第2 3章其他数学建模实例

CHAPTERWritingSystem ByHuiWangOutlineofthischapterBasicLinearMatrixAlgebra&StateMechanicalRotationalThermalHydraulicLinearLiquid-Level AnalogousExamplethesystemshowninnode f=fK=K(xa-xb)node fK=fM+fB=MD2

Node i=v

If

NodeNodei= +12=b+vbRi=xbDxRComparingagain!!Find55KMB(b)CorrespondingmechanicalKMB(b)CorrespondingmechanicalAnalogouscircuitsrepresentsystemsforwhichthedifferentialequationshavethesameform.Thecorrespondingvariablesandparametersintwocircuitsrepresentedbyequationsofthesameformarecalledanalogs.Anelectriccircuitcanbedrawnthatlookslikethemechanicalcircuitandisrepresentedbynodeequations.TheanalogsarelistedinTable2.4.Thereisaphysicalsimilaritybetweenforcefandcurrenti,…seriesNodesinthemechanicalnetworkareanalogoustonodesintheelectricAnalogousf(f(t)-f(t)-f(t)12Md2dtf(t)=Bf2(t)=AnalogousCircuits:Example:S-M-DSystem WritecircuitEq.of +i1+ = 2

1 =

Dy=1Li i2

=1RNote MDMD2y+ +Ky=OutlineofthischapterBasicLinearMatrixAlgebra&StateMechanicalRotationalThermalHydraulicLinearLiquid-Level OtherMathematicModeling-----MechanicalRotationalTheequationcharacterizingrotationalsystemsaresimilartothosefortranslationsystems,wherethedisplacement,velocity,andaccelerationtermsarenowangularquantities.aceJKBbdfaceJKBbdfFig.2.14NetworkelementsmechanicalrotationalThethreeelementsinarotationalsystemareinertia(惯量)，thespring,andthedashpot.Themechanical-networkrepresentationoftheseelementsisshowninFig.2.14.ThetorqueappliedtoabodyhavingamomentofinertiaJproducesanangularacceleration. TJ=Ja=JDw=JD2q(2.73)Whenatorqueisappliedtoaspring,thespringistwistedbyanangleq. Toproducemotionofthebody,atorquemustbeappliedtoovercomethereactiondampingtorque.ThedampingtorqueTB=B(we-wf)=B(Dqe-Dqf)Thetorqueequationiswrittenforeachnodebyequatingthesumofthetorqueateachnodetozero. =Ja=JDw= =Ma= =Ma=MDv=MD2fK=K(xc-xdifxd= = TB=B(we-wf)=B(Dqe-Dqf) fB=B(ve-vf)=B(Dxe

-DxfaaceJKBbdfFig.2.14NetworkelementsmechanicalrotationalThesystemshowninFig.2.15hasamass,withamomentofinertiaJ,immersedinafluid.AtorqueTisappliedtothemass.KBFig.2.15(a)SimplerotationalThereisaonenodehavingadisplacementq;thereforeonlyoneequationisnecessaryAnelectricalanalogcanbeobtainedthustorqueKBFig.2.15(a)SimplerotationalJBK2.15(b)CorrespondingmechanicalJD2q+BDq+KqJBK2.15(b)Correspondingmechanical

d +

JfmechanicalJsJs2q(s)+fsq(s)=M\q(s)M1s(Js+fMultiple-elementmechanicalrotationalThesystemrepresentedbyFig.2.16ahastwodisksthathavedampingbetweenthemandalsobetweeneachofthemandtheframe.ThecorrespondingmechanicalnetworkisdrawninFig.2.16b.Multiple-elementmechanicalrotationalThesystemrepresentedbyFig.2.16ahastwodisksthathavedampingbetweenthemandalsobetweeneachofthemandtheframe.ThecorrespondingmechanicalnetworkisdrawninFig.2.16b.Node1:K1q1-K2q2=T(t) Node2:-K1q1+[J1D2+(B1+B3)D+K1]q2-B3Dq3=0(2.78)Node3:-B3Dq2+[J2D2+(B2+B3)D+K2]q3= J1B1J2K2Fig.2.16(b)Rotationalsystem‘scorrespondingmechanicalMultiple-elementmechanicalrotationalThesethreeequationscanbesolvedsimultaneouslyforq1,q2,andq3asafunctionoftheappliedtorque.G1(D)=TG(D)G(D)G1(D)=TG(D)G(D)AndtheoveralltransferfunctionofthesystemisG=G=GG =q1q2q3=Tq Question2.17:Stateequationforthissystem? Question2.17:Stateequationforthissystem?Figure2.17DetailedandoverallrepresentationsofMoremechanicalrotationalsystemexamplesaresimilar,forEffectivemomentofinertiaanddampingofageartrainshowninFig.2.18ainP.45EffectivemomentofinertiaanddampingofageartrainEffectivemomentofinertiaanddampingofageartrainOtherMathematicModelingThermalSystems（热力系统Alimitednumberofthermalsystemscanberepresentedbydifferentialequations.Thebasicrequirementisthatthetemperatureofabodybeconsidereduniform.Thenecessaryconditionofequilibriumrequiresthattheheataddedtothesystemequaltheheatstoredplustheheatcarriedaway.Thisrequirementcanalsobeexpressedintermsofrateofheatflow.CRFig.2.19NetworkofCRFig.2.19NetworkofthermalAthermalsystemnetworkisdrawnbythermalcapacitanceandthermalresistance.Theadditionalheatstoredinabodywhosetemperatureisraisedfromq1toq2isgivenbyIntermsofrateofheat q=CD(q2-q1)Thethermalcapacitancedeterminestheamountofheatstoredinabody,-----likeacapacitorinanelectriccircuit.q=q3-qRRateofheatflowthroughabodyintermsoftheq=q3-qRThethermalThethermalresistancedeterminestherateofheatflowthroughthebody,-----likearesistorinanelectriccircuit.Considerathinglass-walledthermometer(haveacapacitanceCandaresistanceR)filledwithmercurythathasstabilizedatatemperatureq1.Itisplungedintoabathoftemperaturesq0att=0.Thetemperatureofthemercuryisqm.Theflowofheatintothethermometer q=q0-qmRTheheatenteringthethermometerstoredintheC,isgivenby

h=D

Theseequationscanbecombinedtoh=q0-qm =C(q-q RCDqm+qm=q0 (2.90)ThethermalnetworkisdrawninFig.2.20.Thenodeequationforthiscircuit,withthetemperatureconsideredasavoltage,givesEq.(2.89)directly.Then,thetransferfunctionG=qm/q0maybe SimplerepresentationofRCD SimplerepresentationofG(D)=qmq

RCD+ G(s)=qm(s) x1uLetx=q,u= x1u

q0(

RCs+Objective:heatingcold-liquidtothetemperatureqa,Wqcqa,Wqc,environmenttemperatureInput(controlvariable)canW,qc,qcandq,etc..ThemostsuitablevariableisW.Othersareasdisturbvariables.StepStep2:AssumptionandStep3:developmathematics •LetQexpressesquantityofqa,qa,Wqc,environmenttemperatureQQ+Firstly,considersteadystate, =Q +Q =Q +QQQ +Q =QQ +Q =QQ =qcccqc,Q =WH QQ +Q =Qqa,Wqa,Wqc,environmenttemperatureAssumingspecificvolumeofc =c =∵qa=qc+ »qwhereWisvery\\qa=qcWThisThisisasystem’ssteadystatemodel. y=a+bx Qc+Qs=ThermalSystems:DirectsteamheaterSecondly,considersystemdynamicmodel,itismore

+Q

Q

qa, hot- andVisavailable isfluid’sdensity,environmenttemperature

qc,qc

whereCiscalledcapacitycoefficient.Itrepresentsacapabilitytostoringenergyofthetank.\

+q

cqa=

ccqc+\C dqa+q cqa=qcqc+WHt Sc

OtherMathematicModelingDirectsteam R=qa

Rrepresentsaresistancetopreventheatenergydepartfromthetank.Itiscalledheat∵q =qc+ »q q, Cdqa

1qaR

1qc

+

dqa+q

=qc+ dqa+qa=qc+qc,qcenvironmenttemperatureThermalSystems:Directsteam

+qa=qc+qa,Wqa,Wqc,environmenttemperature (s)=qa(s)= W(s Ts+qa(s qGd(s)qcIfthereis

(s Ts+IncrementformdifferentialInprocesscontrol,weusuallyconsiderincrementequationofvariables,ForEx.Systemdynamicequation dqa+qa

=qc+ ** \qa =qc0+qa

W dqa+qa

qa

=qc+

KWWeobtainincrementFirst-orderConsideringEqs.labeled**,whichisfirst-orderdifferentialequation,thoughthemodelandRCcircuitmodelrepresenteddifferentsystems,theirtransferfunctionsbetweentheinputandoutputhavesameform.Tde0+e0=

E0(s)(Ts+1)=Ei dqa+qa=qc+

E0(sTsWKqTsWKqa(s)

Ts+First-orderControlpathFirst-order

TsTs1qa(s)First-ordersystem’sstepG(s)=Y(s)= G(s)=Y(s)= U(sTs+Tdy(t)

y(t)= TtT

y(t)t

OtherMathematicModelingLiquid-LevelSystem（液位系统Two-tankliquid-levelcontrolsystemconsistsoftwofirst-orderdependentplantsthatareconnectedinseries.Noted,heretheheightsh1andh2ofthetanksarecoupling（耦合）(seeP53Fig.2.23).Objective:holdh2unchanged,whichrelatedtoqoutandqinqObjective:holdh2unchanged,whichrelatedtoqoutandqinA=cross-sectionaltankObjective:holdh2unchanged,whichrelatedtoqoutandTankTank

dh=-q 1TankTankd2hdt+(T1+T2+A1R2 2+h2=R2q -R2q Notethisiscoupling1TankTankd2hdt+(T1+T2+A1R2 2+h2=R2q -R2q LiquidlevelSystem-1:transfercontrolGcontrolG(D)u(t= h2(t)-R2-T1R2T1T2D2++A1R2)D+ (D)y(t =h2(t)(tqin(tRT1T2D2+ + +A1R2)D+LiquidlevelSystem-1:stateAssignedstateAssignedstatevariablesx=h1x=h1 h2x-1x21= R +x0-R -1T-1AandinputvariablesLetoutputvariables h2y=hy=1y=1x0x1 2LiquidlevelSystem-Two-tankliquidlevelcontrolsystemasFig.below.Definitions A1TankA1TankTankA=cross-sectionaltankObjective:Objective:holdh2unchanged,whichrelatedtoqoutandqinNotedthattheheightsandh2herearemeans:controlqout( LiquidlevelSystem-2whereTank Disturb

=q1

- Tank h2

eliminateinternaleliminateinternalvariablesq1,h1,etc.thenwegot

Tank

R1RR = (R 1

-h2Tank Controlinput A1TankA1TankTankThisisaninput/outputmodel,expressestherelationshipofoutputh2andd