Bus Suspension Modeling in Simulink.docx
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Bus Suspension Modeling in Simulink.docx
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BusSuspensionModelinginSimulink
Example:
BusSuspensionModelinginSimulink
Physicalsetup
Designinganautomaticsuspensionsystemforabusturnsouttobeaninterestingcontrolproblem.Whenthesuspensionsystemisdesigned,a1/4busmodel(oneofthefourwheels)isusedtosimplifytheproblemtoaonedimensionalspring-dampersystem.Adiagramofthissystemisshownbelow:
Where:
*bodymass(m1)=2500kg,
*suspensionmass(m2)=320kg,
*springconstantofsuspensionsystem(k1)=80,000N/m,
*springconstantofwheelandtire(k2)=500,000N/m,
*dampingconstantofsuspensionsystem(b1)=350Ns/m.
*dampingconstantofwheelandtire(b2)=15,020Ns/m.
*controlforce(u)=forcefromthecontrollerwearegoingtodesign.
Designrequirements:
Agoodbussuspensionsystemshouldhavesatisfactoryroadholdingability,whilestillprovidingcomfortwhenridingoverbumpsandholesintheroad.Whenthebusisexperiencinganyroaddisturbance(i.e.potholes,cracks,andunevenpavement),thebusbodyshouldnothavelargeoscillations,andtheoscillationsshoulddissipatequickly.SincethedistanceX1-Wisverydifficulttomeasure,andthedeformationofthetire(X2-W)isnegligible,wewillusethedistanceX1-X2insteadofX1-Wastheoutputinourproblem.Keepinmindthatthisisanestimation.
Theroaddisturbance(W)inthisproblemwillbesimulatedbyastepinput.Thisstepcouldrepresentthebuscomingoutofapothole.Wewanttodesignafeedbackcontrollersothattheoutput(X1-X2)hasanovershootlessthan5%andasettlingtimeshorterthan5seconds.Forexample,whenthebusrunsontoa10cmhighstep,thebusbodywilloscillatewithinarangeof+/-5mmandreturntoasmoothridewithin5seconds.
BuildingtheModel
Thissystemwillbemodeledbysummingtheforcesactingonbothmasses(bodyandsuspension)andintegratingtheaccelerationsofeachmasstwicetogivevelocitiesandpositions.Newton'slawwillbeappliedtoeachmass.OpenSimulinkandopenanewmodelwindow.First,wewillmodeltheintegralsoftheaccelerationsofthemasses.
∙InsertanIntegratorblock(fromtheLinearblocklibrary)anddrawlinestoandfromitsinputandoutputterminals.
∙Labeltheinputline"a1"(foracceleration)andtheoutputline"v1"(forvelocity)Toaddsuchalabel,doubleclickintheemptyspacejustabovetheline.
∙InsertanotherIntegratorblockconnectedtotheoutputofthefirst.
∙Drawalinefromitsoutputandlabelit"x1"(forposition).
∙InsertasecondpairofIntegratorsbelowthefirstwithlineslabeled"a2","v2",and"x2".
Next,wewillstarttomodelNewton'slaw.Newton'slawforeachofthesemassescanbeexpressedas:
Theseequationscanberepresentedwithgainblocks(for1/M1and1/M2)andtwosummationblocks.
∙InserttwoGainblocks,(fromtheLinearblocklibrary)oneattachedtotheinputsofeachoftheintegratorpairs.
∙EditthegainblockcorrespondingtoM1bydouble-clickingitandchangingitsvalueto"1/m1".
∙ChangethelabelofthisGainblockto"Mass1"byclickingontheword"Gain"underneaththeblock.
∙Similarly,edittheotherGain'svalueto"1/m2"andit'slabelto"Mass2".(Youmaywanttoresizethegainblockstoviewthecontents.Todothis,singleclickontheblocktohighlightit,anddragoneofthecornerstothedesiredsize.)
TherearethreeforcesactingonM1(onespring,onedamper,andtheinput,u)andfiveforcesactingonM2(twosprings,twodampers,andtheinput,u).
∙InserttwoSumblocks(fromtheLinearblocklibrary),oneattachedbyalinetoeachoftheGainblocks.
∙EditthesignsoftheSumblockcorrespondingtoM1to"+--"torepresentthethreeforces(twoofwhichwillbenegative)
∙EditthesignsoftheotherSumblockto"++-++"torepresentthefiveforces,oneofwhichwillbenegative.
Now,wewilladdintheforcesactingoneachmass.First,wewilladdintheforcefromSpring1.Thisforceisequaltoaconstant,k1timesthedifferenceX1-X2.
∙Insertasumblockaftertheupperpairofintegrators.
∙Edititssignsto"+-"andconnectthe"x1"signaltothepositiveinputandthe"x2"signaltothenegativeinput.
∙DrawalineleadingfromtheoutputoftheSumblock.
∙InsertaGainblockabovethe"Mass1"block.
∙Flipitleft-to-rightbysingle-clickingonitandselectingFlipBlockfromtheFormatmenu(orhitCtrl-F).
∙Editthevalueofthisgainto"k1"andlabeltheblock"Spring1".
∙TapalineofftheoutputofthelastSumblockandconnectittotheinputofthisgainblock.
∙Connecttheoutputofthisgainblock(thespringforce)tothesecondinputoftheMass1Sumblock.ThisinputshouldbenegativesincetheSpring1pullsdownonMass1whenX1>X2.
∙TapalineoffthespringforcelineandconnectittothesecondinputoftheMass2Sumblock.ThisinputispositivesinceSpring1pullsuponMass2.
Now,wewilladdintheforcefromDamper1.Thisforceisequaltob1timesV1-V2.
∙InsertasumblockbelowtheMass1'sfirstintegrator.
∙Flipitleft-to-right,andeditit'ssignsto"+-".
∙Tapalineoffthe"v1"lineandconnectittothepositiveinputofthisSumblock.
∙Tapalineoffthe"v2"lineandconnectittothenegativeinputofthisSumblock.
∙InsertaGainblocktotheleftofthisSumblockandflipitleft-to-right.
∙Editit'svalueto"b1"andlabelit"Damper1".
∙ConnecttheoutputofthenewSumblocktotheinputofthisgainblock.
∙Connecttheoutputofthisgainblock(thedamperforce)tothethirdinputoftheMass1Sumblock.Thisinputisnegative,similartoSpring1'sforceonMass1.
∙TapalineoffDamper1'sforcelineandconnectittothefirstinput(whichispositive)ofMass2'sSumblock.
NowwewilladdintheforcefromSpring2.ThisforceactsonlyonMass2,butdependsonthegroundprofile,W.Spring2'sforceisequaltoX2-W.
∙InsertaStepblockinthelowerleftareaofyourmodelwindow.Labelit"W".
∙Editit'sStepTimeto"0"andit'sFinalValueto"0".(Wewillassumeaflatroadsurfacefornow).
∙InsertaSumblocktotherightoftheWStepblockandedititssignsto"-+".
∙ConnecttheoutputoftheStepblocktothepositiveinputofthisSumblock.
∙Tapalineoffthe"x2"signalandconnectittothenegativeinputofthenewSumblock.
∙InsertaGainblocktotherightofthisSumblockandconnecttheSum'soutputtothenewGain'sinput.
∙Changethevalueofthegainto"k2"andlabelit"Spring2".
∙Connecttheoutputofthisblock(Spring2'sforce)tothefourthinputofMass2'sSumblock.Thisforceaddsininthepositivesense.
Next,wewilladdintheforcefromDamper2.Thisforceisequaltob2timesV2-d/dt(W).SincethereisnoexistingsignalrepresentingthederivativeofWwewillneedtogeneratethissignal.
∙InsertaDerivativeblock(fromtheLinearblocklibrary)totherightoftheWstepblock.
∙TapalineoftheStep'soutputandconnectittotheinputoftheDerivativeblock.
∙InsertaSumblockaftertheDerivativeblockandeditit'ssignsto"+-".
∙ConnecttheDerivative'soutputtothepositiveinputofthenewSumblock.
∙Tapalineoffthe"v2"lineandconnectittothenegativeinputofthisSumblock.
∙ConnecttheoutputofthisSumblock(Damper2'sforce)tothefifthinputofMass2'sSumblock.Thisforcealsoaddsinwithpositivesign.
ThelastforceintheinputUactingbetweenthetwomasses.
∙InsertaStepblockintheupperleftofthemodelwindow.
∙Connectit'soutputtotheremaininginputofMass1'sSumblock(withpositivesign).
∙TapalineoffthissignalandconnectittotheremaininginputofMass2'sSumblock(withnegativesign).
∙EditthisStepblock'sStepTimeto"0"andleaveitsFinalValue"1".
∙LabelthisStepblock"U".
∙Finally,toviewtheoutput(X1-X2)insertaScopeconnectedtotheoutputoftherightmostSumblock.
Open-loopresponse
Tosimulatethissystem,first,anappropriatesimulationtimemustbeset.SelectParametersfromtheSimulationmenuandenter"50"intheStopTimefield.50secondsislongenoughtoviewtheopen-loopresponse.Thephysicalparametersmustnowbeset.RunthefollowingcommandsattheMATLABprompt:
m1=2500;
m2=320;
k1=80000;
k2=500000;
b1=350;
b2=15020;
Runthesimulation(Ctrl-torStartontheSimulationmenu).Whenthesimulationisfinished,double-clickonthescopeandhititsautoscalebutton.Youshouldseethefollowingoutput.
ExtractingaLinearModelintoMATLAB
Alinearmodelofthesystem(instatespaceortransferfunctionform)canbeextractedfromaSimulinkmodelintoMATLAB.ThisisdonethroughtheuseofInandOutConnectionblocksandtheMATLABfunctionlinmod.WewillextractonlythemodelfromtheinputUtotheoutputX1-X2.
∙First,replacetheUStepblockwithanInConnectionBlock.
∙Also,replacetheScopeblockwithanOutConnectionBlock.(TheseblockscanbefoundintheConnectionsblocklibrary).Thisdefinestheinputandoutputofthesystemfortheextractionprocess.
Saveyourfileas"suspmod.mdl"(selectSaveAsfromtheFilemenu).MATLABwillextractthelinearmodelfromthesavedmodelfile,notfromtheopenmodelwindow.AttheMATLABprompt,enterthefollowingcommands:
[A,B,C,D]=linmod('suspmodel')
[num,den]=ss2tf(A,B,C,D)
Youshouldseethefollowingoutput,providingbothstate-spaceandtransferfunctionmodelsofthesystem.
A=
1.0e+003*
0000.0010
000.00100
0.2500-1.8125-0.04800.0011
-0.03200.03200.0001-0.0001
B=
0
0
-0.0031
0.0004
C=
1-100
D=
0
num=
00.00000.00350.01880.6250
den=
1.0e+004*
0.0001
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