in service is needed to be taken into account due tothe dynamic performance of the analysis When studying theinfluence of these parameters on the dynamic overloads anamplitude 119860 = 025mm a wavelength 120582 = 05m and a speed119881= 50 kmh are setThen with two of these parameters beingfixed the third one will be modified in order to analyze itsinfluence on dynamic overloads

It must be highlighted that overload values given for eachamplitude speed and wavelength correspond to the maxi-mum absolute value of the stationary part of the overloadfunction The reason for this choice is to consider the mostunfavorable case from the point of view of track designmaintenance and security

6 Mathematical Problems in Engineering

0

20

40

60

80

100

120

0 20 40 60 80 100

Dyn

amic

ove

rload

(kN

)

Vehicle speed (kmh)

Figure 5

0 02 04 06 08 1 12

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

000E + 00

200E + 02

400E + 02

600E + 02

800E + 02

100E + 03

120E + 03

140E + 03

Figure 6

Figure 5 shows the influence of the speed of the tramon dynamic overloads being the amplitude and wavelengthof corrugation 119860 = 025mm and 120582 = 05m From thisanalysis it can be affirmed that dynamic overloads growwhenincreasing the speed of the vehicle

On the other hand the influence of wavelength ondynamic overloads being 119860 = 025mm and 119881 = 50 kmhis shown in Figure 6 In this graph the peak obtained whenthe wavelength is about 01mmust be highlighted Following[19] natural frequencies of the system are calculated as theroot of the eigenvalues of

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198961+ 1198962

1198981

minus1198962

1198981

minus1198962

1198982

1198962

1198982

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then the natural frequencies are 8949302 rads and94815 rads the wavelength being associated with firstnatural frequency 120582 = 00975m which corresponds to thewavelength estimated from Figure 6Then it can be said thatfor this wavelengthmdashbeing the speed of the vehicle and trainand track features constantsmdasha resonance phenomenon willappear

FromFigure 7 it can be noticed that far from thosewave-lengths affected by the resonance phenomenon dynamicoverloads decrease when the wavelength of the irregularityincreases That is the farther the distance between twoconsecutive irregularities the lower the forces transmitted tothe track

35

36

37

38

39

40

41

42

0 05 1 15 2 25 3 35

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

Figure 7

Figure 8 shows the influence of wear amplitude ondynamic overloads being 120582 = 05m and 119881 = 50 kmh Thisleads us to affirm that the higher the amplitude the greaterthe overload Furthermore it must be noticed that overloadsand corrugation amplitude are linearly related

Taking into account the strong nonlinear behavior of thevehicle-track system the linear behavior of the amplitudeof corrugation must be explained In order to assess thevalidity of the results obtained with the quarter car model amultibody model has been implemented with the VAMPIREcommercial software and the influence of the amplitude hasbeen studied The results are exposed in Figure 9

From Figure 9 two different behaviours of the relationamplitude-dynamic overloads are shown On one handwhen the amplitude of corrugation is lower than 1mm theoverloads generated by this pathology show a quasilinearbehaviour for both models On the other hand when theamplitude exceeds this value a nonlinear behaviour canbe appreciated for the multibody model but its behaviourremains quasilinear when dynamic overloads are calculatedwith the quarter car model

Thus it can be concluded that the quarter car modeldescribed in this paper only is able to be used when theamplitude of the defect considered is lower than 1mm

In conclusion since it is not feasible that amplitudeshigher than 1mm appear in a tram track as the one proposedin this study and the quarter carmodel presents the advantageof its simplicity when comparing with the multibody modelthe use of this model has been justified Nevertheless beforeusing this model for other types of railroads where higheramplitudes may be reached a preliminary study must becarried out

33 Definition of Different Scenarios Below a simulation ofdifferent scenarios will be performed Irregularities simulatedby the model will be added to the imperfections alreadypresent in the calibrated track The values used in thesesimulations are shown in Table 3

Note that in themodel a punctual contact betweenwheelsand rails is assumed Therefore when simulating wear ofa very short wavelength and high amplitude it would benecessary to check the veracity of this hypothesis Thus it is

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the influence of pad stiffness when corrugation appeared inthe low rail of a curve in Bilbaorsquos undergroundThose authorsconsidered a multibody model for the vehicle and applying[7] proposals predicted frequencies at which corrugation ismore likely to occur Besides they demonstrated that elasticpads reduce corrugation by suppressing some wavelengthsthat otherwise will be developed

Different prediction models for corrugation have beendescribed until now acquiring special relevance to know howimperfections in rails surface affect the vibrations inducedto the track Dynamic overloads originated by corrugationwere calculated by [8] and then implemented in a discreteFEM to assess the vibrations derived from itThey concludedthat frequencies associated with those continuous defects canbe determined by the quotient between train speed and thewavelength associated with this imperfection

By using a three-dimensional model of wheel-rail inter-action [9] predicted the evolution of wear as well as its effecton the dynamic behavior of the track According to this studybeing the speed of the vehicle and the wavelength associatedwith corrugation constants the larger the amplitude thehigher the dynamic overloads and wear derived from corru-gation Moreover with the rest of parameters being constantthe greater the corrugation wavelength the lower dynamicoverloads and wear In addition the vertical stiffness of thedifferent elements of the track was noted as a determiningfactor on the corrugation properties (ie amplitude andwavelength)

Reference [10] by using a FEM in a moving referencesystem analyzed the vehicle-track interaction On the surfaceof a rail a sinusoidal corrugation was simulated concludingthat all the dynamic responses (displacement loads andacceleration) induced by this irregularity keep this sinusoidalshape It was also found that the greater the severity of thedefect the higher the dynamic overloads generated As forthe speed its influence on the dynamic response of the trackwas demonstrated When forces in the wheel-rail contactwere measured it was found that both displacements andaccelerations grew with speed

In the present paper an analytical model based on thatused by [11] was implemented to reproduce the vibratorybehavior of a tram track Vibration data obtained in acampaign of measurements were used to calibrate the model

Once proved the ability of the model to reproduce thevibratory phenomenon from the real track corrugation willbe introduced in the model and its effect on the vibrationstransmitted to the ground will be analyzed This will requirethe use of an auxiliary model as that presented by [12]is able to calculate those dynamic overloads generated bycorrugation and transmitted to the track Thus once thedynamic overloads are obtained they will be introducedin the main model to calculate the vibrations generated bythis irregularity Furthermore a sensibility analysis of thedifferent parameters involved in the generation of dynamicoverloadswill be carried out to unravel thosewho are relevantin the vibration phenomenon and thus act on them

This model may be useful to predict vibrations generatedby the track corrugation and after establishing acceptablevibratory levels it may allow determining the optimum

RailElastomer

155 cm Concrete blocks

22 cm Reinforced concrete

25 cm Lean concrete

Ground

Figure 1

moment to carry out the maintenance operations Further-more the present analytical model can also be employedto calculate those track overloads due to corrugation whichgenerate not only an aggravation of the rail irregularities butalso premature deterioration of the rest of the track as statedby [3]

2 Analytical Modelization of the Track

Theaimof this section is to describe themodelization processof the stretch of the tramway track studied as well as itsanalytical resolution In addition the data obtained in ameasurement campaign and used to calibrate the model willbe presented Thus a model able to reproduce accurately thevibratory behavior of the studied track has been obtained andthen discussed

21 Description of the Tramway Track Stretch The cross-section of the track studied is constituted by a Ph37Nrail embedded in elastomeric material which are housedin a 155 cm thick surface layer made of concrete blocksThe surface layer is on a reinforced concrete slab of 22 cmthickness which rests on a 25 cm thick layer of lean concreteas shown in Figure 1

The tram in service on this stretch of track is a Voss-loh 4100 series as shown in Figure 2 Its service speed is35 kmhmdashwith 100 kmh being its maximum speedmdashandtransmits an axle load of 100 kN The vehicle is composed oftwo motorized passenger cars at both extremes and anotherplaced in the middle of them The distance between axles ofa same bogie is 2m and the distances between bogies are inturn 1254m 470m and 1254m

22 Description of the Track Model A bidimensional modelable to calculate stresses and displacements is consideredfollowing [11 13]

A system integrated by three layers representing thedifferent materials that compose the track is developed Asstated in the previous section a Phoenix rail and the elas-tomeric material are placed in the first layer and composed ofconcrete blocksThe second layer represents the concrete slaband the third one the lean concrete Note that an indefinitedepth for ground located under the last of these three layers(Boussinesq half space) is assumed

Mathematical Problems in Engineering 3

Figure 2

The rail has been represented as a Timoshenko beamfollowing [14] with its movement equations being thosedescribed by [3]

1205881198601205972119908

1205971199052=120597

120597119909(119860119896119866(

120597119908

120597119909minus 120579)) + 119902 (119909 119905)

1205881198681205972120579

1205971199052=120597

120597119909(119864119868

120597120579

120597119909) + 119860

119896119866(

120597119908

120597119909minus 120579)

(1)

where 119860 is the cross-sectional area 119860119896is the shear cross-

sectional area 119864 corresponds to Young modulus 119868 is theinertia in 119910-axis direction 120588 is the rail mass density 119908 is thevertical rail displacement 120579 is the angular displacement and119902(119909 119905) is the applied load depending on position and time

Otherwise the features of materials composing the layersare shown in Table 1

By assuming a low viscosity between layers and followingthe researchmade by [15] the displacements can be expressedby means of the following vectorial equation

( + 120583)nabla119909119911(nabla119909119911d) + 120583nabla2

119909119911d = 120588120597

2d1205971199052 (2)

while d = (119906(119909 119911 119905) 0 V(119909 119911 119905)) represents the displacementof each layer with 120588 being the mass density for each layerMeanwhile and 120583 are the parameters employed to describethe viscoelastic behavior of each layer as stated by [15]Thoseparameters will be calibrated by using experimental data

23 Load Modeling When modeling loads two differentsituationsmust be distinguishedOnonehand the quasistaticloads which are due to the weight of the train traveling at aspeed119881- must be taken into account On the other hand thedynamic overloads which are the result of irregularities in therail-wheel contact will be discussedThese irregularities maybe caused by defects in rails or wheels joints between rails orcorrugation

The introduction of the harmonic loads which representthe dynamic overloads in the model is made following [16]These harmonic loads can be written as 119875

119895(119905) = 119875

0119895cos(120603

119895119905)

where 1198750119895and 120603

119895represent respectively the magnitude and

the characteristic angular frequency for each loadThe effect of quasistatic loads can be introduced as a

dynamic overload with a null characteristic angular fre-quency as stated by [13] or following the Zimmermannformulation as described by [17] Because of computationaltime reasons in this paper the second procedure will befollowed

Table 1

119864 (Pa) ] 120588 (kgm3)Paving blocks 225 lowast 10and9 025 2400Reinforced concrete 273 lowast 10and10 025 2400Lean concrete 225 lowast 10and10 02 2300

According to the Zimmermann method described by[17] the displacement of the rail due to a quasistatic load canbe expressed as

1198894119911

1198891199094+119896

119864119868119911 = 0 (3)

Imposing the appropriate boundary conditions (point loadsymmetry of the deformed shape of the rail and disappear-ance of loading effect with distance) leads to the followingexpression for the vertical displacement developed by [17]

119911 =119876

2119887119862

4radic119887119862

4119864119868119890minus119909119871V (cos 119909

119871V+ sin 119909

119871V) (4)

where 119876 is the transmitted load 119887 is the gauge 119862 is anequivalent ballast coefficient and 119871V is an elastic lengthdefined as

119871V =4radic119887119862

4119864119868 (5)

Once rail vertical displacements have been calculated byrenaming 119909 = 119881 sdot 119905 where 119881 is the tram speed and 119905 is timethey are differentiated twice with respect to time to obtain thedesired accelerations

24 Analytical Solution Following [16] in order to solve thehigh complexity that often implies integration of movementequations the Lame potentials and Fourier Transformwill beused Thus the movement of the beam representing the railcan be written as

(1198601198701198661198962(1 +

119860119870119866

1198681205881205962 minus 119860119870119866 minus 1198641198681198962

) minus 1205881198601205962) 119908 (119896 120596)

= 119902 (119896 120596) minus 1198861205901198851198851(119896 0 120596)

(6)

And the expressions for layers motion expressed in terms ofLame potentials are

1198892 120593

1198891199112minus 1198772

119871

120593 = 0

1198892 120595

1198891199112minus 1198772

119879

120595 = 0

(7)

where 120593 and 120595 are the Lame potentials expressed in thefrequency-wave number domain The 119877

119871

2 and 119877119879

2 functions

4 Mathematical Problems in Engineering

represent the longitudinal and shear wave transmissionspeed respectively with their expressions being as follows

1198772

119871= 1198962minus

1205962

1198882119871minus 119894120596 (120582lowast + 2120583lowast) 120588

1198772

119879= 1198962minus

1205962

1198882119879minus 119894120596120583lowast120588

(8)

where 120582lowast and 120583lowast are the parameters which define the

damping of the track and 119888119871and 119888119879represent the velocities

of the longitudinal and shear waves in the layerThen the expressions for the displacements (9) and (10)

and stresses (11) and (12) can be written as

119906 (119896 119911 120596) = 119894119896120593 minus119889120595

119889119911 (9)

V (119896 119911 120596) =119889120593

119889119911+ 119894119896120595 (10)

120590119911119911(119896 119911 120596) =

120582(

1198892 120593

1198891199112minus 1198962 120593) + 2120583(

1198892 120593

1198891199112minus 119894119896

119889120595

119889119911) (11)

120590119911119909(119896 119911 120596) = 120583(2119894119896

119889120593

119889119911+1198892 120595

1198891199112+ 1198962 120595) (12)

Solving the system (7) and replacing 120593 and 120595 in (9)ndash(12)

119906119895(119896 119911 120596) = 119894119896 (119860

1119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 119877119879119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

V119895(119896 119911 120596) = 119877

119871119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

minus 119894119896 (1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119885119895(119896 119911 120596) = 119862

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198622119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119909119895(119896 119911 120596) = 119863

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198632119895

(1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

(13)

where

1198621119895

= (120582119895+ 2120583119895)1198772

119871119895minus120582119895119896

1198622119895

= minus2119894119896120583119895119877119879119895

1198631119895

= 2119894119896120583119895119877119871119895

1198632119895

= 120583119895(1198962+ 1198772

119879119895

)

(14)

Boundary conditions are set so that the displacementsbetween adjacent layers are equal and stresses between themare balanced Furthermore no horizontal rail movement andthe vanishing of radiated vibrations as the soil depth increasesare assumed

Therefore replacing the boundary condition of each layerin (13) follows the expression M sdot A = q where M isa coefficient matrix A is the vector of coefficients to becalculated and q is the vector of loads acting on the system

Following [15] once the motion and stresses expressionshave been obtained in terms of deepness and in frequency-wave number domain they are returned to the space-timedomain by using the inverse Fourier transform In order tooptimize this process the displacements are written in termsof the wave number domain and then the expression ofthe inverse Fourier transform is converted into an additionfollowing [14]

The accelerations in each layer are obtained by dif-ferentiating twice with respect to time the displacementexpressions from the inverse Fourier transform

Total accelerations generated and transmitted to thesurrounding ground are obtained through superposition ofquasistatic and dynamic overloads

In order to provide reliability to the model a measure-ment camping was carried out as aforementioned Dataobtained by triaxial accelerometers FastTracer Sequoia havebeen processed and compared with those calculated in themodel After calibrating the damping parameters of thelayers the result of comparison between model and real datais shown in Figure 3

Figure 3 shows the ability of the model to represent thepeaks of vibrations and clearly reproduce the effect of passingpassenger cars and their bogies through the stretch of thetrack However as shown in Figure 3 measured data (green)is much noisier than the modeled one (red) These noisieraccelerogram recordings may be due to the numerous factorsintervening in the real vibratory phenomenon such as railsand wheels imperfections In contrast only few harmonicloads have been considered in the model to simulate theseirregularities Amplitudes and angular frequencies of theseharmonic loads have been obtained after analyzing theFourier spectrum of the registered data taking their moresignificant values To more accurate results a larger numberof harmonics might be considered with the drawback of anincreased computational time

It must be highlighted that due to the typology of trackstudied it was not possible to set the accelerometers in therail so they were placed next to the elastomeric materialwhere the rail was housed Then the data registered withthe accelerometers were influenced by the presence of thiselastomeric material

3 Analysis of Corrugation

Simulation and analysis of the effect of corrugation in themodel presented in the previous section will be addressedThe procedure will be as follows dynamic overloads causedby the presence of corrugation on the rail will be calculated byusing an auxiliary quarter car model and then the influence

Mathematical Problems in Engineering 5

minus3minus2minus1

01234

minus6 minus4 minus2 0 2 4 6 8 10t (s)

a(m

s2)

Figure 3

in dynamic overloads of parameters which define rail irregu-larities will be discussed

Dynamic overloads generate an increase of vibrations andwear with the first one being the objective of this researchThus overloads will feed the model described in Section 2as harmonic forces in order to obtain the accelerationsgenerated by rail irregularities

31 Quarter Car Model The vehicle can be modeled in dif-ferent ways depending on the number of degrees of freedomgiven to the system it can range from hundreds of degrees offreedom allowed by commercial multibody systems (MBS) tosystems with a single degree of freedom constituted by amassattached to the track by a spring

Between both extreme cases the following can be found(i) the quarter car model which will be presented below (ii)the half car system it takes into account the sprung massesthe semisprung masses and the unsprung masses and (iii)the full car models these allow us to take into account notonly the displacements of masses but also their rotation withrespect to each axis

Nevertheless from the point of view of the main modeland the fact that the loads introduced only have a verticalcomponent using a full bogie model would be an unneces-sary computational expense Moreover [18] experimentallydemonstrated that the influence on the dynamic stresses ofthe sprung and semisprung masses is negligible compared tothe unsprung masses

Therefore to calculate the dynamic overloads generatedby corrugation a quarter car system will be implementedusing MATLAB software In this model the masses of thevehicle are discomposed in sprung and unsprung massesThe compatibility of strengths and displacements betweenboth masses are solved by a spring and a damping elementMeanwhile the strength transmission between the unsprungmasses and the rail is performed by an equivalent springwhich takes into account not only the stiffness of the rail-wheelHertzian contact but also the stiffness of the underlyingtrack

The expressions governing the behavior of the quarter carmodel are as follows

119898211990910158401015840

2+ 1198962(1199092minus 1199091) + 1198882(1199091015840

2minus 1199091015840

1) = 0

119898111990910158401015840

1minus 11988821199091015840

2+ 11988821199091015840

1minus 11989621199092+ (1198962+ 1198961) 1199091minus 1198961119911 = 0

(15)

x2

x1

m1

k1

k2 c2

m2

z

Figure 4

Table 2

Quarter car model values1198981(kg) 500

1198961(Nm) 32000000

1198982(kg) 4000

1198962(Nm) 501745

1198882(Nsdotsm) 875

where 1198981 unsprung mass per wheel 119896

1 contact stiffness

1198982 sprung mass per wheel 119896

2 primary stiffness 119888

2 primary

damping 119909119894 displacement of the 119894th mass 1199091015840

119894 velocity of the

119894th mass and 11990910158401015840119894 acceleration of the 119894th mass

In Figure 4 a quarter car model sketch can be seenIn Table 2 the values for the mass stiffness and damping

are shownProblem resolution is performed by the Laplace transfer-

ence function and once the accelerations of bothmasses havebeen obtained dynamic overloads are calculated as follows

119865dyn = 119898111990910158401015840

1+ 119898211990910158401015840

2 (16)

32 SensitivityAnalysis Real rail profilesmay be compared toa sum of sinusoids of different amplitudes and wavelengthsTherefore in order to analyze the effect of a particularcorrugation phenomenon the profile of the affected track ismodeled as a sinusoidal function which is defined by wearamplitude and wavelength Furthermore the speed of thetrain in service is needed to be taken into account due tothe dynamic performance of the analysis When studying theinfluence of these parameters on the dynamic overloads anamplitude 119860 = 025mm a wavelength 120582 = 05m and a speed119881= 50 kmh are setThen with two of these parameters beingfixed the third one will be modified in order to analyze itsinfluence on dynamic overloads

It must be highlighted that overload values given for eachamplitude speed and wavelength correspond to the maxi-mum absolute value of the stationary part of the overloadfunction The reason for this choice is to consider the mostunfavorable case from the point of view of track designmaintenance and security

6 Mathematical Problems in Engineering

0

20

40

60

80

100

120

0 20 40 60 80 100

Dyn

amic

ove

rload

(kN

)

Vehicle speed (kmh)

Figure 5

0 02 04 06 08 1 12

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

000E + 00

200E + 02

400E + 02

600E + 02

800E + 02

100E + 03

120E + 03

140E + 03

Figure 6

Figure 5 shows the influence of the speed of the tramon dynamic overloads being the amplitude and wavelengthof corrugation 119860 = 025mm and 120582 = 05m From thisanalysis it can be affirmed that dynamic overloads growwhenincreasing the speed of the vehicle

On the other hand the influence of wavelength ondynamic overloads being 119860 = 025mm and 119881 = 50 kmhis shown in Figure 6 In this graph the peak obtained whenthe wavelength is about 01mmust be highlighted Following[19] natural frequencies of the system are calculated as theroot of the eigenvalues of

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198961+ 1198962

1198981

minus1198962

1198981

minus1198962

1198982

1198962

1198982

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then the natural frequencies are 8949302 rads and94815 rads the wavelength being associated with firstnatural frequency 120582 = 00975m which corresponds to thewavelength estimated from Figure 6Then it can be said thatfor this wavelengthmdashbeing the speed of the vehicle and trainand track features constantsmdasha resonance phenomenon willappear

FromFigure 7 it can be noticed that far from thosewave-lengths affected by the resonance phenomenon dynamicoverloads decrease when the wavelength of the irregularityincreases That is the farther the distance between twoconsecutive irregularities the lower the forces transmitted tothe track

35

36

37

38

39

40

41

42

0 05 1 15 2 25 3 35

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

Figure 7

Figure 8 shows the influence of wear amplitude ondynamic overloads being 120582 = 05m and 119881 = 50 kmh Thisleads us to affirm that the higher the amplitude the greaterthe overload Furthermore it must be noticed that overloadsand corrugation amplitude are linearly related

Taking into account the strong nonlinear behavior of thevehicle-track system the linear behavior of the amplitudeof corrugation must be explained In order to assess thevalidity of the results obtained with the quarter car model amultibody model has been implemented with the VAMPIREcommercial software and the influence of the amplitude hasbeen studied The results are exposed in Figure 9

From Figure 9 two different behaviours of the relationamplitude-dynamic overloads are shown On one handwhen the amplitude of corrugation is lower than 1mm theoverloads generated by this pathology show a quasilinearbehaviour for both models On the other hand when theamplitude exceeds this value a nonlinear behaviour canbe appreciated for the multibody model but its behaviourremains quasilinear when dynamic overloads are calculatedwith the quarter car model

Thus it can be concluded that the quarter car modeldescribed in this paper only is able to be used when theamplitude of the defect considered is lower than 1mm

In conclusion since it is not feasible that amplitudeshigher than 1mm appear in a tram track as the one proposedin this study and the quarter carmodel presents the advantageof its simplicity when comparing with the multibody modelthe use of this model has been justified Nevertheless beforeusing this model for other types of railroads where higheramplitudes may be reached a preliminary study must becarried out

33 Definition of Different Scenarios Below a simulation ofdifferent scenarios will be performed Irregularities simulatedby the model will be added to the imperfections alreadypresent in the calibrated track The values used in thesesimulations are shown in Table 3

Note that in themodel a punctual contact betweenwheelsand rails is assumed Therefore when simulating wear ofa very short wavelength and high amplitude it would benecessary to check the veracity of this hypothesis Thus it is

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Figure 2

The rail has been represented as a Timoshenko beamfollowing [14] with its movement equations being thosedescribed by [3]

1205881198601205972119908

1205971199052=120597

120597119909(119860119896119866(

120597119908

120597119909minus 120579)) + 119902 (119909 119905)

1205881198681205972120579

1205971199052=120597

120597119909(119864119868

120597120579

120597119909) + 119860

119896119866(

120597119908

120597119909minus 120579)

(1)

where 119860 is the cross-sectional area 119860119896is the shear cross-

sectional area 119864 corresponds to Young modulus 119868 is theinertia in 119910-axis direction 120588 is the rail mass density 119908 is thevertical rail displacement 120579 is the angular displacement and119902(119909 119905) is the applied load depending on position and time

Otherwise the features of materials composing the layersare shown in Table 1

By assuming a low viscosity between layers and followingthe researchmade by [15] the displacements can be expressedby means of the following vectorial equation

( + 120583)nabla119909119911(nabla119909119911d) + 120583nabla2

119909119911d = 120588120597

2d1205971199052 (2)

while d = (119906(119909 119911 119905) 0 V(119909 119911 119905)) represents the displacementof each layer with 120588 being the mass density for each layerMeanwhile and 120583 are the parameters employed to describethe viscoelastic behavior of each layer as stated by [15]Thoseparameters will be calibrated by using experimental data

23 Load Modeling When modeling loads two differentsituationsmust be distinguishedOnonehand the quasistaticloads which are due to the weight of the train traveling at aspeed119881- must be taken into account On the other hand thedynamic overloads which are the result of irregularities in therail-wheel contact will be discussedThese irregularities maybe caused by defects in rails or wheels joints between rails orcorrugation

The introduction of the harmonic loads which representthe dynamic overloads in the model is made following [16]These harmonic loads can be written as 119875

119895(119905) = 119875

0119895cos(120603

119895119905)

where 1198750119895and 120603

119895represent respectively the magnitude and

the characteristic angular frequency for each loadThe effect of quasistatic loads can be introduced as a

dynamic overload with a null characteristic angular fre-quency as stated by [13] or following the Zimmermannformulation as described by [17] Because of computationaltime reasons in this paper the second procedure will befollowed

Table 1

119864 (Pa) ] 120588 (kgm3)Paving blocks 225 lowast 10and9 025 2400Reinforced concrete 273 lowast 10and10 025 2400Lean concrete 225 lowast 10and10 02 2300

According to the Zimmermann method described by[17] the displacement of the rail due to a quasistatic load canbe expressed as

1198894119911

1198891199094+119896

119864119868119911 = 0 (3)

Imposing the appropriate boundary conditions (point loadsymmetry of the deformed shape of the rail and disappear-ance of loading effect with distance) leads to the followingexpression for the vertical displacement developed by [17]

119911 =119876

2119887119862

4radic119887119862

4119864119868119890minus119909119871V (cos 119909

119871V+ sin 119909

119871V) (4)

where 119876 is the transmitted load 119887 is the gauge 119862 is anequivalent ballast coefficient and 119871V is an elastic lengthdefined as

119871V =4radic119887119862

4119864119868 (5)

Once rail vertical displacements have been calculated byrenaming 119909 = 119881 sdot 119905 where 119881 is the tram speed and 119905 is timethey are differentiated twice with respect to time to obtain thedesired accelerations

24 Analytical Solution Following [16] in order to solve thehigh complexity that often implies integration of movementequations the Lame potentials and Fourier Transformwill beused Thus the movement of the beam representing the railcan be written as

(1198601198701198661198962(1 +

119860119870119866

1198681205881205962 minus 119860119870119866 minus 1198641198681198962

) minus 1205881198601205962) 119908 (119896 120596)

= 119902 (119896 120596) minus 1198861205901198851198851(119896 0 120596)

(6)

And the expressions for layers motion expressed in terms ofLame potentials are

1198892 120593

1198891199112minus 1198772

119871

120593 = 0

1198892 120595

1198891199112minus 1198772

119879

120595 = 0

(7)

where 120593 and 120595 are the Lame potentials expressed in thefrequency-wave number domain The 119877

119871

2 and 119877119879

2 functions

4 Mathematical Problems in Engineering

represent the longitudinal and shear wave transmissionspeed respectively with their expressions being as follows

1198772

119871= 1198962minus

1205962

1198882119871minus 119894120596 (120582lowast + 2120583lowast) 120588

1198772

119879= 1198962minus

1205962

1198882119879minus 119894120596120583lowast120588

(8)

where 120582lowast and 120583lowast are the parameters which define the

damping of the track and 119888119871and 119888119879represent the velocities

of the longitudinal and shear waves in the layerThen the expressions for the displacements (9) and (10)

and stresses (11) and (12) can be written as

119906 (119896 119911 120596) = 119894119896120593 minus119889120595

119889119911 (9)

V (119896 119911 120596) =119889120593

119889119911+ 119894119896120595 (10)

120590119911119911(119896 119911 120596) =

120582(

1198892 120593

1198891199112minus 1198962 120593) + 2120583(

1198892 120593

1198891199112minus 119894119896

119889120595

119889119911) (11)

120590119911119909(119896 119911 120596) = 120583(2119894119896

119889120593

119889119911+1198892 120595

1198891199112+ 1198962 120595) (12)

Solving the system (7) and replacing 120593 and 120595 in (9)ndash(12)

119906119895(119896 119911 120596) = 119894119896 (119860

1119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 119877119879119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

V119895(119896 119911 120596) = 119877

119871119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

minus 119894119896 (1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119885119895(119896 119911 120596) = 119862

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198622119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119909119895(119896 119911 120596) = 119863

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198632119895

(1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

(13)

where

1198621119895

= (120582119895+ 2120583119895)1198772

119871119895minus120582119895119896

1198622119895

= minus2119894119896120583119895119877119879119895

1198631119895

= 2119894119896120583119895119877119871119895

1198632119895

= 120583119895(1198962+ 1198772

119879119895

)

(14)

Boundary conditions are set so that the displacementsbetween adjacent layers are equal and stresses between themare balanced Furthermore no horizontal rail movement andthe vanishing of radiated vibrations as the soil depth increasesare assumed

Therefore replacing the boundary condition of each layerin (13) follows the expression M sdot A = q where M isa coefficient matrix A is the vector of coefficients to becalculated and q is the vector of loads acting on the system

Following [15] once the motion and stresses expressionshave been obtained in terms of deepness and in frequency-wave number domain they are returned to the space-timedomain by using the inverse Fourier transform In order tooptimize this process the displacements are written in termsof the wave number domain and then the expression ofthe inverse Fourier transform is converted into an additionfollowing [14]

The accelerations in each layer are obtained by dif-ferentiating twice with respect to time the displacementexpressions from the inverse Fourier transform

Total accelerations generated and transmitted to thesurrounding ground are obtained through superposition ofquasistatic and dynamic overloads

In order to provide reliability to the model a measure-ment camping was carried out as aforementioned Dataobtained by triaxial accelerometers FastTracer Sequoia havebeen processed and compared with those calculated in themodel After calibrating the damping parameters of thelayers the result of comparison between model and real datais shown in Figure 3

Figure 3 shows the ability of the model to represent thepeaks of vibrations and clearly reproduce the effect of passingpassenger cars and their bogies through the stretch of thetrack However as shown in Figure 3 measured data (green)is much noisier than the modeled one (red) These noisieraccelerogram recordings may be due to the numerous factorsintervening in the real vibratory phenomenon such as railsand wheels imperfections In contrast only few harmonicloads have been considered in the model to simulate theseirregularities Amplitudes and angular frequencies of theseharmonic loads have been obtained after analyzing theFourier spectrum of the registered data taking their moresignificant values To more accurate results a larger numberof harmonics might be considered with the drawback of anincreased computational time

It must be highlighted that due to the typology of trackstudied it was not possible to set the accelerometers in therail so they were placed next to the elastomeric materialwhere the rail was housed Then the data registered withthe accelerometers were influenced by the presence of thiselastomeric material

3 Analysis of Corrugation

Simulation and analysis of the effect of corrugation in themodel presented in the previous section will be addressedThe procedure will be as follows dynamic overloads causedby the presence of corrugation on the rail will be calculated byusing an auxiliary quarter car model and then the influence

Mathematical Problems in Engineering 5

minus3minus2minus1

01234

minus6 minus4 minus2 0 2 4 6 8 10t (s)

a(m

s2)

Figure 3

in dynamic overloads of parameters which define rail irregu-larities will be discussed

Dynamic overloads generate an increase of vibrations andwear with the first one being the objective of this researchThus overloads will feed the model described in Section 2as harmonic forces in order to obtain the accelerationsgenerated by rail irregularities

31 Quarter Car Model The vehicle can be modeled in dif-ferent ways depending on the number of degrees of freedomgiven to the system it can range from hundreds of degrees offreedom allowed by commercial multibody systems (MBS) tosystems with a single degree of freedom constituted by amassattached to the track by a spring

Between both extreme cases the following can be found(i) the quarter car model which will be presented below (ii)the half car system it takes into account the sprung massesthe semisprung masses and the unsprung masses and (iii)the full car models these allow us to take into account notonly the displacements of masses but also their rotation withrespect to each axis

Nevertheless from the point of view of the main modeland the fact that the loads introduced only have a verticalcomponent using a full bogie model would be an unneces-sary computational expense Moreover [18] experimentallydemonstrated that the influence on the dynamic stresses ofthe sprung and semisprung masses is negligible compared tothe unsprung masses

Therefore to calculate the dynamic overloads generatedby corrugation a quarter car system will be implementedusing MATLAB software In this model the masses of thevehicle are discomposed in sprung and unsprung massesThe compatibility of strengths and displacements betweenboth masses are solved by a spring and a damping elementMeanwhile the strength transmission between the unsprungmasses and the rail is performed by an equivalent springwhich takes into account not only the stiffness of the rail-wheelHertzian contact but also the stiffness of the underlyingtrack

The expressions governing the behavior of the quarter carmodel are as follows

119898211990910158401015840

2+ 1198962(1199092minus 1199091) + 1198882(1199091015840

2minus 1199091015840

1) = 0

119898111990910158401015840

1minus 11988821199091015840

2+ 11988821199091015840

1minus 11989621199092+ (1198962+ 1198961) 1199091minus 1198961119911 = 0

(15)

x2

x1

m1

k1

k2 c2

m2

z

Figure 4

Table 2

Quarter car model values1198981(kg) 500

1198961(Nm) 32000000

1198982(kg) 4000

1198962(Nm) 501745

1198882(Nsdotsm) 875

where 1198981 unsprung mass per wheel 119896

1 contact stiffness

1198982 sprung mass per wheel 119896

2 primary stiffness 119888

2 primary

damping 119909119894 displacement of the 119894th mass 1199091015840

119894 velocity of the

119894th mass and 11990910158401015840119894 acceleration of the 119894th mass

In Figure 4 a quarter car model sketch can be seenIn Table 2 the values for the mass stiffness and damping

are shownProblem resolution is performed by the Laplace transfer-

ence function and once the accelerations of bothmasses havebeen obtained dynamic overloads are calculated as follows

119865dyn = 119898111990910158401015840

1+ 119898211990910158401015840

2 (16)

32 SensitivityAnalysis Real rail profilesmay be compared toa sum of sinusoids of different amplitudes and wavelengthsTherefore in order to analyze the effect of a particularcorrugation phenomenon the profile of the affected track ismodeled as a sinusoidal function which is defined by wearamplitude and wavelength Furthermore the speed of thetrain in service is needed to be taken into account due tothe dynamic performance of the analysis When studying theinfluence of these parameters on the dynamic overloads anamplitude 119860 = 025mm a wavelength 120582 = 05m and a speed119881= 50 kmh are setThen with two of these parameters beingfixed the third one will be modified in order to analyze itsinfluence on dynamic overloads

It must be highlighted that overload values given for eachamplitude speed and wavelength correspond to the maxi-mum absolute value of the stationary part of the overloadfunction The reason for this choice is to consider the mostunfavorable case from the point of view of track designmaintenance and security

6 Mathematical Problems in Engineering

0

20

40

60

80

100

120

0 20 40 60 80 100

Dyn

amic

ove

rload

(kN

)

Vehicle speed (kmh)

Figure 5

0 02 04 06 08 1 12

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

000E + 00

200E + 02

400E + 02

600E + 02

800E + 02

100E + 03

120E + 03

140E + 03

Figure 6

Figure 5 shows the influence of the speed of the tramon dynamic overloads being the amplitude and wavelengthof corrugation 119860 = 025mm and 120582 = 05m From thisanalysis it can be affirmed that dynamic overloads growwhenincreasing the speed of the vehicle

On the other hand the influence of wavelength ondynamic overloads being 119860 = 025mm and 119881 = 50 kmhis shown in Figure 6 In this graph the peak obtained whenthe wavelength is about 01mmust be highlighted Following[19] natural frequencies of the system are calculated as theroot of the eigenvalues of

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198961+ 1198962

1198981

minus1198962

1198981

minus1198962

1198982

1198962

1198982

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then the natural frequencies are 8949302 rads and94815 rads the wavelength being associated with firstnatural frequency 120582 = 00975m which corresponds to thewavelength estimated from Figure 6Then it can be said thatfor this wavelengthmdashbeing the speed of the vehicle and trainand track features constantsmdasha resonance phenomenon willappear

FromFigure 7 it can be noticed that far from thosewave-lengths affected by the resonance phenomenon dynamicoverloads decrease when the wavelength of the irregularityincreases That is the farther the distance between twoconsecutive irregularities the lower the forces transmitted tothe track

35

36

37

38

39

40

41

42

0 05 1 15 2 25 3 35

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

Figure 7

Figure 8 shows the influence of wear amplitude ondynamic overloads being 120582 = 05m and 119881 = 50 kmh Thisleads us to affirm that the higher the amplitude the greaterthe overload Furthermore it must be noticed that overloadsand corrugation amplitude are linearly related

Taking into account the strong nonlinear behavior of thevehicle-track system the linear behavior of the amplitudeof corrugation must be explained In order to assess thevalidity of the results obtained with the quarter car model amultibody model has been implemented with the VAMPIREcommercial software and the influence of the amplitude hasbeen studied The results are exposed in Figure 9

From Figure 9 two different behaviours of the relationamplitude-dynamic overloads are shown On one handwhen the amplitude of corrugation is lower than 1mm theoverloads generated by this pathology show a quasilinearbehaviour for both models On the other hand when theamplitude exceeds this value a nonlinear behaviour canbe appreciated for the multibody model but its behaviourremains quasilinear when dynamic overloads are calculatedwith the quarter car model

Thus it can be concluded that the quarter car modeldescribed in this paper only is able to be used when theamplitude of the defect considered is lower than 1mm

In conclusion since it is not feasible that amplitudeshigher than 1mm appear in a tram track as the one proposedin this study and the quarter carmodel presents the advantageof its simplicity when comparing with the multibody modelthe use of this model has been justified Nevertheless beforeusing this model for other types of railroads where higheramplitudes may be reached a preliminary study must becarried out

33 Definition of Different Scenarios Below a simulation ofdifferent scenarios will be performed Irregularities simulatedby the model will be added to the imperfections alreadypresent in the calibrated track The values used in thesesimulations are shown in Table 3

Note that in themodel a punctual contact betweenwheelsand rails is assumed Therefore when simulating wear ofa very short wavelength and high amplitude it would benecessary to check the veracity of this hypothesis Thus it is

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

represent the longitudinal and shear wave transmissionspeed respectively with their expressions being as follows

1198772

119871= 1198962minus

1205962

1198882119871minus 119894120596 (120582lowast + 2120583lowast) 120588

1198772

119879= 1198962minus

1205962

1198882119879minus 119894120596120583lowast120588

(8)

where 120582lowast and 120583lowast are the parameters which define the

damping of the track and 119888119871and 119888119879represent the velocities

of the longitudinal and shear waves in the layerThen the expressions for the displacements (9) and (10)

and stresses (11) and (12) can be written as

119906 (119896 119911 120596) = 119894119896120593 minus119889120595

119889119911 (9)

V (119896 119911 120596) =119889120593

119889119911+ 119894119896120595 (10)

120590119911119911(119896 119911 120596) =

120582(

1198892 120593

1198891199112minus 1198962 120593) + 2120583(

1198892 120593

1198891199112minus 119894119896

119889120595

119889119911) (11)

120590119911119909(119896 119911 120596) = 120583(2119894119896

119889120593

119889119911+1198892 120595

1198891199112+ 1198962 120595) (12)

Solving the system (7) and replacing 120593 and 120595 in (9)ndash(12)

119906119895(119896 119911 120596) = 119894119896 (119860

1119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 119877119879119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

V119895(119896 119911 120596) = 119877

119871119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

minus 119894119896 (1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119885119895(119896 119911 120596) = 119862

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911+ 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198622119895

(1198603119895(119896 120596) 119890

119877119879119895119911minus 1198604119895(119896 120596) 119890

minus119877119879119895119911)

120590119885119909119895(119896 119911 120596) = 119863

1119895

(1198601119895(119896 120596) 119890

119877119871119895119911minus 1198602119895(119896 120596) 119890

minus119877119871119895119911)

+ 1198632119895

(1198603119895(119896 120596) 119890

119877119879119895119911+ 1198604119895(119896 120596) 119890

minus119877119879119895119911)

(13)

where

1198621119895

= (120582119895+ 2120583119895)1198772

119871119895minus120582119895119896

1198622119895

= minus2119894119896120583119895119877119879119895

1198631119895

= 2119894119896120583119895119877119871119895

1198632119895

= 120583119895(1198962+ 1198772

119879119895

)

(14)

Boundary conditions are set so that the displacementsbetween adjacent layers are equal and stresses between themare balanced Furthermore no horizontal rail movement andthe vanishing of radiated vibrations as the soil depth increasesare assumed

Therefore replacing the boundary condition of each layerin (13) follows the expression M sdot A = q where M isa coefficient matrix A is the vector of coefficients to becalculated and q is the vector of loads acting on the system

Following [15] once the motion and stresses expressionshave been obtained in terms of deepness and in frequency-wave number domain they are returned to the space-timedomain by using the inverse Fourier transform In order tooptimize this process the displacements are written in termsof the wave number domain and then the expression ofthe inverse Fourier transform is converted into an additionfollowing [14]

The accelerations in each layer are obtained by dif-ferentiating twice with respect to time the displacementexpressions from the inverse Fourier transform

Total accelerations generated and transmitted to thesurrounding ground are obtained through superposition ofquasistatic and dynamic overloads

In order to provide reliability to the model a measure-ment camping was carried out as aforementioned Dataobtained by triaxial accelerometers FastTracer Sequoia havebeen processed and compared with those calculated in themodel After calibrating the damping parameters of thelayers the result of comparison between model and real datais shown in Figure 3

Figure 3 shows the ability of the model to represent thepeaks of vibrations and clearly reproduce the effect of passingpassenger cars and their bogies through the stretch of thetrack However as shown in Figure 3 measured data (green)is much noisier than the modeled one (red) These noisieraccelerogram recordings may be due to the numerous factorsintervening in the real vibratory phenomenon such as railsand wheels imperfections In contrast only few harmonicloads have been considered in the model to simulate theseirregularities Amplitudes and angular frequencies of theseharmonic loads have been obtained after analyzing theFourier spectrum of the registered data taking their moresignificant values To more accurate results a larger numberof harmonics might be considered with the drawback of anincreased computational time

It must be highlighted that due to the typology of trackstudied it was not possible to set the accelerometers in therail so they were placed next to the elastomeric materialwhere the rail was housed Then the data registered withthe accelerometers were influenced by the presence of thiselastomeric material

3 Analysis of Corrugation

Simulation and analysis of the effect of corrugation in themodel presented in the previous section will be addressedThe procedure will be as follows dynamic overloads causedby the presence of corrugation on the rail will be calculated byusing an auxiliary quarter car model and then the influence

Mathematical Problems in Engineering 5

minus3minus2minus1

01234

minus6 minus4 minus2 0 2 4 6 8 10t (s)

a(m

s2)

Figure 3

in dynamic overloads of parameters which define rail irregu-larities will be discussed

Dynamic overloads generate an increase of vibrations andwear with the first one being the objective of this researchThus overloads will feed the model described in Section 2as harmonic forces in order to obtain the accelerationsgenerated by rail irregularities

31 Quarter Car Model The vehicle can be modeled in dif-ferent ways depending on the number of degrees of freedomgiven to the system it can range from hundreds of degrees offreedom allowed by commercial multibody systems (MBS) tosystems with a single degree of freedom constituted by amassattached to the track by a spring

Between both extreme cases the following can be found(i) the quarter car model which will be presented below (ii)the half car system it takes into account the sprung massesthe semisprung masses and the unsprung masses and (iii)the full car models these allow us to take into account notonly the displacements of masses but also their rotation withrespect to each axis

Nevertheless from the point of view of the main modeland the fact that the loads introduced only have a verticalcomponent using a full bogie model would be an unneces-sary computational expense Moreover [18] experimentallydemonstrated that the influence on the dynamic stresses ofthe sprung and semisprung masses is negligible compared tothe unsprung masses

Therefore to calculate the dynamic overloads generatedby corrugation a quarter car system will be implementedusing MATLAB software In this model the masses of thevehicle are discomposed in sprung and unsprung massesThe compatibility of strengths and displacements betweenboth masses are solved by a spring and a damping elementMeanwhile the strength transmission between the unsprungmasses and the rail is performed by an equivalent springwhich takes into account not only the stiffness of the rail-wheelHertzian contact but also the stiffness of the underlyingtrack

The expressions governing the behavior of the quarter carmodel are as follows

119898211990910158401015840

2+ 1198962(1199092minus 1199091) + 1198882(1199091015840

2minus 1199091015840

1) = 0

119898111990910158401015840

1minus 11988821199091015840

2+ 11988821199091015840

1minus 11989621199092+ (1198962+ 1198961) 1199091minus 1198961119911 = 0

(15)

x2

x1

m1

k1

k2 c2

m2

z

Figure 4

Table 2

Quarter car model values1198981(kg) 500

1198961(Nm) 32000000

1198982(kg) 4000

1198962(Nm) 501745

1198882(Nsdotsm) 875

where 1198981 unsprung mass per wheel 119896

1 contact stiffness

1198982 sprung mass per wheel 119896

2 primary stiffness 119888

2 primary

damping 119909119894 displacement of the 119894th mass 1199091015840

119894 velocity of the

119894th mass and 11990910158401015840119894 acceleration of the 119894th mass

In Figure 4 a quarter car model sketch can be seenIn Table 2 the values for the mass stiffness and damping

are shownProblem resolution is performed by the Laplace transfer-

ence function and once the accelerations of bothmasses havebeen obtained dynamic overloads are calculated as follows

119865dyn = 119898111990910158401015840

1+ 119898211990910158401015840

2 (16)

32 SensitivityAnalysis Real rail profilesmay be compared toa sum of sinusoids of different amplitudes and wavelengthsTherefore in order to analyze the effect of a particularcorrugation phenomenon the profile of the affected track ismodeled as a sinusoidal function which is defined by wearamplitude and wavelength Furthermore the speed of thetrain in service is needed to be taken into account due tothe dynamic performance of the analysis When studying theinfluence of these parameters on the dynamic overloads anamplitude 119860 = 025mm a wavelength 120582 = 05m and a speed119881= 50 kmh are setThen with two of these parameters beingfixed the third one will be modified in order to analyze itsinfluence on dynamic overloads

It must be highlighted that overload values given for eachamplitude speed and wavelength correspond to the maxi-mum absolute value of the stationary part of the overloadfunction The reason for this choice is to consider the mostunfavorable case from the point of view of track designmaintenance and security

6 Mathematical Problems in Engineering

0

20

40

60

80

100

120

0 20 40 60 80 100

Dyn

amic

ove

rload

(kN

)

Vehicle speed (kmh)

Figure 5

0 02 04 06 08 1 12

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

000E + 00

200E + 02

400E + 02

600E + 02

800E + 02

100E + 03

120E + 03

140E + 03

Figure 6

Figure 5 shows the influence of the speed of the tramon dynamic overloads being the amplitude and wavelengthof corrugation 119860 = 025mm and 120582 = 05m From thisanalysis it can be affirmed that dynamic overloads growwhenincreasing the speed of the vehicle

On the other hand the influence of wavelength ondynamic overloads being 119860 = 025mm and 119881 = 50 kmhis shown in Figure 6 In this graph the peak obtained whenthe wavelength is about 01mmust be highlighted Following[19] natural frequencies of the system are calculated as theroot of the eigenvalues of

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198961+ 1198962

1198981

minus1198962

1198981

minus1198962

1198982

1198962

1198982

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then the natural frequencies are 8949302 rads and94815 rads the wavelength being associated with firstnatural frequency 120582 = 00975m which corresponds to thewavelength estimated from Figure 6Then it can be said thatfor this wavelengthmdashbeing the speed of the vehicle and trainand track features constantsmdasha resonance phenomenon willappear

FromFigure 7 it can be noticed that far from thosewave-lengths affected by the resonance phenomenon dynamicoverloads decrease when the wavelength of the irregularityincreases That is the farther the distance between twoconsecutive irregularities the lower the forces transmitted tothe track

35

36

37

38

39

40

41

42

0 05 1 15 2 25 3 35

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

Figure 7

Figure 8 shows the influence of wear amplitude ondynamic overloads being 120582 = 05m and 119881 = 50 kmh Thisleads us to affirm that the higher the amplitude the greaterthe overload Furthermore it must be noticed that overloadsand corrugation amplitude are linearly related

Taking into account the strong nonlinear behavior of thevehicle-track system the linear behavior of the amplitudeof corrugation must be explained In order to assess thevalidity of the results obtained with the quarter car model amultibody model has been implemented with the VAMPIREcommercial software and the influence of the amplitude hasbeen studied The results are exposed in Figure 9

From Figure 9 two different behaviours of the relationamplitude-dynamic overloads are shown On one handwhen the amplitude of corrugation is lower than 1mm theoverloads generated by this pathology show a quasilinearbehaviour for both models On the other hand when theamplitude exceeds this value a nonlinear behaviour canbe appreciated for the multibody model but its behaviourremains quasilinear when dynamic overloads are calculatedwith the quarter car model

Thus it can be concluded that the quarter car modeldescribed in this paper only is able to be used when theamplitude of the defect considered is lower than 1mm

In conclusion since it is not feasible that amplitudeshigher than 1mm appear in a tram track as the one proposedin this study and the quarter carmodel presents the advantageof its simplicity when comparing with the multibody modelthe use of this model has been justified Nevertheless beforeusing this model for other types of railroads where higheramplitudes may be reached a preliminary study must becarried out

33 Definition of Different Scenarios Below a simulation ofdifferent scenarios will be performed Irregularities simulatedby the model will be added to the imperfections alreadypresent in the calibrated track The values used in thesesimulations are shown in Table 3

Note that in themodel a punctual contact betweenwheelsand rails is assumed Therefore when simulating wear ofa very short wavelength and high amplitude it would benecessary to check the veracity of this hypothesis Thus it is

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

minus3minus2minus1

01234

minus6 minus4 minus2 0 2 4 6 8 10t (s)

a(m

s2)

Figure 3

in dynamic overloads of parameters which define rail irregu-larities will be discussed

Dynamic overloads generate an increase of vibrations andwear with the first one being the objective of this researchThus overloads will feed the model described in Section 2as harmonic forces in order to obtain the accelerationsgenerated by rail irregularities

31 Quarter Car Model The vehicle can be modeled in dif-ferent ways depending on the number of degrees of freedomgiven to the system it can range from hundreds of degrees offreedom allowed by commercial multibody systems (MBS) tosystems with a single degree of freedom constituted by amassattached to the track by a spring

Between both extreme cases the following can be found(i) the quarter car model which will be presented below (ii)the half car system it takes into account the sprung massesthe semisprung masses and the unsprung masses and (iii)the full car models these allow us to take into account notonly the displacements of masses but also their rotation withrespect to each axis

Nevertheless from the point of view of the main modeland the fact that the loads introduced only have a verticalcomponent using a full bogie model would be an unneces-sary computational expense Moreover [18] experimentallydemonstrated that the influence on the dynamic stresses ofthe sprung and semisprung masses is negligible compared tothe unsprung masses

Therefore to calculate the dynamic overloads generatedby corrugation a quarter car system will be implementedusing MATLAB software In this model the masses of thevehicle are discomposed in sprung and unsprung massesThe compatibility of strengths and displacements betweenboth masses are solved by a spring and a damping elementMeanwhile the strength transmission between the unsprungmasses and the rail is performed by an equivalent springwhich takes into account not only the stiffness of the rail-wheelHertzian contact but also the stiffness of the underlyingtrack

The expressions governing the behavior of the quarter carmodel are as follows

119898211990910158401015840

2+ 1198962(1199092minus 1199091) + 1198882(1199091015840

2minus 1199091015840

1) = 0

119898111990910158401015840

1minus 11988821199091015840

2+ 11988821199091015840

1minus 11989621199092+ (1198962+ 1198961) 1199091minus 1198961119911 = 0

(15)

x2

x1

m1

k1

k2 c2

m2

z

Figure 4

Table 2

Quarter car model values1198981(kg) 500

1198961(Nm) 32000000

1198982(kg) 4000

1198962(Nm) 501745

1198882(Nsdotsm) 875

where 1198981 unsprung mass per wheel 119896

1 contact stiffness

1198982 sprung mass per wheel 119896

2 primary stiffness 119888

2 primary

damping 119909119894 displacement of the 119894th mass 1199091015840

119894 velocity of the

119894th mass and 11990910158401015840119894 acceleration of the 119894th mass

In Figure 4 a quarter car model sketch can be seenIn Table 2 the values for the mass stiffness and damping

are shownProblem resolution is performed by the Laplace transfer-

ence function and once the accelerations of bothmasses havebeen obtained dynamic overloads are calculated as follows

119865dyn = 119898111990910158401015840

1+ 119898211990910158401015840

2 (16)

32 SensitivityAnalysis Real rail profilesmay be compared toa sum of sinusoids of different amplitudes and wavelengthsTherefore in order to analyze the effect of a particularcorrugation phenomenon the profile of the affected track ismodeled as a sinusoidal function which is defined by wearamplitude and wavelength Furthermore the speed of thetrain in service is needed to be taken into account due tothe dynamic performance of the analysis When studying theinfluence of these parameters on the dynamic overloads anamplitude 119860 = 025mm a wavelength 120582 = 05m and a speed119881= 50 kmh are setThen with two of these parameters beingfixed the third one will be modified in order to analyze itsinfluence on dynamic overloads

It must be highlighted that overload values given for eachamplitude speed and wavelength correspond to the maxi-mum absolute value of the stationary part of the overloadfunction The reason for this choice is to consider the mostunfavorable case from the point of view of track designmaintenance and security

6 Mathematical Problems in Engineering

0

20

40

60

80

100

120

0 20 40 60 80 100

Dyn

amic

ove

rload

(kN

)

Vehicle speed (kmh)

Figure 5

0 02 04 06 08 1 12

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

000E + 00

200E + 02

400E + 02

600E + 02

800E + 02

100E + 03

120E + 03

140E + 03

Figure 6

Figure 5 shows the influence of the speed of the tramon dynamic overloads being the amplitude and wavelengthof corrugation 119860 = 025mm and 120582 = 05m From thisanalysis it can be affirmed that dynamic overloads growwhenincreasing the speed of the vehicle

On the other hand the influence of wavelength ondynamic overloads being 119860 = 025mm and 119881 = 50 kmhis shown in Figure 6 In this graph the peak obtained whenthe wavelength is about 01mmust be highlighted Following[19] natural frequencies of the system are calculated as theroot of the eigenvalues of

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198961+ 1198962

1198981

minus1198962

1198981

minus1198962

1198982

1198962

1198982

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then the natural frequencies are 8949302 rads and94815 rads the wavelength being associated with firstnatural frequency 120582 = 00975m which corresponds to thewavelength estimated from Figure 6Then it can be said thatfor this wavelengthmdashbeing the speed of the vehicle and trainand track features constantsmdasha resonance phenomenon willappear

FromFigure 7 it can be noticed that far from thosewave-lengths affected by the resonance phenomenon dynamicoverloads decrease when the wavelength of the irregularityincreases That is the farther the distance between twoconsecutive irregularities the lower the forces transmitted tothe track

35

36

37

38

39

40

41

42

0 05 1 15 2 25 3 35

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

Figure 7

Figure 8 shows the influence of wear amplitude ondynamic overloads being 120582 = 05m and 119881 = 50 kmh Thisleads us to affirm that the higher the amplitude the greaterthe overload Furthermore it must be noticed that overloadsand corrugation amplitude are linearly related

Taking into account the strong nonlinear behavior of thevehicle-track system the linear behavior of the amplitudeof corrugation must be explained In order to assess thevalidity of the results obtained with the quarter car model amultibody model has been implemented with the VAMPIREcommercial software and the influence of the amplitude hasbeen studied The results are exposed in Figure 9

From Figure 9 two different behaviours of the relationamplitude-dynamic overloads are shown On one handwhen the amplitude of corrugation is lower than 1mm theoverloads generated by this pathology show a quasilinearbehaviour for both models On the other hand when theamplitude exceeds this value a nonlinear behaviour canbe appreciated for the multibody model but its behaviourremains quasilinear when dynamic overloads are calculatedwith the quarter car model

Thus it can be concluded that the quarter car modeldescribed in this paper only is able to be used when theamplitude of the defect considered is lower than 1mm

In conclusion since it is not feasible that amplitudeshigher than 1mm appear in a tram track as the one proposedin this study and the quarter carmodel presents the advantageof its simplicity when comparing with the multibody modelthe use of this model has been justified Nevertheless beforeusing this model for other types of railroads where higheramplitudes may be reached a preliminary study must becarried out

33 Definition of Different Scenarios Below a simulation ofdifferent scenarios will be performed Irregularities simulatedby the model will be added to the imperfections alreadypresent in the calibrated track The values used in thesesimulations are shown in Table 3

Note that in themodel a punctual contact betweenwheelsand rails is assumed Therefore when simulating wear ofa very short wavelength and high amplitude it would benecessary to check the veracity of this hypothesis Thus it is

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

20

40

60

80

100

120

0 20 40 60 80 100

Dyn

amic

ove

rload

(kN

)

Vehicle speed (kmh)

Figure 5

0 02 04 06 08 1 12

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

000E + 00

200E + 02

400E + 02

600E + 02

800E + 02

100E + 03

120E + 03

140E + 03

Figure 6

Figure 5 shows the influence of the speed of the tramon dynamic overloads being the amplitude and wavelengthof corrugation 119860 = 025mm and 120582 = 05m From thisanalysis it can be affirmed that dynamic overloads growwhenincreasing the speed of the vehicle

On the other hand the influence of wavelength ondynamic overloads being 119860 = 025mm and 119881 = 50 kmhis shown in Figure 6 In this graph the peak obtained whenthe wavelength is about 01mmust be highlighted Following[19] natural frequencies of the system are calculated as theroot of the eigenvalues of

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198961+ 1198962

1198981

minus1198962

1198981

minus1198962

1198982

1198962

1198982

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then the natural frequencies are 8949302 rads and94815 rads the wavelength being associated with firstnatural frequency 120582 = 00975m which corresponds to thewavelength estimated from Figure 6Then it can be said thatfor this wavelengthmdashbeing the speed of the vehicle and trainand track features constantsmdasha resonance phenomenon willappear

FromFigure 7 it can be noticed that far from thosewave-lengths affected by the resonance phenomenon dynamicoverloads decrease when the wavelength of the irregularityincreases That is the farther the distance between twoconsecutive irregularities the lower the forces transmitted tothe track

35

36

37

38

39

40

41

42

0 05 1 15 2 25 3 35

Dyn

amic

ove

rload

(kN

)

Wavelength (m)

Figure 7

Figure 8 shows the influence of wear amplitude ondynamic overloads being 120582 = 05m and 119881 = 50 kmh Thisleads us to affirm that the higher the amplitude the greaterthe overload Furthermore it must be noticed that overloadsand corrugation amplitude are linearly related

Taking into account the strong nonlinear behavior of thevehicle-track system the linear behavior of the amplitudeof corrugation must be explained In order to assess thevalidity of the results obtained with the quarter car model amultibody model has been implemented with the VAMPIREcommercial software and the influence of the amplitude hasbeen studied The results are exposed in Figure 9

From Figure 9 two different behaviours of the relationamplitude-dynamic overloads are shown On one handwhen the amplitude of corrugation is lower than 1mm theoverloads generated by this pathology show a quasilinearbehaviour for both models On the other hand when theamplitude exceeds this value a nonlinear behaviour canbe appreciated for the multibody model but its behaviourremains quasilinear when dynamic overloads are calculatedwith the quarter car model

Thus it can be concluded that the quarter car modeldescribed in this paper only is able to be used when theamplitude of the defect considered is lower than 1mm

In conclusion since it is not feasible that amplitudeshigher than 1mm appear in a tram track as the one proposedin this study and the quarter carmodel presents the advantageof its simplicity when comparing with the multibody modelthe use of this model has been justified Nevertheless beforeusing this model for other types of railroads where higheramplitudes may be reached a preliminary study must becarried out

33 Definition of Different Scenarios Below a simulation ofdifferent scenarios will be performed Irregularities simulatedby the model will be added to the imperfections alreadypresent in the calibrated track The values used in thesesimulations are shown in Table 3

Note that in themodel a punctual contact betweenwheelsand rails is assumed Therefore when simulating wear ofa very short wavelength and high amplitude it would benecessary to check the veracity of this hypothesis Thus it is

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

000E + 00

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

160E + 02

180E + 02

Figure 8

0

50

100

150

200

250

300

0 1 2 3 4 5

Dyn

amic

ove

rload

(kN

)

Amplitude (mm)

Multibody (kN)Quarter car (kN)

Figure 9

Table 3

119881 (kmh) 35 50 80119860 (mm) 01 025 05120582 (m) 03 05 1

established that the curvature of the sinusoid representing therail must be greater than wheel curvature

Wheel curvature is 1119877119908

The curvature of the sinusoid representing the rail can beexpressed as 119910 = minus119860 sin(119896119909) so 119910 = minus119860 sdot sin(2120587119909120582) Bydifferentiating twice with respect to the position

11991010158401015840=41205872

1205822sdot 119860 sdot sin(2120587119909

120582) (18)

So the maximum curvature of the rail is (412058721205822)119860mminus1Relating both curvatures

1

119877119908

le41205872

1205822119860 (19)

Then the wavelength of the irregularity considered mustsatisfy

120582 ge 2120587radic119860119877119908 (20)

After verifying that all of them satisfy (20) the scenarios arediscussed in Table 4

Table 4

Scenario 119881

(kmh)119860

(mm)120582

(m) 119865din max (N)119886max(ms2)

Case 1 35 025 05 17783119890 + 04 244511Case 2 50 025 05 38228119890 + 04 477442Case 3 80 025 05 10621119890 + 05 123465Case 4 50 01 05 15291119890 + 04 462642Case 5 50 05 05 76456119890 + 04 492609Case 6 50 025 03 11547119890 + 05 569606Case 7 50 025 1 84703119890 + 03 401358

From the results for the dynamic overloads the behaviorpredicted by earlier Figures 5ndash8 is confirmed high speedhigh wavelengths and low amplitudes generate higher valuesfor dynamic overloads

Furthermore in general the higher the dynamic overloadgenerated by corrugation the greater the maximum acceler-ations generated and transmitted to the track However thisaffirmation is not satisfied in all cases (see cases 1 and 7) Asdiscussed in Section 2 this is due to the fact that the wholebehavior of the system is determined by not only dynamicoverloads but also the quasistatic ones which depend onspeedThus although in case 7 lower dynamic overloads havebeen obtained compared to case 1 the influence of the vehiclespeed causes the greater accelerations generated in case 7

4 Conclusions

The aim of this paper was to implement a model ableto reproduce the vibratory behavior of a tram railroadwhere a measurement campaign was conducted Then afterconfirming its correct behavior a simulation of corrugationin that track was performed

For this purpose an analytical model has been imple-mented in theMathematica software based on that developedby [16] Subsequently a morphological calibration was car-ried out

An auxiliary quarter car model was implemented toevaluate the dynamic overloads generated by corrugationand the following conclusions were obtained the greater thespeed of the vehicle and the amplitude of the defect thegreater the dynamic overloads transmitted to the track Onthe other hand the higher corrugation wavelength the loweroverloads generated

Dynamic overloads obtained as stated before feed themain model to obtain the vibrations generated by rail cor-rugation From this analysis it may be concluded that theoverloads transmitted to the track due to corrugation arehighly influential in the maximum acceleration generatedby the vehicle but also the effect of vehicle speed is verysignificant

The model described in this paper may be used as anaid to maintenance of railway infrastructure Overall for agiven stretch of track tram vehicles usually travel at the samespeed Moreover according to [2] given the main featuresof a stretch of track and of those vehicles traveling through

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

it wavelength of rail corrugation may be predicted Thenbeing the maximum allowable values for vibrations fixed fora particular stretch of track model can predict from whichcorrugation amplitude maintenance operations are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper The aim of this paperis to publish the results of the research conducted consideringonly academic purposes

References

[1] DThomson Railway Noise and Vibration Mechanisms Model-ing and Means of Control Elsevier 2009

[2] S L Grassie and J Kalousek ldquoRail corrugation characteristicscauses and treatmentsrdquo Journal of Rail and Rapid Transit vol207 no 1 pp 57ndash68 1993

[3] S L Grassie ldquoRail corrugation advances in measurementunderstanding and treatmentrdquoWear vol 258 no 7-8 pp 1224ndash1234 2005

[4] O Oyarzabal N Correa E G Vadillo J Santamarıa and JGomez ldquoModelling rail corrugation with specific-track param-eters focusing on ballasted track and slab trackrdquo InternationalJournal of Vehicle Dynamics and Mobility vol 49 no 11 pp1733ndash1748 2011

[5] N Correa O Oyarzabal E G Vadillo J Santamarıa and JGomez ldquoRail corrugation development in high speed linesrdquoWear vol 271 no 9-10 pp 2438ndash2447 2011

[6] J I Egana J Vinolas and M Seco ldquoInvestigation of theinfluence of rail pad stiffness on rail corrugation on a transitsystemrdquoWear vol 261 no 2 pp 216ndash224 2006

[7] E Tassilly and N Vincent ldquoA linear model for the corrugationof railsrdquoWear vol 150 no 1 pp 25ndash45 1991

[8] R Ferrara G Leonardi and F Jourdan ldquoNumerical modellingof train induced vibrationsrdquo in Proceedings of the 5th Interna-tional Congress on Sustainability of Road Infrastructures p 155Elsevier Rome Italy 2012

[9] X S Jin Z F Wen K Y Wang Z R Zhou Q Y Liu and CH Li ldquoThree-dimensional train-track model for study of railcorrugationrdquo Journal of Sound and Vibration vol 293 no 3ndash5pp 830ndash855 2006

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control pp 1ndash17 2013

[11] J I Real T Asensio L Montalban and C Zamorano ldquoAnalysisof vibrations in a modeled ballasted track using measured raildefectsrdquo Journal of Vibroengineering vol 14 no 2 pp 880ndash8932012

[12] M Melis Introduccion a la Dinamica Vertical de la vıa y SenalesDigitales en Ferrocarriles UPM Madrid Spain 2008

[13] J Real P Martınez L Montalban and A Villanueva ldquoMod-elling vibrations caused by tram movement on slab track linerdquoMathematical and ComputerModelling vol 54 no 1-2 pp 280ndash291 2011

[14] P Salvador J Real C Zamorano and A Villanueva ldquoA proce-dure for the evaluation of vibrations induced by the passing ofa train and its application to real railway trafficrdquo Mathematicaland Computer Modelling vol 53 no 1-2 pp 42ndash54 2011

[15] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000

[16] P M Koziol C Mares and I Esat ldquoWavelet approach tovibratory analysis of surface due to a load moving in the layerrdquoInternational Journal of Solids and Structures vol 45 no 7-8 pp2140ndash2159 2008

[17] A Lopez Pita Infraestructuras ferroviarias Edicions UPCBarcelona Spain 2006

[18] A ProudrsquoHomme ldquoLa Voierdquo Revue Generale des Chemins deFer no 1 pp 56ndash72 1970

[19] S Timoshenko and D Young Vibration Problems in Engineer-ing D Van Nostrand New York NY USA 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of