Dimensional Analysis and Similitude Applied to Soi

1DIMENSIONALANALYSISANDSIMILITUDEAPPLIEDTOSOIL-MACHINESYSTEMSS.K.UpadhyayaAlthoughanalyticalmodelsbasedonfundamentalorbasicprinciplesarecriticaltoprovideagoodinsightintoanyphysicalorbiologicalprocess,somesystemsaretoocomplexandpresentagreatchallengeforpropermathematicaltreatmentoftheproblem.Inspiteofthestridesmadeinelastoplasticsoilmechanics(Wulfsohn,2002;WulfsohnandAdams,2002)andnumericalanalysisofagriculturalsoil-machinesystems(Upadhyayaetal.,2002),soil-machine/soil-plantinteractioncontinuestobeaverychallenginganalyticalproblem.Dimensionalanalysisandsimilitudemethodologycanbeveryusefultoolstodevelopempiricalpredictionequationstopredictsystembehavior.Webrieflyoutlinethebasicsofthesetwotechniques,beforeconsideringtheirapplicationtosoildynamics.DIMENSIONALANALYSISEverymeasurementhastwocharacteristics:quantitativeandqualitative.Dimensionalanalysisdealswiththequalitativeaspectsofmeasurements.Qualitativeaspectsofmeasurementsareexpressedintermsofeitherprimaryorsecondaryunits.Theprimaryunitsareinternationallyacceptedreferencequantitiesintermsofwhichotherquantitiesarespecified(Skoglund,1967).Forexamplekg,m,s,andKarerespectivelytheprimaryunitsofmass,length,time,andtemperatureinSIunits.Thederivedorsecondaryunitsareexpressedintermsofprimaryunitsbasedonmathematicalrelationships(Murphy,1950).Thusspeed,whichisdistance/timehastheunitm/s.Fundamentalorbasicquantitiesareasetofquantitiesthatarechosentorepresentotherquantitiesbasedonconvenience.Fundamentalquantitiesmaynotbethesameasprimaryquantities.Oftenforceischosenasafundamentalquantityinengineering,althoughitisnotaprimaryquantity.Thedimensionsofthesebasicquantitiesareusedinobtainingthedimensionsofotherquantities.Thus,ifforce(F),length(L),time(T),andtemperature(θ)areusedasfundamentalquantities,thenmasswillhavethefollowingdimensions:FromNewton'ssecondlaw:force=mass×acceleration(1)acceleration=d2x/dt2(2)wherexisdisplacement,andtistime.Fromequations1and2,massisgivenby:22forcemassdtxd=(3)or,intermsofdimensions:212TFLTLFmass−==(4)Inthisdiscussion,weuseforce,length,time,andtemperatureasfundamentalquantities.Thepowerofdimensionalanalysisresidesinitsabilitytoclassifyequations,convertequationsfromonesystemofunitstoanother,developpredictionequations,reducethenumberofvariablestobeinvestigatedinanexperiment,andprovidethebasisforthetheoryofsimilitude(Murphy,1950).Soildynamicsasadisciplinehasextensivelybenefitedfromthesepowerfulfeaturesofdimensionalanalysis.Thistechniquehasbeenwidelyusedduringthe1960sand1970stodeveloppredictionequations,reducethenumberofvariablestobeinvestigated,andconductmodelstudies.2DEVELOPMENTOFPREDICTIONEQUATIONSTherequirementthatageneralhomogeneousequationisdimensionallyconsistentmakesitpossibletodeveloppredictionequationsasfollows:Considerthegravitationalcomponentofstresswithinahomogeneoussoilmassatadepth(d)belowthesurfaceintheabsenceofsurcharge.Letσbethestressandγbethespecificweight.Thenthestress(σ)isgivenby:21),(CCCddfγ=γ=σ(5)or,intermsofdimensions:32FLL,,FL−−=γ==σd(6)whereCisadimensionlessconstant,and=representsdimensionalequivalenceandnotnecessarilynumericalequivalence.Therefore:()()()22121FLFLFLLFL)3(232CCCCC−−−−==(7)Sincethedimensionsmustbeconsistentforthisequationtobegeneral,exponentsofeachdimensionshouldmatch,i.e.:1=C2(8a)-2=C1-3C2(8b)Fromequations8aand8b,wehaveC1=1andC2=1.Therefore,equation5becomes:dCdCγ=γ=σ11(9)Inthiscase,dimensionalanalysisyieldedtheformoftheequationcompletely,exceptforthedimensionlessconstantC.Thisconstant,whichisequaltounity(i.e.,C=1),canbedeterminedbyconductinglimitedexperiments.Nowconsiderthecaseofawideverticalblade(i.e.,rakeangleis90°)ofwidthw,operatingatdepthd,inacohesionless,homogeneoussoilintheabsenceofanysurcharge.Thedraft(D)neededtoovercomethegravitationalcomponentofthesoilcuttingresistance(i.e.,thesoilcuttingforcenecessarytoresistthepassivepressureonthetillagetoolduetothesoilweight)underquasistaticconditions(i.e.,verylowgroundspeed)isdependentonthespecificweightofthesoil,operatingdepth(d),bladewidth(w),andsoilinternalfrictionangle(φ)intheabsenceofsoilmetalfriction.Therefore,wehave:()4321,,,CCCCwCdwdfDφγ=φγ=(10)or,intermsofdimensions:F=DL=dL=w3FL−=γessdimensionl=φTherefore,intermsofdimensions,equation10becomes:3321)FL()L()L(FCCC−=(11)or:33321)3()(CCCCFLF−+=(12)Therefore:C1+C2-3C3=0(13a)C3=1(13b)Fromequations13aand13b,weget:C1+C2=3orC2=3-C1(13c)Therefore,fromequation10,wehave:411)3(CCCwCdDγφ=−(14)Simplifyingandrearrangingequation14,weobtain:413CCwdCwDφ⎟⎠⎞⎜⎝⎛=γ(15)Althoughequation15providestheform,constantsC,C1,andC4areunknownandshouldbedeterminedexperimentally.Notethat(D/γw3),(d/W),andφarealldimensionlesstermsandareoftencalledPiterms.Amoregeneralwayofwritingequation15istoexpressthedependentPiterm(D/γw3)asafunctionofindependentPiterms(d/w)andφ(Murphy,1950).Thus,equation15canbewritteninamoregeneralformas:⎥⎦⎤⎢⎣⎡φ⎟⎠⎞⎜⎝⎛Φ=γ,3wdwD(16)Ingoingfromequation10toequation15,afive-variableproblemhasbeenreducedtoathree-variableorPitermsproblem.Thisreductioninthenumberofvariablesisamajoradvantageprovidedbydimensionalanalysis.Thisreductioninth