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TrimSize:7.375inx9.25in Klue ffirs.tex V1-01/05/2015 4:22P.M. Pagei TrimSize:7.375inx9.25in Klue ffirs.tex V1-01/05/2015 4:22P.M. Pagei Dynamic Systems: Modeling, Simulation, and Control Craig A. Kluever UniversityofMissouri-Columbia TrimSize:7.375inx9.25in Klue ffirs.tex V1-01/05/2015 4:22P.M. Pageii VP&ExecutivePublisher DonFowley ExecutiveEditor LindaRatts SeniorMarketingManager ChristopherRuel AssociateProductionManager JoycePoh Coverdesigner KenjiNgieng ThisbookwassetbyLaserwordsPrivateLimited. MATLAB®andSimulinkareregisteredtrademarksofTheMathWorks,Inc.andareusedwithpermission.See www.mathworks.com/trademarksforalistofadditionaltrademarks.TheMathWorksdoesnotwarranttheaccuracyofthetextor exercisesinthisbook.Thisbook’suseordiscussionofMATLAB®softwareorrelatedproductsdoesnotconstituteendorsementor sponsorshipbyTheMathWorksofaparticularpedagogicalapproachorparticularuseoftheMATLAB®software. Foundedin1807,JohnWiley&Sons,Inc.hasbeenavaluedsourceofknowledgeandunderstandingformorethan200years,helping peoplearoundtheworldmeettheirneedsandfulfilltheiraspirations.Ourcompanyisbuiltonafoundationofprinciplesthatinclude responsibilitytothecommunitiesweserveandwhereweliveandwork.In2008,welaunchedaCorporateCitizenshipInitiative,a globalefforttoaddresstheenvironmental,social,economic,andethicalchallengeswefaceinourbusiness.Amongtheissuesweare addressingarecarbonimpact,paperspecificationsandprocurement,ethicalconductwithinourbusinessandamongourvendors,and communityandcharitablesupport.Formoreinformation,pleasevisitourwebsite:www.wiley.com/go/citizenship. Copyright©2015JohnWiley&Sons,Inc.Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrieval system,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanningorotherwise,exceptas permittedunderSections107or108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthe Publisher,orauthorizationthroughpaymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter,Inc.,222Rosewood Drive,Danvers,MA01923(Website:www.copyright.com).RequeststothePublisherforpermissionshouldbeaddressedtothe PermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030-5774,(201)748-6011,fax(201)748-6008, oronlineat:www.wiley.com/go/permissions. Evaluationcopiesareprovidedtoqualifiedacademicsandprofessionalsforreviewpurposesonly,foruseintheircoursesduringthe nextacademicyear.Thesecopiesarelicensedandmaynotbesoldortransferredtoathirdparty.Uponcompletionofthereview period,pleasereturntheevaluationcopytoWiley.Returninstructionsandafreeofchargereturnshippinglabelareavailableat: www.wiley.com/go/returnlabel.Ifyouhavechosentoadoptthistextbookforuseinyourcourse,pleaseacceptthisbookasyour complimentarydeskcopy.OutsideoftheUnitedStates,pleasecontactyourlocalsalesrepresentative. LibraryofCongressCataloging-in-PublicationData Kluever,CraigA.(CraigAllan) Dynamicsystems:modeling,simulation,andcontrol/CraigKluever. pagescm Includesbibliographicalreferencesandindex. ISBN978-1-118-28945-7(hardback) 1.Systemanalysis.2.Dynamics–Mathematicalmodels.3.Mechanicalengineering–Mathematicalmodels.I.Title. T57.6.K482014 620.001’185–dc23 2014023118 PrintedintheUnitedStatesofAmerica 10987654321 TrimSize:7.375inx9.25in Klue ftoc.tex V1-01/19/2015 5:10P.M. Pageiii Contents Preface vi 1 IntroductiontoDynamicSystemsandControl 1 1.1 Introduction 1 1.2 ClassificationofDynamicSystems 2 1.3 ModelingDynamicSystems 4 1.4 ObjectivesandTextbookOutline 5 References 6 2 ModelingMechanicalSystems 7 2.1 Introduction 7 2.2 MechanicalElementLaws 7 2.3 TranslationalMechanicalSystems 13 2.4 RotationalMechanicalSystems 24 Summary 30 References 30 Problems 30 3 ModelingElectricalandElectromechanicalSystems 43 3.1 Introduction 43 3.2 ElectricalElementLaws 43 3.3 ElectricalSystems 46 3.4 Operational-AmplifierCircuits 54 3.5 ElectromechanicalSystems 57 Summary 68 References 69 Problems 69 4 ModelingFluidandThermalSystems 80 4.1 Introduction 80 4.2 HydraulicSystems 80 4.3 PneumaticSystems 93 4.4 ThermalSystems 99 Summary 104 References 105 Problems 105 TrimSize:7.375inx9.25in Klue ftoc.tex V1-01/19/2015 5:10P.M. Pageiv iv Contents 5 StandardModelsforDynamicSystems 115 5.1 Introduction 115 5.2 State-VariableEquations 115 5.3 State-SpaceRepresentation 119 5.4 Linearization 128 5.5 Input-OutputEquations 133 5.6 TransferFunctions 136 5.7 BlockDiagrams 139 5.8 StandardInputFunctions 143 Summary 145 Problems 146 6 NumericalSimulationofDynamicSystems 156 6.1 Introduction 156 6.2 SystemResponseUsingMATLABCommands 157 6.3 BuildingSimulationsUsingSimulink 162 6.4 SimulatingLinearSystemsUsingSimulink 165 6.5 SimulatingNonlinearSystems 170 6.6 BuildingIntegratedSystems 178 Summary 183 References 184 Problems 184 7 AnalyticalSolutionofLinearDynamicSystems 197 7.1 Introduction 197 7.2 AnalyticalSolutionstoLinearDifferentialEquations 197 7.3 First-OrderSystemResponse 205 7.4 Second-OrderSystemResponse 213 7.5 Higher-OrderSystems 230 7.6 State-SpaceRepresentationandEigenvalues 232 7.7 ApproximateModels 235 Summary 237 Reference 238 Problems 238 8 SystemAnalysisUsingLaplaceTransforms 247 8.1 Introduction 247 8.2 LaplaceTransformation 247 8.3 InverseLaplaceTransformation 254 8.4 AnalysisofDynamicSystemsUsingLaplaceTransforms 259 Summary 271 References 271 Problems 271 9 Frequency-ResponseAnalysis 277 9.1 Introduction 277 9.2 FrequencyResponse 277 TrimSize:7.375inx9.25in Klue ftoc.tex V1-01/19/2015 5:10P.M. Pagev Contents v 9.3 BodeDiagrams 290 9.4 Vibrations 306 Summary 313 References 313 Problems 313 10 IntroductiontoControlSystems 322 10.1 Introduction 322 10.2 FeedbackControlSystems 322 10.3 FeedbackControllers 329 10.4 Steady-StateAccuracy 344 10.5 Closed-LoopStability 350 10.6 Root-LocusMethod 352 10.7 StabilityMargins 373 10.8 ImplementingControlSystems 381 Summary 385 References 385 Problems 386 11 CaseStudiesinDynamicSystemsandControl 397 11.1 Introduction 397 11.2 VibrationIsolationSystemforaCommercialVehicle 397 11.3 SolenoidActuator–ValveSystem 409 11.4 PneumaticAir-BrakeSystem 417 11.5 HydraulicServomechanismControl 427 11.6 FeedbackControlofaMagneticLevitationSystem 444 Summary 455 References 455 AppendixA Units 457 AppendixB MATLABPrimerforAnalyzingDynamicSystems 459 B.1 Introduction 459 B.2 BasicMATLABComputations 459 B.3 PlottingwithMATLAB 462 B.4 ConstructingBasicM-files 463 B.5 CommandsforLinearSystemAnalysis 464 B.6 CommandsforLaplaceTransformAnalysis 465 B.7 CommandsforControlSystemAnalysis 467 AppendixC SimulinkPrimer 469 C.1 Introduction 469 C.2 BuildingSimulinkModelsofLinearSystems 469 C.3 BuildingSimulinkModelsofNonlinearSystems 478 C.4 SummaryofUsefulSimulinkBlocks 482 Index 485 TrimSize:7.375inx9.25in Klue f03.tex V1-12/22/2014 9:58P.M. Pagevi Preface This textbook is intended for an introductory course in dynamic systems and control, typically required in undergraduate mechanical engineering andsomeaerospace engineering curricula.Suchacourseisusually takeninthejuniororsenioryear,afterthestudenthascompletedcoursesinmechanics,differentialequations, andelectricalcircuits.Themajortopicsofadynamicsystemsandcontrolcourseinclude(1)mathematical modeling, (2) system-response analysis, and (3) an introduction to feedback control systems. The primary objectiveofthistextbookisacomprehensiveyetconcisetreatmentofthesemajortopicswithanemphasis ondemonstratingphysicalengineeringapplications.Ithasbeenmyexperiencethatundergraduatestudents remainengagedinasystemdynamicscoursewhentheconceptsarepresentedintermsofrealengineering systems (such as a hydraulic actuator) instead of academic examples. This textbook is a distillation of 20 years of course notes and strategies for teaching system dynamics in the Mechanical and Aerospace EngineeringDepartmentattheUniversityofMissouri-Columbia.Itisthusbasedonmyextensiveclassroom experienceandstudentfeedback,andtheendresultisatextwhosekeyfeaturesdifferfromcurrentsystem dynamicstextbooks. Chapter1introducesdynamicsystemsandcontrol,includingdefinitionsoftherelevanttermsandcat- egoriesofdynamicsystems.Thefollowingthreechaptersalldealwithdevelopingmathematicalmodelsof physicalengineeringsystems.Chapter2introducesthefundamentaltechniquesusedtoderivethemodeling equationsformechanicalsystems.Mechanicalsystems,perhapsmoreintuitivelyunderstoodbyundergradu- atestudentsastheyinvolveNewton’slawsofmotion,aretreatedfirst.Chapter3introducesthefundamental methodsofdevelopingthemathematicalmodelsforelectricalandelectromechanicalsystems.Modelsinthis chapterarederivedbyapplyingKirchhoff’sandFaraday’slawsandtheelectricalelementlawsthatgovernthe interactionsamongelectricalcharge,current,magneticflux,andvoltage.Chapter4presentsthefundamental techniquesinvolvedindevelopingmodelsforfluidandthermalsystems.Itisthelastchapterdedicatedtothe derivationofmathematicalmodels.Fluid(hydraulicandpneumatic)modelsarebasedontheconservationof mass,whilethermalmodelsarederivedusingtheconservationofenergy. Chapter5presentsstandardformsforrepresentingthemathematicalmodelsofdynamicsystems:state- variableequations,state-spacerepresentation,input-outputequations,transferfunctions,andblockdiagrams. Thelinearizationprocessisalsodescribedinthischapter.ThekeyconceptemphasizedinChapter5isthat eachstandardformissimplyaconvenientrepresentationofthesystemmodel(i.e.,thedifferentialequations) that lends itself to analysis of the system’s dynamic response. Therefore Chapter 5 serves as a transition betweendevelopingthemathematicalmodel(Chapters2–4)andobtainingthesystem’sresponseusingeither numericalsimulationoranalyticaltechniques(Chapters6–10). Chapter6presentsnumericalsimulationmethodsforobtainingtheresponseofdynamicsystems.Here, theMATLABsimulationsoftwareisusedexclusivelyasithasbecomethestandardcomputationalplatform foracademiaandindustry.SimulatingtheresponseoflinearsystemsusingMATLABcommandsispresented first.ThegraphicalsoftwareSimulinkispresentednext,anditisthefocusofthischapterandtheprimarysim- ulationtoolusedthroughouttheremainderofthetextbook.Simulinkisusedtosimulatelinearandnonlinear systemsusingthestandardformspresentedinChapter5. The next three chapters involve the analytical solution of linear systems. Chapter 7 covers analytical methodsforobtainingthesystem’sresponseinthetimedomainwithanemphasisonfirst-andsecond-order system response. Here, the two key concepts are (1) the correlation between the roots of the characteristic equationandtheformofthefree(ortransient)response;and(2)theequivalenceofthecharacteristicroots, polesofthetransferfunction,andeigenvaluesofthesystemmatrix.Chapter8presentsabriefoverviewof Laplacetransformtheoryanditsuseinobtainingtheresponseoflineardynamicsystems.Chapter9involves TrimSize:7.375inx9.25in Klue f03.tex V1-12/22/2014 9:58P.M. Pagevii Preface vii frequencyresponse,orthesystemresponsetoperiodicinputfunctions.Inthischapter,theemphasisisonthe Bodediagramasagraphicaldepictionoftheinformationrequiredforcompletefrequency-responseanalysis. Chapter10introducesfeedbackcontrolsystemswherethePIDcontroller(anditsvariants)isempha- sized.Twographicaltechniques,theroot-locusmethodandtheBodediagram,areusedtoanalyzetheclosed- loop response, design controllers, and assess stability. The chapter closes with a brief discussion of how controllersareimplementedasdiscrete-timealgorithmsinadigitalcomputer. Servingasacapstoneforthetextbook,Chapter11presentscasestudiesindynamicsystemsandcontrol. Mixed-discipline,integratedengineeringsystems,inspiredbytheresearchliterature,serveasthecasestudies. The five case studies illustrate the major topics of the textbook: (1) developing mathematical models, (2) predicting the system’s behavior using analytical and numerical methods, and (3) selecting the important system parameters in order to improve performance. Simulink is used extensively to obtain the dynamic responseofthesesystemsthatofteninvolvenonlinearitiesandothercomplexities. NumerousexamplesareprovidedatkeylocationsthroughoutChapters2to10inordertoillustratethe topic discussed by the particular section. Chapters 2–10 contain end-of-chapter problems that are grouped intothreecategories:(1)conceptualproblems,(2)MATLABproblems,and(3)engineeringapplications.In manycases,physicalengineeringsystems(suchasasuspensionsystem,solenoidactuator,orfiltercircuit)are revisited throughout the textbook in the chapter examples and problems. As with the capstone Chapter 11, many of the example and end-of-chapter problems illustrate concepts in dynamic systems and control by presenting“real-world,”physicalengineeringsystems. Appendix A presents the basic and derived units used in this textbook. Appendix B provides a brief introduction to MATLAB, M-files, and the commands that pertain to solving problems involving dynamic systemsandcontrol.AppendixCisaprimeronSimulinkandexpandsonthebriefSimulinktutorialcovered inChapter6. As previously mentioned, this textbook is an outgrowth of 20 years of teaching a system dynamics course. In the long time that I have taught this course, I have often employed a trial-and-error approach to determininganoptimumsetofstrategiesformaximizingstudentunderstandingofkeytopics.Someofthese strategies are unique to my teaching, and I outline them here so that the reader can see how this textbook marksadeparturefromothersinthefield. 1. Because system analysis and control system design begin with developing the appropriate math- ematical models, the foundations for modeling all physical systems are presented in sequence by Chapters 2–4.Thegoal ofthesechapters istopresentmodeling explanations thatareconcise and clear, and this objective is achieved by focusing on essential, representative systems. Chapter 5 succinctly presents all standard forms for representing system models so that the fundamentals of systemmodelingarecoveredearlyinthetextbookandnotspreadthroughoutthemanuscript. 2. NumericalsimulationoflinearandnonlinearsystemsusingMATLABandSimulink(Chapter6)are coveredbeforeanalyticalmethods(Chapters7–9).Myexperienceisthatpresentingmodelingand simulation of real engineering applications early in the semester engages the students in the sub- jectmaterial.IndustryhaslongusedSimulinktomodelandanalyzerealengineeringsystems.My classroomexperiencehasshownthatundergraduatestudentsarequitecapableof – andenthusias- tic about! – using Simulink to simulate complex, integrated systems that involve real effects such as nonlinear friction,turbulent flow, and modeling discontinuities.Instructors whowishtopresent analyticalmethodsbeforenumericalsimulationmaysimplycoverChapter6afterChapters7and8. 3. ThemethodsforobtaininganalyticalsolutionsdonotrelyonLaplacetransformtheory.InChapter7 (time-domainresponse)thecharacteristicequationandtheassociatedcharacteristicroots(ortransfer functionpolesoreigenvalues)areusedtodeterminethetransientresponse.Thefrequencyresponse isderivedinChapter9bydeterminingtheforcedresponsetotheinputestwheres=𝜎+j𝜔isacom- plexvariable.Transferfunctionsareusedextensivelyinthistextbookandarederivedwithoutusing Laplace transforms. I believe that using Laplace transforms (and the associated inversion process) toobtaintheresponseisundulytedious;furthermore,myexperienceisthatmoststudentsfindtheir TrimSize:7.375inx9.25in Klue f03.tex V1-12/22/2014 9:58P.M. Pageviii viii Preface use nonintuitive. Therefore, my approach is to focus on the characteristic roots r and the form of i the system response in terms of erit. Chapter 8 presents the Laplace transformation and its use to obtainthedynamicresponse.Therefore,instructorswhowishtoutilizeLaplacetransformmethods maydoso.However,Chapter8maybeomittediftheinstructordoesnotcaretoincludetheLaplace transformmethod. 4. Chapter 11 presents five engineering case studies in dynamic systems and control. The five case studies, inspired by research articles, are (1) vibration isolation in a vehicle suspension system, (2) an electromechanical (solenoid) actuator, (3) a pneumatic air-brake system, (4) hydraulic ser- vomechanismcontrol,and(5)feedbackcontrolofamagneticlevitationsystem.Thesecasesillustrate topicsfromthepreviouschapters:modeling,simulation,linearization,analyticalmethods,andcon- trol.InstructorsmaywishtointroducethevariousapplicationscontainedinChapter11earlyinthe semester as certain topics are covered and then revisit the cases studies as the course progresses. Again, I cannot overstate the importance of presenting analysis of real, complex engineering sys- tems to undergraduate students in order to motivate them, engage their interest, and illustrate key conceptsindynamicsystemsandcontrol. Several people have contributed to the production of this textbook. I would like to thank Roger C. Fales,mycolleagueattheUniversityofMissouri,forhisexpertknowledgeandcommentsonfluidsystems. TwostudentsattheUniversityofMissouriprovidedtremendousassistance:AnnemarieHoyersearchedthe engineering literature and helped with the solution manual, and James Smith created illustrations for the draft manuscript. Many reviewers provided valuable suggestions for improving this textbook and they are listedhere: BradleyT.Burchett,Rose-HulmanInstituteofTechnology MarkB.Colton,BrighamYoungUniversity EricConstans,RowanUniversity R.ReesFullmer,UtahStateUniversity RajeshGanesan,GeorgeMasonUniversity SeonHan,TexasTechUniversity S.GrahamKelly,TheUniversityofAkron KathleenLamkin-Kennard,RochesterInstituteofTechnology JavadMohammadpour,UniversityofGeorgia NejatOlgac,UniversityofConnecticut AndrewR.Plummer,UniversityofBath ErnestW.Tollner,TheUniversityofGeorgia Hwan-SikYoon,TheUniversityofAlabama It was a sincere pleasure to work with the editorial and production staff at Wiley and I particularly thank Linda Ratts and Hope Ellis. I also thank my wife Nancy M. West for her skilled editorial work and heartfeltencouragementthroughoutthisproject.Finally,Iacknowledgeandthankmymentorswhohavehad animmenseimpactonmylife:myadvisorBionL.PiersonofIowaStateUniversity,JohnP.RiehlofNASA GlennResearchCenter,andmyparentsAllanandJanetKluever. Andfinally,Ithankmyson,SilasWestKluever,towhom,alongwithmyparents,Idedicatethisbook. At 12, he has all the makings of a mechanical engineer, although at present he is much more interested in becomingamajorleaguebaseballplayer. CraigA.Kluever Mechanical&AerospaceEngineering UniversityofMissouri-Columbia January2015

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