Ballast mats for the reduction of railway traffic vibrations. Numerical study

2.5维有限元经典SCI,将该方法计算土工材料对振动的影响分析。

SoilDynamicsandEarthquakeEngineering42(2012)137–150

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SoilDynamicsandEarthquakeEngineering

journalhomepage:http://wendang.chazidian.com/locate/soildyn

Ballastmatsforthereductionofrailwaytraf?cvibrations.Numericalstudy

P.AlvesCostan,R.Calc-ada,A.SilvaCardoso

UniversityofPorto,FacultyofEngineering,Porto,Portugal

articleinfo

Articlehistory:

Received9December2011Receivedinrevisedform11June2012

Accepted13June2012

Availableonline7July2012

abstract

Thispaperpresentsanumericalstudydevelopedinordertounderstandthedynamicbehaviorofballastedtrackswithmatsincludingthetrain–track–groundinteraction.Inordertoachievethatgoal,acasestudyismodeledbya2.5DFEM–BEMformulation.Acomprehensiveapproachispresentedandtheeffectsofthematstiffnessandlocationindeptharediscussed.Thecomparisonbetweenisolatedandnon-isolatedscenariosallowedconcludingthattheballastmathasadualeffect,focusingonthetrain–trackdynamicbehaviorandonthereductionofhigh-frequencyvibrationsthataretransmittedtotheground.Furthermore,itwasfoundthatglobalef?ciencycanbereachedbyplacingthematbeneaththesubballastinsteadofbelowtheballastlayer.

&2012ElsevierLtd.Allrightsreserved.

1.Introduction

Thegeometricalimperfectionsofwheelsandtracksarethemajorcauseofdynamicloadsdevelopedduringtherollingoftrains.Thesedynamicloadsactandexcitewavespropagatingfromthetracktothefree-?eldandtonearbybuildings,givingrisetothecommonlycalledtrain-inducedgroundvibrations.

Theenvironmentalproblemsrelatedtovibrationsinducedbytrainpassagesarebecominganimportantconcerninrecentyears,mainlyduetothelatesttechnologicaladvancesanddevelop-ments,whichallowforanincreaseofspeed.Thegrowthoftheintensityofurbanoccupationleadstoaworseningofthesurroundingconditions.So,thenuisanceforpeoplewholiveorworkinbuildingsnearrailwaylinesmustbemitigated,demand-ingforthedesignanddevelopmentofef?cientsolutions.

Thecountermeasuresagainstvibrationsinducedbytraf?ccanbegroupedintothreedistinctgroups,dependingonthelocation:i)atthesource;ii)alongthewavepropagationpath;iii)atthereceiver.Thelatter,comprisingtheisolationofthebuildings,doesnotchangethepatternortheamplitudeofthewavesthatpropagatefromthetrack.Alternatively,itisalsopossibletointerruptthewavepath,i.e.,topreventthewavesfromreachingthebuildings[1].

Countermeasurescanalsobeadoptedatthesourcelevel,i.e.,atthetrack.Generally,theobjectiveisachievedbychangingthedynamicbehaviorofthetrackthroughtheinsertionofresilientelementsoradditionalmassestothetrackstructure[2].Theformer,moreusualthanthelatter,canbeplacedatdifferentlocationlevelsoftherailwaytrackcross-section.Themainideais

Correspondingauthor.Tel.:þ351962934245.E-mailaddresses:pacosta@fe.up.pt(P.AlvesCosta),ruiabc@fe.up.pt(R.Calc-ada),scardoso@fe.up.pt(A.SilvaCardoso).0267-7261/$-seefrontmatter&2012ElsevierLtd.Allrightsreserved.http://wendang.chazidian.com/10.1016/j.soildyn.2012.06.014

n

toinducealowernaturalfrequencytotherailwaytrack,whichisconditionedbythemassabovetheresilientelementandbyitsstiffness(lowerthanthatoftheremainingelements).Asisevident,theeffectivenessofthisisolationtechniqueisrelatedtothevalueoftheresonancefrequency.Insertionofresilientelementsimmediatelybeneaththerailsorthesleepers,suchasrailpadsorundersleeperpads,givesrisetohighvaluesfortheresonantfrequencyduetothelowermassabovetheisolation.Ontheotherhand,lowerresonantfrequenciescanbeachievedinheavysystems,suchasin?oatingslabtracksorballastedtrackswithballastmats,whichallowincreasingtheeffectivenessoftheisolationforfrequenciesabovefewtensofHertz.

Ballastmatsareanef?cientcountermeasuretoreducevibrationsinducedinthemiddlefrequencyrange[3].Infact,theintroductionoftheresilientelementsbeneaththeballastorthesub-ballast,allowsachievingalowresonantfrequency,i.e.,from20Hzto50Hz,duetotheheaviermassabovetheresilientelement.Thereductionoftheresonantfrequencyforvalueslowerthan20Hzisdif?cult,especiallyforhighspeedrailways,sincetheballastmatresilienceislimitedbythemaximumallowablestaticrailde?ection.Auersch[4]pointedouttherangeoffrequenciesbetween20Hzand50Hzasusualvaluesfortheresonancefrequencyintroducedbythepre-senceoftheballastmat.Regardingtheeffectivenessoftheisolationprovidedbytheballastmat,Muller[5]showed,basedonexperi-mentaltests,thattheinsertionlossprovidedbythematisconsiderablyhigherifthesupportisstiffer.Therefore,ballastmatsinstalledintunnelsorinstiffersupports(suchasbridges,forinstance)aremuchmoreef?cientthanmatsinstalledinopenlines.SimilarconclusionswerealsoachievedbyLombaertandDegrande[6]fromthenumericalstudyof?oatingslabtracks.

Nevertheless,despitetheresultsobtainedbytheauthorsmentionedabove,thisisoneofthecaseswheretheconstructionindustryisonestepaheadoftheresearch,i.e.,thereareseveral

2.5维有限元经典SCI,将该方法计算土工材料对振动的影响分析。

138P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering42(2012)137–150

productsandproceduresincontrasttothelackofcomprehensivestudiesaboutitsparticularperformance.Intheauthors’knowl-edge,theexistingnumericalstudiesaboutthistopicarereducedtosimpleone-dimensionalformulations[3,7],withsomeexcep-tionsconcerningthestudiesperformedon?oatingslabtracks[6,8],whichcanberegardedasasimilarsystem,andtoaresearchpaperdedicatedtothetopicpresentedbyAuersch[4].However,effectsofthetrainmovementandtheconsequentdynamictrain–trackinteraction,aswellastheDoppler’seffectinducedbythemovingcharacteristicsoftheload,havenotyetbeenaddressedinanystudy,thusjustifyinganddemandingforfurthercontribu-tionsonthisimportanttopic.

Theaimofthepresentpaperistocontributeforthestudyofthedynamicbehaviorofballastedtrackswithmats,includingthetrain–trackdynamicinteractionandthewavepropagationinthefree-?eld.A2.5DFEM–BEMmodel,developedbytheauthors,isadoptedforthesimulationofthetrack–groundsystem,withtherollingstockmodeledbyamulti-bodymodel.BothmodelswereintegratedinacomputationalcodedevelopedinMatlab,whichallowstakingintoaccountthetrain–trackinteractiondynamicloadsinducedbythetrackunevenness.

Inordertoobtainarealisticscenario,acasestudypreviouslyanalyzedandpresentedbytheauthorswasusedasastartingpointforthepresentnewstudies[9].Thescenarioofthenon-isolatedtrack,whichwaspreviouslyvalidatedbythecomparisonbetweenmeasuredandcomputedresults,isusedasreferencesolutiontogetacomprehensiveunderstandingoftheeffectivenessoftheballastmatonthemitigationofvibrationsinducedbytraf?c.

Thepaperisorganizedasfollows.Firstly,abriefoverviewofthemaincharacteristicsofthecomputationalcodeispresented,comprisingthemodelingofthetrack–groundsystemandthesimulationoftherollingstockaswellasitsinteractionwiththeremainingdomain.Subsequently,someapplicationsaredevel-opedwiththeaimofdiscussingthein?uenceoftheballastmatsonthemechanismofexcitationandpropagationofseismicwaves.Thesecomprehensiveapplicationsaredividedintwomainparts:thediscussionofthein?uenceofthematsonthedynamicbehaviorofthetrackwhensubmittedtostand-stillloadsiscarriedoutinthe?rstpart,whiletheanalysisoftheef?ciencyoftheisolationsystemformovingloadsisevaluatedinthesecondpart.Finally,someconclusionsarepresentedwithfocusontheef?ciencyofthiskindofisolationsystem.

2.Numericalpredictionmodel2.1.Thetrack–groundsystem

Forthecomputationofthe3Dtrack–grounddynamicresponse

inducedbythetrainpassage,anumericalprocedurebasedonthe2.5Dcouplingof?nite(FEM)andboundaryelements(BEM)formulationwasadopted[9–12].The2.5Dformulationcorre-spondstoanattemptforreducingthecomputationaleffortsince,assumingthelinearityandtheinvariabilityofthedomain,theequilibriumcanbeestablishedinthewavenumber-frequencydomain,employingFourierexpansionsforspace(inthetrackdevelopmentdirection)andtime.Therefore,the3Dsolutionisobtainedwithouttheneedofnumericaldiscretizationalongthedevelopmentdirectionofthetrack.

Domaindecompositionisusedtosolvethedynamicproblem,beingthetrackmodeledbythe2.5DFEMandthelayeredgroundsimulatedthrough2.5DBEM,asshowninFig.1.

Thecouplingbetweenbothdomainsisdonebya?niteelementsformulation,comprisingthetransformationofthe?ex-ibilitymatrixthatgovernsthedynamicbehavioroftheBEMdomainintoadynamicstiffnessmatrix.

Fig.1.2.5DFEM–BEMcoupling.

Followingtheproperformalismofthe2.5DFEM,thedynamic

equilibriumequationsofthemediumcanbereachedbyavariationalformulation.So,afterintroducingthediscretizationofthecross-sectioninto2.5D?niteelementsandapplyingaFouriertransformregardingthexcoordinate,thedynamicequili-briumcanbedescribedbythefollowingsystemofequations:ðKglobalþik2

4

1Kglobalþkglobal1Kþkglobal1231Kglobal4

Ào2MþKglobal5

ðk1,oÞÞunðk1,oÞ¼pnðk1,oÞð1Þ

whereKglobal1toKglobal4

arestiffnessmatricesofthedomaindescribedby?niteelements,Mglobalisthemassmatrix,k1istheFourierimageofthecoordinatex,oisthefrequency,unisthevectorofthenodaldisplacements,pnisthevectoroftheexternalforces;and,?nally,Kglobal5

isthematrixthatcollectstheimpe-dancetermsofthelayeredground.

Complementaryinformationaboutthedeductionoftheabovementionedmatricescanbeconsultedelsewhere[13–15].Strate-giesformodelingparticularaspects,asforinstancethesleepersandtherail,havealsobeenreportedbyAlvesCostaetal.[14].ThematrixKglobal5

iscomputedfromthe?exibilitymatrixthatgovernsthedynamicbehaviorofthedomaindescribedby2.5DBEM,accordinglytotheprocedureproposedbyBodeetal.[16].AsusualintheBEMformulation,theselectionoffundamentalsolutions,i.e.,theGreen’sfunctions,isakeyaspect.Indeed,severalalternativeprocedurescanbefollowedforcomputingthe?exibilityfora2.5Dformulation,andfundamentalsolutionsdevelopedinthe2.5Ddomainarealreadyavailable[17–19].However,inthepresentapproach,the3DGreen’sfunctionsforalayeredhalf-spacearecomputedbytheHaskell-Thomsonprocedure,aspreviouslysuggestedbyShengetal.[20].

SolvingthesystemofEq.(1),thenodaldisplacementsinthetransformeddomainareobtained,aswellasthepressuresalongthecouplingboundary.OncethesepressuresandtheGreen’sfunctionsofthedisplacementsareknown,thecomputationofthefree?eldresponseisasimplestep.

2.2.ThetrainanditsinteractionwiththeremainingsystemTheloadappliedbythetrainonthetrackcanbedividedintotwocomponents:i)thestaticload,resultingfromtheweightofthetrain;ii)thedynamicload,duetothedynamicinteractionbetweenthetrainandthetrack.Asthe?rstcomponentisaninputdataandnotanunknownvariable,itsconsiderationonanumericalmodelistrivial.Onthecontrary,thelattercomponentdemandsforthesolutionofthedynamictrain–trackinteractionproblem,whichinthepresentstudyissolvedbyacomplianceprocedureformulatedinaframeofreferencethatmoveswiththetrain,assuggestedbyseveralauthors[21–23].

2.5维有限元经典SCI,将该方法计算土工材料对振动的影响分析。

P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering42(2012)137–150139

A2Dvehiclemodelwasadoptedinthepresentstudy.Conse-quently,onlytheverticalmovementofthetrainwastakenintoaccount,i.e.,thedynamicloadsinducedbythemovementofthetraininthedirectionsdifferentfromtheverticalwereneglected.Assumingperfectcontactbetweenthetrainandthetrack,thefollowingrelationshipmustbeadheredtoatanytemporalinstantforallconnectingpoints:

uu??a PðtÞ

c,i¼rðx¼ctþaiÞþDutþicþikð2Þ

Hwhereuc,irepresentstheverticaldisplacementsofthecontact

pointiofthevehicle,uristheverticaldisplacementsofthetrackatthesamelocation,Duistherailunevenness,tisthetime,aiisthelocationofcontactpointiatt¼0sandcisthevehiclespeed,kHistheHertzianstiffnessandPiisthedynamicinteractionloaddevelopedattheconnectionpointi.TheinclusionoftheHertzianspringbetweenthewheelandtherailenablestotakeintoaccountthecontactdeformationbyalinearizationapproach[24,25].Notethatthetrainhasseveralcontactpointswiththetrack,incorrespondencetothenumberofwheelsets.

Eq.(2)canbewritteninmatrixforminthefrequencydomainusingthetransformationoftheunevennessforthatdomain.So,thetrain–trackinteractionforceatthefrequencydomainisgivenbypðOÞ¼ÀðFþFHþAÞÀ1DuðOÞ

ð3Þ

whereFisthetraincomplianceatthecontactpointswiththetrack,FHisadiagonalmatrixwiththetermsequalto1/kH,AisthecompliancematrixofthetrackandOisthedrivenfrequency,i.e.,theoscillationfrequencyofthewheelsetduetotheunevennesswithwavelengthl(O¼2pc/l).Allmatricesaresquarewithadimensionequivalenttothenumberofwheelsets.

FromEq.(3)itispossibletoconcludethatthedynamicstiffnessmatrix,whichestablishesarelationshipbetweentheunevennessperceivedbyeachcontactpointbetweenthetrainandthetrackandthedynamicloaddevelopedateachwheelset,herecalledastrain–trackdynamicstiffnessmatrix,isgivenbyKtÀt¼ðFþFHþAÞÀ1

ð4Þ

Thetrack–groundmodelisusedtocomputethetermsofmatrixAbythefollowingexpression:

A1Zþ1ijðOÞ¼2uGðk1,o¼OÀk1cÞUeiðaiÀaÀ1cjÞk1

dk1ð5ÞwhereuGccorrespondstotheverticaldisplacementoftherailin

thetransformeddomainduetoahalf-unitloadappliedintheheadofeachrail.

Ontheotherhand,matrixF,whichcomprisesthevehiclecompliance,iscomputedthroughasimpli?edvehiclemodel,wherethedynamicsofthesprungmass(carbody)isdiscarded(Fig.2).Actually,arecentstudydevelopedbytheauthorsofthepresentpapershowedthatasimpli?edmodel,whereonlythe

Bogies

PrimaryWheelsetsFig.2.2Dvehiclemodeladoptedonnumericalsimulation.

motionoftheaxlesandbogiesistakenintoaccount,suf?cesfortheevaluationoftrain–trackinteractionloads,sincethesecond-arysuspensioninducesanef?cientisolationofthesprungmassforfrequencieshigherthanafewHertz.

3.Thecasestudy3.1.Generaldescription

Inordertovalidatethenumericalmodeldevelopedbytheauthors,anexperimentaltestsitewasselectedandimplementedinthePortugueserailwaynetwork,neartothetownofCarre-gado[9].Sincethepurposeofthepresentstudyistoevaluatetheeffectoftheballastmatsinareliablescenarioofasurfaceline,thegeometryandmaterialpropertiesregardingthiscasestudywereconsidered.Thecasestudyisassumedasabasicscenario,aroundwhichaparametricstudyisdevelopedinordertostudytheeffectsinducedbytheintroductionofballastmats.

ThetrackpresentsastraightalignmentandcorrespondstoarenewedpartoftherailwayconnectionbetweenPortoandLisbon.ThereadershouldrefertoAlvesCostaetal.[9,26]foradetaileddescriptionoftheexperimentsperformedaswellasforthediscussionofthenumericalcalibrationprocedure.3.2.Soildynamicproperties

Thedynamicpropertiesofthesoilwereobtainedfromtwocross-holetestperformedinthesite.Fig.3showsthepro?leofmeasuredwavevelocities(PandS),aswellastheaveragevalues.Thelayeringofthesoilwasincludedinthenumericalanalysisincorrespondencewiththepro?lesofaveragevaluesdepictedinFig.3.

Regardingthedampingpropertiesoftheground,itsevaluationwasperformedbyaninversionprocedurebymatchingthenumericalandthemeasuredtransferfunctions[9].Followingthatapproach,thedampingpro?ledepictedinFig.4wasassumedasrepresentativeofthehystereticmaterialdampingofthegroundforbothvolumetricandsheardeformation.

Moreover,laboratorytestsdevelopedonsamplesrecoveredinboreholesallowed?ndingthat1900kg/m3isareasonablevalueforthemassdensityofthesoil.

3.3.Trackdynamicpropertiesanditsunevenness

TherailwayconsistsofUIC60railssupportedbyconcretesleepersspacedof0.60m,eachonewith300kgofmass.Thedynamiccharacteristicsoftheremainingelementsofthe

CH 20

CH 2CH 1

CH1

Average values

Average values

55

)

m( htp1010

eD1515

20

100

200300400

20500

1000150020002500

Cs (m/s)

Cp (m/s)

Fig.3.Seismicwavevelocitypro?le:a)Swaves;b)Pwaves.

2.5维有限元经典SCI,将该方法计算土工材料对振动的影响分析。

140P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering42(2012)137–150

0246Profundidade(m)

81012141618200

0.02

0.04

0.06

0.08

0.1

Fig.4.Dampingpro?le.

non-isolatedtrackwereestimatedbyreceptancetests[9].Fromthosetestsitwasfoundthatthemechanicalpropertiesoftheembankmentaresimilartothepropertiesofthesubballastlayer.Fig.5showsthetrackgeometryandthemechanicalpropertiesobtainedaftertheinversionofthereceptancetests.Theseproper-tiesareassumedforthenon-isolatedscenario.Forisolatedsystems,thematisintroducedimmediatelybelowthesubballastlayer(scenario1),asdepictedinFig.5a,orbeneaththeballastlayer(Fig.5b;scenario2).Sincetheembankmenthasathicknessof0.30mandthesamemechanicalpropertiesasthesubballastlayer,aproperfoundationforthematisguaranteedwhenitisplacedbeneaththesubballastlayer.

Differentvaluesfortheballastmatstiffnesswereconside-red,alwaysassumingaconstantvalueof0.1foritshystereticdamping.

Unlikeothercasestudies,wherethetrackunevennesswasarti?ciallygenerated,inthepresentstudyitwasmeasured.Fig.6ashowsthemeasuredunevennesspro?leoftherailfortherangeofwavelengthsbetween0.4mand25m.Thesameinformation,butinaspectralrepresentation,isalsoillustratedinFig.6b,wherethespectraldensityfunctionproposedbyBraunandHellenbroich[27]fortrackswithgoodormediumqualityarealsorepresented.

Railpad:k=600 kN/mm

Railpad:k=600 kN/mm

Fig.5.Crosssectionofthetrack:a)scenario1—matbeneathsubballastlayer;b)scenario2—matbeneathballastlayer.

2.5维有限元经典SCI,将该方法计算土工材料对振动的影响分析。

P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering42(2012)137–1500.01

0.005

141

Unevenness (m

)

PSD (m3/rad

)

-0.005

-0.01

50.54175

25

00

75

50

25

00

.5

.7

.7

.6

.6

.6

.6

41

41

41

41

41

41

41

41.525

Location(km)

Wavenumber (rad/m)

Fig.6.Railunevennesspro?le:a)spatialrepresentation;b)powerspectrumdensity.

Fig.7.GeometryofthetrainAlfa-Pendular.

Table1

Mechanicalpropertiesofthetrain.Axles

PrimarysuspensionBogiesCarbody

Mw(kg)Kp(kN/m)Cp(kNs/m)Mb(kg)Jb(kg/m2)Mc(kg)

1538–1884342036

4712–49325000–515032900–35710

3.4.Rollingstock

Inthefollowinganalysis,thepassageoftrainAlfa-Pendularisconsidered.TheAlfa-PendularisthefastesttrainoperatinginPortugal.Itisaconventionaltraincomposedby6vehicles,asindicatedinFig.7.

Themainmechanicalpropertiesofthetrainwereprovidedbytheoperator.Inspiteofthisinformation,someidenti?cationmodaltestsforthestructuralcharacterizationofthetrainwerealsodevelopedandthepropertiesadoptedinthenumericalmodelwereadjustedinordertoobtainagood?tbetweennumericalandexperimentalnaturalfrequencies[28].Table1summarizesthevaluesofthemainpropertiesofthetrain,adoptedinthenumericalanalyses.Thevehiclesofthetrainarenotexactlyequal;Table1indicatestherangeofvaluesfoundforthedistinctvehiclesofthetrain.

4.Dynamicbehaviorofnon-isolatedandisolatedtracks4.1.Receptanceanalysis

Asimplebutcomprehensiveapproachforunderstandinghowthemataffectsthedynamicbehaviorofthetrain–groundsystemisbasedontheanalysisofitsreceptance.Fig.8showsthereceptanceoftherailforbothscenariosunderconsiderationandfortwovaluesoftheballastmatstiffness.Thereceptancecurvesarecomputedassumingahalf-unitstand-stillloadappliedineachrail,andthedisplacementsareevaluatedattherailundertheloadingposition.

Consideringthedynamicresponseofthenon-isolatedscenarioasreferencesolution,whichisalsodepictedinbothFig.8aandb,

itispossibletoidentifytwomainresonancesofthetrack–groundsystemasfollows:i)the?rstpeakofthereceptance,around9Hz,isrelatedtothedynamicresponseinducedbythelayeringoftheground;ii)thehighestfrequencypeak,around82Hz,correspondstotheresonanceofthetracksystemaboveitsfoundation.AspreviouslydiscussedbyKnothandWu[29],thefull-trackresonancecanbehighlydampedwhenitssupportissoftandhomogeneous,beingmorepronouncedifthegroundislayered,asinthepresentcase.Thevaluefoundforthefrequencyofthefull-trackresonance,82Hz,iswithinthetypicalrangeofvalues.AccordingtoMann[30],ballastedtrackshavefull-trackresonantfrequenciesintherangebetween40Hzand140Hz.ThesameconclusionwasalsoachievedbyDalhberg[31].

Asexpected,theintroductionofthematgivesrisetoanewnaturalfrequencyofthesystem,herecalledascut-onfrequency.Actually,theresonanceinducedbythepresenceofthematisverydampedingeneralsituations,butnotfullyperceptiblewhenthestiffermatislocatedbeneaththesubballastlayer(Fig.8a).However,forfrequenciessuf?cientlyhigherthantheresonanceinducedbythematandforallthescenariosunderanalysis,areductionofthereceptanceoftherailisclear,denotingtheef?ciencyoftheisolationinducedbytheballastmat,whichattenuatestheresonanceofthetrackontheground.

Amoredetailedanalysisallowstopointoutthefollowingmainremarks:i)independentlyofthematposition,theincreaseofitsstiffnessleadstoanincreaseofthefrequencyforwhichtheattenuationeffectsstartstobeeffective;ii)forthesamematstiffness,thecut-onfrequencyinducedbythepresenceofthematisaslowasdeepitspositionis;iii)forthesamematstiffnessandintherangeoflowerfrequencies,theresponseoftherailisemphasizedwhenthematislocatedbeneaththeballast.Thislast?ndingisparticularlyimportantsincetheusageofsoftermatsisusuallynotrecommendedduetotherequiredlimitationofthede?ectionoftherail.However,itseemsthatthislimitationcanbeovercomebyameticulousselectionofthematpositiononthecross-sectionofthetrack.

Inwhatconcernstheisolationofvibrations,amoreinterestingeffectisillustratedinFig.9,wherethecross-receptance,com-putedatthecontactbetweentheembankmentandthenatural