机械控制理论复习资料.ppt

FundamentalsofMechanicalControlTheory 2008 5 Review1Definition2WritingSystemEquation3Solutionofdifferentialequations4LaplaceTransform5SystemRepresentation6Controlsystemcharacteristics7Frequencyresponsecharacteristics 1Definition open loopcontrolsystemsclosed loopcontrolsystemsfunctionalblockdiagramsystemtransferfunctiongeneralsolution Particularintegral Complementaryfunctionorsteady statecomponentc t ss transientcomponentc t t DampingRatio Theundampednaturalfrequency ntimeconstantTPeakovershootcpTimetomaximumovershoottpTimetofirstzeroerrort0SettlingtimetsRisetimetrFrequencyofoscillation d LaplacetransformTheorem1 LinearityTheorem2 SuperpositionL f1 t f2 t L f1 t L f2 t F1 s F2 s 3 TranslationintimeL f t a u 1 t a e asF s 4 Complexdifferentiation5 Translationinthes6 Realdifferentiation7 Realintegration8 Finalvalue9 Initialvalue10 Complexintegration11 Resembletheorem forwardtransferfunctionfeedbacktransferfunctionH s controlratio oroveralltransferfunctionopen looptransferfunction Frequencyresponse Thesteady stateresponseofthesystemforsininputs Frequencyresponsemethod Amethodtostudythesystem sperformanceusingthefrequencyresponse Magnitude frequencycharacteristic theratioofamplitudesoftheoutputtotheinput A Xo Xi It salsorepresentedas G j Phase frequencycharacteristic thedifferenceofphasebetweentheoutputandinputsignal It srepresentedas G j It spositivewhenitrotatedcounterclockwise CCW Frequencycharacteristic thecomprehensivenessofA and It srepresentedasA orA ej Nyquistplotistheplotwhichtheoutput inputratiomaybeplottedinpolarcoordinateswithfrequencyasaparameter Bodeplots Logarithmicplots ThefrequencycharacteristicisrepresentedbyA and curvesrespectively WritingSystemEquation Inordertowritethedifferentialequation thebasiclawsgoverningperformancearefirststatedforelectrical mechanical thermal andhydraulicsystems Thesebasiclawsarethenappliedtospecificdevices andtheirdifferentialequationsofperformanceareobtained Electriccircuits I LoopMethodII NodeMethod1 Thenumberofequationsrequiredisequaltothenumberofunknownnodevoltages 2 Anequationiswrittenforeachnode 3 Eachequationincludesthefollowing a Thenodevoltagemultipliedbythesumofalltheadmittancesthatareconnectedtothisnode Thistermispositive b Thenodevoltageattheotherendofeachbranchmultipliedbytheadmittanceconnectedbetweenthetwonodes Thistermisnegative Mechanicaltranslationsystems Beforethedifferentialequationsofacompletesystemcanbewritten themechanicalnetworkmustfirstbedrawn Thisisdonebyconnectingtheterminalsofelementsthathavethesamedisplacement Thentheforceequationiswrittenforeachnodeorpositionbyequatingthesumoftheforcesateachpositiontozero Theequationsaresimilartothenodeequationsinanelectriccircuit Thereferencepositionshouldbetakenfromthestaticequilibriumpositions Solutionofdifferentialequations c t c t ss c t tsteady statecomponentc t sswhichhasthesameformasthedrivingfunctiontheformofthetransientcomponentc t toftheresponseisdependedonthecharacteristicequation transientcomponentc t tisthesolutionofhomogeneousequationsinusoidalinput polynomialinput Thecoefficientsofthetransienttermsaredeterminedbytheinstantaneousvalueofthesteady statecomponent therootsofthecharacteristicequation andtheinitialconditions ThestandardformsofthequadraticfactorThecorrespondingrootsare LaplaceTransform Solvingordinaryconstant coefficientdifferentialequationstheLaplacetransformtodifferentialequationsInversetransformations partialfractions F s canbeexpressedasThecoefficientsA1 A2 AnaretermedtheresiduesofF s atthecorrespondingpoless1 s2 sn ToevaluateresiduesAkofF s Case1 unrepeatedPolesCase2 repeatedPoles Partial fractionshortcutsTheprocedureistoevaluateonlythosecoefficientsforrealpolesthatcanbeobtainedreadilywithoutdifferentiation Thecorrespondingpartialfractionsarethensubtractedfromtheoriginalfunction LocationofpolesandstabilitySystemstabilityrequiresthatallrootsofthecharacteristicequation allpolesofcontrolsystem belocatedinthelefthalfofthesplane SystemRepresentation Blockdiagram feedbackSignalflowgraph SFG intercoupledrelationshipsThesimplificationofblockdiagramsTheMasonGainRulewhereTnisthetransmittanceofeachforwardpathbetweenasourceandasinknode 1 L1 L2 L3 nisthecofactorofTn Controlsystemcharacteristics Routh sstabilitycriterionAnecessarybutnotsufficientconditionforstablerootsisthatallthecoefficientsincharacteristicsequationofsystembepositive ThenumberofrootsofthecharacteristicequationwithpositiverealpartsisequaltothenumberofchangesofsignofthecoefficientsinthefirstcolumnofRouthianarray Theorem1 DivisionofarowTheorem2 AzerocoefficientinthefirstcolumnTheorem3 Azerorow FeedbackSystemTypesthestandardformofthetransferfunctionasm 0 1 2 denotesthetransferfunctiontypeAnalysisofSystemTypes Controlsystemcharacteristics FindingthefrequencycharacteristicSubstitutingsinG s withj s j frequencycharacteristicu real frequencycharacteristicv imaginary frequencycharacteristicMagnitude frequencycharacteristic A G j Xo Xi Phase frequencycharacteristic NyquistplotsNyquistplotistheplotwhichtheoutput inputratiomaybeplottedinpolarcoordinateswithfrequencyasaparameter 1 FindthefrequencycharacteristicsG j fromthetransferfunctionG s G j u jv 2 Findthevalueofu andv When 0and 3 Findthesituationofu andv When 0and 4 DeterminethequatantofNyquistlocus 5 FindtheNyquistlocuspassingthroughtheofrealandimaginaryaxis 6 FindtheasymptotesofNyquistlocus BodeplotsThefrequencycharacteristicisrepresentedbyA G j and G j curvesrespectively 1 TheBodeplotofmagnituderesponse Horizontalaxis abscissa equallydividedbylg Longitudinalaxial equallydividedby20lg G j 1db decibel 20lg G j 2 TheBodeplotofphaseresponse Horizontalaxis abscissa equallydividedbylg Longitudinalaxial j equallydividedby j 1 FindthefrequencycharacteristicsG j fromthetransferfunctionG s 2 ConverttheG j tothecombinationofseveraltypicalfactors generalizedform multiplicationordivision 3 Findthecornerfrequency Tofeachtypicalfactor 4 Drawtheasymptotesofeachtypicalfactor 5 Superposition addingallthemagnitudeofeachfactorexceptthegainKm 6 Movethecurveup ordown ward vertically bythevalueof20lgKm 7 Drawthephaseplotofeachfactorandthenaddthemtoformthecurveofoverallsystem Nyquist sstabilitycriterionIfthenumberofloopsthroughwhichGk s G s H s encirclesthepoint 1 j0 Counter clockwiseinthe GH planewhen changesfrom to isequaltothenumberofpolesofGk s locatedattherighthalf s plane theclosed loopsystemwillbestable 1 P 0 IfthelocusofGk s inthe GH planewhen cha