Variable Stiffness Suspension System

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Title:
Variable Stiffness Suspension System
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1 online resource (191 p.)
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english
Creator:
Anubi, Olugbenga M
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Crane, Carl D, Iii
Committee Members:
Dixon, Warren E
Barooah, Prabir
Crisalle, Oscar Dardo
Hager, William Ward
Roberts, Rodney G
Rico Martinez, Jose Maria

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Subjects / Keywords:
adaptive -- controls -- nonlinear -- suspension -- system
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Improvements over passive suspension designs is an active area of research. Past approaches utilize one of three techniques; adaptive, semi-active, or fully active suspension. An adaptive suspension utilizes a passive spring and an adjustable damper with slow response to improve the control of ride comfort and road holding. A semi-active suspension is similar, except that the adjustable damper has a faster response and the damping force is controlled in real-time. A fully active suspension replaces the damper with a hydraulic actuator, or other types of actuators like electromagnetic actuators, which can achieve optimum vehicle control, but at the cost of design complexity. The fully active suspension is also not fail-safe in the sense that performance degradation results whenever the control fails, which may be due to either mechanical, electrical, or software failures. Recently, research in semi-active suspensions has continued to advance with respect to capabilities, narrowing the gap between semi-active and fully active suspension systems. Today, semi-active suspensions (e.g using Magneto-Rheological (MR), Electro-Rheological (ER) etc) are widely used in the automobile industry due to their small weight and volume, as well as low energy consumption compared to purely active suspension systems. However, most semi-active design concepts are focused on only varying the damping coefficient of the shock absorber while keeping the stiffness constant. Meanwhile, in suspension optimization, both the damping coefficient and the spring rate of the suspension elements are usually used as optimization arguments. Therefore, a semi-active suspension system that varies both the stiffness and damping of the suspension element could provide more flexibility in balancing competing design objectives. This work considers the design, analyses, and experimentation of a new variable stiffness suspension system. The design is based on the concept of a variable stiffness mechanism. The mechanism, which is a simple arrangement of two springs, a lever arm, and a pivot bar, has an effective stiffness that is a rational function of the horizontal position of the pivot. The effective stiffness is varied by changing the position of the pivot while keeping the point of application of the external force constant. The overall suspension system consists of a horizontal control strut and a vertical strut. The main idea is to vary the load transfer ratio by moving the location of the point of attachment of the vertical strut to the car body. This movement is controlled passively, semi-actively, and actively using the horizontal strut. The system is analyzed using an L2-gain analysis based on the concept of energy dissipation. The analyses, simulation, experimental results, show that the variable stiffness suspension achieves better performance than the constant stiffness counterpart. The performance criteria used are; ride comfort, characterized by the car body acceleration, suspension deflection, and road holding, characterized by tire deflection.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Olugbenga M Anubi.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Crane, Carl D, Iii.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-05-31

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VARIABLESTIFFNESSSUSPENSIONSYSTEMByOLUGBENGAMOSESANUBIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013OlugbengaMosesAnubi 2

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TomyLORDandsaviorJesusChrist.TomylovelyandbeautifulwifeSerenaLeahAnubi.TomysweetmotherVictoriaAnubi 3

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ACKNOWLEDGMENTS Iexpressmygratitudetomysupervisorycommittee:Dr.CarlCrane,Dr.WarrenDixon,Dr.PrabirBarooah,Dr.WilliamHagger,Dr.OscarD.Crisalle,Dr.RicoJose,andDr.RodneyRobertsfortheirtime,efforts,andinvaluablecontributionstomyacademicgrowthduringmytimeattheUniversityofFlorida.ManythankstoDarsanPatelforhishelpwithexperimentsanddatacollection.IalsothankeverymemberoftheCenterforIntelligentMachinesandRobotics(CIMAR)fortheirinvaluablesupport.Also,myappreciationgoestomyparentin-law,CarlandBernadetteSealy,fortheirinvaluablesupport,prayersandmentorship.Finally,Iexpressmygratitudetothemembersofmymicrochurch;SerenaLeahAnubi,OlawaleAdeleye,AdriaMcKire(Adeleyetobe),OluwatosinAdeladan,OluwabusayoFawole(Adeladantobe),DavidWalker,KerlinandKettyBien,EthelPorras,EyitayoOwoeye,BrendaNelson,QwamelHanks,OreceCarty,ConradCole,ChelseaBrown,OtonyeBraids,andConradColefortheirprayers,counseling,andencouragementinmywalkwithourLordJesus. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 NOMENCLATURE ..................................... 13 ABSTRACT ......................................... 16 CHAPTER 1INTRODUCTION ................................... 18 1.1SomeHistoricalNotesonVehicleSuspension ................ 18 1.2TypesofSuspension .............................. 19 1.2.1IndependentSuspension ........................ 19 1.2.2DependentSuspension ........................ 20 1.2.3Semi-independentSuspension .................... 21 1.3SuspensionGeometry ............................. 22 1.4ControlledSuspension ............................. 24 1.4.1ActiveSuspensionDesign ....................... 25 1.4.2AdaptiveSuspensionDesign ..................... 25 1.4.3Semi-ActiveSuspensionDesign ................... 26 2VARIATIONOFSTIFFNESSINSUSPENSIONDESIGN ............. 30 2.1GeneralSemi-ActiveSuspension ....................... 30 2.2SingleModulationOptimalSemi-ActiveControlLaws ............ 33 2.3SpringandDamperModulation ........................ 37 2.4DoubleModulationOptimalSemi-ActiveControlLaws ........... 41 2.4.1SequentialModulation ......................... 41 2.4.2SimultaneousModulation ....................... 45 2.5Simulation .................................... 47 2.5.1TimeDomainSimulation ........................ 48 2.5.2FrequencyResponse .......................... 50 3VARIABLESTIFFNESSMECHANISM ....................... 54 3.1ForwardAnalysis ................................ 54 3.1.1EffectofronKandl0 ......................... 61 3.1.2SpecialCases .............................. 64 3.2ReverseAnalysis ................................ 65 3.3DynamicalAnalysis .............................. 67 5

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4VARIABLESTIFFNESSSUSPENSIONSYSTEM:PASSIVECASE ....... 70 4.1SystemDescription .............................. 71 4.1.1VariableStiffnessConcept ....................... 71 4.1.2MechanismDescription ........................ 72 4.1.3EquationsofMotion .......................... 73 4.2SystemAnalysis ................................ 77 4.2.1PerformanceObjective ......................... 79 4.2.2ConstantStiffnessCase ........................ 81 4.2.3PassiveVariableStiffnessCase .................... 87 4.2.4Experiment ............................... 90 4.2.5Simulation ................................ 94 4.2.5.1TimeDomainSimulation .................. 94 4.2.5.2FrequencyDomainSimulation ............... 95 5VARIABLESTIFFNESSSUSPENSIONSYSTEMSUSINGNONLINEARENERGYSINKS:ACTIVEANDSEMI-ACTIVECASES ................... 100 5.1OrthogonalNonlinearEnergySink ...................... 101 5.2ActiveCase ................................... 103 5.2.1ControlMassesandActuatorDynamics ............... 105 5.2.2ControlDevelopment .......................... 106 5.2.3StabilityAnalysis ............................ 109 5.2.4Simulation ................................ 111 5.3Semi-activeCase ................................ 113 5.3.1MR-damperModeling ......................... 115 5.3.2ControlDevelopment .......................... 120 5.3.2.1OpenLoopTrackingErrorDevelopment .......... 121 5.3.2.2ClosedLoopErrorSystemDevelopment ......... 123 5.3.3StabilityAnalysis ............................ 127 5.3.4SimulationResults ........................... 128 6ROLLSTABILIZATIONENHANCEMENTUSINGVARIABLESTIFFNESSSUSPENSION .................................... 135 6.1MechanismDescription ............................ 136 6.2Modeling .................................... 137 6.2.1YawDynamics ............................. 137 6.2.2RollDynamics .............................. 140 6.3KinematicControl ............................... 142 6.3.1ControlAllocation ............................ 144 6.3.2StabilityAnalysis ............................ 145 6.3.3Simulation ................................ 148 6.3.3.1FishhookManeuver ..................... 149 6.3.3.2DoubleLaneChangeManeuver .............. 152 6.4DynamicControl ................................ 153 6

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6.4.1ControlMassesandActuatorDynamics ............... 154 6.4.2ControlDesign ............................. 155 6.4.2.1VehicleBodyRoll ...................... 155 6.4.2.2ControlMasses ....................... 159 6.4.2.3HydraulicActuators ..................... 160 6.4.3StabilityAnalysis ............................ 163 6.4.4Simulation ................................ 167 7CONCLUSIONSANDFUTUREWORK ...................... 171 7.1Conclusion ................................... 171 7.2FutureWork ................................... 172 APPENDIX:PROOFOFTHEOREMS 2.1 AND 2.2 ................... 174 A.1ProofofTheorem 2.1 ............................. 174 A.2ProofofTheorem 2.2 ............................. 176 REFERENCES ....................................... 180 BIOGRAPHICALSKETCH ................................ 191 7

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LISTOFTABLES Table page 2-1Dynamicparametervalues ............................. 48 4-1RMSgainvaluesofexperimentalresults ...................... 91 5-1Dynamicparametervalues ............................. 111 5-2Hydraulicparametervalues ............................. 112 5-3Variancegainvalues ................................. 113 5-4MR-damperparametervalues ............................ 121 5-5Variancegainvalues ................................. 130 8

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LISTOFFIGURES Figure page 1-1Frontwheeldoublewishbonesuspensions .................... 20 1-2McPhersonsuspensionsystem ........................... 21 1-3Rigidaxlesuspensionsystem ............................ 22 1-4Twistbeamsuspensionsystem ........................... 23 2-1Quartercar-modulatedspring/damper ...................... 31 2-2Quartercar-modulatedspringanddamper .................... 37 2-3Timeresponse-carbodyacceleration ....................... 48 2-4Timeresponse-suspensiondeection ....................... 49 2-5Timeresponse-tiredeection ........................... 49 2-6Timeresponse-performanceindex ........................ 50 2-7Frequencyresponse-carbodyacceleration .................... 52 2-8Frequencyresponse-suspensiondeection ................... 53 2-9Frequencyresponse-tiredeection ........................ 53 3-1Schematics ...................................... 55 3-2Freebodydiagram .................................. 56 3-3Effectivestiffnessagainstd ............................. 59 3-4Effectivestiffnessagainst1 d ............................. 59 3-5Overallfreelengthagainstd ............................ 60 3-6Effectivestiffnessagainstroverd ......................... 62 3-7Variationofoverallfreelengthagainstroverd .................. 63 3-8Effectivestiffnessagainstdoverr ......................... 63 3-9Variationofoverallfreelengthagainstdoverr .................. 64 3-10Effectofaspectratioonachievablestiffnesslowerbound ............ 67 3-11Naturalfrequency .................................. 68 4-1Variablestiffnessmechanism ............................ 72 9

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4-2Variablestiffnesssuspensionsystem ........................ 73 4-3Quartercarmodel-passivecase .......................... 74 4-4Quartercarexperimentalsetup ........................... 91 4-5Sinusoidaltest-xedcase ............................. 92 4-6Sinusoidaltest-passivecase ............................ 93 4-7Droptest-carbodyacceleration .......................... 93 4-8Droptest-tiredeectionacceleration ....................... 94 4-9Solidworksquartercarmodel ............................ 95 4-10Timedomainsimulation-carbodyacceleration .................. 96 4-11Timedomainsimulation-suspensiondeection ................. 96 4-12Timedomainsimulation:tiredeection ....................... 96 4-13Timedomainsimulation-controlmassposition .................. 97 4-14Frequencydomainsimulation-carbodyacceleration .............. 98 4-15Frequencydomainsimulation-suspensiondeection .............. 98 4-16Frequencydomainsimulation-tiredeection ................... 99 5-1Orthogonalnonlinearenergysink(NES) ...................... 101 5-2VarianceGain .................................... 104 5-3Quartercarmodel-activecase ........................... 105 5-4Simmechanicmodel ................................. 111 5-5Carbodyacceleration(CBA) ............................ 114 5-6Suspensiontravel(ST) ................................ 115 5-7Tiredeection(TD) .................................. 116 5-8Controlmassdisplacement ............................. 116 5-9Actuatorforces .................................... 117 5-10Quartercarmodel-Semi-activecase ....................... 118 5-11NonparametricMR-dampermodel ......................... 119 5-12Polynomialapproximation .............................. 124 10

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5-13Carbodyacceleration(CBA)-semi-activecase .................. 130 5-14Suspensiontravel(ST)-semi-activecase ..................... 130 5-15Tiredeection(TD)-semi-activeCase ....................... 131 5-16Controlmassdisplacement-semi-activecase .................. 131 5-17Parameterestimates-semi-activecase ...................... 132 5-18Controlcurrents ................................... 133 5-19MR-damperforces .................................. 134 6-1Halfcarmodel .................................... 137 6-2Modelingschematics ................................. 138 6-3Bicyclemodel ..................................... 138 6-4Idealizedhalfcarmodelforrolldynamicsmodeling ................ 140 6-5Snapshotduringdatacollectionprocess ..................... 149 6-6Parameterestimationvalidation-snakedata ................... 150 6-7Fishhook-steeringcommand ............................ 150 6-8Fishhook-rollresponse ............................... 151 6-9Fishhook-controlmassdisplacement ....................... 151 6-10Doublelanechange-steeringcommand ..................... 152 6-11Doublelanechange-rollresponse ......................... 152 6-12Doublelanechange-controlmassdisplacement ................. 153 6-13Halfcarmodel .................................... 153 6-14Lateraltireforceapproximation ........................... 157 6-15Rollresponse ..................................... 167 6-16Controlmassdisplacement ............................. 167 6-17Voltagecommand .................................. 168 6-18Spoolvalveresponse ................................ 168 6-19Hydraulicforceoutput ................................ 168 6-20Adaptiveparameterestimationhistory,^Q ..................... 169 11

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6-21Adaptiveparameterestimationhistory,^ ...................... 169 6-22Adaptiveparameterestimationhistory,^ ...................... 169 6-23Adaptiveparameterestimationhistory,^ ...................... 170 6-24Vehicletrajectory ................................... 170 12

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NOMENCLATURE ls Verticalstrutlength. l0s Naturallengthofsuspensionspring l0s Naturallengthofverticalstrut. T Intersectionofsets S Unionofsets Frontwheelsteeringangle Verticaldisplacementofthepointofapplicationofforce maxfAg ThemaximumeigenvalueofthematrixA minfAg TheminimumeigenvalueofthematrixA Ls(q1,q2) Thesetofpointsthatlieonthelinesegmentjoiningthevectorsq1andq2 R Thesetofrealnumbers R Setofrealnumbers Vehiclebodyrollangle Vehicleyawangle ei,n Theithcolumnoftheidentitymatrixofdimensionn Angulardisplacementofleveraboutpivot Ai:j,k:l Thesub-matrixofmatrixAformedbyrowsitojandcolumnsktol Ai:j Thesub-matrixofmatrixAformedbyrowsitojandallcolumns bs suspensiondampingcoefcient d Horizontaldisplacementofthepivotfromthecenteroflever dL Leftcontrolmassdisplacement dR Rightcontrolmasdisplacement detfAg ThedeterminantofthematrixA eigfAg SetoftheeigenvaluesofmatrixA F Externalforce. H Heightofthecontrolmassfromthepivotpointofthelowerwishbone. 13

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H Heightofthepivotbar hu HalfdistancebetweenpointsCandD. I Identitymatrix Ic Momentofinertiaofcontrolarm. Is Vehiclerollmomentofinertia Iz Vehicleyawmomentofinertia K(d) Effectivestiffnessofthemechanismexpressedasafunctiond k1,k2 Springconstants ks stiffnessofsuspensionspring ks,bs VerticalStrutstiffnessanddampingcoefcient kt,bt Tirespringconstantanddampingcoefcient. ku,bu Control(Horizontal)Strutstiffnessanddamping L1,L2 Horizontaldistancesoftheverticalsprings(k1andk2)fromcenteroflever lD Lengthofthelowerwishbone. lf distanceoffrontaxlefromthecenterofmass lr distanceofrearaxlefromthecenterofmass l01,l02 Springfreelengths m Vehicletotalmass ms,mu,md Sprung,unsprungandcontrolmasses. r Yawrate Refg Therealpartofthecomplexnumber roots((i)) Thesetofrootsofthepolynomial(i) trfAg ThetraceofthematrixA vx Longitudinalvelocity vy Lateralvelocity x DistancebetweenpointsOandAalongthelowerwishbone. ys Verticaldisplacementofthesprungmass. 14

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yu Verticaldisplacementoftheunsprungmass. fg Emptyset 15

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyVARIABLESTIFFNESSSUSPENSIONSYSTEMByOlugbengaMosesAnubiMay2013Chair:CarlCraneMajor:MechanicalEngineeringImprovementsoverpassivesuspensiondesignsisanactiveareaofresearch.Pastapproachesutilizeoneofthreetechniques;adaptive,semi-active,orfullyactivesuspension.Anadaptivesuspensionutilizesapassivespringandanadjustabledamperwithslowresponsetoimprovethecontrolofridecomfortandroadholding.Asemi-activesuspensionissimilar,exceptthattheadjustabledamperhasafasterresponseandthedampingforceiscontrolledinreal-time.Afullyactivesuspensionreplacesthedamperwithahydraulicactuator,orothertypesofactuatorslikeelectromagneticactuators,whichcanachieveoptimumvehiclecontrol,butatthecostofdesigncomplexity.Thefullyactivesuspensionisalsonotfail-safeinthesensethatperformancedegradationresultswheneverthecontrolfails,whichmaybeduetoeithermechanical,electrical,orsoftwarefailures.Recently,researchinsemi-activesuspensionshascontinuedtoadvancewithrespecttocapabilities,narrowingthegapbetweensemi-activeandfullyactivesuspensionsystems.Today,semi-activesuspensions(e.gusingMagneto-Rheological(MR),Electro-Rheological(ER)etc)arewidelyusedintheautomobileindustryduetotheirsmallweightandvolume,aswellaslowenergyconsumptioncomparedtopurelyactivesuspensionsystems.However,mostsemi-activedesignconceptsarefocusedononlyvaryingthedampingcoefcientoftheshockabsorberwhilekeepingthestiffnessconstant.Meanwhile,insuspensionoptimization,boththedampingcoefcientandthespring 16

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rateofthesuspensionelementsareusuallyusedasoptimizationarguments.Therefore,asemi-activesuspensionsystemthatvariesboththestiffnessanddampingofthesuspensionelementcouldprovidemoreexibilityinbalancingcompetingdesignobjectives.Thisworkconsidersthedesign,analyses,andexperimentationofanewvariablestiffnesssuspensionsystem.Thedesignisbasedontheconceptofavariablestiffnessmechanism.Themechanism,whichisasimplearrangementoftwosprings,aleverarm,andapivotbar,hasaneffectivestiffnessthatisarationalfunctionofthehorizontalpositionofthepivot.Theeffectivestiffnessisvariedbychangingthepositionofthepivotwhilekeepingthepointofapplicationoftheexternalforceconstant.Theoverallsuspensionsystemconsistsofahorizontalcontrolstrutandaverticalstrut.Themainideaistovarytheloadtransferratiobymovingthelocationofthepointofattachmentoftheverticalstruttothecarbody.Thismovementiscontrolledpassively,semi-actively,andactivelyusingthehorizontalstrut.ThesystemisanalyzedusinganL2-gainanalysisbasedontheconceptofenergydissipation.Theanalyses,simulation,experimentalresults,showthatthevariablestiffnesssuspensionachievesbetterperformancethantheconstantstiffnesscounterpart.Theperformancecriteriausedare;ridecomfort,characterizedbythecarbodyacceleration,suspensiondeection,androadholding,characterizedbytiredeection. 17

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CHAPTER1INTRODUCTIONSuspensionisacollectivetermgiventothesystemofsprings,damperandlinkagesthatisolatesavehiclebody(sprungmass)fromthewheelassembly(unsprungmass).Thevehicleinteractswiththeroadviathedirectcontactbetweentireandtheroad.Thesuspensionsystemservestoisolatethepassengerfromtheroadnoiseasmuchaspossiblewhilekeepinggoodroadcontactforimprovedhandlingandmobility.Theautomotivesuspensionsystemconsistsofthetires,guidingelementswhichincludecontrolarmsandlinks(A-arms),struts,leafsprings,andforceelementswhichincludesprings(coilspring,airspring,orleafspring),torsionbar,anti-rollbar,damper(passiveorsemi-active),bushings,etc. 1.1SomeHistoricalNotesonVehicleSuspensionThehistoryofvehiclesuspensiondatesbacktotheeraofhorsedrawnvehicles.Bytheearly19thcentury,mostBritishhorsecarriageswereequippedwithwoodenspringsinlightone-horsevehicles,andsteelspringsinlargervehicles.Thesteelspringsweremadeoflow-carbonsteelandweredesignedinformofmultiplelayerleafsprings[ 1 ].TheBritishsteelspringswerenotwellsuitedforuseonAmerica'sroughroadsofthattime.Asaresult,inthe1820's,theAbbotDowningCompanyofConcord,NewHampshiredevelopedasystemwherebythebodiesofstagecoachesweresupportedonleatherstrapscalledthoroughbraces,whichgaveaswingingmotioninsteadofthejoltingupanddownofaspringsuspension.Automotiveswereinitiallydesignedasself-propelledversionsofhorsedrawnvehicles.However,thehorse-drawnvehiclesuspensiondesignedforslowspeedswerenotsuitableforhigherspeedspermittedbytheinternalcombustionengine.In1901,MorsofParisrstttedanautomobilewithshockabsorbers.HenriFournierlaterwontheprestigiousParis-to-BerlinraceonJune20,1901withtheaidofhis'MorsMachine'[ 2 ].Leylandusedtorsionbarsinasuspensionsystemin1920.In1922, 18

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independentfrontsuspensionwaspioneeredontheLanciaLambdaandbecamemorecommoninmass-producedcarsby1932[ 3 ].EarlyindependentsuspensionswerealsoproducedbyAndreDubonnetinFranceinlate1920's[ 4 ].Alsoin1932,twoexperimentalCadillaccarswerebuit,oneusingDubonnet'stypeofsuspension,theotherwithadouble-wishbonesuspensionofGM'sdesign.Duringthegreatdepression,therewereheavynancialconstraintsoncarmanufacturingandretailpriceswerepressing.However,independentfrontsuspensiondesignswereenthusiasticallyaccepted,andshowntothepublicin1934.In1935ChevroletandPontiachadcarsavailablewithDubonnetsuspensions,whileCadillac,Buick,andOldsmobilehaddoublewishbonesuspensions.Bythattimetherigidfrontaxlewasbeginningtofazeoutinpassengercars. 1.2TypesofSuspensionSuspensionsystemsaredividedintothreeclasses:independent,dependentandsemi-independentsuspensions. 1.2.1IndependentSuspensionAsthenameimplies,theindependentsuspensionhasnomechanicallinkagesbetweenthetwohubsofthesameaxle;theforceactingononewheeldoesnotaffecttheother.Thelinkagesmustbedesignedtoconstrainveoutofthesixdegreesoffreedomofthewheelhub.Theunconstraineddegreeoffreedomisthetranslationinadirectionperpendiculartotheground.Noneofthemanydeviceswhicharecurrentlyusedfulllsthisrequirementexactly[ 5 ].IndependentsuspensionsareusuallyeitheradoublewishbonetypeoraMcPhersonsuspensiontype.Doublewishbonesuspensionsareappliedtoluxurysedansandsportscarsbecausetheyallowadesignoftheelasto-kinematicparametersthatprovidesanoptimumcompromisebetweenhandlingandcomfort.TheyhavetwoA-arms(wishbones),connectedtothetopandbottomofthewheelhubviaaballandsocketjoint.Figure 1-1 showsdoublewishbonesuspensionsofthehighandlowtypes.IftheupperA-armisreplacebyaprismatic 19

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Figure1-1. Frontwheeldoublewishbonesuspensions[ 5 ].Left:hightype,Right:lowtype. joint,aMcPhersonsuspensionresultsasshowninFigure 1-2 .Itissimpleandallowsmoreroomfortheengine.Asaresult,ithasbecomeacommonsolutionforautomotivefrontaxles,particularlyinsmallcars.Otherformsofindependentsuspensiongenerallyusedforrearsuspensionduetotheirminimalinvasivenessintothechassisincludethetrailingarmsuspension,semi-trailingarmsuspension,guided-trailingarmsuspension,andmultilinksuspension. 1.2.2DependentSuspensionDependentsuspensionshaverigidaxleswhichprovidearigidlinkagebetweenthetwowheelsofthesameaxle(seeFigure 1-3 ).Thedynamicresponseofthewheels 20

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Figure1-2. McPhersonsuspensionsystem[ 6 ]. causedbyroadexcitationsarecoupledwitheachother.Thissuspensioniswidelyusedinindustrialvehiclesandoffroadvehicles. 1.2.3Semi-independentSuspensionThistypeofsuspensionhasintermediatecharacteristicsbetweenthersttwocategories.Anexampleisthetwistbeamsuspensionwhichisessentiallycomprisedoftwotrailingarmsxedtothechassiswithanelasticbushingandconnectedbyacrossbeam.Thespringsandshockabsorbersarexedbetweenthearmsandthecarbody.ThissuspensionsystemisdepictedinFigure 1-4 21

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Figure1-3. Rigidaxlesuspensionsystem[ 5 ]. 1.3SuspensionGeometryThisdescribesthekinematicrelationshipsbetweenthevarioussuspensionelements,thesprung,andunsprungmasses.Someoftheterminologiesusedtodescribetheserelationshipsaredescribedasfollows: Bump: Theverticaldisplacementoftheentiresprungmass. BodyRoll: Therotationofthesprungmassaboutthebodylongitudinalaxis,arisingfromcorneringactivityandroadroughness.ThelongitudinalaxisisforwardinbothISOandSAEsystems.Thus,clockwiserotationasseenfromthereardenespositiverollangle. SuspensionRoll: AsformallydenedbySAE,suspensionrollistherotationofthesprungmassaboutafore-aftaxiswithrespecttoatransverselinejoiningapairof 22

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Figure1-4. Twistbeamsuspensionsystem[ 5 ]. wheelcenters.Ifthegroundisatandfrontandrearwheelscentershaveparalleltransverselines,thedenitionisstraightforward.Otherwise,somemeangroundplanemustbeadopted. Pitch: Therotationofthesprungmassaboutatransverseaxis,resultingineitheranose-uporanose-downconguration.Thismotionisusuallyassociatedwithaccelerationandbraking.IntheSAEaxissystem,thenose-upcongurationdenespositivepitchwhileintheISOaxissystem,thenose-downcongurationdenespositiveroll. RollCenter: Thisisthepointaboutwhichthesprungmasspivotsduringarollsituation.Itisadynamicpoint-itmovesaroundthroughoutthesuspensiontravel. 23

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PitchCenter: Thisisthepointaboutwhichthesprungmasspivotsduringapitchsituation.Itisadynamicpoint-itmovesaroundthroughoutthesuspensiontravel. Camber: Thetiltingofthetireasseenfromeitherthefrontorrearview.Leaningofthetireinboardtowardsthechassisdenesanegativecamber. Toe: Thetiltingofthetireinastaticsituationasseenfromeitherthetoporbottomview.Turninginofthefrontofthetireisreferredtoastoe-in. SteeringAxis: Theaxisaboutwhichthewheel/tirerotatesduringsteering.ItisalsoknownastheKingPinAxis. Caster: Thetiltingofthesteeringaxisasseenfromsideview.Itcreatescamberchangewithsteeringinput.Italsocreatesarestoringtorque(aligningmoment)forcenteringthesteeringwheel. CasterTrail: Thedistancebetweenthecenterofthetirecontactpatchandthepointofintersectionofthesteeringaxisandthegroundplaneasseenfromthesideview.Thisalsogeneratesselfaaligningmomentforthesteeringwheel. ScrubRadius: Thedistancebetweenthecenterofthetirecontactpatchandthepointofintersectionofthesteeringaxisandthegroundplaneasseenfromeitherthefrontvieworrearview. SteeringArm: Thelinebetweenthesteeringaxisandthesteeringlinkage(tierod). BumpTravel: Themaximumpossibleverticalupwarddisplacementofthewheelfromtheequilibriumpositionrelativetothesprungmass. DroopTravel: Themaximumpossibleverticaldownwarddisplacementofthewheelfromtheequilibriumpositionrelativetothesprungmass. 1.4ControlledSuspensionToimproveontheperformanceofsuspensionsystem,researchershaveattemptedtosystematicallymodulatethesuspensionforce.Improvementsoverpassivesuspensiondesignsisanactiveareaofresearch[ 7 17 ].Pastapproachesutilizeoneofthreetechniques[ 18 ],adaptive[ 19 ],semi-active[ 9 20 ]orfullyactivesuspension[ 19 21 ]. 24

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1.4.1ActiveSuspensionDesignAfullyactivesuspensionreplacesthedamperwithaforcegeneratorwhichcouldbehydraulic,electric,orpneumatic.Thiscanachieveoptimumvehiclecontrol,butatthecostofdesigncomplexity,expensiveactuators,etc.IntheworkofFialhoandBalas[ 19 ],anovelapproachtothedesignofroadadaptiveactivesuspensionsviaacombinationoflinearparameter-varyingcontrolandnonlinearbacksteppingtechniqueswaspresented.Twolevelsofadaptationwereconsidered:thelowerlevelcontroldesignshapesthenonlinearcharacteristicsofthevehiclesuspensionasafunctionofroadconditions,whilethehigherleveldesigninvolvesadaptiveswitchingbetweenthesedifferentnonlinearcharacteristics,basedontheroadconditions.In[ 22 ],anactivesuspensioncontrolapproachcombiningalteredfeedbackcontrolschemeandaninputdecouplingtransformationwasusedforafull-vehiclesuspensionsystem.Recently,BoseCorp.hasdevelopedanautomobileactivesuspensionsystemusinganelectromagneticactuator[ 23 ].TheBosesystemequipseachwheelwithaseparateelectromagneticmotorsimilartothoseusedinrollercoasters.Ratherthanrevolving,theelectromagneticmotorstelescopeupordowntoimitatethebehaviorofatypicalshockabsorber.Thisactivesystemhavebeenshowntohaveenormousimprovementwithregardtoridecomfortandhandling. 1.4.2AdaptiveSuspensionDesignAnadaptivesuspensionutilizesapassivespringandanadjustabledamperwithslowresponsetoimprovethecontrolofridecomfortandroadholding.In[ 24 ],theconceptofadaptivesuspension,inwhichthepassivesuspensionparameterswerecontrolledactivelyinresponsetovariousmeasuredsignals,wasappliedtoroadvehicles.In[ 25 ],avehiclesuspensionsysteminwhichacomputercontrolsdampingandspringforcestooptimizerideandhandlingcharacteristicsunderawiderangeofdrivingconditionswaspresented. 25

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1.4.3Semi-ActiveSuspensionDesignTheoriginalconceptofsemi-activesuspensiondatesbacktoKarnopp[ 13 15 ],whereitwasintroducedasanalternativetothecostly,highlycomplicated,andpower-demandingactivesystems.Whilefullyactivesuspensionsystemsaretheoreticallyunrestrictedenergywise,semi-activeelementsmustbeeitherdissipativeorconservativeintheirenergydemand.Sofar,semi-activedesignsfallintoageneralclassofvariabledamper,variableleverarm,andvariablestiffness[ 12 ].Variabledampertypesemi-activedevicesarecapableofvaryingthedampingcoefcientsacrosstheirterminals.Initialpracticalimplementationswereachievedusingavariableoriceviscousdamper.Byclosingoropeningtheorice,thedampingcharacteristicschangefromsofttohardandviceversa.Withtime,theuseofelectro-rheological(ER)andmagneto-rheological(MR)uidsreplacedtheuseofvariableorices[ 18 26 27 ].ERandMRuidsarecomposedofasuspensionofpolarizedsolidparticlesdispersedinanonconductingliquid.Whenanelectric(ormagneticforMR)eldisimposed,theparticlesbecomealignedalongthedirectionoftheimposedeld.Whenthishappens,theyieldstressoftheuidchanges,hencethedampingeffect.ThecontrollablerheologicalpropertiesmakeERandMRuidssuitableforuseassmartmaterialsforactivedevices,transformingelectricalenergytomechanicalenergy.Variableleverarmtypesemi-activesuspensionsconserveenergybetweenthesuspensionandspringstorage.Theyarecharacterizedbycontrolledforcevariationwhichconsumesminimalpower.Themainideabehindtheiroperationisthevariationoftheforcetransferratiowhichisachievedbymovingthepointofforceapplication[ 16 17 28 30 ].Ifthispointmovesorthogonallytotheactingforce,theoreticallynomechanicalworkisinvolvedinthecontrol.Variablestiffnesssemi-activesuspensionsexhibitavariablestiffnessfeature.Thisisachievedeitherbychangingthefreelengthofaspringorbyamechanismwhichchangesitseffectivestiffnesscharacteristicsasaresultofoneormoremovingparts.In 26

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[ 12 ],anexampleofahydro-pneumaticspringwithavariablestiffnesscharacteristicwasgiven.Ontheotherhand,Semi-activesuspensioncontrolmethodsarevaried.Skyhook(SH)controlisoneoftherstapproachestobeappliedincommercialvehicles[ 31 ],inwhichactitiousdamper(calledskyhookdamper)isplacedbetweenthesprungmassandtheinertiaframe,thesuspensiondampingcoefcientisthenmodulatedtomimicthebehavioroftheskyhookdamper.Inthislinearmodel-basedcontroldesign,thedampingcoefcientisswitchedcontinuouslybetweentheminimumandmaximumvalues.Asimilarconceptcalledground-hookhasalsobeendevelopedforroadfriendlysuspensions[ 11 ].Thiscontrolconceptshasalsobeenappliedtosemi-activesuspensions.Also,theAccelerationDrivenDamping(ADD)techniquewasdevelopedfromanoptimalcontrolapproach[ 32 ].SHandADDhavecomplementarycharacteristics:SHprovideslargebenetsaroundtherattlespacefrequencywhiletheADDprovideslargebenetsaroundthetirehopfrequency.Theybothperformsimilarlytothepassivesuspensionotherwise.Intheirspecicdomains,SHandADDprovidequasi-optimalperformances[ 33 ].Thatis,itisimpossibletoachieve(withthesamesemi-activeshock-absorber)betterperformances.Theresultprovidedalower-boundtothelteringcapabilitiesofacontrollablesemi-activesuspension.In[ 33 ],amixedSHandADD(SH-ADD)controlmethodwasintroduced.SH-ADDprovidesanoptimalmixofSHandADDtechniques.Theoptimalityanalysisin[ 33 ]indicatesthattheSH-ADDisacloseapproximationtothebestpossiblealgorithmforsemi-activesuspensionsforagivencomfort-basedobjectivefunction.InbothSHandADD,andsubsequentlySH-ADD,thedamperismodeledasalineardamperwhosedampingcoefcientisadjustedusingthecorrespondingalgorithm.TheseapproachesdonotallowtheuseofmorerealisticMR-dampermodels.In[ 20 ],aquartervehiclemodelequippedwithasemi-activedamperwasreformulatedintheLinearParameterVarying(LPV)frameworkusinganonlinearstaticsemi-activedamper 27

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model.ThemethodincorporatetheMRdamperdissipativityconstraint.However,themodelofsuspensionusedisanidealizedlinearmodelwhichunderminesthekinematicdetailsofthesuspensionmechanismaswellastherotationalcharacteristicsoftheunsprungmass.AnewmodeloftheMacPhersonsuspensionsystemwasintroducedin[ 34 ]andlaterusedin[ 35 ].Themodelincorporatesthekinematicdetailsofthesuspensionsystemaswellastherotationalmotionoftheunsprungmass.TheconventionalidealizedlinearquartercarmodelwasshowntobeaspecialcaseoftheMacPhersonmodelsincethetransferfunctionofthelinearizedMacPhersonmodelcoincideswiththeconventionalmodelifthelowersupportpointofthedamperislocatedatthemasscenteroftheunsprungmass.TheresonancefrequenciesoftheMacPhersonmodelwerealsoshowntoagreebetterwithexperimentalresultsthantheconventionallinearmodel.Anubiet.al[ 36 37 ]considersthecontroldesignandanalysisofasuspensionsystemusingthenewnonlinearmodelingoftheMacPhersonSuspensionsystemequippedwithanMRdamper.Thedamperforcewasmodeledusingthenonlinearstaticsemi-activedampermodelgivenin[ 38 ].ThecontrollerwasdesignedusinganL2-gainanalysisbasedontheconceptofenergydissipation.ThecontrolleriseffectivelyasmoothsaturatedPIDwhichallowsthedissipativityconstraintoftheMRdampertobesatised.Theperformanceoftheclosed-loopsystemiscomparedwithapurelypassiveMacPhersonsuspensionsystemandasemi-activedamper,whosedampingcoefcientistunedbytheSH-ADDmethod.ItwasshownviasimulationthatthedevelopedcontrolleroutperformsthepassivecaseatboththerattlespaceandtirehopfrequenciesandtheSH-ADDattirehopfrequencywhileshowingcloseperformancetotheSH-ADDattherattlespacefrequency.Timedomainsimulationresultsconrmedthatthedevelopedcontrollersatisesthedissipativeconstraint. 28

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Othercontrolconceptsthathavebeenappliedtosemi-activeandactivesuspensionsinclude;optimalcontrol[ 9 10 39 40 ],robustcontrol[ 41 ],androbustoptimalcontrol[ 19 20 42 ],etc. 29

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CHAPTER2VARIATIONOFSTIFFNESSINSUSPENSIONDESIGNVariationofthedampingcoefcienthasbeenthemainfocusofresearchesinsemi-activesuspensiondesignsinthepast.Inthischapter,acombinedvariationofstiffnessanddampingcoefcientisconsidered.First,twofundamentaltheoremsintheoptimalcontrolofsemi-activesuspensionareextendedtocoverstiffnessvariationaswell.Itwasthenshownthatabetterperformance,intermsofridecomfortandhandling,isachievablebyvaryingthestiffnessalongsidethedampingcoefcientovervaryingeitherdampingorstiffnessalone.Twoadditionalcontrollawsarepresentedforvaryingthedampingandstiffnessofasemi-activesuspensionbasedonaquartercarmodel.Therstvariesthedampingandstiffnesssequentiallywhilethesecondvarythemsimultaneously. 2.1GeneralSemi-ActiveSuspensionThetermgeneralsemi-activereferstoanysemi-activedevicewhichmodulateseitherthedampingcoefcientorthestiffnessofthesuspensionelement.Figure 2-1 showsaquartercarmodelofageneralsemi-activesuspension.Itisatwodegreeoffreedommodelwhichcapturesthebasicelementoftheverticaldynamicsofthecar.Thesprungmassmsisthemassofthecarbody(chassis).Theunsprungmassmuisthemassofthewheelassembly.ksandbsarethestiffnessandthedampingcoefcientofthepassivesuspensionelementrespectively.zrdenotestheroaddisturbanceandvdenotesthevalueofthemodulatedvariableofthesemi-activesuspension.Itisusedheregenericallytorepresenteitherthestiffnessordampingcoefcient,dependingonwhetherthesemi-activedeviceisavariabledamperratetypeorvariablespringratetype. 30

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Figure2-1. Quartercar-modulatedspring/damper Letx=266666664x1x2x3x4377777775=266666664zs)]TJ /F3 11.955 Tf 11.95 0 Td[(zu_zszu)]TJ /F3 11.955 Tf 11.96 0 Td[(zr_zu377777775 (2)bethestatevectorofthesystem,theverticaldynamicsofthecarisgivenbythefollowingstateequation_x=Ax+(x)v+L_zr=Ax)]TJ /F7 11.955 Tf 11.95 0 Td[(b)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(TTxv+L_zr, (2) 31

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whereA=266666664010)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F5 7.97 Tf 11.85 4.71 Td[(ks ms)]TJ /F5 7.97 Tf 11.73 4.71 Td[(bs ms0bs ms0001ks mubs mu)]TJ /F5 7.97 Tf 12.2 4.7 Td[(kt mu)]TJ /F5 7.97 Tf 11.99 4.7 Td[(bs mu377777775b=01 ms0)]TJ /F4 7.97 Tf 13.73 4.71 Td[(1 muTL=00)]TJ /F6 11.955 Tf 9.3 0 Td[(10T,andT=010)]TJ /F6 11.955 Tf 9.3 0 Td[(1Tifthemodulatedelementisadamper,inwhichcasethecontrolvariablevisthevariabledampingcoefcientoftheshockabsorber,orT=1000Tifthemodulatedelementisaspringinwhichcasevbecomesthevariablestiffnessofthespring.Thefollowingassumptionsaremade: 1. Thehorizontalmovementofthesprungmass,ms,isneglected,i.eonlytheverticaldisplacementzsisconsidered. 2. Thevaluesofzsandzuaremeasuredfromtheirstaticequilibriumpoints.Hence,theeffectofgravityisneglectedinthismodel 3. Thespringanddampingforcesareinthelinearregionsoftheiroperatingranges.ItisalsoassumedthattheroadinputzrisastationaryWienerprocessanditsderivative_zrisawhitenoisewithintensity.Forthetheoreticalanalysispartofthispaper,itwillassumedthat_zr=0.Inotherwords,theanalysisiscarriesoutforthedeterministiccase.PerformanceCharacterization:Theperformanceofthesuspensionsystemischaracterizedbytheridecomfort,suspensiontravelandroadholdingcapability.Theseperformancecriteriaaremeasuredintermsofthechassisacceleration,zs,suspensiondeectionzs)]TJ /F3 11.955 Tf 12.34 0 Td[(zu,andtiredeectionzu)]TJ /F3 11.955 Tf 12.34 0 Td[(zrrespectively.Thus,theperformanceindex 32

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J(x,v),expressedastheweightedsumoftheparametersaboveisdenedasfollows:J(x,v)=Ztft0)]TJ /F6 11.955 Tf 7.31 -9.69 Td[(zs2+21(zs)]TJ /F3 11.955 Tf 11.96 0 Td[(zu)2+22(zu)]TJ /F3 11.955 Tf 11.95 0 Td[(zr)2dt (2)=Ztft0g(x,v)dt. (2)=Ztft0xTQx)]TJ /F6 11.955 Tf 11.95 0 Td[(2w(x)aTxv ms+w(x)2 m2sv2dt (2)wherew(x)=TTx (2)a=)]TJ /F5 7.97 Tf 11.85 4.71 Td[(ks ms)]TJ /F5 7.97 Tf 11.72 4.71 Td[(bs ms0bs msT (2)Q=26666666421+k2s m2sksbs m2s0)]TJ /F5 7.97 Tf 10.49 4.7 Td[(ksbs m2sksbs m2sb2s m2s0)]TJ /F5 7.97 Tf 12.07 5.7 Td[(b2s m2S00220)]TJ /F5 7.97 Tf 10.5 5.11 Td[(kSbs m2s)]TJ /F5 7.97 Tf 11.73 5.7 Td[(b2s m2s0b2s m2s377777775. (2)1,22
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conditionin( 2 )._x=Ax+(x)v (2)0vv (2)x(0)=x0. (2)First,thesaturationconstraint( 2 )isignoredanditisshownthatforthecasewherevisallowedtovaryboundlessly,theoptimalcontrolmakesthesemi-activesuspensionachievethesameperformanceastheoptimalfullyactivesuspensionsystem.Thefollowingtheoremexpressesthefactthatanysemi-activedevice,ifmodulatedboundlessly,canachievethesameperformanceasanoptimalactiveone. Theorem2.1. Iftheconstraint( 2 )isignored,theoptimalcontrolthatminimizestheperformanceindex( 2 )isv=8><>:ms 2w(x))]TJ /F6 11.955 Tf 5.48 -9.68 Td[(2aT+msbTPxifTTx6=00ifTTx=0 (2)whereP2<44isapositivedenitesolutiontothericattiequation_P+PAT+AP)]TJ /F3 11.955 Tf 11.96 0 Td[(PBP+Q=0 (2)whereA=AT)]TJ /F3 11.955 Tf 11.95 0 Td[(msabTQ=Q)]TJ /F13 11.955 Tf 11.96 0 Td[(aaTB=1 2m2sbbT.ThevaluefunctionJ(x,v)isgivenbyJ(x,v)=1 2xT(t0)P(t0)x(t0) (2) 34

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whichisthesameforanoptimalfullyactivesuspensionsystem[ 9 ].Hence,theuncon-strainedoptimalmodulationofanysingleelementsemi-activesuspensionsystemisequivalenttotheoptimalactivecounterpart.TheproofofthistheoremisgivenintheAppendix A.1 .Thefollowingtheoremputsthesaturationconstraintintoconsideration. Theorem2.2. Thesolutionvtotheoptimalcontrolproblemstatementwithconstraint( 2 )isgivenbyv=8>>>><>>>>:0ifv00m2s 2w(x)2v0if0>>><>>>>:Aifv00ATif0>>><>>>>:0ifv00Bif0
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andQr(x,P)=8>>>><>>>>:2Qifv00QTif00isthesolutiontothealgebraicriccatiequationPaAT+APa)]TJ /F3 11.955 Tf 11.96 0 Td[(PaBPa+Q=0 (2)whichcorrespondstotheunconstrainedoptimalactivesuspensioncontrollaw[ 9 ].TheproofofthistheoremisgiveninAppendix A.2 Remark2.1. Itcanbeshown,bytakinglimitsfromleftandright,thatthecontrollaw( 2 )iscontinuousandthatit'sderivative@v @x=M1(x)x (2)whereM1(x)=8>>>><>>>>:ms 2(2a+msPb)TT)]TJ /F16 7.97 Tf 6.58 0 Td[(T(2aT+msbTP) xTTTTxif0
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Figure2-2. Quartercar-modulatedspringanddamper Differentcontrolactionsaretakenasthestatetrajectorychangesfromoneregiontotheother.Ifx2R1,nocontrolactionistaken.Thisisbecausesemi-activedevicesonlydissipateenergyandarenotcapableofsupplyingenergy.Thisbehaviorisimposedonthemodelbytheinequalityconstraintv0andisignoredinTheorem 2.1 .Ifx2R2,optimalenergydissipationisachievedbecausethephysicallimitofthedeviceisnotexceeded.Moreover,ifthestatetrajectoryisinregionR3,theunconstrainedoptimalenergydissipationrequirementexceedsthephysicallimitofthesemi-activedevice,thusthecontrolactionisclippedasv=v.Thenextsectiondescribesthevariationofbothstiffnessanddampingandpossibleimprovementsinoverallenergydissipation. 2.3SpringandDamperModulationFigure 2-2 showsthequartercarmodelofasemi-activesuspensionsystemwithvariablestiffnessanddampingsuspensionelements,showninparallelwithtraditional 37

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springanddamperelements.Sinceboththespringanddampercanbemodulated,letv=264v1v2375 (2)andTi=8>>>>>><>>>>>>:010)]TJ /F6 11.955 Tf 9.29 0 Td[(1Tiftheithelementisadamper1000Tiftheithelementisaspring (2)i=1,2.Then,theequationofmotionisexpressedasthestateequation_x=Ax+(x)v+L_zr (2)where:<47!<21isgivenby(x)=)]TJ /F7 11.955 Tf 9.3 0 Td[(bxTT=)]TJ /F7 11.955 Tf 9.3 0 Td[(bwT(x) (2)withT=T1T2,andthesaturationconstraintisgivenby0vv. (2)FollowingRemark 2.2 ,thequestionarises,ofhowtoimprovetheenergydissipationintheregionR3withoutnecessarilychangingthelimitofthesemi-activedevice.Theorem 2.3 ,whichisoneofthemainresultsofthischapter,showsthatbymodulating 38

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asecondsuspensionelement1,itispossibletoimproveperformanceifsaturationoccursintherst. Theorem2.3. Supposethatthestiffnessanddampingcoefcientofasemi-activesuspensionareboundedinaccordancetotheinequality( 2 ).Theperformance,denedintermsoftheperformanceindex( 2 ),achievedbyoptimallyvaryingboththestiffnessanddampingisbetter(lowerperformanceindex)thanthatachievedbyvaryingeitherthestiffnessordampingalone. Proof. Itissufcienttoshowthatthevaluefunctionassociatedwiththeoptimalmodulationofstiffnessanddampingislessthanthatassociatedwiththeoptimalmodulationofeitherstiffnessordampingalone.Letv2=v1. (2)Then,theperformanceindex( 2 )iswrittenasJ(x,v)=J(x,v1,)=Ztft0g(x,v1,)dt. (2)Claim :Supposev1=visgivenbyTheorem 2.2 .Thereexistssatisfying0v1v2 (2)andx:<7!<4suchthatgivenJ(x,v1,),min0v1v1J(x,v1,) (2)thenJ(x,v1,)J(x,v1,)8suchthat0v1v2. (2) 1Thisiseitheradamperoraspringdependingonwhethertheoriginalisaspringoradamperrespectively 39

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ProofofClaim :Thedynamics( 2 )isexpandedas_x=Ax)]TJ /F7 11.955 Tf 11.95 0 Td[(bxTTv+L_zr=Ax)]TJ /F7 11.955 Tf 11.95 0 Td[(bxT(T1+T2)v1+L_zr=Ax)]TJ /F7 11.955 Tf 11.95 0 Td[(bw(x,)v1+L_zr (2)wherew(x,)=(T1+T2)Tx. (2)LetJa,xT(t0)Pax(t0)bethevaluefunctionfortheactivesuspension,then( 2 )becomesJ(x,v1,)=Ja+Ztft0w(x,)v1 ms)]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(msbTPa+aTx2dt.=Ja+1 m2sZtft0)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(v1TT2x()]TJ /F8 11.955 Tf 11.96 0 Td[(s)2dt (2)wheres=ms)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(msbTPa+aTx v1TT2x)]TJ /F7 11.955 Tf 13.15 8.09 Td[(TT1x TT2x. (2)ThusJ(x,v1,)isconvexin,andtheminimizer=argmin0v1v2Ztft0w(x,)v1 ms)]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(msbTPa+aTx2dt (2)existsforallt0,tf2<+,tf>t0.Followinganengineeringapproach,theintegrandisminimizedateveryinstantoftimeandisgivenby=8>>>><>>>>:0ifs0sif0sv2 v1v2 v1ifsv2 v1. (2) 40

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ThusJ(x,k,))]TJ /F3 11.955 Tf 11.95 0 Td[(J(x,k,)=8>>>><>>>>:1 m2sRtft0)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(v1TT2x2()]TJ /F6 11.955 Tf 11.96 0 Td[(2s)dtifs01 m2sRtft0)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(v1TT2x2()]TJ /F8 11.955 Tf 11.96 0 Td[(s)2dtif0s,v2 v11 m2sRtft0)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(v1TT2x2)]TJ /F4 7.97 Tf 14.04 4.89 Td[(v2 v1+v2 v1)]TJ /F6 11.955 Tf 11.96 0 Td[(2sdtifsv2 v1=)J(x,v1,))]TJ /F3 11.955 Tf 11.96 0 Td[(J(x,v1,)08suchthat0v1v2. 2.4DoubleModulationOptimalSemi-ActiveControlLawsWhiletheengineeringapproachissufcienttoshowtheresultintheabovetheorem,theresultingsteepestgradientcontrolisnottrulyoptimalunlessnosaturationoccursalongthewholetrajectory[ 10 ].Asaresult,twoadditionalcontrollawsarepresented.Therstminimizestheresidualperformanceindexsubjecttotheresultingclosedloopdynamicsfromtheoptimalmodulationoftherstsuspensionelementwhilethesecondsimultaneouslymodulatesthetwosuspensionelementstominimizetheoverallperformanceindexsubjecttothecombineddynamics. 2.4.1SequentialModulationThemainideahereistooptimallymodulatetherstsuspensionelement,keepingthesecondconstant,andthenoptimallymodulatethesecondelementwithrespecttotheresidualperformanceindexoftherstsubjecttotheresultingclosedloopdynamicsfromtherstmodulation.Thedeterministicopenloopdynamicsisgivenby_x=Ax)]TJ /F3 11.955 Tf 11.95 0 Td[(v1w1(x)b)]TJ /F3 11.955 Tf 11.95 0 Td[(v2w2(x)b (2) (2) 41

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wherewi(x)=TTix,i=1,2. (2)Letv1begivenbyTheorem 2.2 ,i.ev1=v1=8>>>><>>>>:0ifv010m2s 2w1(x)2v01if0
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whereg2(x,v1,v2)=xTa1aT1x+xT266666664210000000002200000377777775x, (2)a1(x,P1)=a)]TJ /F6 11.955 Tf 17.12 8.09 Td[(1 msv1T1. (2)Thenewobjectiveisnowtominimizetheperformanceindex( 2 )subjecttothedynamicconstraint( 2 )andtheinequalityconstraint( 2 ).0v2v2, (2)TheresultingcontrollawisgivenbyTheorem 2.4 Theorem2.4. Giventhedynamicconstraint( 2 )andtheinequalityconstraint( 2 ),theoptimalvalueofv2thatminimizestheperformanceindex( 2 )isgivenbyv2=8>>>><>>>>:0ifv020m2s 2w2(x)2v02if0
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whereA1(x,P1,P2)=A)]TJ /F13 11.955 Tf 11.95 0 Td[(b)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(v1TT1+v2TT2 (2)Q1(x,P1)=a1)]TJ /F6 11.955 Tf 17.12 8.09 Td[(1 msM1(x,P1)xxTT1aT1+266666664210000000002200000377777775,P1isthesolutiontothericcatiequation( 2 )andM1(x,P1)isgivenby( 2 ). Proof. Usingthecalculusofvariation,theHamiltonianH=g2(x,v1,v2)+pT2(A1(x,P1)x)]TJ /F3 11.955 Tf 11.96 0 Td[(v2w2(x)b))]TJ /F8 11.955 Tf 11.96 0 Td[(1v2+2(v2)]TJ /F6 11.955 Tf 12.25 0 Td[(v2) (2)isdened,where1,2aretheLagrangemultipliersfortheinequalityconstraint( 2 )andp2istheLagrangemultiplierfortheresidualclosedloopdynamicconstraint( 2 ),andthenecessaryoptimalityconditionsaregivenby)]TJ /F6 11.955 Tf 11.21 .88 Td[(_p2=@g2(x,v1,v2) @x+@ @xpT2(A1(x,P1)x)]TJ /F3 11.955 Tf 11.95 0 Td[(v2w2(x)b)=2Q1(x,P1)x+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(AT)]TJ /F6 11.955 Tf 11.96 -.17 Td[((v1T1+v2T2)bT)]TJ /F6 11.955 Tf 14.56 2.66 Td[(M1xxTT1bTp2 (2)0=@g2(x,v1,v2) @v2)]TJ /F7 11.955 Tf 11.96 0 Td[(pT2bTT2x)]TJ /F6 11.955 Tf 11.95 -.16 Td[((1)]TJ /F8 11.955 Tf 11.95 0 Td[(2)=)]TJ /F6 11.955 Tf 14.46 8.09 Td[(2 msw2(x)aT1x+2 m2sw2(x)2v2)]TJ /F3 11.955 Tf 11.95 0 Td[(w2(x)bTp2)]TJ /F6 11.955 Tf 11.95 -.17 Td[((1)]TJ /F8 11.955 Tf 11.95 0 Td[(2) (2)Ifw2(x)6=0,v2isobtainedfrom( 2 )asv2=m2s 2w2(x)2(v02+1)]TJ /F8 11.955 Tf 11.95 0 Td[(2). (2) 44

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Now,@H @1=@H @2=0,togetherwiththethreecases;(1>0,2=0),(1=2=0)and(1=0,2>0),yieldsv2=8>>>><>>>>:0ifv020m2s 2w2(x)2v02if00,P22<44,then( 2 )becomes_p2=_P2x+P2_x=)]TJ /F6 11.955 Tf 9.3 0 Td[(2Q1(x,P1)x)]TJ /F10 11.955 Tf 11.96 9.69 Td[()]TJ /F6 11.955 Tf 6.37 -7.03 Td[(AT1(x,P1,P2))]TJ /F3 11.955 Tf 11.95 0 Td[(w1(x)bxTMT1(x,P1)P2x (2)or)]TJ /F6 11.955 Tf 10.14 -7.03 Td[(_P2+P2A1(x,P1,P2)+AT1(x,P1,P2)P2)]TJ /F3 11.955 Tf 11.96 0 Td[(w1(x)bxTMT1(x,P1)P2+2Q1(x,P1)x=0,which,providedthatx6=0andP2(tf)=0,yieldsthericcatiequation( 2 ). 2.4.2SimultaneousModulationHere,theoverallperformanceindexisminimizedsubjecttotheoveralldynamics.First,thecarbodyaccelerationisgivenbyzs=aTx)]TJ /F6 11.955 Tf 17.12 8.09 Td[(1 msxTTv. (2) Thentheobjectivefunction( 2 )isexpandedasfollowsg(x,v)=zs2+21(zs)]TJ /F3 11.955 Tf 11.96 0 Td[(zu)2+22(zu)]TJ /F3 11.955 Tf 11.95 0 Td[(zr)2=xTQx)]TJ /F6 11.955 Tf 17.12 8.09 Td[(2 msaTxxTTv+1 m2svTTTxxTTv (2) =xTQx+v21w1(x)2 m2s+v22w2(x)2 m2s)]TJ /F6 11.955 Tf 11.96 0 Td[(2v1w1(x)aTx ms)]TJ /F6 11.955 Tf 9.3 0 Td[(2v2w2(x)aTx ms)]TJ /F6 11.955 Tf 11.95 0 Td[(2v1v2w1(x)w2(x) m2s. (2) 45

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However,)]TJ /F6 11.955 Tf 9.29 0 Td[(2v1v2w1(x)w2(x) m2sv21w1(x)2 m2s+v22w2(x)2 m2s. (2)Therefore,g(x,v)isupperboundedasg(x,v)g(x,v) (2)whereg(x,v)=xTQx+2v21w1(x)2 m2s+2v22w2(x)2 m2s)]TJ /F6 11.955 Tf 9.3 0 Td[(2v1w1(x)aTx ms)]TJ /F6 11.955 Tf 11.95 0 Td[(2v2w2(x)aTx ms. (2) Inordertoavoidthesingularityassociatedwiththerank1matrixTTxxTT,theperformanceindexJ(x,v)isredenedasJ(x,v)=Ztft0g(x,v)dt (2)andthesimultaneousdoublemodulationoptimalcontrollawisgiveninTheorem 2.5 Theorem2.5. Theoptimalcontrollawv=v1v2Tthatminimizestheperformanceindex( 2 )subjecttothedeterministicdynamicconstraint( 2 )andthesaturationconstraint( 2 )isgivenbyvi=8>>>><>>>>:0ifv0i0m2s 4wi(x)2v0iif0
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andP>0isthesolutiontothericcatiequation_P+PA1(x,P)+AT1(x,P)P+Q2(x,P))]TJ /F6 11.955 Tf 17.12 8.09 Td[(4 msavTTT+TvaT=0P(tf)=0 (2)whereQ2(x,P)=2Q+4 m2s)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(v1TTT1+v2TTT2. (2)Furthermore,thevaluefunctionisgivenbyJ(x,v)=1 2x(t0)TP(t0)x(t0). (2) Proof. TheproofofthistheoremfollowsbydeningtheHamiltonianH=g(x,v)+pT)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Ax)]TJ /F7 11.955 Tf 11.96 0 Td[(bxTTv)]TJ /F12 11.955 Tf 11.95 0 Td[(T1v+T2(v)]TJ /F6 11.955 Tf 12.14 .71 Td[(v), (2)wherepistheLagrangemultiplierforthedynamicconstraint( 2 )and1,2aretheLagrangemultipliervectorsforthesaturationconstraint( 2 )andfollowingsimilarprocedureusedinAppendix A.2 Remark2.3. Thesequentialmodulationlawisthetrueoptimal.Thisassertionfollowsfromthefollowingfactmin0vvJ(x,v)=min0v1v1min0v2v2J(x,v)=min0v2v2min0v1v1J(x,v). (2)However,whilethesimultaneousmodulationisnotthetrueoptimal,itisoptimalinthesenseofminimizingthemoreconservativeperformanceindex( 2 ).Theconser-vationisintroducedin( 2 )anditisshownlater,usingsimulationresults,thattheperformancedegradationresultingfromthisconservatismisnegligible. 2.5SimulationTheperformancesofthecontrollersdevelopedinthispaperareevaluatedviasimulation.Thesimulationparametervaluesaregivenintable 2-1 .Themass,stiffness 47

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Table2-1. Dynamicparametervalues ParameterValue ms315kgmu37.5kgbs1500N=m=sks29500N=mkt210000N=m Figure2-3. Timeresponse-carbodyacceleration anddampingvaluesaretheonesgivenintheRenaultMeganeCoupemodel[ 43 ].Forthemodulatedelements,thedampingcoefcientandthespringconstantareallowedtovaryintheinterval[9001670]N=m=sand[1700030600]N=mrespectively.Thevaluesof1and2arebothtakentobe1s)]TJ /F4 7.97 Tf 6.58 0 Td[(2forthissimulation. 2.5.1TimeDomainSimulationForthetimedomainsimulation,thevehicletravelingatasteadyhorizontalspeedof40mphissubjectedtoaspeedhumpofheight25cmandlengthofabout4m.Theroadproleisgeneratedusingagaussianfunctionofheight25cmandspread3.5m.Theresponses2arecomparedforthepassivesuspension,singlemodulationdamper,doublemodulationsuspensioncontrolledbythesequentiallaw,anddoublemodulation 2Thisincludesthecarbodyacceleration,suspensiondeection,andtiredeection. 48

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Figure2-4. Timeresponse-suspensiondeection Figure2-5. Timeresponse-tiredeection suspensioncontrolledbythesimultaneouslaw.Figures 2-3 2-4 and 2-5 showthecarbodyacceleration,suspensiondeection,andtiredeectionrespectively,fromwhichitseenthatacombinedmodulationofdamperandspringallowsgettinggloballymuchbetterperformancethanasinglemodulationofeitherthedamperorspringalone.Itisalsoseenthattheperformancesoftheproposeddoublemodulationlawsareveryclose.Figure 2-6 showstheperformanceindexplots,fromwhichitisalsoseenthattheperformancedegradationresultingfromtheconservatisminthesimultaneouslawisnegligible. 49

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Figure2-6. Timeresponse-performanceindex 2.5.2FrequencyResponseAnapproximatefrequencyresponseisobtainedusingtheconceptofasinusoidalinputdescribingfunction[ 44 ].Theclosedloopsystemisgivenby_x=Ax)]TJ /F3 11.955 Tf 11.95 0 Td[(f(x)b+L_zr (2)wheref(x)=v1w1(x)+v2w2(x). (2)Letx(j!)axcos(!t)+bxsin(!t) (2)bethenearlysinusoidalresponseof( 2 )totheinputsignal_zr=arcos(!t),ar2<+. (2)Then,thenonlinearfunctionf(x)isthenapproximatedasf(x)a0(ax,bx)+a1(ax,bx)cos(!t)+b1(ax,bx)sin(!t) (2)=a0+MT(ax,bx)x+N(ax,bx)_zr (2) 50

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whereM(ax,bx)=b1(ax,bx)bx bTxbx (2)N(ax,bx)=a1(ax,bx)bTxbx)]TJ /F3 11.955 Tf 11.95 0 Td[(b1(ax,bx)bTxax arbTxbx. (2)Thebasisforthisapproximationisthattheuniqueequilibriumpoint,x=0,isasymptoticallystablewhichimpliesthattherstharmonicoftheresponseisdominantoverhigherharmonics.Asaresult,thethenonlinearfunctionisapproximatedbyit'ssinusoidalinputdescribingfunction.Substituting( 2 ),( 2 ),and( 2 )into( 2 )anddoingharmonicbalancing[ 44 ]yieldsba0(ax,bx)=0Abx)]TJ /F8 11.955 Tf 11.95 0 Td[(!ax)]TJ /F7 11.955 Tf 11.95 0 Td[(bb1(ax,bx)=0Aax)]TJ /F8 11.955 Tf 11.95 0 Td[(!bx)]TJ /F7 11.955 Tf 11.95 0 Td[(ba1(ax,bx)+Lar=0 (2)Now,evaluatingtheFouriercoefcientsa0,a1andb1forthefunction( 6 )isdifcult,oratleasttedious.SincetheFouriercoefcientsminimizesthemeansquaredapproximationerror,numericestimatesateachfrequencyareobtainedbynumericallyminimizingtheobjectivefunctionJ(x)=1 2nXk=1(f(axcos(!tk)+bxsin(!tk)))]TJ /F3 11.955 Tf 11.95 0 Td[(a0)]TJ /F3 11.955 Tf 11.96 0 Td[(a1cos(!tk))]TJ /F3 11.955 Tf 11.96 0 Td[(b1sin(!tk))2 (2)subjecttotheconstraints( 2 ).ThegainofthefrequencyresponseisthengivenbyG(!,ar)=^Cj!I)]TJ /F6 11.955 Tf 12.85 2.66 Td[(^A)]TJ /F4 7.97 Tf 6.58 0 Td[(1^L)]TJ /F6 11.955 Tf 13.72 2.66 Td[(^D (2) 51

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Figure2-7. Frequencyresponse-carbodyacceleration where^A=A)]TJ /F7 11.955 Tf 11.95 0 Td[(bMT(ax,bx)^B=L)]TJ /F7 11.955 Tf 11.96 0 Td[(bN(ax,bx)^C=266664aT)]TJ /F4 7.97 Tf 16.12 4.71 Td[(1 msMT(ax,bx)10000020377775^D=1 ms266664N(ax,bx)00377775.Figures 2-7 2-8 ,and 2-9 showthefrequencyresponsesforthecarbodyacceleration,suspensiondeection,andtiredeectionrespectively.Thefrequencyresponseisobtainedforthepassive,singlemodulationdamper,singlemodulationspring,anddoublemodulationcontrolledbythesimultaneouslaw.Aroundtherattlespacefrequency,thevariabledamperperformsbetterthanthevariablespring.Thisperformancerelationisreversedaroundthetirehopfrequency.Thereasonforthisisbecauseofthedifferenceinstiffness/dampingrequirementsatbothfrequencies.Itisseenalsothat 52

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Figure2-8. Frequencyresponse-suspensiondeection Figure2-9. Frequencyresponse-tiredeection thedoublemodulationresponsestrackthesinglemodulationdamperresponsesatlowfrequenciesandtrackthesinglemodulationspringresponsesathighfrequencies. 53

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CHAPTER3VARIABLESTIFFNESSMECHANISMThedesignandanalysisofamechanismwithvariablestiffnessisexamined.Themechanism,whichisasimplearrangementoftwosprings,aleverarmandapivotbar,hasaneffectivestiffnessthatisarationalfunctionofthehorizontalpositiondofthepivot.Theexternalpureforceactingonthesystemisconstrainedtoalwaysremainvertical.Theeffectivestiffnessisvariedbychangingdwhilekeepingthepointofapplicationoftheexternalloadconstant.Theexpressionfortheeffectivestiffnessisderived.Areverseanalysisisalsocarriedoutonthemechanism.Specialdesigncasesareconsidered.Thedynamicequationofthesystemisderivedandusedtodeducethenaturalfrequencyofthemechanismfromwhichsomeinsightsweregainedonthedynamicbehaviorofthemechanism.TheschematicsforthesystemisshowninFigure 3-1 .TheforceFisconstrainedtomoveverticallyandthepivotbarisconstrainedtomovehorizontally.Theleftandrightsprings,ofspringconstantsk1andk2respectively,canonlybedeectedvertically(thereisaslidingmotionallowedbetweenthespringandthepivotbar).l01andl02arethefreelengthsoftheleftandrightspringsrespectively.Theeffectivestiffnessisvariedbychangingd,thehorizontalpositionoftheleverpivotpoint,whilekeepingthepointofapplicationofexternalloadconstant. 3.1ForwardAnalysisGivenallthesystemparameters,k1,k2,L1,L2,l01,l02,theexternalforceF,andthehorizontaldistancedofthepivotbarfromthepointofapplicationofF,itisrequiredtondtheexpressionfortheeffectivestiffnessKandtheeffectivefreelengthl0ofthemechanism.LetF1andF2bethespringforcesactingontheleveratpointsAandBwithheightsx1andx2fromthegroundrespectively(Figure 3-2 ).Letthefunctions(x,lb,lc) 54

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Figure3-1. Schematics and(x,lb,lc)bedenedas(x,lb,lc)=8>><>>:1,x2lblc0,x=2lblc (3)and(x,lb,lc)=8>><>>:0,x2lblc1,x=2lblc. (3)Thus,F1andF2canbewrittenasF1=(x1,lb1,lc1)(x1)]TJ /F3 11.955 Tf 11.96 0 Td[(l01)k1+(x1,lb1,lc1)P1 (3)F2=(x2,lb2,lc2)(x2)]TJ /F3 11.955 Tf 11.96 0 Td[(l02)k2+(x2,lb2,lc2)P2 (3)where 55

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Figure3-2. Freebodydiagram lb1andlb2aretheblocklengths1oftheleftandrightspringsrespectively lc1andlc2aretheopenlengths2oftheleftandrightspringsrespectively P1andP2arethepurereactionforcesoftheblockedsprings.Equations( 3 )and( 3 )capturethecaseswhenthespringsbehaveasrigidbars(blockedoropen)orascompliantmembers(x12(lb1lc1),x22(lb2lc2)).TakingmomentsaboutpointOanddividingbydgives F=)]TJ /F3 11.955 Tf 9.3 0 Td[(F1L1+d d+F2L2)]TJ /F3 11.955 Tf 11.95 0 Td[(d d(3) 1theblocklengthofacompressionspringisdenedasthemaximallengthofthespringaftertotalblockingi.ewhenitisfullycompressed2theopenlengthofatensionspringisdenedasthelengthofthespringwhenitisfullystretched 56

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withx1=H)]TJ /F3 11.955 Tf 13.15 8.09 Td[(L1+d dx2=H+L2)]TJ /F3 11.955 Tf 11.95 0 Td[(d d (3)Substituting( 3 ),( 3 )and( 3 )in( 3 )yields F=K)]TJ /F3 11.955 Tf 11.96 0 Td[(C(3)whereK=k11(L1+d)2 d2+k22(L2)]TJ /F3 11.955 Tf 11.96 0 Td[(d)2 d2 (3)C=k11(H)]TJ /F3 11.955 Tf 11.95 0 Td[(l01)(L1+d) d+k22(H)]TJ /F3 11.955 Tf 11.95 0 Td[(l02)(L2)]TJ /F3 11.955 Tf 11.95 0 Td[(d) d+1P1L1+d d+2P2L2)]TJ /F3 11.955 Tf 11.96 0 Td[(d d (3)where1=(x1,lb1,lc1)2=(x2,lb2,lc2)1=(x1,lb1,lc1)2=(x2,lb2,lc2)Considerwhentheleftspringisblockedi.ex1=lb1,1=0)1=1,thesystemofFigure 3-2 becomesstaticallyindeterminateandrigidprovidedthatx2lc2.However,anydecreaseinFwillcausethesystemtoreverttothestatewherebothspringsareneitherblockednoropen.Asimilarargumentexistsforthecasewheretheleftspringisopeninstead,andalsoforwhentherightspringisopenorblocked.Thus( 3 ) 57

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becomesK=8>>>><>>>>:k1(L1+d)2 d2+k2(L2)]TJ /F5 7.97 Tf 6.59 0 Td[(d)2 d2x12(lb1lc1),x22(lb2lc2)1Otherwise (3)Equation( 3 )istheexpressionfortheoverallstiffnessofthesystemfromwhichitiseasilyseenthatthesystemisrigidwheneitherorboththeleftandrightspringsbecomeblockedoropenord=0.Itis,however,possibleindesigntorestrictx1andx2intherangewhereKnevergoesunboundedexceptintheneighborhoodofd=0.Thisispossiblebyusingspringsofzerofreelengthandalsosatisfyingthecondition H1+L2 L1lc2(3)TheratioL2 L1istermedtheaspectratioandthespacef(lb1lc1)(lb1lc1)gnfd=0gtheusefulspaceofthemechanism.Now,considerthemechanismofFigure 3-1 restrictedtotheusefulspaceandwhoseaspectratioissuchthattheconditionin( 3 )issatised.TheplotoftheeffectivestiffnessKisshowninFigures 3-3 and 3-4 fromwhichitiseasilyseenthattheminimumstiffnessoccursattheboundaryoftheparameterdandaregivenby: Kmin=8>>>>>>>>>><>>>>>>>>>>:k2(r+1)2,r<14min(k1,k2),r=1k1)]TJ /F4 7.97 Tf 6.67 -4.98 Td[(1 r+12,r>1(3)wherer=L2 L1istheaspectratioofthemechanism.Figure 3-4 showsthevariationofKwithrespectto1 dwhichisagoodwaytovisualizethebehaviorofthesystemasd!1.Letl0betheoverallfreelengthofthesystem,then l0=H)]TJ /F8 11.955 Tf 11.96 0 Td[(0(3) 58

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Figure3-3. Effectivestiffnessagainstd Figure3-4. Effectivestiffnessagainst1 d where0isthedeectionwhenF=0whichisgivenby0=C K 59

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Figure3-5. Overallfreelengthagainstd Thustheoverallfreelengthbecomesl0=H)]TJ /F3 11.955 Tf 13.35 8.09 Td[(C K (3)=N2d2+N1d+N0 D2d2+D1d+D0 (3)whereN0=H(k1L21+k2L22)N1=k1L1(H+l01))]TJ /F3 11.955 Tf 11.96 0 Td[(k2L2(H+l02)N2=k1l01+k2l02D0=k1L21+k2L22D1=2(k1L1)]TJ /F3 11.955 Tf 11.96 0 Td[(k2L2)D2=k1+k2Figure 3-5 showsthevariationoftheoverallfreelengthwithrespecttod.The 60

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maximumfreelengthoccurswhend=dmax,wheredmaxisthesolutionto D1D2N1N2d2max+D0D2N0N2dmax+D0D1N0N1=0.(3)However,fromapracticalpointofview,itmightbedesiredtokeepthemechanismoverallfreelengthconstantforallvaluesofd.Inthatcase,anadditionalconstrainton( 3 )canbewrittenasN2d2+N1d+N0=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(D2d2+D1d+D0,2
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Figure3-6. Effectivestiffnessagainstroverd setasfollows:k1=3N=cmk2=4.5N=cml01=0.2cml02=0.5cmH=0.35cmranddwerevarieduniformlyintheintervals[03]and[0.11]respectively.Foreachvalueofd,theeffectivestiffnessandoverallfreelengthwereplottedagainstr.Figure 3-6 showsaparabolicrelationshipbetweentheoverallstiffnessandtheaspectratio.Italsoshowsthatthisvariationvanishesasd!1asKbecomesfairlyconstantwithrespecttor.Thisobservationagreeswith( 3 )andFigure 3-3 fromwhichitisalsoeasilyseenthatlimd!1K=k1+k2. 62

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Figure3-7. Variationofoverallfreelengthagainstroverd Figure 3-7 showsthevariationoftheoverallfreelengthwithrespecttotheaspectratiorovertheoffsetdistanced.Figure 3-8 showsthattheaspectratiocontrolsthecurvature Figure3-8. Effectivestiffnessagainstdoverr ofK.ThisisveryusefulindesignbecauseithelpstoshapethesensitivityofKtothedvalueoveragiveninterval.Inthesamevein,Figure 3-9 showsthevariationofthe 63

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Figure3-9. Variationofoverallfreelengthagainstdoverr overallfreelengthwithrespecttodoverr.Inthenextsection,theadditionalconstraintimposedbysomespecialdesigncasesareexamined. 3.1.2SpecialCasesThespecialcasesconsideredherearethosethatarisebypre-stressingtheleftandrightspringsinsomeways.First,thecasesarehighlightedasfollows 1. Pre-stressed:Here,bothspringsarealwayseitherintensionorcompression.Therearefoursub-casesunderthis. a.bothspringsintension.l01,l02H. b.bothspringsincompression.l01,l02H. c.leftspringincompressionandrightspringintension.l01H,l02H.ThiscongurationresultsinanunstablesystemexceptforsomevaluesofdwhichdependsontheexternalforceF d.rightspringincompressionandleftspringintension.l01H,l02H.ThiscongurationalsoresultsinanunstablesystemasinCase1-cabove. 2. PartiallyStressed:Here,onlyoneofthespringsispre-stressedwhiletheotherremainsunstressed.Therearealsofoursub-casesunderthiscase.AsnotedinCase1-c,allthesub-caseshereareonlystableforsomevaluesofdwhichdependsonF a.leftspringintension.l01H,l02=H. b.leftspringincompression.l01H,l02=H. c.rightspringintension.l01=H,l02H. 64

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d.rightspringincompression.l01=H,l02H 3. Unstressed:Here,neitherofthespringsisstressed.i.el01=l02=H.ThiscaseisexactlytheexamplegivenearlierasoneofthemembersoftheclassthatsatisfyEqn.( 3 )Oneinterestingthingaboutallthesecasesisthatnoneofthemchangestheeffectivestiffnessofthemechanism.Theyonlychangetheoverallfreelength.Thisshowsthatnomatterhowthesystemispre-stressed,theeffectivestiffnessremainsthesame. 3.2ReverseAnalysisGiventhedesiredoverallspringconstant,thegoalofthissectionistondthecorrespondingcontrolparameterdrequiredtoachievethegivenstiffness.Judgingfromtheformofthestiffnessequation( 3 ),allthatneedstobedoneissetKtothedesiredvaluesandsolvetheresultingquadraticequationford.However,thesolutionisnotguaranteedtobealwaysreal.Thus,inadditiontosolvingthequadraticequation,thissectionalsodetailsthederivationofaspecialconstraintthatmustbeimposedonthesetvalueforK.Thisisachievedbyconstrainingthediscriminantofthequadraticequationtobealwayspositive.LetKdbethedesiredoverallstiffnessofthemechanism.SettingK=Kdin( 3 )yieldsKdd2=k1(L1+d)2+k2(L2)]TJ /F3 11.955 Tf 11.96 0 Td[(d)2or(Kd)]TJ /F3 11.955 Tf 11.96 0 Td[(k1)]TJ /F3 11.955 Tf 11.96 0 Td[(k2)d2)]TJ /F6 11.955 Tf 11.96 0 Td[(2(k1L1)]TJ /F3 11.955 Tf 11.96 0 Td[(k2L2)d)]TJ /F6 11.955 Tf 11.95 0 Td[((k1L21+k2L22)=0 (3)whosesolutionisgivenbyd=Bp B2+AC A (3) 65

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whereA=Kd)]TJ /F3 11.955 Tf 11.96 0 Td[(k1)]TJ /F3 11.955 Tf 11.95 0 Td[(k2B=k1L1)]TJ /F3 11.955 Tf 11.95 0 Td[(k2L2C=k1L21+k2L22.Butd2
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Figure3-10. Effectofaspectratioonachievablestiffnesslowerbound fromwhichitisconcludedthatthemaximumcomplianceachievableintheassemblyistheequivalentcomplianceoftheparallelconnectionoftwosimplespringsofspringconstants(1+r)2k2and(1+1 r)2k1. 3.3DynamicalAnalysisInthissection,thenaturalfrequencyofthemechanismisdeducedfromtheanalysisofthesystemdynamics.First,theequationofmotionisderivedusingNewton'slawsofmotion.TakingamomentaboutpointOofFigure 3-2 yields)]TJ /F3 11.955 Tf 9.3 0 Td[(Kd2tan+mgd+Fd)]TJ /F6 11.955 Tf 11.95 0 Td[((md2sec2+I)=0 (3)whereIisthecentralmomentofinertiaoftheleverandmisthemassoftheleverarm.Linearizingabout=0yields)]TJ /F3 11.955 Tf 9.3 0 Td[(Kd2+mgd+Fd)]TJ /F6 11.955 Tf 11.96 0 Td[((md2+I)=0 (3)fromwhichthenaturalfrequencyisdeducedas !n=r Kd2 I+md2.(3) 67

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Figure3-11. Naturalfrequency SubstitutingtheexpressionofKfrom( 3 ),theabovebecomes !n=r k1(L1+d)2+k2(L2)]TJ /F3 11.955 Tf 11.96 0 Td[(d)2 I+md2.(3)Figure 3-11 showsaplotofthenaturalfrequencyagainsttheparameterdovertheaspectratior.Theplotand( 3 )showthatthemaximumvalueoccurswhend=0andthevalueisgivenby!nmax=r k1L21+k2L22 I.ThepracticalinterpretationofthisobservationisthatthesystembehaveslikeacompoundpendulumwhosecenterofmasshasadistanceLfromthepointofapplicationoftheforcegivenbyL=k1L21+k2L22 mg.Also,asdtendstoinnity,thenaturalfrequencyasymptoticallyapproachesq k1+k2 m,whichistheresultantnaturalfrequencyofparallelconnectionoftwosimplespringsofspringconstantsk1andk2.Moreover,asshowningure 3-11 ,theaspectratior 68

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changestheshapeofthenaturalfrequencycharacteristic.Thedomeattensoutwithreducedr. 69

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CHAPTER4VARIABLESTIFFNESSSUSPENSIONSYSTEM:PASSIVECASEMostsemi-activesuspensionsystemsaredesignedtoonlyvarythedampingcoefcientoftheshockabsorberwhilekeepingthestiffnessconstant.Meanwhile,insuspensionoptimization,boththedampingcoefcientandthespringrateofthesuspensionelementsareusuallyusedasoptimizationarguments.Therefore,asemi-activesuspensionsystemthatvariesboththestiffnessanddampingofthesuspensionelementcouldprovidemoreexibilityinbalancingcompetingdesignobjectives.Suspensiondesignsthatexhibitvariablestiffnessphenomenonarefewinliteratureconsideringthevastamountofresearchesthathasbeendoneonsemi-activesuspensiondesigns.Knaapet.al[ 16 29 30 ]designedavariablegeometryactuatorforvehiclesuspensioncalledtheDelftactivesuspension(DAS).Although,theintentionofthedesignwasnottovarythestiffnessofthesuspensionsystem,thedesignusedavariablegeometryconcepttovarythesuspensionforcebyeffectivelychangingthestiffnessofthesuspensionsystem.ThebasicideabehindtheDASconceptisbasedonawishbonewhichcanberotatedoveranangleandisconnectedtoapretensionedspringatavariablelocation.Thespringpretensiongeneratesaneffectiveactuatorforce,whichcanbemanipulatedbychangingtheposition.Thiswasachievedusinganelectricmotor.Jerz[ 45 ]inventedavariablestiffnesssuspensionsystemwhichincludestwospringsconnectedinseries.Oneofthespringsisstifferthantheother.Undernormalloadconditions,thesofterspringisresponsibleforkeepingagoodridecomfort.Upontheimpositionofheavierloadforces,thevehicleissupportedmorestifyandprimarilybythestrongerspring.Conversionbetweenthetwoconditionswasdoneautomaticallybyengagementunderheavyloadconditionsofapairofstopshouldersactingtolimitthecompressionofthelightspring.Similarly,uponexcessiveextensionofthesprings,anadditionalsetofstopshouldersareengagedautomaticallytolimittheamountofextensionofthesofterspringandcausesthestifferspringtoresistfurther 70

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extension.Koboriproposedavariablestiffnesssystemtosuppressbuildings'responsestoearthquakes[ 46 ].Theaimwastoachieveanon-stationaryandnon-resonantstateduringearthquakes.YounandHacusedanairspringinasuspensionsystemtovarythestiffnessamongthreediscretevalues[ 47 ].Liuet.alproposedasuspensionsystemwhichusestwocontrollabledampersandtwoconstantspringstoachievevariablestiffnessanddamping[ 48 ].AVoigtelementandaspringinseriesareusedtocontrolsystemstiffness.TheVoigtelementiscomprisedofacontrollabledamperandaconstantspring.TheequivalentstiffnessofthewholesystemischangedbycontrollingthedamperintheVoigtelement.Thevariationofstiffnessconceptusedinthischapterusesreciprocalactuation[ 49 ]toeffectivelytransferenergybetweenaverticaltraditionalstrutandahorizontaloscillatingcontrolmass,therebyimprovingtheenergydissipationoftheoverallsuspension.Duetotherelativelyfewernumberofmovingparts,theconceptcaneasilybeincorporatedintoexistingtraditionalfrontandrearsuspensiondesigns.Animplementationwithadoublewishboneisshowninthischapter. 4.1SystemDescriptionThissectiongivesadetaileddescriptionofthevariablestiffnessconcept,overallsystem,itsincorporationinavehiclesuspension,andtheresultingsystemdynamicmodel. 4.1.1VariableStiffnessConceptThevariablestiffnessmechanismconceptisshowninFig 4-1 .TheLeverarmOA,oflengthL,ispinnedataxedpointOandfreetorotateaboutO.ThespringABispinnedtotheleverarmatAandisfreetorotateaboutA.TheotherendBofthespringisfreetotranslatehorizontallyasshownbythedoubleheadedarrow.Withoutlossofgenerality,theexternalforceFisassumedtoactverticallyupwardsatpointA.disthehorizontaldistanceofBfromO.Theideaistovarytheoverallstiffnessofthesystembylettingdvarypassivelyundertheinuenceofahorizontalspring-dampersystem 71

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Figure4-1. Variablestiffnessmechanism (notshowninthegure).Letkandl0bethespringconstantandthefreelengthofthespringABrespectively,andtheverticaldisplacementofthepointA.Theoverallfreelength0ofthemechanismisdenedasthevalueofwhennoexternalforceisactingonthemechanism. 4.1.2MechanismDescriptionThesuspensionsystemconsideredisshowninFig 4-2 .TheschematicdiagramisshowninFig 4-3 .Themodeliscomposedofaquartercarbody,wheelassembly,twospring-dampersystems,roaddisturbance,lowerandupperwishbones.ThepointsO,A,andBarethesameasshowninthevariablestiffnessmechanismofFig 4-1 .Thehorizontalcontrolforceucontrolsthepositiondofthecontrolmassmdwhich,inturn,controlstheoverallstiffnessofthemechanism.Thetireismodeledasalinearspringofspringconstantkt.TheassumptionsadoptedinFig 4-3 aresummarizedasfollows: 1. Thelateraldisplacementofthesprungmassisneglected,i.eonlytheverticaldisplacementysisconsidered. 2. Thewheelcamberangleiszeroattheequilibriumpositionanditsvariationisnegligiblethroughoutthesystemtrajectory. 72

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Figure4-2. Variablestiffnesssuspensionsystem 3. Thespringsandtiredeectionsareinthelinearregionsoftheiroperatingranges. 4.1.3EquationsofMotionLetq=266664ysd377775, (4)bedenedasthegeneralizedcoordinates.Theequationsofmotion,derivedusingLagrange'smethod,arethengivenbyM()q+C(,_)+B()_q)]TJ /F7 11.955 Tf 11.96 0 Td[(K(q)+G()=e3,3u+Wd()dr (4) 73

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Figure4-3. Quartercarmodel-passivecase whereM()=266664ms+mu+mdmulDcos0mulDcosIc+mul2Dcos2000md377775,C(,_)=)]TJ /F3 11.955 Tf 9.3 0 Td[(mulD_2sinw(), 74

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w()=2666641lDcos0377775,B()=266664btbtlDcos0btlDcosbtl2Dcos2+bsgbs 2gd0bs 2gdbsgd377775,gd(d,)=(d)]TJ /F3 11.955 Tf 11.95 0 Td[(lAcos)2 H2+d2+l2A)]TJ /F6 11.955 Tf 11.96 0 Td[(2lAdcos)]TJ /F6 11.955 Tf 11.96 0 Td[(2HlAsin,gd(d,)=2lA(d)]TJ /F3 11.955 Tf 11.96 0 Td[(lAcos)(dsin)]TJ /F3 11.955 Tf 11.96 0 Td[(Hcos) H2+d2+l2A)]TJ /F6 11.955 Tf 11.96 0 Td[(2lAdcos)]TJ /F6 11.955 Tf 11.96 0 Td[(2HlAsin,g(d,)=l2A(dsin)]TJ /F3 11.955 Tf 11.96 0 Td[(Hcos)2 H2+d2+l2A)]TJ /F6 11.955 Tf 11.96 0 Td[(2lAdcos)]TJ /F6 11.955 Tf 11.96 0 Td[(2HlAsin,K(q)=266664kt(t)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(ys+lDsin)kt(t)]TJ /F6 11.955 Tf 11.96 0 Td[(1)lDcos(ys+lDsin)ks(s)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(d)]TJ /F3 11.955 Tf 11.95 0 Td[(lAcos)377775+2666640ks(s)]TJ /F6 11.955 Tf 11.96 0 Td[(1)lA(dsin)]TJ /F3 11.955 Tf 11.95 0 Td[(Hcos)0377775,G()=266664ms+mu+mdmulDcos0377775g, 75

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Wd()=266664kt(t)]TJ /F6 11.955 Tf 11.96 0 Td[(1)btktlD(t)]TJ /F6 11.955 Tf 11.95 0 Td[(1)cosbtlDcos00377775,dr=264r_r375.r(t)istheroaddisplacementsignal.Itisafunctionoftheroadproleandthevehiclevelocity.Thetermssandtcharacterizethecompressionoftheverticalstrutandtirespringsrespectively.Theyaredenedastheratioofthefreelengthandinstantaneouslengthofthecorrespondingspring.Properties:Thefollowingpropertiesofthedynamicsgivenin( 4 )areexploitedinsubsequentanalyses: 1. TheinertiamatrixM()issymmetric,positivedenite.Also,sinceeachelementofM()canbeboundedbelowandabovebypositiveconstants,itfollowsthattheeigenvalues,hencethesingularvaluesofM()canalsobeboundedbyconstants.Thus,thereexistsm1,m22R+suchthatm1kxk2xTM()xm2kxk2and (4)1 m2kxk2xTM)]TJ /F4 7.97 Tf 6.59 0 Td[(1()x1 m1kxk2,8x2R2 (4) 2. C(,_)canbeupperboundedasfollowsC(,_)c1_2,c12R+. (4)Also,thereexistamatrixVm(,_)suchthatC(,_)=Vm(,_)_qandxT1 2_M())]TJ /F3 11.955 Tf 11.96 0 Td[(Vm(,_)x=0,8x2R2 (4)Thepropertyin( 4 )istheusualskewsymmetricpropertyoftheCoriolis/centripetalmatrixofLagrangedynamics[ 50 ]. 76

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3. ThedampingmatrixB()issymmetricandpositivesemidenite.Also,thereexistspositivedenitematricesB andBsuchthat0
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whereisatruncationofgivenas(t)=8><>:(t)0t0t>.. (4)Forthepurposeofthispaper,theL2-spaceisconsidered,hencethenite-gainL-stabilityconditionin( 4 )isrewrittenas([ 54 ])kMH(w)k2kwk2+, (4)wherek.k2denotestheL2normofasignalgivenbykk2=Z10T(t)(t)dt1 2. (4)=inffjkMH(w)k2kwk2+gisthegainofthesystem,and,inthecaseoflinearquadraticproblems,istheH1normofthesystem.Givenanattenuationlevel>0,andthecorrespondingsystemdynamics,theobjectiveistoshowthat( 4 )issatisedforsome>0.Thissolutionisapproachedfromtheperspectiveofdissipativesystems([ 51 54 ]).Thefollowingdenitiondescribestheconceptofdissipativitywithrespecttothesystemin( 4 ). Denition4.2(Dissipativity). Thedynamicssystem( 4 )isdissipativewithrespecttoagivensupplyrates(w,z)2R,ifthereexistsanenergyfunctionV(x)0suchthat,forallx(t0)=x0andtft0,V(x(tf))V(x(t0))+Ztft0s(w,z)dt,8w2L2. (4)Ifthesupplyrateistakenass(w,z)=2kwk2)-222(kzk2, (4) 78

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thenthedissipationinequalityin( 4 )impliesnite-gainL-stability[ 54 ],andthesystemissaidtobe-dissipative.Thedissipativityinequalityisthenwrittenas_V2kwk2)-222(kzk2. (4) 4.2.1PerformanceObjectiveAsusualwithsuspensionsystemsdesigns,theperformancecriterionisexpressedintermsoftheridecomfort,suspensiondeection,anddynamictireforce.Theperformancevectorz=266664!1ycba!2ysd!3ydtf377775 (4)characterizestheridecomfort,suspensiondeection,androadholdingperformances,where!1,!2,and!3aretherespectiveuserspeciedperformanceweightsforcarbodyaccelerationycba,suspensiondeectionysd,anddynamictireforceydtf.Theridecomfortischaracterizedbythecarbodyaccelerationyswhichisapproximatedusingthefollowinghighgainobserver([ 55 ]):"_=A+b_ys,(0)=0ycba=1 "cT (4)whereA=264)]TJ /F6 11.955 Tf 9.3 0 Td[(11)]TJ /F6 11.955 Tf 9.3 0 Td[(10375,b=26411375,c=26401375.TheL2-normofthecarbodyaccelerationcanbeupperboundedas([ 55 ])kycbak2c1k_ysk2c1k_ek2, (4) 79

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wherec1=22max(P)kbk2kck2 min(P)andPisthesolutionoftheLyapunovequationPA+ATP+I=0,whichisobtainedasP=1 "26411 21 23 2375.Thesuspensiondeectionisgivenasysd(t)=p l2s(0))]TJ /F3 11.955 Tf 11.95 0 Td[(l2s(t)=d(0)2)]TJ /F3 11.955 Tf 11.96 0 Td[(d(t)2)]TJ /F6 11.955 Tf 11.96 0 Td[(2Hx(sin(0))]TJ /F6 11.955 Tf 11.95 0 Td[(sin(t)))]TJ /F6 11.955 Tf 9.3 0 Td[(2x(d(0)cos(0))]TJ /F3 11.955 Tf 11.95 0 Td[(d(t)cos(t))g1 2 (4)0k41k42266664jy0s)]TJ /F3 11.955 Tf 11.96 0 Td[(ysjj)]TJ /F8 11.955 Tf 11.96 0 Td[(0jjd)]TJ /F3 11.955 Tf 11.96 0 Td[(d0j377775, (4)UsingtheCauchy-Schwarzinequality,ysd(t)canbeupperboundedasysd(t)k4kek, (4)wherek41,k42,andk4arepositiveconstants,andk4p k241+k242.Thedynamictireforceischaracterizedusingthetiredeectionandisgivenbyydtf(t)=yu(0))]TJ /F3 11.955 Tf 11.95 0 Td[(yu(t)=y0s)]TJ /F3 11.955 Tf 11.96 0 Td[(ys+lD(sin0)]TJ /F6 11.955 Tf 11.95 0 Td[(sin) (4)1k50266664jy0s)]TJ /F3 11.955 Tf 11.95 0 Td[(ysjj)]TJ /F8 11.955 Tf 11.96 0 Td[(0jjd)]TJ /F3 11.955 Tf 11.95 0 Td[(d0j377775, (4) 80

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wherek5isapositiveconstant.UsingtheCauchy-Schwarzinequality,ydtf(t)canbeupperboundedasydtfq 1+k25kek=k6kek. (4)Finally,theL2-normoftheperformancevectorin( 4 )canbeupperboundedaskzk21k_ek2+2kek2 (4)where1=!1c12=!2k4+!3k6. 4.2.2ConstantStiffnessCaseNow,considertheconstantstiffnesscaseinwhichthecontrolmassislockedatagivenpositiond.Asaresult,theoverallstiffnessisconstantfortheentiretrajectoryofthesystem.Forthiscase,thedynamicsin( 4 )reducestoM1()q1+C1(,_)+B1()_q1)]TJ /F7 11.955 Tf 11.96 0 Td[(K1(q1)+G1()=w, (4)whereM1=M1:2,1:2,C1=C1:2,K1=K1:2,B1=B1:2,1:2,w=Wd1dr,Wd1=Wd1:2,1:2Here,thecorrespondingdynamicsofthecontrolmasshasbeeneliminated.Lete1=q1)]TJ /F7 11.955 Tf 11.95 0 Td[(q01 (4) 81

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whereq01=264ys00375 (4)betheequilibriumvalueofthereducedstatevectorq1.AfterusingtheMeanValueTheorem,theclosed-loopdynamicsin( 4 )isexpressedasM1e1+Vm1_e1+K1e1+B1_e1=w (4)whereK1=)]TJ /F8 11.955 Tf 13.68 8.09 Td[(@K1 @q1q1=1+@G1 @q1q1=21,2,2Ls(q01,q1). Lemma1. ThematrixP=264Im1Im1IM1375 (4)ispositivedenite,wherem21
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whichimpliesthat=1 21+mq (1+m)2)]TJ /F6 11.955 Tf 11.96 0 Td[(4(m)]TJ /F3 11.955 Tf 11.95 0 Td[(m21), (4)fromwhichitfollowsthat>0.SincePissymmetric,theconclusionfollows. Remark4.1. ItfollowsfromRayleigh-RitzInequalitythatp1kk2TPp2kk2, (4)wherep1=minfPg,andp2=maxfPg. Theorem4.1. IfthematrixH1=1 2264)]TJ /F6 11.955 Tf 11.08 2.65 Td[(^K1)]TJ /F6 11.955 Tf 13.73 2.65 Td[(^KT1)]TJ /F3 11.955 Tf 9.29 0 Td[(KT1)]TJ /F3 11.955 Tf 11.96 0 Td[(m1M)]TJ /F4 7.97 Tf 6.59 0 Td[(11B1)]TJ /F3 11.955 Tf 9.3 0 Td[(K1)]TJ /F3 11.955 Tf 11.95 0 Td[(m1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M)]TJ /F4 7.97 Tf 6.59 0 Td[(11B1T)]TJ /F6 11.955 Tf 9.3 0 Td[(2^B1375, (4)where^K1=m1M)]TJ /F4 7.97 Tf 6.59 0 Td[(11K1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(c1k_ek 2I (4)^B1=B1)]TJ /F10 11.955 Tf 11.95 16.85 Td[(m1+c1k_ek 2I, (4)isnegativedenitealongtheentiretrajectoryoftheclosed-looperrorsystemin( 4 ),thentheL2-normoftheperformancevectorin( 4 )canbeupperboundedaskzk21kwk2+1, (4)where1=p2 p1h1, (4)1=p 2p2 p p1h1, (4) 83

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and=maxf1,2g (4)=max8><>:264m1M)]TJ /F4 7.97 Tf 6.59 0 Td[(11I3759>=>; (4)h1=jminfH1gj. (4) Proof. ConsidertheenergyfunctionV(e1,_e1)=1 2T1P1, (4)where1=264e1_e1375. (4)Takingtimederivativeof( 5 )andusingtheskewsymmetricpropertyin( 4 )yields_V=)]TJ /F6 11.955 Tf 10.88 .89 Td[(_eT1(B1)]TJ /F3 11.955 Tf 11.96 0 Td[(m1I)_e1)]TJ /F6 11.955 Tf 16 .89 Td[(_e1TK1e1+_eT1w+m1eT1M)]TJ /F4 7.97 Tf 6.59 0 Td[(11w)]TJ /F3 11.955 Tf 11.96 0 Td[(m1eT1M)]TJ /F4 7.97 Tf 6.58 0 Td[(11Vm_e1)]TJ /F3 11.955 Tf 11.95 0 Td[(m1eT1M)]TJ /F4 7.97 Tf 6.58 0 Td[(11B1_e1)]TJ /F3 11.955 Tf 11.96 0 Td[(m1eT1M)]TJ /F4 7.97 Tf 6.59 0 Td[(11K1e1. (4)Usingthepropertyin( 4 )yields_VT1H11+T1264m1M)]TJ /F4 7.97 Tf 6.59 0 Td[(11I375w, (4)whichafterusingthenegativedenitenessofH1yields_V)]TJ /F3 11.955 Tf 21.92 0 Td[(h1k1k2+k1kkwk. (4)TakeW(t)=p V(1).WhenV(1)6=0,_W=_V=(2p V)yields_W)]TJ /F3 11.955 Tf 26.26 8.09 Td[(h1 2p2W+ 2p p1kwk. (4) 84

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WhenV(1)=0,itcanbeveried([ 55 ])thatD+W 2p p1kwk, (4)whereD+denotestheupperrighthanddifferentiationoperator.HenceD+W)]TJ /F3 11.955 Tf 26.25 8.08 Td[(h1 2p2W+ 2p p1kwk (4)forallvaluesofV(1).Nextusingcomparison(Lemma3.4,[ 55 ])yieldsW(t)W(0)exp)]TJ /F3 11.955 Tf 10.91 8.09 Td[(h1t 2p2+ 2p p1Zt0kwkexp)]TJ /F3 11.955 Tf 10.49 8.09 Td[(h1(t)]TJ /F8 11.955 Tf 11.95 0 Td[() 2p2d, (4)whichimpliesthatk1(t)kr p2 p1k1(0)kexp)]TJ /F3 11.955 Tf 10.9 8.09 Td[(h1t 2p2+ 2p1Zt0kwkexp)]TJ /F3 11.955 Tf 10.5 8.09 Td[(h1(t)]TJ /F8 11.955 Tf 11.95 0 Td[() 2p2d. (4)Thusk1(t)k2p2 p1h1kwk2+p 2p2 p p1h1k1(0)k.Lastly,afterusingtheinequalityin( 4 ),theL2-normoftheperformancevectorcanbeupperboundedaskzk2p2 p1h1kwk2+p 2p2 p p1h1k1(0)k. (4) Remark4.2. TheL2-gainofthesystemdecreaseswithincreasingh1.ThismeansthatthemorethenegativedenitenessofH1,themorethedisturbancerejectionachievablebythesystem.Thefollowingtheoremgivestheboundsonachievable. 85

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Theorem4.2. Givenanattenuationlevel,andprovidedthattheperformanceweightsareselectedtosatisfythesufcientcondition=maxf1,2g


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architecture.Thelowerboundin( 4 )istermedbest-case-gain.Itdenesthesmallestgainachievablebythesystem.ThestiffnessanddampingmatricesK1,andB1containboundedfunctionsofstateanduncertaindynamicparameterswhichrangebetweenboundedvalues.Thusthebest-casegainofthesystemwithconstantstiffnesscanbelowerboundedas 10.5 p h1)]TJ /F8 11.955 Tf 11.95 0 Td[(2. (4)whereh1isthesmallestpositivenumberlargerthanthesmallestsingularvalueofH1,and 1istermedtherobustbest-casegain. 4.2.3PassiveVariableStiffnessCaseHere,thecontrolmassisallowedtomoveundertheinuenceofarestoringspringanddamperforces.Thereisnoexternalforcegeneratoraddedtothesystem.Asaresult,thesystemresponseispurelypassive.Letkuandbubethespringconstantanddampingcoefcientoftherestoringspringanddamperrespectively.Thecontrolforceuisthengivenbyu=)]TJ /F3 11.955 Tf 9.3 0 Td[(bu_d)]TJ /F3 11.955 Tf 11.96 0 Td[(ku(d)]TJ /F3 11.955 Tf 11.96 0 Td[(l0d), (4)andtheresultingdynamicsofthecontrolmassisgivenbymdd+bu_d+ku(d)]TJ /F3 11.955 Tf 11.95 0 Td[(l0d)+ks(s)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(d)]TJ /F3 11.955 Tf 11.95 0 Td[(xcos)+bs 2gd_+bsgd_d=0, (4)andthestaticequilibriumequationforthecontrolmassisgivenbyku(d0)]TJ /F3 11.955 Tf 11.95 0 Td[(l0d)+ks(s0)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(d0)]TJ /F3 11.955 Tf 11.95 0 Td[(xcos0)=0, (4) 87

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whered0istheequilibriumpositionofthecontrolmass,andl0disthefreelengthoftherestoringspring.Leted=d)]TJ /F3 11.955 Tf 11.96 0 Td[(d0 (4)bethedisplacementofthecontrolmassfromitsequilibriumposition.Substituting( 4 )into( 4 )andusingtheMeanValueTheoremyieldsmded+BTd_e+KTde=0, (4)wheree=264e1ed375, (4)Bd=2666640bs 2gdbsgd+bu377775, (4)Kd=26666640ks@(s)]TJ /F4 7.97 Tf 6.58 0 Td[(1)(d)]TJ /F5 7.97 Tf 6.59 0 Td[(xcos) @2Ls(0,)ku+ks@(s)]TJ /F4 7.97 Tf 6.58 0 Td[(1)(d)]TJ /F5 7.97 Tf 6.59 0 Td[(xcos) @dd2Ls(d0,d)3777775. (4)Now,considertheenergyfunctionV2(e,_e)=.T2P22, (4)where,2=264e_e375, (4) 88

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andP2=264ImImIM375 (4)ispositivedenite,withm2
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system.Thisclaimissupportedsubsequentlybyexperimentalandsimulationre-sults.Thisisaveryappealingresultduetoitspracticability.Noadditionalelectronicallycontrolledorforcegeneratingdeviceisrequired,onlymechanicalelementslikethespringanddamperareused. 4.2.4ExperimentTheexperimentalsetupisshowninFig. 4-4 .Itisaquartercartestrigscaleddowntoaboutaratioof1:10comparedtoanaveragepassengercarin2004[ 56 ].Thequartercarbodyisallowedtotranslateup-and-downalongarigidframe.Thiswasmadepossiblethroughtheuseoftwopairsoflinearmotionball-bearingcarriages,witheachpaironseparateparallelguiderails.Theguiderailsarexedtotherigidframeandthecarriageisattachedtothequartercarframe.Thequartercarframeismadeof80/20aluminiumframingandthenloadedwithasolidsteelcylinderweighingapproximately80lbs.Thehorizontalandverticalstrutsare2011HondaPCXscooterfrontsuspensions.Theroadgeneratorisasimpleslider-crankmechanismactuatedbySmartmotorRSM3440Dgeareddowntoaratioof49:1usingCMIRgearheadP/N34EP049.Threeaccelerometersareattachedoneeachtothequartercarframe,thewheelhub,andtheroadgenerator.DataacquisitionisdoneusingtheMATLABdataacquisitiontoolboxviaaNIUSB-6251.Experimentswereperformedforthepassivecase,wherethehorizontalstrutisjustapassivespring-dampersystem,andalsoforthexedstiffnesscase,wherethetopoftheverticalstrutislockedinaxedposition.Thispositionistheequilibriumpositionofthepassivecasewhenthesystemisnotexcited.Twotestswerecarriedout;Sinusoidal,anddroptest.FortheSinusoidaltest,theroadgeneratorisactuatedbyaconstanttorquefromtheDCmotor.Asaresult,thequartercarframemovesupanddowninasinusoidalfashion.Forthedroptest,thesuspensionsystemwasdroppedfromaxedheight.Theresultingquartercarbodyaccelerationandtiredeectionaccelerationswererecorded.Thetiredeection 90

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Figure4-4. Quartercarexperimentalsetup Table4-1. RMSgainvaluesofexperimentalresultsCBA:carbodyacceleration.TDA:tiredeectionacceleration FixedPassive DropCBA(g)0.62060.5864TDA(g)0.97590.9685SinusoidalCBA(g)0.61810.5240TDA(g)1.31521.0460 accelerationisobtainedasthedifferencebetweenthewheelaccelerationandtheroadgeneratoracceleration.Figure 4-5 andFigure 4-6 showstheresultsofthesinusoidaltestforthexedandpassivecasesrespectively.Theresultsarenotplottedtogetherbecauseofthe 91

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Figure4-5. Sinusoidaltest-xedcaseCBA:carbodyacceleration,TDA:tiredeectionaccleration.RA:roadacceleration differenceintheroadaccelerationforbothcases,evenastheDCmotorwasrunatthesameconstanttorqueforbothcases.Oneofseveralreasonsforthisobservedphenomenonistheinteractionandenergytransferbetweenthehorizontalandverticalstrutsforthepassivecase.Tofacilitateagoodcomparisonoftheobservations,thermsgainofthesystemforagivenresponseiscomputedasrmsgain=rmsvalueoftheresponsesignal rmsvalueoftheroadaccelerationsignal. (4)Table 5-1 showsthermsgainsforthesinusoidalandthedroptest.However,forthedroptest,theresponsesforthexedandpassivecasesareplottedtogetherbecausetheDCmotorwasnotusedandtheapparatuswasdroppedfromthesameheightforboth 92

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Figure4-6. Sinusoidaltest:passivecaseCBA:carbodyacceleration,TDA:tiredeectionaccleration.RA:roadacceleration Figure4-7. Droptest-carbodyacceleration 93

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Figure4-8. Droptest-tiredeectionacceleration cases.Figure 4-7 andFigure 4-8 showsthecarbodyaccelerationresponsesandtiredeectionaccelerationresponsesforthexedandactivecases. 4.2.5SimulationInordertostudythebehaviorofthequartercarsystematfullscaleaswellasresponseslikesuspensiondeection,whicharedifculttomeasureexperimentally,andexcitationscenariosthataredifculttoimplementexperimentally,realisticsimulationswerecarriedoutusingMATLABSimmechanicsSecondGeneration.First,thesystemwasmodeledinSolidworksasshowninFig. 4-9 .Next,aSimmehanicsmodelwasdeveloped.Themass/inertiapropertiesusedaretheonesgeneratedfromtheSolidworksmodel.TheverticalstrutandtiredampingandstiffnessusedaretheonesgivenintheRenaultMeganeCoupemodel[ 43 ].ThevaluesaregiveninTable 2-1 4.2.5.1TimeDomainSimulationInthetimedomainsimulation,thevehicletravelingatasteadyhorizontalspeedof40mphissubjectedtoaroadbumpofheight8cm.TheCarBodyAcceleration,SuspensionDeection,andTireDeectionresponsesarecomparedbetweentheconstantstiffnessandthepassivevariablestiffnesscases.Fortheconstantstiffnesscase,thecontrolmasswaslockedatthreedifferentlocations(d=40cm,d=45.56cmandd=50cm).Thevalued=45.56cmistheequilibriumpositionofthe 94

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Figure4-9. Solidworksquartercarmodel controlmass.Next,asimulationisperformedforthepassivecase.TheresultsarereportedinFigures 4-10 4-11 ,and 4-12 whicharethethecarbodyacceleration,suspensiondeection,andtiredeectionresponses,respectively.Figure 4-13 showsthepositionhistoryofthecontrolmassforboththepassivevariablestiffnesscase. 4.2.5.2FrequencyDomainSimulationForthefrequencydomainsimulation,anapproximatefrequencyresponsefromtheroaddisturbanceinputtotheperformancevectorgivenin( 4 ),iscomputedusingthe 95

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Figure4-10. Timedomainsimulation-carbodyacceleration Figure4-11. Timedomainsimulation-suspensiondeection Figure4-12. Timedomainsimulation:tiredeection 96

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Figure4-13. Timedomainsimulation-controlmassposition notionofvariancegain[ 57 58 ].TheapproximatevariancegainisgivenbyG(j!)=vuuuuuut Z2N=!0z2dt Z2N=!0A2sin2(!t)dt, (4)wherezdenotestheperformancemeasureofinterestwhichistakentobecarbodyacceleration,suspensiondeection,andtiredeection.Theclosedloopsystemisexcitedbythesinusoidr=Asin(!t),t2[0,2N=!],whereNisanintegerbigenoughtoensurethatthesystemreachesasteadystate.Thecorrespondingoutputsignalswererecordedandtheapproximatevariancegainswerecomputedusing( 5 ).Figures 4-14 4-15 ,and 4-16 showthevariancegainplotsforthecarbodyacceleration,suspensiondeection,andtiredeectionrespectively.Theguresshowthatthevariablestiffnesssuspensionachievesbettervibrationisolationinthehumansensitivefrequencyrange[ 59 ](4-8Hz),andbetterhandlingbeyondthetirehopfrequency[ 19 ](>59Hz). 97

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Figure4-14. Frequencydomainsimulation-carbodyacceleration Figure4-15. Frequencydomainsimulation-suspensiondeection 98

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Figure4-16. Frequencydomainsimulation-tiredeection 99

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CHAPTER5VARIABLESTIFFNESSSUSPENSIONSYSTEMSUSINGNONLINEARENERGYSINKS:ACTIVEANDSEMI-ACTIVECASESNonlinearenergysinks(NES)areessentiallynonlineardampedoscillatorswhichareattachedtoaprimarysystem1forthesakeofvibrationabsorptionandmitigation.Suchattachmentshavebeenusedextensivelyinengineeringapplications,particularlyinvibrationsuppressionoraeroelasticinstabilitymitigation.Thevibrationofsystemswithessential(stronglyorweakly)couplednonlinearityhasbeenstudiedextensivelyinliterature[ 60 64 ].Itwasshownin[ 61 ]thatsuchattachmentscanbedesignedtoactasasinkforunwantedvibrationsgeneratedbyexternalimpulsiveexcitations.Theunderlyingdynamicalphenomenongoverningthepassiveenergypumpingfromaprimaryvibratingsystemtotheattachednonlinearenergysinkhasbeshowntobearesonancecaptureona1:1manifold[ 64 69 ].Itwasshown[ 68 70 ]thatundercertainconditions,vibrationenergygetspassivelypumpedfromdirectlyexcitedprimarysystemtothenonlinearsecondarysysteminaone-wayirreversiblefashion.Nonlinearpassiveabsorberscanbedesignedwithfarsmalleradditionalmassesthanthelinearabsorbers[ 62 ],thankstotheenergypumpingphenomenon.Thiscorrespondstoacontrolledone-waychannelingofthevibrationenergytoapassivenonlinearstructurewhereitlocalizesanddiminishesintimeduetodampingdissipation.Thisallowsnonlinearenergypumpingtobeusedincoupledmechanicalsystems,wheretheessentialnonlinearityoftheattachedabsorberenablesittoresonatewithanyofthelinearizedmodesofthesubstructure[ 64 ].Inthepreviouschapter,reciprocalactuationconceptwasusedtodesignavariablestiffnesssuspensionsystemforisolatingacarbodyfromroaddisturbance[ 71 ].Thesystemisessentiallyapassivevibrationisolationsysteminwhichthemotionofthe 1Thisreferstothemainsystemwhosevibrationisintendedtobeabsorbed 100

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Figure5-1. Orthogonalnonlinearenergysink(NES) secondarylinearattachmentismadeorthogonaltotheprimarysystem.Theprimaryandsecondarysystemsarecoupledthroughthetraditionalsuspensionsystem.Inthischapter,theconceptisextendedbyusingactiveandsemi-activelineargenerators,controlledtomimicthebehaviorofnonlinearenergysinks,todrivetheorthogonalsecondarysystem2.ThemotivationfortheuseofNESisprimarilyduetotheirprovencapabilitytoachieveone-wayirreversibleenergypumpingfromthelinearprimarysystemtothenonlinearattachment.Thegoalthereforeistoachieveaone-wayirreversibleenergypumpingoftheroaddisturbancetothesecondarysystemwhosevibrationisorthogonaltothecarbodymotion.Afairlygeneralnonlinearfunctionisusedinthiswork,insteadofcubicnonlinearitythatisgenerallyused. 5.1OrthogonalNonlinearEnergySinkFig. 5-1 showstheNESconsideredinthiswork.ThetermorthogonalNESisusedtodescribetheconceptbecausethedirectionofmotionofthesecondarysystem 2Thisreferstothevibrationabsorberorisolator 101

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isorthogonaltotheprimarysystem.Thisissuitable,structurally,fortheapplicationinquestion.ThesubsystemsS1,C,andS2constitutetheprimarysubsystem,andareallowedtoslideverticallytogetherasaunitoftotalsprungmassms+md.ThesubsystemCistermedthecontrolmass(orcontrolsubsystem).It,togetherwiththenonlinearspringandthedashpotofdampingcoefcientbd,constitutethesecondarysubsystem.ThenonlinearfunctionisdenedasF=g(d)=)]TJ /F3 11.955 Tf 9.3 0 Td[(k1(l0d)]TJ /F3 11.955 Tf 11.95 0 Td[(d))]TJ /F3 11.955 Tf 11.95 0 Td[(k2sinh(1(l0d)]TJ /F3 11.955 Tf 11.96 0 Td[(d)), (5)wherel0disthefreelengthoftheidealizednonlinearspring.Thenonlinearfunctionusedisfairlymoregeneralcomparedtothepurecubicnonlinearitythathavebeenusedinthepast[ 72 ].TheTaylorseriesexpansionisF=)]TJ /F3 11.955 Tf 9.3 0 Td[(k1(l0d)]TJ /F3 11.955 Tf 11.96 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(k21Xi=1k2i)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)(l0d)]TJ /F3 11.955 Tf 11.96 0 Td[(d)2i)]TJ /F4 7.97 Tf 6.59 0 Td[(1,k2i)]TJ /F4 7.97 Tf 6.58 0 Td[(1(1)>0 (5)fromwhichthegeneralityobvious.ThemasslabeledUistheunsprungmass,whosedisplacementyuisusedasthesourceofdisturbancetothesystem.Anapproximatefrequencyresponsefromtheinputyutothesprungmassaccelerationysandtherattlespacedeectionys)]TJ /F3 11.955 Tf 12.77 0 Td[(yu,iscomputedusingthenotionofvariancegain([ 57 58 ]).TheapproximatevariancegainisgivenbyG(j!)=vuuuuuut Z2N=!0z2dt Z2N=!0A2sin2(!t)dt, (5)wherezdenotestheperformancemeasureofinterest(sprungmassaccelerationandrattlespacedeectioninthiscase).Thesystemisexcitedbythesinusoidr=Asin(!t),t2[0,2N=!],whereNisanintegerbigenoughtoensurethatthesystemreachesasteadystate.Thecorrespondingoutputsignalswererecordedandtheapproximatevariancegainswerecomputedusing( 5 ).Theresultingvariance 102

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gainresponsesareshowninFigs. 5-2A ,and 5-2B fortheconstantstiffnesssuspension(CSS)3,thevariablestiffnesssuspensionwithlinearenergysink(VSS:LES)4,andthevariablestiffnesssuspensionwithnonlinearenergysink(VSS:NES).Theguresshowthatthevariablestiffnesssuspensionachievesbettervibrationisolation,withasignicantimprovementfromthelinearenergysinkcasetothenonlinearenergysinkcase.AsshowninFig. 5-2B ,theimprovementgainedinvibrationisolationresultsinacorrespondingperformancedegradationintherattlespacedeection.However,whencomparedwiththeimprovementinthevibrationisolation,thereisanoverallimprovementinperformanceassociatedwiththeuseofthevariablestiffnesssuspensionwithnonlinearenergysink.Thisagreeswiththeusualtrade-offinsuspensiondesign[ 19 ].TheperformanceimprovementcanfurtherbeincreasedbytransitioningfromLESinlowfrequencyrange(<8Hz)toNESinhighfrequencyrange(>8Hz). 5.2ActiveCaseIntheactivecase,ahydraulicactuatorisusedtodrivethecontrolmass.ThequartercarmodelofthesuspensionsystemconsideredisshowninFig 5-3 .Itiscomposedofaquartercarbody,wheelassembly,twospring-dampersystems,roaddisturbance,andlowerandupperwishbones.ThepointsO,A,andBarethesameasshowninthevariablestiffnessmechanismofFig 4-1 .Thehorizontalcontrolforce,exertedbythehydraulicactuatorH,controlsthepositiondofthecontrolmassmdwhich,inturn,controlstheoverallstiffnessofthemechanism.Thecontrolforceis 3Here,thepositionofthecontrolmassisxed.Thiscorrespondstothetraditionalconstantstiffnesssuspensionsystem4Inthiscase,thecontrolmassisallowedtomoveundertheinuenceofalinearhorizontalspringanddamper.Thisisthecasereportedin[ 71 ] 103

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ACarbodyacceleration BRattlespacedisplacementFigure5-2. VarianceGain designedinsubsequentsectionstomimictheorthogonalNESintroducedintheprevioussection.Thetireismodeledasalinearspringofspringconstantkt.TheassumptionsadoptedinFig 5-3 aresummarizedasfollows: 1. Thelateraldisplacementofthesprungmassisneglected,i.eonlytheverticaldisplacementysisconsidered. 2. Thewheelcamberangleiszeroattheequilibriumpositionanditsvariationisnegligiblethroughoutthesystemtrajectory. 3. Thespringsandtiredeectionsareinthelinearregionsoftheiroperatingranges. 104

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Figure5-3. Quartercarmodel-activecase 5.2.1ControlMassesandActuatorDynamicsThedynamicsofofthehydraulicactuatorisgivenby[ 73 74 ]_PL=)]TJ /F8 11.955 Tf 9.3 0 Td[(Avp)]TJ /F8 11.955 Tf 11.95 0 Td[(PL+xvp Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(xv)PL, (5)_xv=)]TJ /F6 11.955 Tf 10.56 8.09 Td[(1 xv+K u, (5)F=APL, (5)whereAisthepressureareaintheactuator,PListheloadpressure,vp=_distheactuatorpistonvelocity,Fistheoutputforceoftheactuator,,,andarepositiveparametersdependingontheactuatorpressurearea,effectivesystemoilvolume, 105

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effectiveoilbulkmodulus,oildensity,hydraulicloadow,totalleakagecoefcientofthecylinder,dischargecoefcientofthecylinder,andservovalveareagradient,xvisthespoolvalveposition,istheactuatorelectricaltimeconstant,KistheDCgainofthefour-wayspoolvalve,anduistheinputcurrenttotheservovalve. 5.2.2ControlDevelopmentIntermsoftheForce,Fexertedbytheactuator,theactuatordynamicsin( 6 )iswrittenas_F=)]TJ /F8 11.955 Tf 9.3 0 Td[(F)]TJ /F8 11.955 Tf 11.95 0 Td[(A2_d+Au, (5)whereu=xvr Ps)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn(xv)F A (5)isactitiouscontrolvariable,fromwhichtheslowcomponent(orenvelop)ofthecontrolisobtained,aftersingularperturbationofthevalvedynamics.Lettheactuatorforcetrackingerrorbedenedase=F)]TJ /F3 11.955 Tf 11.95 0 Td[(Fd, (5)whereFd=)]TJ /F3 11.955 Tf 9.3 0 Td[(k1(l0d)]TJ /F3 11.955 Tf 11.96 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(k2sinh((l0d)]TJ /F3 11.955 Tf 11.95 0 Td[(d)))]TJ /F3 11.955 Tf 11.95 0 Td[(bd_d, (5)isthedesiredforcetobetrackedbytheactuatorforcedynamicsin( 5 ).Takingthederivativeof( 5 )yieldstheactuatorforcetrackingerrordynamics_e=)]TJ /F8 11.955 Tf 9.3 0 Td[(FA2_d+Au)]TJ /F6 11.955 Tf 14.57 2.66 Td[(_Fd (5)=)]TJ /F8 11.955 Tf 9.3 0 Td[(e)]TJ /F10 11.955 Tf 12.99 5.28 Td[(e_Fd+Au)]TJ /F8 11.955 Tf 13.15 8.09 Td[(A _d)]TJ /F8 11.955 Tf 16.91 8.09 Td[( AFd)]TJ /F6 11.955 Tf 17.41 8.09 Td[(1 A^_Fd (5)=)]TJ /F8 11.955 Tf 9.3 0 Td[(e)]TJ /F10 11.955 Tf 12.99 5.28 Td[(e_Fd+A)]TJ /F6 11.955 Tf 5.72 -9.69 Td[(u)]TJ /F3 11.955 Tf 11.96 0 Td[(YT, (5) 106

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wheretheregressionmatrixYandtheunknownparametervectoraregivenbyY=_dFd^_FdT, (5)=A A1 AT, (5)and^_Fdisanapproximationofthedesiredforce_Fdobtainedusingthehighgainobserver[ 55 ]"2_p=Ahgp+bhgFd (5)^_Fd=1 "2cThgp,a,b (5)wherethesaturationfunction(...)isgivenby(,a,b)=8>>>><>>>>:a,ifb, (5)andAhg=264)]TJ /F6 11.955 Tf 9.3 0 Td[(11)]TJ /F6 11.955 Tf 9.3 0 Td[(10375,bhg=26411375,chg=26401375,"21.Theisdonebecause,ascanbeseenin( 5 ),_Fdcontainsunmeasurablesignald.Itcanbeshown(see[ 55 ])thattheestimationerror,e_Fd=_Fd)]TJ /F6 11.955 Tf 13.41 4.79 Td[(^_Fddecays,inthefasttimescale,totheballje_Fdj
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obtainedbysubstituting( 6.4.2 )into( 5 ),isgivenby_e=)]TJ /F6 11.955 Tf 11.29 -.16 Td[((+k1A)e)]TJ /F10 11.955 Tf 12.99 5.28 Td[(e_Fd)]TJ /F3 11.955 Tf 11.96 0 Td[(c1Asgn(e))]TJ /F8 11.955 Tf 11.96 0 Td[(AYTe, (5)wheretheparameterestimationerroreisgivenbye=)]TJ /F6 11.955 Tf 13.7 2.65 Td[(^. (5)Inordertosimplifythecontrollerdesignfortheactuators,thespoolvalvedynamicsisreduced,usingasingularperturbationtechnique[ 75 ].Thecontrolinputisdesignedasu=)]TJ /F3 11.955 Tf 9.3 0 Td[(Kfxv+1+KKf Kus, (5)whereusisaslowcontrolintimeandKfisapositivedesigncontrolgain.Consequently,thevalvepsuedo-closedloopdynamicsisgivenby"_xv+xv=us, (5)where"= 1+KKf (5)istheperturbationconstant.Thepseudo-closedloopin( 6 )hasaquasi-steadystatesolution,xvi("=0),xvi,givenbyxv=us. (5)Usingthefasttimescale=t "andTichonov'sTheorem[ 75 ],thevalvedynamicsisdecomposedintofastandslowtimescalesasfollowsxv=xv++O("), (5)d d=)]TJ /F8 11.955 Tf 9.3 0 Td[(, (5) 108

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where()isaboundarylayercorrectionterm.Itisseenthat()decaysexponentiallyinthefasttimescale.Typically,thetimeconstantintheactualsystemisdesignedtosatisfy0<"1[ 76 ].Therefore,bychoosingthecontrolgainKflargeenough,theperturbationconstantcanbemadeassmallaspossible.Asaresult,+O(")becomesnegligiblysmall,andthectitiouscontrolbecomesu=usr Ps)]TJ /F6 11.955 Tf 11.95 0 Td[(sgn(us)F A. (5)Assumingsufcientpressureforthehydraulicpump,theterminsidethesquarerootoperatoristakenaspositive.Thussgn(u)=sgn(us), (5)whichimpliesthatu=uPs)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn(u)F A)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2. (5) 5.2.3StabilityAnalysisThissectionpresentstheLyapunovbasedstabilityanalysisoftheclosedlooperrordynamicsin( 5 ).Theadaptationlawfortheparameterestimationisdesigned.Itisalsoshownthatifthecontrolgainsarechosentosatisfycertainsufcientconditions,thentheactuatorforcetrackingerrorwillapproachzeroasymptotically. Theorem5.1. Giventheadaptiveupdatelaw_^=)]TJ /F6 11.955 Tf 9.3 0 Td[()]TJ /F3 11.955 Tf 6.78 0 Td[(Ye, (5)where)]TJ /F15 11.955 Tf 10.09 0 Td[(isapositivedeniteadaptationgainmatrix.Ifthecontrolgainc1ischosentosatisfythefollowingsufcientconditionsc1jO("2)j Aje_Fdj A, (5) 109

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thentheactuatortrackingerrorin( 5 )approacheszeroasymptotically.i.ee(t)!0,ast!1. Proof. ConsiderthefollowingpositivedeniteLyapunovfunctioncandidateV=1 2e2+A 2eT)]TJ /F14 7.97 Tf 6.78 4.93 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(1e, (5)Takingthersttimederivativeandsubstitutingtheclosedlooperrordynamicsin( 5 )yields_V=e_e)]TJ /F8 11.955 Tf 11.96 0 Td[(AeT)]TJ /F14 7.97 Tf 6.78 4.94 Td[()]TJ /F4 7.97 Tf 6.58 0 Td[(1_^ (5)=e)]TJ /F6 11.955 Tf 11.29 -.17 Td[((+k1A)e)]TJ /F10 11.955 Tf 12.99 5.28 Td[(e_Fd)]TJ /F3 11.955 Tf 11.96 0 Td[(c1Asgn(e))]TJ /F8 11.955 Tf 11.96 0 Td[(AYTe)]TJ /F8 11.955 Tf 11.96 0 Td[(AeT)]TJ /F14 7.97 Tf 6.78 4.93 Td[()]TJ /F4 7.97 Tf 6.58 0 Td[(1_^, (5)which,afterapplyingtheupdatelawsin( 5 ),becomes_V)]TJ /F6 11.955 Tf 23.92 -.17 Td[((+k1A)e2+je_Fdjjej)]TJ /F3 11.955 Tf 17.93 0 Td[(c1Ajej. (5)Usingthesufcientconditionin( 5 ),theinequalityin( 5 )yields_V)]TJ /F6 11.955 Tf 23.91 -.16 Td[((+k1A)e20. (5)From( 5 )and( 5 ),itfollowsthatV(t)isbounded,whichalsoimpliesthate(t),ande(t)arebounded.Usingtheboundednessof^_Fd(t),from( 5 ),itfollowsfrom( 5 )that_e(t)isbounded,whichimpliesthatthesignale(t)isuniformlycontinuous.Integrating( 5 )yieldslimt!1Zt0e(t)2dt=V(0))]TJ /F3 11.955 Tf 11.95 0 Td[(V(1) +k1A2L1. (5)Thus,usingBarbalat'sLemma(Section8.3,[ 55 ]),itcanbeshownthate(t)!0,ast!1. 110

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Figure5-4. Simmechanicmodel Table5-1. Dynamicparametervalues ParameterValue ms315kgmu37.5kgbs1500N=m=sks29500N=mkt210000N=m 5.2.4SimulationInordertostudythebehaviorofthequartercarsystemtodifferentroadexcitationscenarios,aswellasmeasureresponseslikesuspensiondeection,realisticsimulationswerecarriedoutusingMATLABSimmechanics.First,thesystemwasmodeledinSolidworks,andthentranslatedtoaSimmechanicmodel(Fig. 5-4 ).TheverticalstrutandtiredampingandstiffnessusedaretheonesgivenintheRenaultMeganeCoupemodel([ 43 ]).ThevaluesaregiveninTable 5-1 .Thevaluesofthehydraulicparameterswereobtainedempericallyin[ 73 ],andaregiveninTable 5-4 111

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Table5-2. Hydraulicparametervalues ParameterValue 4.5151013N=m51sec)]TJ /F4 7.97 Tf 6.58 0 Td[(11.545109N=m5=2kg1=21=30secPs10342500PaA3.3510)]TJ /F4 7.97 Tf 6.58 0 Td[(4m2 Inthesimulation,thevehicletravelingatasteadyhorizontalspeedof40mphissubjectedtoaroadbumpofheight10cm.TheCarBodyAcceleration,SuspensionTravel,andTireDeectionresponsesaremeasured.Thesuspensiontravelisdenedastheverticaldistancebetweenthecentersofmassofthesprungandunsprungmasses,andthetiredeectionasthedifferencebetweenthecenterofmassoftheunsprungmassandtheroadheight.Simulationswerecarriedoutfortheconstantstiffnessandthevariablestiffnesssuspensionsystems.Fortheconstantstiffnesssuspension,thecontrolmasswaslockedataxedpositioncorrespondingtotheequilibriumpositionofthecontrolmassforthevariablestiffnesssystem.Moreover,foreachstiffnesstype,bothpassiveandactivecaseswereconsidered.Thepassivecaseoftheconstantstiffnesssuspensionisthetraditionalpassivesuspension,whileintheactivecase,thepassivespringdamperisreplacedwithahydraulicactuatorcontrolledtotrackaskyhook[ 31 ]suspensionforce.Ontheotherhand,thepassivecaseofthevariablestiffnesssuspensioncorrespondstotheLES,whiletheactivecasecorrespondstotheNES.TheresultsobtainedarereportedinFigures 5-5 ,through 5-9 .Table 5-3 showsthevariancegainsforthedifferentresponses.Fig 5-5 showsthecarbodyacceleration,whichisusedheretodescribetheridecomfort.Thelowerthecarbodyacceleration,thebettertheridecomfort.Asseeninthegure,theNESisthemostridefriendlysuspension,outperformingtheskyhookcontrol.AsshowninFig 5-6 ,associatedwiththisimprovementisacorrespondingdegradationinthesuspensiontravel.Thisagreeswiththeobservationmadeinearliersections,aswellasthewellknowtradeoffbetween 112

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Table5-3. Variancegainvalues ConstantStiffnessPassiveConstantStiffnessActiveVariableStiffnessPassiveVariableStiffnessNES CarBodyAcceleration(s)]TJ /F4 7.97 Tf 6.59 0 Td[(1)109.038964.281865.612742.9737SuspensionDefelction80.881780.872584.383482.6723TireTravel1.05621.01001.01881.0152 ridecomfortandsuspensiondeection.Fortunately,thedegradationinsuspensiondeectionisnotasmuchastheimprovementgainedintheridecomfort,resultinginanoverallbetterperformance.Moreover,thesuspensiontravelperformancecanbeimprovedbydesigningtheagainscheduledcontroller,usinganobservedfrequencyofthesprungmassastheschedulingvariable.ASaresult,theNEScanbeturnedonandoffdependingonthefrequency,asdescribedpreviously.Figure 5-8 showsthepositionhistoryofthecontrolmassforthevariablestiffnesssuspension,fromwhichtheboundednessofthemotionofthecontrolmassisseen.Themaximumdisplacementofthecontrolmassfromtheequilibriumpositionisabout7cm.Thisimpliesthatthespacerequirementforthecontrolmassissmall,whichfurtherdemonstratesthepracticalityofthesystem.Fig 5-7 showsthatthereisnosignicantreductioninthetiredeection.Thus,thesuspensionsystemsareapproximatelyequallyroadfriendly.Itisalsoseen,inFig 5-9 ,thatthehydraulicforcefromtheNESisabout60%ofthatfromtheskyhookcounterpart.Thistranslatestoalowerpowerrequirementfortheproposedsystem. 5.3Semi-activeCaseInthesemi-activecasetwosemi-activedevicesareused;avertical,mountedalongtheverticalstrut,andahorizontal,mountedalongthehorizontalstrut,semi-activedevices.Thesemi-activedeviceconsideredistheMR-damper.ThequartercarmodelofthesuspensionsystemconsideredisshowninFig 5-10 .Itiscomposedofaquartercarbody,wheelassembly,twospring-MRdampersystems, 113

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Figure5-5. Carbodyacceleration(CBA) roaddisturbance,andlowerandupperwishbones.ThepointsO,A,andBarethesameasshowninthevariablestiffnessmechanismofChapter3.Themotionofthecontrolmass,whichinturnsdeterminetheeffectivestiffnessofthesuspensionsystem,isinuencedbytheMR1.TheMR1damperforceisdesignedinsubsequentsectionstomimictheorthogonalNESintroducedintheprevioussectionandtheMR2damperforceisdesignedtomimicthetraditionalskyhook[ 31 ]dampingforce.Thetireismodeledasalinearspringofspringconstantkt.TheassumptionsadoptedinFig 5-10 aresummarizedasfollows: 1. Thelateraldisplacementofthesprungmassisneglected,i.eonlytheverticaldisplacementysisconsidered. 2. Thewheelcamberangleiszeroattheequilibriumpositionanditsvariationisnegligiblethroughoutthesystemtrajectory. 114

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Figure5-6. Suspensiontravel(ST) 3. Thespringsandtiredeectionsareinthelinearregionsoftheiroperatingranges.Thedampingcharacteristicsftheconsideredsemi-activedevicecanbechangedbyacontrolcurrent.However,thereisnocorrespondingenergyinputintothesystemasaresultofthecontrolcurrent.ThisimpliesapassivityconstraintontheMR-dampermodel.Thecontrolcurrentisdesignedtomimicadesiredforceascloseaspossible,whileenforcingthepassivityconstraint.Thisapproachhasbeenusedinthepastforsemi-activecontroldesign[ 9 10 77 ]. 5.3.1MR-damperModelingTherelationshipbetweentheMR-dampercontrolcurrentandthedampingforceexhibitanonlinearphenomenon,andasaresult,MR-damperbasedvibrationcontrolisachallengingtasks.DifferentdampermodelshavebeendevelopedtocapturethebehaviorofMR-dampers.Generally,theapproachesthatexistsinliteraturecanbe 115

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Figure5-7. Tiredeection(TD) Figure5-8. Controlmassdisplacement groupedintoparametricandnonparametric[ 78 79 ].TheparametricmodelingtechniquecharacterizestheMR-damperdeviceasacollectionof(linearand/ornonlinear)springs,dampers,andotherphysicalelements.Anumberofstudieshaveaddressedthe 116

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Figure5-9. Actuatorforces parametricmodelingofMR-dampers.OneoftheearlymodelsisBouc-Wenmodel[ 80 ]whichwasderivedfromaMarkov-vectorformulationtomodelnonlinearhystericsystems.Later,theBinghamviscoelastic-plasticmodelwasdescribedbyShamesandCozzarelli[ 81 ].Spencerandco-workers[ 26 ]developedaphenomenologicalmodelthataccuratelyportraystheresponseofanMR-damperinresponsetocyclicexcitations.ThisisamodiedBouc-Wenmodelgovernedbyordinarydifferentialequations.Bouc-Wenbasedmodelsinsemi-activeseismicvibrationcontrolhaveproventobeeasytouseandnumericallyamenable.OtherauthorshavestudiedparametricmodelofMR-dampers,emphasizingthedifferencebetweenthepre-yieldviscoelasticregionandthepost-yieldviscousregionasakeyaspectofthedamper.Oneofsuchmodelisgivenin[ 38 ],wherethedamperforceismodeledusingthenonlinearstaticsemi-activedampermodel.TheallowsfulllingthepassivityconstraintofMR-damper.Ontheotherhand,nonparametricmodelingemploysanalyticalexpressionstodescribethecharacteristicsofthemodeleddevicebasedonbothtestingdataanalysisanddeviceworkingprinciple[ 78 ].AlthoughparametricmodelseffectivelycharacterizestheMR-dampersatxedvaluesofthecontrolcurrent,theydonotincludethemagneticeldsaturationthatisinherentinMR-dampers.TherepresentationofthemagneticeldsaturationiscrucialinaccuratelyusingtheMR-dampermodelfordesignanalysis 117

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Figure5-10. Quartercarmodel-Semi-activecase andcontroldevelopment.Recently,Songet.al[ 78 ]proposedanonparametricmodelwherethecharacteristicsofacommercialMR-damperarerepresentedbyaseriesofcontinuousfunctionsanddifferentialequations,whicharetractableusingnumericalsimulationtechniques.Thismodelwasusedin[ 77 ]andwillalsobeusedinthisworktorepresentthedynamicsoftheMR-damper.ThenonparametricmodelisshownschematicallyinFigure 5-11 .Theinputtothemodelistherelativevelocity,v(t),acrossthedamperterminals,andtheoutputisthedamperforce,F(t).ThemodeledaspectoftheMR-damperare: 118

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Figure5-11. NonparametricMR-dampermodel MaximumDampingForce,P2(i(t)) Thisisdescribedusingapolynomialfunctionofthecontrolcurrent,i(t),asP2(i)=A0+A1i+A2i2+A3i3+A4i4, (5)whereAj,j=0)]TJ /F6 11.955 Tf 11.96 0 Td[(4arethepolynomialcoefcientswithappropriateunits.ShapeFunctionSb(v(t)) Thisisusedtopreservethewave-shapecorrelationbetweenthedamperforceandtherelativevelocityacrossthedamper,andisgivenbySb(v)=(b0+b1jvrj)b2vr)]TJ /F6 11.955 Tf 11.96 -.16 Td[((b0+b1jvrj))]TJ /F5 7.97 Tf 6.58 0 Td[(b2vr bb2vr0+b)]TJ /F5 7.97 Tf 6.59 0 Td[(b2vr0, (5)wherevr=v)]TJ /F3 11.955 Tf 11.96 0 Td[(v0, (5)andb0>0,b1>0,b2>0,v0areconstants.DelayDynamicsG(s,i(t)) Arst-orderlterisusedtocreatethehysteresisloopobservedinexperimentaldata.Itisgiveninstatespaceformas_x=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1(i)x+Fs=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1(i)x+P2(i)Sb(v)Fh=P1(i)x, (5) 119

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wherexistheinternalstateofthelterandP1(i(t))isapolynomialfunctionofthecontrolcurrentgivenbyP1(i)=h0+h1i+h2i2, (5)wherehj,j=0)]TJ /F6 11.955 Tf 12.28 0 Td[(2arepolynomialcoefcientswithappropriateunits.ItisworthnotingthattheconditionP1(i)>0 (5)isimposedonP1(i(t))inordertoguaranteeadecayingsolution.OffsetFunction,Fbias Insomecases,thedampingforceisnotcenteredatzerobecauseoftheeffectofthegas-chargedaccumulatorinthedamper.TheforcebiasFbiasisincludedinthemodeltocapturethiseffect,andasresult,theoveralldamperforceisgivenbyF(t)=Fh+Fbias. (5)Table 5-4 showstheoptimalvaluesoftheMR-dampermodelobtainedfromexperimentaldataviaanoptimizationprocess[ 78 ].Intermsoftheinputv(t)andoutputF(t),theoveralldynamicsoftheMR-damperisgivenby_Fh=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1(i)Fh+P1(i)Sb(v)P2(i) (5)F(t)=Fh+Fbias (5) 5.3.2ControlDevelopmentTheschemesusedforthedesireddamperforcesaretheNES,andskyhookbasedcontrolforcesfdH=k1sinh(1(l0d)]TJ /F3 11.955 Tf 11.96 0 Td[(d)), (5)fdV=bsky_ys, (5) 120

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Table5-4. MR-damperparametervalues ParameterValue A0164.8b05.8646h2566A11316.5b10.0060v00.6248A21407.8b20.2536Fbias0A3-1562.8h0299.7733A4388.8h1-210.32 wherefdi,i=fH,Vgarethecorrespondinghorizontalandverticaldesiredforcesrespectively,k1and1arepositiveconstantsusedtotunetheperformanceoftheNEScontrol,andbskyisthedampingcoefcientoftheskyhookdamper5. 5.3.2.1OpenLoopTrackingErrorDevelopmentAlthoughtheMRdamperparametervaluesgiveninTable 5-4 weredeterminedexperimentally,theycanchangeovertimeduetousageandothercauses.Asaresult,anadaptivetrackingcontrolfortheMRdampingforceisdeveloped.Tothiseffect,itisassumedthatthecoefcientsofthepolynomialP2(i)areunknown.Also,thedesiredcontrolforcemaynotgenerallysatisfythepassivityconstraintatagiveninstant.Attheinstanceswhenthepassivityconstraintisviolated,thedesireddampingforceliesoutsidethetrackablepassivityregionoftheMRdamper.Inordertoensureavalidtrackeddesireddampingforce,theforcefdigivenin( 5 )isclippedinthepassivityregion.UsingtheFinalValueTheorem,thesteadystateMRdamperforce,from( 5 ), 5averticalctitiousdamperbetweenthesprungmassandinertialframe 121

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isgivenbyFss=Sb(v)P2(i). (5)Thus,thetrackeddesireddamperforceisobtainedbyclippingfdasfollowsFd(fdi,v)=8>>>><>>>>:Sb(v)P 2ifvrfdvrSb(v)P 2fdiifvrSb(v)P 2
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where(i)=)]TJ /F3 11.955 Tf 9.3 0 Td[(Fd+Sb^P2, (5)and^P2isanadaptiveestimateofthepolynomialP2(i).Theupdatelawisdesignedsubsequently. 5.3.2.2ClosedLoopErrorSystemDevelopmentFirst,acloseapproximationofthepolynomialP2(i)isgivenwithintheoperatinginterval.GiventheboundsP 2and P2,thepolynomialP2(i)isapproximatedintheinterval[0imax]asP2(i)=P 2+i imax)]TJ ET q .478 w 240.11 -229.73 m 248.96 -229.73 l S Q BT /F3 11.955 Tf 240.11 -239.71 Td[(P2)]TJ /F3 11.955 Tf 11.96 0 Td[(P 2+(i), (5)where(i)=i(i)]TJ /F3 11.955 Tf 11.95 0 Td[(imax))]TJ /F8 11.955 Tf 5.48 -9.68 Td[(0+i1+i22, (5)ischosentosatisfytheconstraints(0)=(imax)=0,whichimpliesthatP2(0)=P 2andP2(imax)= P2.TheapproximationislargelydependentonthemonotonicityandtheontopropertiesofP2(i). Lemma2. ThereexistsuniqueidealparametersP 2, P2,0,1,and2suchthattheapproximatedpolynomialgivenin( 5-12 )matchestheoriginalpolynomialin( 5 )exactly. Proof. TheapproximatedpolynomialiswritteninanexpandedformasfollowsP2(i)=P 2+i P2)]TJ /F3 11.955 Tf 11.95 0 Td[(P 2 imax)]TJ /F8 11.955 Tf 11.96 0 Td[(0imax+i2(0)]TJ /F8 11.955 Tf 11.95 0 Td[(1imax)+i3(1)]TJ /F8 11.955 Tf 11.95 0 Td[(2imax)+i42. (5) 123

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Figure5-12. Polynomialapproximation Comparing( 5 )with( 5 )yieldsthesystemoflinearequation266666666664100001)]TJ /F6 11.955 Tf 9.29 0 Td[(1)]TJ /F3 11.955 Tf 9.3 0 Td[(i2max00001)]TJ /F3 11.955 Tf 9.3 0 Td[(imax00001)]TJ /F3 11.955 Tf 9.3 0 Td[(imax00001377777777775266666666664P 2 P2012377777777775=266666666664A0A1imaxA2A3A4377777777775. (5)Thedeterminantofthecoefcientmatrixin( 5 )is)]TJ /F6 11.955 Tf 9.3 0 Td[(1,whichimpliesthatthecoefcientmatrixisfullranked.As,aresult,thereexistsauniquevectorP 2 P2012Tthatsatises( 5 ). Fig 5-12 showstheplotoftheactualandtheapproximatedpolynomialswith0,1,2determinedusingleastsquaremethod,givenP 2and P2.ThePolynomialin 124

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( 5 )islinearintheunknownparametersj,j=0)]TJ /F6 11.955 Tf 11.95 0 Td[(2.Thus( 5 )becomes_e=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1e+SbP1YT)]TJ /F6 11.955 Tf 13.69 2.66 Td[(^+P1(i) (5)=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1e+SbP1YT~+P1(i), (5)where=012T (5)istheparametervectortobeestimated,withacorrespondingparameterestimationerrorvector~=)]TJ /F6 11.955 Tf 13.7 2.66 Td[(^, (5)Y=i(i)]TJ /F3 11.955 Tf 11.96 0 Td[(imax)2666641ii2377775T (5)isthecurrentdependentregressionmatrix,and^P2(i)=P 2+i imax)]TJ ET q .478 w 229.95 -414.07 m 238.79 -414.07 l S Q BT /F3 11.955 Tf 229.95 -424.05 Td[(P2)]TJ /F3 11.955 Tf 11.95 0 Td[(P 2+Y(i)T^. (5)Thefollowinglemmaisusedtoguaranteetheexistenceofavalidcontrolcurrentintheinterval[0imax]. Lemma3. Iftheparameterupdatelawisdesignedsuchthattheestimate^iscon-tinuous,thenthepolynomial(i)givenin( 5 )hasatleastonerootintheoperatinginterval[0imax]. 125

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Proof. From( 5 ),itisseenthattheclippeddesireddampingforcesatisesthefollowingpassivityconstraint(v)]TJ /F3 11.955 Tf 11.95 0 Td[(v0)Sb(v)P 2(v)]TJ /F3 11.955 Tf 11.96 0 Td[(v0)Fd(u,v)(v)]TJ /F3 11.955 Tf 11.96 0 Td[(v0)Sb(v) P2. (5)Also,from( 5 ),itcanbeshownthattheterm(v)]TJ /F3 11.955 Tf 12.22 0 Td[(v0)Sb(v)ispositive.Thusdividingthroughby(v)]TJ /F3 11.955 Tf 11.96 0 Td[(v0)Sb(v)in( 5 )yieldsP 2Fd(fd,v) Sb(v) P2, (5)whichimpliesthatFd(fd,v) Sb(v)2[P 2 P2]. (5)Since^iscontinuousbythehypothesis,itimpliesthat^P2(i)iscontinuous.Also,since^P2(0)=P 2and^P2(imax)= P2,usingtheIntermediateValueTheorem,itfollowsthatthereexistsatleastoneic2[0imax]suchthat^P2(ic)=Fd(fd,v) Sb(v), (5)whichimpliesthat(ic)=Fd(fd,v))]TJ /F3 11.955 Tf 11.95 0 Td[(Sb(v)^P2(ic)=0. (5)Thusic2roots((i))and,sinceic2[0imax],theproofiscomplete. Next,Supposethat^iscontinuous,then,usingLemma 3 ,itfollowsthatthereexistsacontrolcurrentic2[0imax]suchthat(ic)=0.Consequently,theclosedlooperrorsystemisgivenby_e=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1e+SbP1Y(ic)T~. (5) 126

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5.3.3StabilityAnalysis Theorem5.2. Giventheupdatelaw_^=LeSb(v)Y(ic),(0)=0, (5)whereLisapositiveconstantadaptationgain,andthecontrollawic=argmin[0imax]roots(()), (5)theclosed-looperrordynamicsin( 5 )isstable,andthetrackingerrore(t)ap-proacheszeroasymptotically.Also,theparameterestimate^iscontinuous,thussatisfyingthehypothesisofLemma 3 Proof. ConsiderthepositivedeniteLyapunovcandidatefunctionVL=1 2e2+1 2~T~. (5)Takingthersttimederivativeof( 5 )alongtheclosedlooptrajectoryin( 5 )yields_VL=e_e)]TJ /F6 11.955 Tf 13.7 2.66 Td[(~_^ (5)=e)]TJ /F3 11.955 Tf 9.3 0 Td[(P1e+SbP1Y(ic)T~)]TJ /F6 11.955 Tf 13.7 2.66 Td[(~T_^. (5)Substitutingtheupdatelawin( 5 )yields_VL=)]TJ /F3 11.955 Tf 9.3 0 Td[(P1e2. (5)SinceP1(i)>0,itimpliesthat_VLisnegativesemi-denite,andsinceVLispositivedenite,itfollowsthatVL2L1.From( 5 ),itfollowsthate,~2L1,whichalsoimpliesthat^2L1-sinceisaconstant.Integrating( 5 )yieldsVL)]TJ /F3 11.955 Tf 11.96 0 Td[(VL(0))]TJ /F10 11.955 Tf 23.91 16.27 Td[(Zt0P1(i())e()2d, (5) 127

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fromwhichitfollowsthate2L2.Alsofrom( 5 ),itfollowsthat_e2L1whichimpliesthateisuniformlycontinuous.Thus,sincee2L2anduniformlycontinuous,itcanbeshownusingBarbalat'slemma[ 55 ]thate(t)!0asymptotically. Remark5.1. Thefollowingalgorithmsummarizesthecontrolandupdatelawsdevel-opedinthelasttwosections: Algorithm5.3.1:CONTROL/UPDATE(fd,v,^)comment:ClippedDesiredForceFd Fd(fd,v)comment:Computetrackingerrore F)]TJ /F3 11.955 Tf 11.96 0 Td[(Fdcomment:Computecontrolcurrentic=min[0imax]roots)]TJ /F3 11.955 Tf 9.3 0 Td[(Fd+Sb(v)^P2(i)comment:ParameterUpdate^ LRt0e()Sb(v)Y(ic)d+^0return(ic,^) 5.3.4SimulationResultsThesimulationissetupasdescribedintheprevioussection,thevehicletravelingatasteadyhorizontalspeedof40mphissubjectedtoaroadbumpofheight10cm.TheCarBodyAcceleration,SuspensionTravel,andTireDeectionresponsesaremeasured.Thesuspensiontravelisdenedastheverticaldisplacementofthecenterofmassofthesprungmasswithrespecttotheunsprungmass,andthetiredeectionastheverticaldisplacementoftheunsprungmasswithrespecttotheroadlevel.Simulationswerecarriedoutfortheconstantstiffnessandthevariablestiffness 128

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suspensionsystems.Fortheconstantstiffnesssuspension,thecontrolmasswaslockedataxedpositioncorrespondingtotheequilibriumpositionofthecontrolmassforthevariablestiffnesssystem.TheresultsobtainedarereportedinFigures 5-13 ,through 5-19 .Table 5-5 showsthevariancegainsforthedifferentresponses.Fig 5-13 showsthecarbodyacceleration,whichisusedheretodescribetheridecomfort.Thelowerthecarbodyacceleration,thebettertheridecomfort.Asseeninthegure,thevariablestiffnesssuspensionisamoreridefriendlysuspension,outperformingthetraditionalverticalskyhookcontrol.AsshowninFig 5-14 ,associatedwiththisimprovementisacorrespondingdegradationinthesuspensiontravel.Thisagreeswiththeobservationmadeinearliersections,aswellasthewellknowntradeoffbetweenridecomfortandsuspensiondeection.Fortunately,the12%degradationinsuspensiondeectionisnotasmuchasthe30%improvementgainedintheridecomfort,resultinginanoverallbetterperformance.Figure 5-16 showsthepositionhistoryofthecontrolmassforthevariablestiffnesssuspension,fromwhichtheboundednessofthemotionofthecontrolmassisseen.Themaximumdisplacementofthecontrolmassfromtheequilibriumpositionislessthan15cm.Thisimpliesthatthespacerequirementforthecontrolmassissmall,whichfurtherdemonstratesthepracticalityofthesystem.Fig 5-15 showsthatthereisnosignicantreductioninthetiredeection.Thus,thesuspensionsystemsareapproximatelyequallyroadfriendly.Fig. 5-17 showstheparameterestimates.Theuppersub-gureshowsthattheparametersarenotupdatedinthecontrolofthehorizontalMRdamper.Thisisbecausethecorrespondingcontrolcurrentisbang-bang,switchingfromic=0toic=imax.Asaresult,theelementsoftheregressionmatrixgivenin( 5 )arezeros,whichfurtherimplies,from( 5 ),that_^=0.Thustheparameterestimatewillremainconstant.Fig. 5-19 showsthehorizontalandverticaldamperforces. 129

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Table5-5. Variancegainvalues ConstantStiffnessVariableStiffness CBA(s)]TJ /F4 7.97 Tf 6.58 0 Td[(1)50.730635.5151ST99.9988112.1389TD1.06691.0450 Figure5-13. Carbodyacceleration(CBA)-semi-activecase Figure5-14. Suspensiontravel(ST)-semi-activecase 130

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Figure5-15. Tiredeection(TD)-semi-activeCase Figure5-16. Controlmassdisplacement-semi-activecase 131

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Figure5-17. Parameterestimates-semi-activecase 132

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Figure5-18. Controlcurrents 133

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Figure5-19. MR-damperforces 134

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CHAPTER6ROLLSTABILIZATIONENHANCEMENTUSINGVARIABLESTIFFNESSSUSPENSIONInthischapter,avariablestiffnessarchitectureisusedinthesuspensionsystemtocounteractthebodyrollmoment,therebyenhancingtherollstabilityofthevehicle.Theproposedsystemcanbeusedinconjunctionwithexistingmethodsthatdonotinterferewiththesuspensionsystem.First,akinematiccontrolusingthepositionofthecontrolmassesasthecontrolinputisdesigned.Then,afullyactuatedsystemfeaturinghydraulicactuatorsisconsidered.Thelateraldynamicsofthesystemisdevelopedusingabicyclemodel.Theaccompanyingrolldynamicsarealsodevelopedandvalidatedusingexperimentaldata.Thepositionsoftheleftandrightcontrolmassesareoptimallyallocatedtoreducetheeffectivebodyrollandrollrate.Simulationresultsshowthattheresultingvariablestiffnesssuspensionsystemhasmorethan50%improvementinrollresponseoverthetraditionalconstantstiffnesscounterparts.Thesimulationscenariosexaminedare;theshhookmaneuverandtheISO3888-2doublelanechangemaneuvers.Rolldynamicsiscriticaltothestabilityofroadvehicles.Alossofrollstabilityresultsinarolloveraccident.Typically,vehiclerolloversareverydangerous.ResearchbytheNationalHighwayTrafcSafetyAdministration(NHTSA)showsthatrolloveraccidentsarethesecondmostdangerousformofaccidentsintheUnitedStates,afterhead-oncollision[ 83 ].In2000,approximately9,882peoplewerekilledintheUnitedStatesinarolloveraccidentinvolvinglightvehicles[ 83 ].Rollovercrasheskillmorethan10,000occupantsofpassengervehicleseachyear.Aspartofitsmissiontoreducefatalitiesandinjuries,sincemodelyear2001,theNationalHighwayTrafcSafetyAdministration(NHTSA)hasincludedrolloverinformationaspartofitsNewCarAssessmentProgram(NCAP)ratings.Oneoftheprimarymeansofassessingrolloverriskisthestaticstabilityfactor(SSF),ameasurementofavehicle'sresistancetorollover[ 84 ].ThehighertheSSF,thelowertherolloverrisk.Rollstability,ontheotherhand,referstothecapability 135

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ofavehicletoresistoverturningmomentsgeneratedduringcornering,thatistoavoidrollover[ 85 ].Severalfactorscontributetorollstability,amongwhichareStaticStabilityFactor(SSF),kinematicandcompliancepropertiesofthesuspensionsystemetc.Anumberofrolloverpreventionandrollstabilityenhancementmethodsexistinliteraturethatarebasedononeormoreofdifferentialbraking,steer-by-wire,differentialdrivetorquedistribution,andactivesteering.In[ 86 ],anoptimalrolloverpreventionsystemusingacombinationofsteer-by-wireanddifferentialbrakingwaspresented.Adifferentialbrakingbasedanti-rollovercontrolalgorithmbasedontheTime-To-RollovermetricwasproposedforSportUtilityVehiclesin[ 87 ]andevaluatedusinghuman-in-the-loopsimulations.In[ 88 ],theauthorsdiscusssomeoftheproblemsrellatedtocommercialvehiclestabilityingeneral,andproposedasolutionfordetectingandavoidingrolloverusingexistingsensorsandactuatorsoftheelectronicbrakesystem(EBS).Theauthorin[ 89 ]proposedamethodofidentifyingreal-timepredictivelateralloadtransferratioforrolloverpreventionsystems.Moreover,engineershaveinventedmechanical/electromechanicalsystemstoimprovetherollstabilityofroadvehicles.Oneoftheearliestbasicinventionistheanti-rollbar(orswaybarorstabilizerbar).Aswaybarisusuallyintheformofatorsionalbarconnectingopposite(left/right)wheelstogether.Itgenerallyhelpsinresistingvehiclebodyrollmotionsduringfastcorneringorroadirregularitiesbyincreasingthesuspension'srollstiffness,independentoftheverticalspringconstants.TherstswaybarpatentwasawardedtoS.L.CColemanonApril22,1919[ 90 ].Afterthen,somemoreinventionshavebeengearedtowardsvehiclerollstabilization.Theseanti-rollsystemsareeitherpassive[ 91 92 ],semi-active[ 93 ],oractive[ 94 98 ]bydesign. 6.1MechanismDescriptionTheschematicdiagramofthehalfcarmodelofthevariablestiffnesssuspensionsystemisshowninFig 6-1 .Themodeliscomposedofahalfcarbody(sprungmass),twoidenticalwheelassemblies(unsprungmasses),twoverticalspring-dampersystems, 136

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Figure6-1. Halfcarmodel leftandrightlowerandupperwishbones,andcontrolmasses.Themainideaofthedesignistovarytheeffectiveverticalreactiveforcesoftheleftandrightsuspensionstocounteractthebodyrollmoments.Thisisachievedbyanappropriatelydesignedcontrolforthevariationofthepointofattachmentofthetopendofthesuspensionstrutstothecarbody.Duringcornering,avehicleexperiencesaradiallyoutwardslateralaccelerationactingatthecenterofmass,aswellascorrespondinglateraltireforcesactingatthetire/roadcontacts.Thisresultsinarollmomentwhichcausesthevehicletoleanoutwards.Tocounteractthisrollmoment,theoutsidesuspensionshouldbecomestifferwhiletheinsidesuspensionshouldbecomesofter.Thisgeneratesacountermomenttoimprovethestabilityoftherolldynamics. 6.2ModelingFig. 6-2 showsaschematicofthemodelingaspectsofthesystem.Eachblockintheschematicisfurtherexpatiatedinthesubsequentsubsections. 6.2.1YawDynamicsTheyawdynamicsofavehiclemaybeeffectivelydecoupledfromtherolldynamicsbymodelingitasarigidbicycleinaplanarmotionasshowninFig. 6-3 .Themodelhasthreedegreesoffreedom.Asaresult,theyawdynamicsaregivenbyasetof 137

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Figure6-2. Modelingschematics Figure6-3. Bicyclemodel threecoupledrstorderordinarydifferentialequations[ 99 101 ].However,sincethemanueversconsideredinthispaperareconstantspeedmaneuvers,thecorrespondingforwardvelocitydynamicisremoveandtheremainingyawdynamicsaregivenasfollows:_x=vxcos )]TJ /F3 11.955 Tf 11.95 0 Td[(vysin (6)_y=vxsin +vycos (6)_ =r (6)_vy=1 m(Fxfsin+Fyfcos+Fyr))]TJ /F3 11.955 Tf 11.95 0 Td[(vxr (6)_r=1 Iz(lf(Fxfsin+Fyfcos))]TJ /F3 11.955 Tf 11.96 0 Td[(lrFyr), (6) 138

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Thetermintheseequationsaredenedinthenomenclaturesectionofthepaper.Tocapturetheeffectofthenonlineartireforcesatlargeslipangles,thewellknownPacejkaMagicFormula[ 102 ]isusedtomodelthetirelateralforces.ThelateralforcesareexpressedasFyj=)]TJ /F8 11.955 Tf 9.3 0 Td[(yjFzj,(j=f,r), (6)whereisthemaximumfrictioncoefcientoftheroadsurface,Fzjisthenormalloadateachtire,andyjisthetire-roadinteractioncoefcientgivenbytheMagicFormulayj=MF(syj)=sin)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Ctan)]TJ /F4 7.97 Tf 6.59 0 Td[(1(Bsyj), (6)wheresyjarethelateralslipratios,givenrespectivelyforthefrontandreartiresassyf=vycos)]TJ /F3 11.955 Tf 11.96 0 Td[(vxsin+rlfcos vxcos+vysin+rlfsin (6)syr=vy)]TJ /F3 11.955 Tf 11.95 0 Td[(lrr vx. (6)Here,vxistheconstantvehicleforwardspeed.Inordertokeepthetotaltireforcesfromexceedingthemaximumfrictionalforce,thefrictionconeconstraintisenforcedasfollowsF2xj+F2yj=2F2zj, (6)whichimpliesthatFxj=xjFzj (6)xj=q 1)]TJ /F8 11.955 Tf 11.96 0 Td[(2yj. (6)Theeffectoflongitudinalloadtransferiscapturedbysummingforcesintheverticaldirection,andtakingmomentsaboutthebodylateralaxis,whileneglectingpitch 139

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dynamics,asfollowsFzf+Fzr=mg (6)lfFzf)]TJ /F3 11.955 Tf 11.95 0 Td[(lrFzr=h(Fxfcos)]TJ /F3 11.955 Tf 11.95 0 Td[(Fyfsin+Fxr), (6)wherehistheheightofthebodycenterofmassfromtheground.Aftersomealgebraicmanipulations,andusing( 6 )and( 6 ),Equations( 6 )and( 6 )yieldtheexpressionsfortherespectivenormallaodsatthefrontandreartiresasFzf=mg(lr+hxr) lf+lr)]TJ /F3 11.955 Tf 11.96 0 Td[(h(xfcos+yfsin)]TJ /F8 11.955 Tf 11.96 0 Td[(xr) (6)Fzr=mg(lf)]TJ /F3 11.955 Tf 11.96 0 Td[(h(xfcos+yfsin)]TJ /F8 11.955 Tf 11.96 0 Td[(xr)) lf+lr)]TJ /F3 11.955 Tf 11.95 0 Td[(h(xfcos+yfsin)]TJ /F8 11.955 Tf 11.95 0 Td[(xr). (6) 6.2.2RollDynamicsThefreebodydiagramofanidealizedhalfcarmodelofthesystemisshowninFig. 6-4 ,wherethesuspensionforceshavebeenreplacedwiththeirhorizontalcomponents,ML,MR,andverticalcomponentsNL,NR.Theassumptionsadoptedfor Figure6-4. Idealizedhalfcarmodelforrolldynamicsmodeling thesubsequentdynamicmodelaresummarizedasfollows: 140

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1. Thehalfcarbodyissymmetricaboutthemid-plane,andasaresultthecenterofmassislocatedonthemid-planeataheighthabovethebaseofthechassis. 2. Theroadislevelandthepointsofcontactofthetiresareonthesamehorizontalplane. 3. Thespringsanddamperforcesareinthelinearregionsoftheiroperatingranges. 4. Thecomplianceeffectsinthejointsarenegligible.Theinstantaneouslengths,lLandlRoftheleftandrightsuspensionsrespectively,aregivenasl2L=(Tcos)]TJ /F3 11.955 Tf 11.96 0 Td[(dLcos)]TJ /F3 11.955 Tf 11.96 0 Td[(Hsin+TL)2+(z)]TJ /F3 11.955 Tf 11.95 0 Td[(dLsin+h2cos)2, (6)l2R=()]TJ /F3 11.955 Tf 9.3 0 Td[(Tcos+dRcos)]TJ /F3 11.955 Tf 11.95 0 Td[(Hsin)]TJ /F3 11.955 Tf 11.96 0 Td[(TR)2+(z+dRsin+h2cos)2, (6)andthecorrespondingsuspensionforcesaregivenbyFsL=ks(l0s)]TJ /F3 11.955 Tf 11.95 0 Td[(lL))]TJ /F3 11.955 Tf 11.96 0 Td[(bs_lL (6)FsR=ks(l0s)]TJ /F3 11.955 Tf 11.95 0 Td[(lR))]TJ /F3 11.955 Tf 11.95 0 Td[(bs_lR. (6)Thusthehorizontalandverticalcomponentsoftheleft,andrightsuspensionforcesaregivenbyML=FsL lL(Tcos)]TJ /F3 11.955 Tf 11.96 0 Td[(dLcos)]TJ /F3 11.955 Tf 11.95 0 Td[(Hsin+TL), (6)MR=FsR lR()]TJ /F3 11.955 Tf 9.29 0 Td[(Tcos+dRcos)]TJ /F3 11.955 Tf 11.96 0 Td[(Hsin)]TJ /F3 11.955 Tf 11.96 0 Td[(TR), (6)NL=FsL lL(z)]TJ /F3 11.955 Tf 11.96 0 Td[(dLsin+h2cos), (6)NR=FsR lR(z+dRsin+h2cos). (6) 141

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Followingtheassumptionsabove,andneglectingthelateraldynamics,theequationsofmotionofthesystemaregivenbythefollowingsetofdifferentialalgebraicequations:NL+NR)]TJ /F3 11.955 Tf 11.96 0 Td[(msg)]TJ /F3 11.955 Tf 11.95 0 Td[(msz=0, (6)Mc)]TJ /F3 11.955 Tf 11.95 0 Td[(Is=0, (6)T2L+(z)]TJ /F3 11.955 Tf 11.96 0 Td[(Tsin)]TJ /F3 11.955 Tf 11.96 0 Td[(hcos)2)]TJ /F3 11.955 Tf 11.96 0 Td[(l2w=0, (6)T2R+(z+Tsin)]TJ /F3 11.955 Tf 11.96 0 Td[(hcos)2)]TJ /F3 11.955 Tf 11.96 0 Td[(l2w=0, (6)whereMc=gL(NL,ML,)dL+gR(NR,MR,)dR)]TJ /F6 11.955 Tf 11.96 -.16 Td[(((NL+NR)sin+(ML+MR)cos)h2+Fyjz, (6)andgL(NL,ML,)=)]TJ /F3 11.955 Tf 9.3 0 Td[(NLcos+MLsin (6)gR(NR,MR,)=NRcos)]TJ /F3 11.955 Tf 11.95 0 Td[(MRsin. (6)Here,thetotalgroundforceFL+FRisequivalenttothelateraltireforcesFyjfromtheyawdynamics. 6.3KinematicControlThepurposeofthissectionistodesignthedesiredtrajectoryforthecontrolmassestogeneratetheappropriatecounterrollmoment,giventhephysicalconstraintsofthesuspensionkinematics.And,usingthesimulationresults,tounderstandhowthemotionofthecontrolmassesaffectrollstability.Inthesubsequentsection,hydraulicactuatorswillbeusedtodrivethecontrolmassesalongthedesiredtrajectorydesignedinthissection,whileimposingthephysicalsaturationlimitsontheactuator.Sinceonlykinematiccontrolisconsideredinthissection,thedynamicsofthecontrolmassesareneglected.TheirpositionsdL,anddRareusedascontrolinputsto 142

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adjusttheeffectiveanti-rollmomentgeneratedbythesuspensions,therebycontrollingtherolldynamicsofthehalfcar.Acontrol-orientedreduced-orderrolldynamicsofthehalfcaristhengivenby:Is=Mc. (6)Addingthestiffnessanddampingtermk1+k2_tobothsidesofEquation( 6 )yieldsIs+k2_+k1)]TJ /F10 11.955 Tf 13 3.15 Td[(eFyjz+u+((NL+NR)sin+(ML+MR)cos)eh2=ed, (6)whereed=gLdL+gRdR+k1+k2_+u)]TJ /F6 11.955 Tf 11.95 -.17 Td[(((NL+NR)sin+(ML+MR)cos)^h2+^Fyjz, (6)^Fyjand^h2areestimatesofthelateraltireforceFyjandtheheighth2respectivelywiththecorrespondingestimationerrorsgivenbyeFyj=Fyj)]TJ /F6 11.955 Tf 13.17 2.65 Td[(^Fyj (6)eh2=h2)]TJ /F6 11.955 Tf 12.06 2.66 Td[(^h2, (6)anduisanauxiliarycontrolwhichisdesignedinthesubsequentsection.Here,itisassumedthatthelateraltireforceestimationerrorcanbeupperboundedbyaknownpositiveconstantasfollows0jeFyjjC. (6)ThecomponentsNL,NR,ML,MRofthespringforcesarealsoassumedtobemeasurableusingforcesensors.Thecontrolgainsk1andk2aredesignedtominimizeJ=Z10!21(t)2+!22_(t)2+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(k1(t)+k2_(t)2dt (6) 143

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subjecttoIs+k2_+k1=0(0)=0_(0)=0, (6)where!1,and!2areperformanceweightsusedtopenalizetheperformanceindexwithrespecttorollandrollraterespectively.Theperformanceindexin( 6 )ischosentoensurefastsmoothandboundedrolldynamicsofthevehiclebody,withtheperformanceweightsspecifyingatrade-offbetweenachievedboundedness(controlledbyk1)andsmoothness(controlledbyk2)oftheride.ThesolutiontotheLQRproblemaboveisobtainedask1=!1 (6)k2=q 2Isk1+!22. (6) 6.3.1ControlAllocationAcontrolallocationapproachisgenerallyusedwhendifferentpossiblecontrolchoicescanproducethesameresult.Thisusuallyhappenswhenthenumberofeffectorsexceedsthestatedimension,asisthecaseinthispaper.Thegeneralcontrolallocationproblem,aswellasexistingsolutionmethods,arewellexpoundeduponin[ 103 104 ].Tothiseffect,letdL=d0+L (6)dR=d0+R, (6) 144

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whered0istheequilibriumpositionofthecontrolmasses,withLandRbeingtheirdesireddisplacementsrespectively.Thened=gLL+gRR)]TJ /F3 11.955 Tf 11.96 0 Td[(f, (6)wheref=)]TJ /F6 11.955 Tf 11.96 -.17 Td[((gL+gR)d0)]TJ /F3 11.955 Tf 11.96 0 Td[(k1)]TJ /F3 11.955 Tf 11.96 0 Td[(k2_)]TJ /F3 11.955 Tf 11.96 0 Td[(u (6)+((NL+NR)sin+(ML+MR)cos)^h2)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fyjz. (6)ThecontrollawisthereforedenedasL,R=argminjedj:d L,Rd, (6)whered anddarephysicallimitsonthepositionofthecontrolmasses.Duetothespecialformof( 6 ),thesolutionto( 6 )isobtainedsequentiallyasfollowsL=clipf gL,d ,d (6)R=clipf)]TJ /F3 11.955 Tf 11.95 0 Td[(gLL gR,d ,d, (6)wherethesaturationfunction,clip(...),isdenedasclip(x,a,b),8>>>><>>>>:a,ifxb (6)=minfmaxfa,xg,bg. (6) 6.3.2StabilityAnalysisLetbetheresidualerroroftheoptimizationin( 6 ),andletasignalr(t)bedenedasr(t)=_(t)+(t), (6) 145

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whereisapositivegainconstant.TheclosedlooprolldynamicsisthengivenbyIs_r=)]TJ /F6 11.955 Tf 11.95 -.16 Td[((k2)]TJ /F8 11.955 Tf 11.96 0 Td[(Is)r)]TJ /F6 11.955 Tf 11.95 -.16 Td[((k1)]TJ /F8 11.955 Tf 11.95 0 Td[((k2)]TJ /F8 11.955 Tf 11.95 0 Td[(Is))+eFyjz+u)]TJ /F3 11.955 Tf 11.96 0 Td[(Yeh2, (6)wheretheregressionsignalYisgivenbyY=(NL+NR)sin+(ML+MR)cos. (6)Theparenthesizedargumentshavebeendroppedunlessotherwiserequiredforclarity. Theorem6.1. Giventheauxiliarycontrolandtheadaptiveupdatelawu=)]TJ /F3 11.955 Tf 9.3 0 Td[(Csgn(r)jzj (6)_^h2=)]TJ /F8 11.955 Tf 9.3 0 Td[(^h2+Yr,^h2(0)=h0; (6)where>0isanadaptationgainconstant.Ifthecontrolgainsarechosentosatisfythefollowingsufcientconditionsk2)]TJ /F8 11.955 Tf 11.95 0 Td[(Is=1+2 (6)k1)]TJ /F8 11.955 Tf 11.95 0 Td[((k2)]TJ /F8 11.955 Tf 11.95 0 Td[(Is)=3, (6)1,2,3>0,thentheclosedlooprolldynamicsin( 6 )isuniformlyultimatelybounded1withrespecttotheclosedballB(r)=8<::kkr 2 1r+1 p 1s 2 42+h22 29=;, (6) 1Asignalx(t)isuniformlyultimatelybounded(UUB)withrespecttoaclosedballB(r)ifforallr>0,thereexistsT(r)suchthatkx(t0)krimpliesthatx(t)2B(r),8t>t0+T 146

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where=min1 Is,, 2,1=minIs,3,1 ,2=maxIs,3,1 Proof. ConsiderthefollowingpositivedenitecandidateLyapunovfunctionV=1 2Isr2+1 232+eh22 2. (6)Takingthersttime-derivativeandusingthesufcientconditionsin( 6 )and( 6 )yields_V=r)]TJ /F6 11.955 Tf 11.95 0 Td[((1+2)r)]TJ /F8 11.955 Tf 11.95 0 Td[(3+eFyjz+u)]TJ /F3 11.955 Tf 11.96 0 Td[(Yeh2+3(r)]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F10 11.955 Tf 13.07 11.24 Td[(eh2_^h2 (6)whichaftersubstitutingtheauxiliarycontrolandtheadaptiveupdatelawyields_V)]TJ /F8 11.955 Tf 21.91 0 Td[(1r2+r()]TJ /F8 11.955 Tf 11.95 0 Td[(2r))]TJ /F8 11.955 Tf 11.95 0 Td[(32+eh2^h2 )]TJ /F8 11.955 Tf 21.91 -.01 Td[(1r2)]TJ /F8 11.955 Tf 11.95 .01 Td[(2r)]TJ /F8 11.955 Tf 19.41 8.09 Td[( 222+2 42)]TJ /F8 11.955 Tf 11.96 0 Td[(32)]TJ /F8 11.955 Tf 13.15 8.09 Td[(eh22 2+h22 21r2)]TJ /F8 11.955 Tf 11.96 0 Td[(32)]TJ /F10 11.955 Tf 13.87 11.24 Td[(eh22 2+2 42+h22 2)]TJ /F6 11.955 Tf 21.91 0 Td[(21 2Isr2+1 232+1 2eh22+2 42+h22 2=)]TJ /F6 11.955 Tf 9.3 0 Td[(2V+2 42+h22 2. (6)UsingtheComparisonLemma(Lemma3.4,[ 55 ]),itfollowsthatV(t)V(0)e)]TJ /F4 7.97 Tf 6.58 0 Td[(2t+1 22 42+h22 2)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(2t, (6)V(0)e)]TJ /F4 7.97 Tf 6.58 0 Td[(2t+1 22 42+h22 2, (6) 147

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whichimpliesthat1 2k(t)k22 2k(0)k2e)]TJ /F4 7.97 Tf 6.59 0 Td[(2t+1 22 42+h22 2, (6)where=266664reh2377775. (6)Takingthesquarerootsandusingtheinequalityp a2+b2a+bfornonnegativenumbersaandbyieldsk(t)kr 2 1k(0)ke)]TJ /F19 7.97 Tf 6.59 0 Td[(t+1 p 12 42+h22 2 (6)r 2 1k(0)k+1 p 12 42+h22 2. (6)Therefore,k(0)kr)(t)2B(r)8t>0 Remark6.1. Itcanbeeasilyveriedthatthesufcientconditionsin( 6 )and( 6 )aresatisedbythecontrolgainsin( 6 )and( 6 )iftheperformanceweightsareselectedas!1=3+(1+2) (6)!2=q (1+2)2+2I2s)]TJ /F6 11.955 Tf 11.96 0 Td[(2Is3, (6)given,1,2,3>0. 6.3.3SimulationTheperformanceoftheproposedcontrolisexaminedviasimulation,usingtheNTSHAshhookanddoublelanechangemaneuvers.First,theparametersoftherolldynamicsareestimatedsothattheresultantrolldynamicsmatchesexperimentaldata.Thevehicleusedforthedatacollectionisa2007ToyotaHighlanderHybridequippedwithanInertialMeasurementUnit,showninFig. 6-5 duringoneofthemaneuvers. 148

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Figure6-5. Snapshotduringdatacollectionprocess Twosetsofdatawerecollected.TherstistermedtheSnakeData,inwhichthecarisdrivenaroundequidistantconesarrangedonastraightlineinasnake-likefashion.ThesecondistermedtheEightData.Here,thevehicleisdrivenseveraltimesalonganeight-shapedpath.Thedatacollectedforeachexperimentincludesthelongitudinalandlateralvelocities,lateralacceleration,rollangleandrollrate.Theparametersofthemodelareestimatedusingthetrust-region-reectivemethodinMATLAB.Figs. 6-6A and 6-6C showvalidationsoftheestimatedparametersagainstanewSnakeDatasetwhichwasnotusedfortheestimationprocess.Figs. 6-6B and 6-6D showsimilarplotsfortheEightDataset.Thevaluesofthecontrolgainsusedforthesubsequentsimulationsaregivenk1=5000,k2=1565.2,=2,=10,=10,C=5. 6.3.3.1FishhookManeuverTheFishhookmaneuver,byNHTSA,isaveryusefultestmaneuverinthecontextofrollover,inthatitattemptstomaximizetherollangleundertransientconditions.Theprocedureisoutlinedasfollows,withanentrancespeedof50mph(22.352m=s): 149

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ASnake-rollangle BEight-rollangle CSnake-rollrate DEight-rollrateFigure6-6. Parameterestimationvalidation-snakedata 1. Thesteeringangleisincreasedatarateof720deg/supto6.5stat,wherestatisthesteeringanglewhichisnecessarytoachieve0.3gstationarylateralaccelerationat50mph 2. Thisvalueisheldfor250ms 3. Thesteeringwheelisturnedintheoppositedirectionatarateof720deg/supto-6.5statThesteeringanglefortheshhookmaneuverisshowninFig. 6-7 Figure6-7. Fishhook-steeringcommand 150

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AFrontaxle BRearaxleFigure6-8. Fishhook-rollresponse AFrontaxle BRearaxleFigure6-9. Fishhook-controlmassdisplacement Figs. 6-15A and 6-8B showtherolldynamicsforthefrontandrearaxlesrespectively,wheretheconstantandvariablestiffnesscasesareplottedtogetherforcomparison.Theseresultsshowthatbyusingthevariablestiffnessmechanismtogetherwiththekinematiccontroldevelopedintheprevioussections,therollangleandrollratesarereducedbymorethan50%.TheassociateddisplacementsoftheleftandrightcontrolmassesareshowninFigs. 6-16A and 6-9B forthefrontandrearaxlesrespectively.Itisseenalsothatthecontrolallocationexhibitsomegangingphenomenon. 151

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6.3.3.2DoubleLaneChangeManeuverTheISO3888Part2DoubleLaneChangecoursewasdevelopedtoobserverthewayvehiclesrespondtohandwheelinputsdriversmightuseinanemergencysituation.Thecourserequiresthedrivertomakeasuddenobstacleavoidancesteertotheleft(orrightlane),brieyestablishpositioninthenewlane,andthenrapidlyreturntotheoriginallane[ 105 ].ThesteeringcommandtothewheelsisshowninFig. 6-10 .ThecorrespondingrollresponsesandcontrolauthoritiesareshowninFigs. 6-11A through 6-12B ,fromwhichitisalsoseenthatthevariablestiffnesssystemsshowsmuchbetterbehaviorduringthesevereobstacleavoidancemaneuver. Figure6-10. Doublelanechange-steeringcommand AFrontaxle BRearaxleFigure6-11. Doublelanechange-rollresponse 152

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AFrontaxle BRearaxleFigure6-12. Doublelanechange-controlmassdisplacement 6.4DynamicControlInthissection,thefulldynamicsofthecontrolmassesaretakenintoconsideration,aswellastheactuatormodel.Theschematicdiagramofthehalfcarmodelofthe Figure6-13. Halfcarmodel variablestiffnesssuspensionsystemisshowninFig 6-13 .Themodeliscomposedof 153

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ahalfcarbody(sprungmass),twoidenticalwheelassemblies(unsprungmasses),twoverticalspring-dampersystems,leftandrightlowerandupperwishbones,hydraulicactuators.Themainideaofthedesignistovarytheeffectiveverticalreactiveforcesoftheleftandrightsuspensionstocounteractthebodyrollmoments.Thisisachievedbyanappropriatelydesignedcontrolforthehydraulicactuators. 6.4.1ControlMassesandActuatorDynamicsSimilarlytopreviouschapter,thedynamicsofofthehydraulicactuatorisgivenby[ 73 74 ]_PL=)]TJ /F8 11.955 Tf 9.3 0 Td[(vp)]TJ /F8 11.955 Tf 11.95 0 Td[(PL+xvp Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(xv)PL, (6)_xv=)]TJ /F6 11.955 Tf 10.56 8.08 Td[(1 xv+K u, (6)Fa=APL, (6)whereAisthepressureareaintheactuator,PListheloadpressure,vpistheactuatorpistonvelocity,Faistheoutputforceoftheactuator,,,andareparametersdependingontheactuatorpressurearea,effectivesystemoilvolume,effectiveoilbulkmodulus,oildensity,hydraulicloadow,totalleakagecoefcientofthecylinder,dischargecoefcientofthecylinder,andservovalveareagradient,xvisthespoolvalveposition,anduistheinputcurrenttotheservovalve.Aftersummingforcesalongthelineofactionoftheactuatorsonthecontrolmasses,theequationsofmotionoftheleftandrightcontrolmasses,togetherwiththeactuatormodel,aregivenbymddi=Fai)]TJ /F3 11.955 Tf 11.95 0 Td[(Micos)]TJ /F3 11.955 Tf 11.96 0 Td[(Nisin (6)_Fai=)]TJ /F8 11.955 Tf 9.3 0 Td[(Fai)]TJ /F8 11.955 Tf 11.95 0 Td[(A_di+Axvir Ps)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(xvi)Fai A (6)_xvi=)]TJ /F3 11.955 Tf 9.3 0 Td[(xvi+Kui. (6)Thesubscripti=fL,Fgisusedtoindicateleftandrightquantitiesrespectively. 154

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6.4.2ControlDesignThissectiondetailsthedesignofcontroldesignforthehydraulicactuatorsgearedtowardsimprovementofthebodyrolldynamics.First,thecontrollawsaredesigned,andtheresultingclosedlooperrorsystemgiven.Thedesiredactuatorforcesrequiredtoachievedadesiredrollbehavioraredesignedusingamodelreferenceadaptivecontrolandslidingmodetechniques[ 55 73 106 108 ],thenthenecessaryservocurrentcommandtothespoolvalveisdesignedfromtheactuatordynamicsusinganadaptivesingularperturbationapproach[ 76 ].Next,aLyapunov-basedstabilityanalysisiscarriedoutfortheoverallclosedlooperrordynamicstoguaranteetheconvergenceofthetrackingerrorandboundednessofthesystemstates.Thecontroldevelopmentisdonehierarchically.Firstforthevehiclebodyroll,thenforthecontrolmasses,andnallyforthehydraulicactuators. 6.4.2.1VehicleBodyRollAgain,thedesiredreferencerollmodelisgivenbyIsm+k2_m+k1m=0, (6) (6)wherek1=!1, (6)k2=s !22 Is+2!1 (6)weredesignedtominimizeJ=Z10!21m(t)2+!22_m(t)2+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(k1m(t)+k2_m(t)2dt (6)subjectto( 6 ),where!1,and!2areperformanceweightsusedtopenalizetheperformanceindexwithrespecttorollandrollraterespectively.Theperformanceindex 155

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in( 6 )ischosentoensuresmoothandboundedrolldynamicsofthevehiclebody,withtheperformanceweightsspecifyingatrade-offbetweenachievedboundedness(controlledbyk1)andsmoothness(controlledbyk2)oftheride.Lete(t)=(t))]TJ /F8 11.955 Tf 11.96 0 Td[(m(t) (6)bethetrackingerrordeninghowwelltherolldynamicsin( 6 )tracksthereferencemodelin( 6 ).Theobjectiveisthentodrivethetrackingerrortoassmallaspossibleusingtheactuatorforces.Takingtherstandsecondderivativesof( 6 )andsubtracting( 6 )from( 6 )yieldsIse+k2_e+k1e=Mc)]TJ /F6 11.955 Tf 11.96 -.17 Td[((k2)]TJ /F3 11.955 Tf 11.96 0 Td[(bsb)_)]TJ /F6 11.955 Tf 11.96 -.17 Td[((k1)]TJ /F3 11.955 Tf 11.96 0 Td[(ksb), (6)whereksb,bsbarethestiffnessanddampingduetotheswaybarandothercomplianceanddampingelementsthathaveindirectordirectinuenceontherolldynamics.Thispart,whichwasneglectedforthekinematiccontrol,isincludedhereforcompletion.Tofacilitatesubsequentcontroldesignandanalyses,thenonlinearlateralforcegivenbythePacejkaformulaisapproximatedasFyj=nXi=1QiLi(sj) (6)=L(sj)TQ, (6)wheretheregressionmatrixR(sj)andtheconstantcoefcientvectorQaregivenbyL(sj)=L1(sj)L2(sj)...Ln(sj)T, (6)Q=Q1Q2...QnT, (6) 156

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withLi(sj)=sin)]TJ /F6 11.955 Tf 5.48 -9.69 Td[((2i)]TJ /F6 11.955 Tf 11.96 0 Td[(1)tan)]TJ /F4 7.97 Tf 6.58 0 Td[(1(sj),i=1,2,...,n (6)beingthesetofbasesfunctions.Otherbasesfunctionscanbeused(e.gpolynomial[ 109 ],rationalfunction[ 110 ]).Thefunctionsin( 6 )areusedasbasisforthelateraltireforceapproximationbecausetheypreservetheformgivenintheMagicformula.Fig. 6-14 showstheresultingapproximationforn=10,wheretheidealweightvectorQwasobtainedusingaleastsquareapproach. Figure6-14. Lateraltireforceapproximation Thus,therollerrordynamicsin( 6 )becomesIse+k2_e+k1e)]TJ /F7 11.955 Tf 11.96 0 Td[(LTeQz+Yeh2=f, (6)wheref=gLdL+gRdR)]TJ /F3 11.955 Tf 11.96 0 Td[(Y^h2+LT^Qz)]TJ /F6 11.955 Tf 11.95 -.17 Td[((k2)]TJ /F3 11.955 Tf 11.96 0 Td[(bsb)_)]TJ /F6 11.955 Tf 11.95 -.17 Td[((k1)]TJ /F3 11.955 Tf 11.96 0 Td[(ksb), (6) 157

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and^h2,^Qaretheadaptiveestimatesoftheunknownsystemconstantparametersh2,Q,withthecorrespondingestimationerrorsgivenbyeh2=h2)]TJ /F6 11.955 Tf 12.06 2.66 Td[(^h2, (6)eQ=Q)]TJ /F6 11.955 Tf 13.47 3.21 Td[(^Q. (6)Theparenthesizedargumentshavebeendroppedunlessotherwiserequiredforclarity.LetdL=d0+L, (6)dR=d0+R, (6)whereL,R=argminfjfj:)]TJ /F6 11.955 Tf 9.3 0 Td[(L,Rg, (6)bethedesireddisplacementofthecontrolmasses.denesthephysicallimitsontheallowablepositionsofthecontrolmasses.Theoptimizationin( 6 )denesacontrolallocationproblem.Controlallocationapproachisgenerallyusedwhendifferentpossiblecontrolchoicescanproducethesameresult.Thisusuallyhappenswhenthenumberofeffectorsexceedsthestatedimension,asthecaseinthispaper.Thegeneralcontrolallocationproblem,aswellasexistingsolutionmethods,arewellexpoundeduponin[ 103 104 ].However,duetothespecialformof( 6 ),thesolutionto( 6 )isobtainedsequentiallyasfollowsL=clipf0 gL,)]TJ /F6 11.955 Tf 9.3 0 Td[(,, (6)R=clipf0)]TJ /F3 11.955 Tf 11.96 0 Td[(gLL gR,)]TJ /F6 11.955 Tf 9.3 0 Td[(,, (6) 158

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wherethesaturationfunction,clip(...),isdenedasclip(x,a,b),8>>>><>>>>:a,ifxb (6)=minfmaxfa,xg,bg, (6)andf0=(gL+gR)d0)]TJ /F3 11.955 Tf 11.96 0 Td[(Y^h2+LT^Qz)]TJ /F6 11.955 Tf 11.95 -.17 Td[((k2)]TJ /F3 11.955 Tf 11.96 0 Td[(bsb)_)]TJ /F6 11.955 Tf 11.96 -.17 Td[((k1)]TJ /F3 11.955 Tf 11.96 0 Td[(ksb). (6)Consequently,let"dbetheresidualvalueoff0aftertheoptimizationabove.Also,letr1(t)=_e(t)+1e(t) (6)denesaslidingsurfacefortherollerrordynamics.Then,thecorrespondingclosedlooprollerrordynamicsisgivenbyIs_r1="d)]TJ /F6 11.955 Tf 11.96 -.17 Td[((k2)]TJ /F8 11.955 Tf 11.96 0 Td[(1Is)r1)]TJ /F6 11.955 Tf 11.95 -.17 Td[((k1)]TJ /F8 11.955 Tf 11.96 0 Td[(1(k2)]TJ /F8 11.955 Tf 11.95 0 Td[(1))e+LTeQz)]TJ /F3 11.955 Tf 11.96 0 Td[(Yeh2. (6) 6.4.2.2ControlMassesInordertoensuresmoothnessoftheensuingmotionofthecontrolmasses,thedesiredtrajectoryofthecontrolmassesisgivenbythefollowingrst-orderlowpasslterdynamics_ddi=)]TJ /F3 11.955 Tf 9.29 0 Td[(ddi+di,i=fL,Rg. (6)Letri=_edi+2edi, (6) 159

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denesaslidingsurfaceforthepositiontrackingerroredi=di)]TJ /F3 11.955 Tf 11.95 0 Td[(ddi (6)ofthecontrolmasses,where2isapositivecontrolgain.Differentiating( 6 )andsubstitutingthecontrolmassdynamicsin( 6 )yieldtheclosedlooptrackingerrordynamicsmd_ri=)]TJ /F6 11.955 Tf 11.29 -.17 Td[((k3)]TJ /F8 11.955 Tf 11.96 0 Td[(2md)ri)]TJ /F8 11.955 Tf 11.95 0 Td[(2mdedi)]TJ /F3 11.955 Tf 11.96 0 Td[(Ni+eFi, (6)wherethedesiredactuatorforceisgivenbyFdi=k3ri+Micos+Nisin, (6)andtheactuatorforcetrackingerrorisgivenbyeFi=Fai)]TJ /F3 11.955 Tf 11.96 0 Td[(Fdi. (6)k3>0isacontrolgain,andthedesiredpositiondynamicsNi=mdddiisassumedtobeupperboundedasfollowsjNijci (6) 6.4.2.3HydraulicActuatorsInordertosimplifythecontrollerdesignfortheactuators,thespoolvalvedynamicsiscanceledbyusingasingularperturbationtechnique[ 75 ].Thecontrolinputisdesignedasui=)]TJ /F3 11.955 Tf 9.3 0 Td[(Kfxvi+1+KKf Kusi,i=fL,Fg (6) 160

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whereusiisaslowcontrolintimeandKfisapositivedesigncontrolgain.Consequently,thevalvepsuedo-closedloopdynamicsisgivenby"_xvi+xvi=usi, (6)where"= 1+KKf (6)istheperturbationconstant.Thepseudo-closedloopin( 6 )hasaquasi-steadystatesolution,xvi("=0),xvi,givenbyxvi=usi. (6)Usingthefasttimescale=t "andTichonov'sTheorem[ 75 ]yieldsxvi=xvi++O("), (6)d d=)]TJ /F8 11.955 Tf 9.3 0 Td[(, (6)where()isaboundarylayercorrectionterm.Itisseenthat()decaysexponentiallyinthefasttimescale.Typically,thetimeconstantintheactualsystemisdesignedtosatisfy0<"1[ 76 ].Therefore,bychoosingthecontrolgainKflargeenough,theperturbationconstantcanbemadeassmallaspossible.Asaresult,+O(")becomesnegligiblysmall.Thus,theactuatordynamicsin( 6 )becomes_Fai=f(Fai,_di)+g(Fai,xvi)usi, (6) 161

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wheref(Fai,_di)=)]TJ /F8 11.955 Tf 9.3 0 Td[(Fai)]TJ /F8 11.955 Tf 11.95 0 Td[(A_di (6),fi,g(Fai,xvi)=Ar Ps)]TJ /F6 11.955 Tf 11.95 0 Td[(sgn(xvi)Fai A (6),gi.Functionsf(Fai,_di)andg(Fai,xvi),hencethedynamicsin( 6 ),containunknownsystemparameters,and.Therefore,anadaptivecontrolapproachisusedtodesignthecontrolusi.Thus,theactuatorforceclosedlooptrackingerrordynamicsisgivenby_eFi=_Fai)]TJ /F6 11.955 Tf 14.56 2.65 Td[(_Fdi, (6)=fi+giusi)]TJ /F6 11.955 Tf 14.57 2.66 Td[(_Fdi (6)=fi)]TJ /F3 11.955 Tf 13.15 8.09 Td[(gi ^gi^fi)]TJ /F3 11.955 Tf 11.96 0 Td[(gikueFi)]TJ /F6 11.955 Tf 14.57 2.66 Td[(_Fdi+gi ^gi^_Fdi+gi usi+^fi ^gi+kueFi)]TJ /F6 11.955 Tf 14.37 12.87 Td[(^_Fdi ^gi!. (6)Theslowcontrolusiisdesignedasfollowsusi=)]TJ /F6 11.955 Tf 11.77 10.75 Td[(^fi ^gi)]TJ /F3 11.955 Tf 11.96 0 Td[(kueFi+^_Fdi ^gi (6)where^fiand^giaretheestimatesoffiandgirespectively,andthederivative_Fdiofthedesiredforceisapproximatedusingthehighgainobserver[ 55 ]"2_p=Ap+bFdi (6)^_Fdi=1 "2cTp, (6)whereA=264)]TJ /F6 11.955 Tf 9.3 0 Td[(11)]TJ /F6 11.955 Tf 9.3 0 Td[(10375,b=26411375,c=26401375,"21. 162

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Itcanbeshown(see[ 55 ])thattheestimationerror,e_Fdi=_Fdi)]TJ /F6 11.955 Tf 12.75 4.78 Td[(^_Fdidecaysveryfasttotheballje_Fdij0. (6) Proof. ConsiderthecandidateLyapunovfunctionV=Is 2r21+3 22+1 2Lheh22+1 2eQTL)]TJ /F4 7.97 Tf 6.59 0 Td[(1QeQ+1 2Xi=fL,Rgmdr2i+2mde2di+e2Fi+1 Le2i+1 Le2i+1 Le2i. (6)It'stimederivativeis_V=Isr1_r1+3(r1)]TJ /F8 11.955 Tf 11.95 0 Td[(1))]TJ /F10 11.955 Tf 13.07 11.24 Td[(eh2_^h2 Lh)]TJ /F10 11.955 Tf 13.28 3.71 Td[(eQTL)]TJ /F4 7.97 Tf 6.58 0 Td[(1Q_^Q+Xi=fL,Rg mdri_ri+2mdedi(ri)]TJ /F8 11.955 Tf 11.95 0 Td[(2edi)+eFi_eFi)]TJ /F10 11.955 Tf 13.91 8.09 Td[(ei_^i L)]TJ /F10 11.955 Tf 14.44 11.25 Td[(ei_^i L)]TJ /F10 11.955 Tf 13.19 8.09 Td[(ei_^i L!, (6) 165

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which,aftersubstitutingtheclosedlooperrordynamics( 6 )-( 6 )andapplyingtheupdatelaws( 6 )-( 6 ),becomes_Vr1()]TJ /F8 11.955 Tf 9.3 0 Td[(1r1)]TJ /F8 11.955 Tf 11.96 0 Td[(2r1+"d))]TJ /F8 11.955 Tf 11.95 0 Td[(12+Xi=fL,Rgri()]TJ /F8 11.955 Tf 9.3 0 Td[(2mdri)]TJ /F8 11.955 Tf 11.96 0 Td[(4ri)]TJ /F3 11.955 Tf 11.96 0 Td[(Ni)+eFi)]TJ /F8 11.955 Tf 9.3 0 Td[(5eFi)]TJ /F8 11.955 Tf 11.96 0 Td[(6eFi+e_Fdi (6))]TJ /F8 11.955 Tf 21.92 0 Td[(1r21)]TJ /F8 11.955 Tf 11.95 0 Td[(12)-222(jr1j(2jr1j)-222(j"dj)+Xi=fL,Rg)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F8 11.955 Tf 9.3 0 Td[(2mdr2i)]TJ /F8 11.955 Tf 11.95 0 Td[(22mde2di)-222(jrij(4jrij)]TJ /F3 11.955 Tf 17.93 0 Td[(ci))]TJ /F8 11.955 Tf 11.95 0 Td[(5e2Fi)-222(jeFij(6jeFij)]TJ /F3 11.955 Tf 17.93 0 Td[(O("2)), (6)Usingtheboundednesspropertyoftheparameterestimationerrorin( 6 ),duetotheprojectionoperator[ 111 ],theinequalityin( 6 )yields_V)]TJ /F8 11.955 Tf 21.92 0 Td[(1r21)]TJ /F8 11.955 Tf 11.96 0 Td[(12+Xi=fL,Rg)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F8 11.955 Tf 9.3 0 Td[(2mdr2i)]TJ /F8 11.955 Tf 11.95 0 Td[(22mde2di)]TJ /F8 11.955 Tf 11.96 0 Td[(5e2Fi)-222(kek2+ (6))]TJ /F8 11.955 Tf 21.92 0 Td[(2kk2+, (6))]TJ /F8 11.955 Tf 23.11 8.08 Td[(2 1V+, (6)where=r1rLedLrRedReTT. (6)UsingtheComparisonLemma[ 55 ],itfollowsthatV(t)1 2+V(t0))]TJ /F8 11.955 Tf 13.15 8.08 Td[(1 2exp)]TJ /F8 11.955 Tf 10.49 8.08 Td[(2 1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(t0). (6)Thus,anytrajectorystartingoutsideofBrwillapproachBrmonotonically,andanytrajectorystartinginsideBrwillremaininBr.Thisshowsthatthesystemisuniformlyultimatelybounded[ 112 ]. 166

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6.4.4SimulationSimilartotheKinematicControl,simulationsarecarriedoutusingthesamesteeringcommandsasintheprevioussection.However,unlikethekinematiccontrolcase,onlytheresponseforthefontaxleisreported.Thisisbecause,ascanbeseeninkinematiccontrolcase,theperformanceofthesystemissimilarforbothaxles.Theresults,shownintheguresbelow,alsoshowthatbyusingtheactuatedvariablestiffnessmechanismtogetherwiththecontroldevelopedintheprevioussections,therollangleandrollratesarereducedbymorethan50%. AFishhook BDoublelanechangeFigure6-15. Rollresponse AFishhook BDoublelanechangeFigure6-16. Controlmassdisplacement 167

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AFishhook BDoublelanechangeFigure6-17. Voltagecommand AFishhook BDoublelanechangeFigure6-18. Spoolvalveresponse AFishhook BDoublelanechangeFigure6-19. Hydraulicforceoutput 168

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AFishhook BDoublelanechangeFigure6-20. Adaptiveparameterestimationhistory,^Q AFishhook BDoublelanechangeFigure6-21. Adaptiveparameterestimationhistory,^ AFishhook BDoublelanechangeFigure6-22. Adaptiveparameterestimationhistory,^ 169

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AFishhook BDoublelanechangeFigure6-23. Adaptiveparameterestimationhistory,^ AFishhook BDoublelanechangeFigure6-24. Vehicletrajectory 170

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CHAPTER7CONCLUSIONSANDFUTUREWORK 7.1ConclusionTheideaofimprovingtheperformanceofvehiclesuspensionsystemsisanactiveareaofresearch.Pastapproachesutilizeoneofthreetechniques;adaptive,semi-active,orfullyactivesuspension.Thisresearchconsideredthedesign,analyses,simulation,andexperimentationofanewvariablestiffnesssuspensionsystem.Thedesignwasbasedontheconceptofavariablestiffnessmechanism.Themechanism,whichisasimplearrangementoftwosprings,aleverarm,andapivotbar,hasaneffectivestiffnessthatisarationalfunctionofthehorizontalpositionofthepivot.Theeffectivestiffnesswasvariedbychangingthepositionofthepivotwhilekeepingthepointofapplicationoftheexternalforceconstant.Theoverallsuspensionsystemconsistsofahorizontalcontrolstrutandaverticalstrut.Themainideawastovarytheloadtransferratiobymovingthelocationofthepointofattachmentoftheverticalstruttothecarbody.Thismovementwascontrolledpassively,semi-actively,andactivelyusingthehorizontalstrut.Atheoreticaljusticationformodulatingthestiffnessofasuspensionsystem,alongsidethedamping,waspresented.Itwasshownthatabetterperformance,intermsofridecomfortandhandling,isachievablebyvaryingthestiffnessalongsidethedampingcoefcientovervaryingeitherdampingorstiffnessalone.Twoadditionalcontrollawswerepresentedforvaryingthedampingandstiffnessofasemi-activesuspensionbasedonaquartercarmodel.Therstvariedthedampingandstiffnesssequentiallywhilethesecondvariedthemsimultaneously.Thenewvariablestiffnessmechanismwasintroduced.Theexpressionfortheeffectivestiffnesswasderived.Areverseanalysiswasalsocarriedoutonthemechanism.Specialdesigncaseswereconsidered.Thedynamicequationofthe 171

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systemwasderivedandusedtodeducethenaturalfrequencyofthemechanismfromwhichsomeinsightsweregainedonthedynamicbehaviorofthemechanism.Theincorporationofthenewvariablestiffnessmechanismintovehiclesuspensiondesignwasconsidered.Theconceptusedreciprocalactuationtoeffectivelytransferenergybetweenaverticaltraditionalstrutandahorizontaloscillatingcontrolmass,therebyimprovingtheenergydissipationoftheoverallsuspension.Duetotherelativelyfewernumberofmovingparts,theconceptcaneasilybeincorporatedintoexistingtraditionalfrontandrearsuspensiondesigns.Animplementationwithadoublewishbonewasshown.AdetailedL2-gainanalysiswasusedtoshowthattheresultingvariablestiffnesssuspensionsystemhasmuchbetterperformancethanthetraditionalconstantstiffnesscounterpart.Thedesignwasextendedtoincorporatesemi-activeandactiveactuators.Thevariablestiffnessarchitecturewasalsousedinthesuspensionsystemtocounteractthebodyrollmoment,therebyenhancingtherollstabilityofthevehicle.Thelateraldynamicsofthesystemwasdevelopedusingabicyclemodel.Theaccompanyingrolldynamicswerealsodevelopedandvalidatedusingexperimentaldata.Thepositionsoftheleftandrightcontrolmasseswereoptimallyallocatedtoreducetheeffectivebodyrollandrollrate.Simulationresultsshowthattheresultingvariablestiffnesssuspensionsystemhasmorethan50%improvementinrollresponseoverthetraditionalconstantstiffnesscounterparts.Thesimulationscenariosexaminedwere;theshhookmaneuverandtheISO3888-2doublelanechangemaneuvers.Acombinedvibrationisolationandrollstabilizationperformanceimprovementwasalsoexamined. 7.2FutureWorkInthisresearch,onlytherollandvibrationisolationperformanceofthenewlydesignedvariablestiffnesssuspensionsystemwereconsidered.Theexaminationofthepitchperformance,andpossibleinuenceontheyawdynamicsofthevehicleis 172

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stillanopenareaofresearch.Also,acombinationofvibration,roll,pitch,andyawperformancesusingafullcarmodelwouldbeaninterestingresearch.Withrespecttovibrationisolationperformanceenhancement,thebehaviorofthefrontsuspensionscanbeusedtobuildadisturbanceobserverfortheroadinput.Theobservedroaddisturbancecanthenbefedforwardtoimprovethebehavioroftherearsuspension.Thiswouldprovidesomesortofpreviewinformationfortherearsuspensionwithoutusingexpensivepreviewequipments.Onemoreinterestingaspectofthisresearchisthenalimplementationofthevariablestiffnessdesigninarealvehicle.Whenthisisdone,morerealisticexperimentscanbeconducted. 173

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APPENDIXPROOFOFTHEOREMS 2.1 AND 2.2 A.1ProofofTheorem 2.1 Proof. DenetheHamiltonianH,g(x,v)+pT(Ax+(x)v) (A)wherepistheLagrangemultipliervectorforthedynamicconstraint( 2 ).Usingcalculusofvariations,thenecessaryconditionsforoptimalityaregivenby)]TJ /F6 11.955 Tf 11.21 .88 Td[(_p=dg dx+AT+v@(x) @xp (A)0=@g @v+pT(x). (A)From( A )w(x))]TJ /F6 11.955 Tf 10.49 8.09 Td[(2aTx ms+2w(x) m2sv)]TJ /F7 11.955 Tf 11.96 0 Td[(bTp=0. (A)Ifw(x)6=0thenv=ms 2w(x))]TJ /F6 11.955 Tf 5.48 -9.69 Td[(2aTx+msbTp. (A)Substituting( A )in( A )yields)]TJ /F6 11.955 Tf 11.21 .88 Td[(_p=2Qx)]TJ /F6 11.955 Tf 13.15 8.08 Td[(2v ms)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(w(x)a+TaTx+2w(x) m2sv2T+ATp)]TJ /F3 11.955 Tf 11.96 0 Td[(vTbTp=)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(AT)]TJ /F3 11.955 Tf 11.96 0 Td[(m2sabTp+2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Q)]TJ /F7 11.955 Tf 11.96 0 Td[(aaTx=Ap+2Qx. (A) 174

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Similarly,susbstituting( A )inthedeterministicdynamicconstraint( 2 )yields_x=Ax)]TJ /F3 11.955 Tf 13.16 8.09 Td[(ms 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(msbTp+2aTxb=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(A)]TJ /F3 11.955 Tf 11.95 0 Td[(m2sbaTx)]TJ /F3 11.955 Tf 13.15 8.09 Td[(m2s 2bbTp=ATx)]TJ /F6 11.955 Tf 13.42 2.66 Td[(Bp. (A)Putting( A )and( A )togetheryieldsthehomogeneouslinearordinarydifferentialequation( A )intermsofthestatexandcostatep.264_x_p375=264AT)]TJ /F6 11.955 Tf 10.76 2.65 Td[(B)]TJ /F6 11.955 Tf 9.3 0 Td[(2Q)]TJ /F6 11.955 Tf 10.2 2.65 Td[(A375264xp375. (A)Now,letp=Px (A)whereP2<44isapositive-denitematrix.Substituting( A )into( A )yields)]TJ /F10 11.955 Tf 11.29 9.68 Td[()]TJ /F6 11.955 Tf 8.16 -7.02 Td[(_Px+P_x=Qx+APx=))]TJ /F6 11.955 Tf 8.16 -7.03 Td[(_P+P(AT)]TJ /F6 11.955 Tf 13.41 2.65 Td[(BTP)+Q+APx=0which,providedx6=0reducestothetheRiccatiequation_P+PAT+AP)]TJ /F3 11.955 Tf 11.96 0 Td[(PBTP+Q=0. (A)Now,aftersomealgebraicmanipulations,g(x,v)becomesg(x,v)=1 2xT)]TJ /F6 11.955 Tf 7.47 -7.03 Td[(Q+PBPx (A)=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2d dt(xTPx). (A) 175

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ThusthevaluefunctionJ,J(x,v)isgivenbyJ=Ztft0g(x,v)dt=)]TJ /F6 11.955 Tf 13.69 8.09 Td[(1 2xTPxtft0.ImposingtheboundaryconditionP(tf)=0ontheRiccatiequation( A )yieldsJ=1 2xT(t0)Px(t0). (A) A.2ProofofTheorem 2.2 Proof. TheHamiltonian( A )ismodiedasH=g(x,v)+pT(Ax+(x)v))]TJ /F8 11.955 Tf 11.95 0 Td[(1v+2(v)]TJ /F6 11.955 Tf 12.24 0 Td[(v), (A)where1,20aretheLagrangemultipliersfortheinequalityconstraint( 2 ).Similarly,thenecessaryconditionsforoptimalityare)]TJ /F6 11.955 Tf 11.21 .88 Td[(_p=dg dx+AT+v@(x) @xp (A)0=@g @v+pT(x))]TJ /F8 11.955 Tf 11.96 0 Td[(1+2. (A)( A )yieldsv=m2s 2w(x)2aTx ms+bTp+m2s 2w(x)2(1)]TJ /F8 11.955 Tf 11.95 0 Td[(2)=m2s 2w(x)2(v0+1)]TJ /F8 11.955 Tf 11.96 0 Td[(2), (A)wherev0=w(x) ms)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(2aTx+msbTp. (A)Therearethreepossibilitiesforthevaluesof1and2.Case1: (1>0,2=0) 176

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@H @1=m2s 2w(x)2(v0+1)=0=)1=8><>:0ifv0>0)]TJ /F3 11.955 Tf 9.29 0 Td[(v0ifv00. (A)Case2: (1=0,2=0)v=m2s 2w(x)2v0 (A)Case3: (1=0,2>0)@H @2=m2s 2w(x)2(v0)]TJ /F8 11.955 Tf 11.95 0 Td[(2))]TJ /F6 11.955 Tf 12.25 0 Td[(v=0=)2=8><>:0ifv0<2w(x)2 m2svv0)]TJ /F4 7.97 Tf 13.15 5.48 Td[(2w(x)2 m2svif2w(x)2 m2svv0. (A)Puttingtheresultsforthecasestogetheryields1=)]TJ /F3 11.955 Tf 9.3 0 Td[(v0,2=0ifv001=2=0if0>>><>>>>:0ifv00m2s 2w(x)2v0if0
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Thedeterministicdynamicconstraint( 2 )and( A )thenbecome264_x_p375=8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:264A0)]TJ /F6 11.955 Tf 9.3 0 Td[(2Q)]TJ /F3 11.955 Tf 9.3 0 Td[(AT375264xp375ifx2R1264AT)]TJ /F6 11.955 Tf 10.76 2.66 Td[(B)]TJ /F6 11.955 Tf 9.3 0 Td[(2Q)]TJ /F6 11.955 Tf 10.2 2.66 Td[(A375264xp375ifx2R2264^A10)]TJ /F3 11.955 Tf 9.3 0 Td[(Q)]TJ /F6 11.955 Tf 10.19 2.66 Td[(^AT1375264xp375ifx2R3 (A)where^A1=A)]TJ /F6 11.955 Tf 12.79 0 Td[(vbTT,Q=Q+2 m2s(vT)]TJ /F3 11.955 Tf 12.5 0 Td[(msa)(vT)]TJ /F3 11.955 Tf 12.5 0 Td[(msa)TandR1,R2andR3areasdenedinsection 2.2 .Thus,usingtherelationship( A )yieldstheRiccatiequation( 2 ).Now,letVa=xTPax (A)where,PaisdenedinTheorem 2.2 .Theassociatedvaluefunctionofthefullyactiveoptimalcontrol[ 9 ]isJa=xT(t0)Pax(t0). (A)Then,forthedeterministicsemi-activesuspension_Va=_xTPax+xTPa_x=xT)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ATPa+PaAx+v)]TJ /F12 11.955 Tf 5.48 -9.68 Td[(T(x)Pax+xTPa(x)which,using( 2 )andg(x,v)denedin( 2 ),becomes_Va=)]TJ /F3 11.955 Tf 9.3 0 Td[(g(x,v)+w(x)v ms)]TJ /F10 11.955 Tf 11.95 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(msbTPa+aTx2. (A) 178

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Integratingbothsidesandusingtheboundaryconditionx(tf)=0yields)]TJ /F3 11.955 Tf 9.29 0 Td[(Va(t0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(J(x,v)+Ztft0w(x)v ms)]TJ /F10 11.955 Tf 11.96 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(msbTPa+aTx2dt (A)fromwhich( 2 )follows. 179

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REFERENCES [1] W.B.AdamsEnglishPleausreCarriages,C.Knight&Co.,1837. [2] chroniclingamerica.loc.gov,TheWashingtonTimes,Sunday30June,;. [3] K.K.JainandR.B.AsthanaAutomobileEngineering,TataMcGraw-Hill,2002. [4] J.C.DixonSuspensionGeometryandComputation,JohnWiley&SonsLtd,2009. [5] G.GentaandL.MorelloTheAutomotiveChassis,Vol.1:ComponentsDesign,Vol.1,Springer,2009. [6] Ingolstadt,AudiTTcoupe'07-SuspensionSystem,,ServiceTraining,SelfStudyProgramme381,2006. [7] R.Alkhatib,G.N.Jazar,andM.F.Golnaraghi,OptimaldesignofPassiveLinearSuspensionUsingGeneticAlgorithm,JournalofSoundandvibration275(2004),pp.665. [8] R.A.Williams,Automotiveactivesuspensions.Part1:basicprinciples,inPro-ceedingsofIMechE,Vol.211,1997,pp.415. [9] T.Butsuen,TheDesignofSemi-activeSuspensionsforAutomotiveVehicles,MassachussetsInstituteofTechnology,1989. [10] H.E.TsengandJ.K.Hedrick,Semi-ActiveControlLaws-OptimalandSub-Optimal,VehicleSystemDynamics23(1994),pp.545. [11] M.ValasekandW.Kortum,Nonlinearcontrolofsemi-activeroad-friendlytrucksuspension,inProceedingsAVEC98,Nagoya,1998,pp.275. [12] M.ValasekandW.Kortum,2001,Semi-ActiveSuspensionSystemsII.inTheMechanicalSystemsDesignHandbook;Modeling,MeasurementandControlCRCPressLLC. [13] D.Karnopp,M.Crosby,andR.Harwood,Vibrationcontrolusingsemi-activeforcegenerators,JournalofEngineeringforIndustry96(1974),pp.619. [14] D.Karnopp,Activedampinginroadvehiclesuspensionsystems,VehicleSystemDynamics12(1983),pp.291. [15] D.KarnoppandG.Heess,Electronicallycontrollablevehiclesuspensions,VehicleSystemDynamics20(1991),pp.207. [16] W.J.Evers,A.Teehuis,A.Knaapvander,I.Besselink,andH.Nijmeijer,TheElectromechanicalLow-PowerActiveSuspension:Modeling,Control,andPrototypeTesting,JournalofDynamicSystems,Measurement,andControl133(2011). 180

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[17] A.C.M.Knaapvander,A.P.Teerhuis,R.B.G.Tinsel,andR.M.A.F.Vershuren,ActiveSuspensionAssemblyforaVehicle,InternationalPatentNo.2008/049845.(2008). [18] A.Ashfak,A.Saheed,K.K.Abdul-Rasheed,andJ.A.Jaleel,Design,FabricationandEvaluationofMRDamper,WorldAcademyofScience,EngineeringandTechnology53(2009). [19] I.FialhoandG.J.Balas,RoadAdaptiveActiveSuspensionDesignUsingLinearParameter-VaryingGain-Scheduling,IEETransactionsoncontrolsystemstechnology10(2002),pp.43. [20] A.L.Do,O.Sename,andL.Dugard,AnLPVControlApproachforSemi-activeSuspensionControlwithActuatorConstraints,in2010AmericanControlConfer-ence,2010. [21] R.A.Williams,A.Best,andI.L.Crawford,Renedlowfrequencyactivesuspen-sion,inProceedingsofIMechEInternationalConference,November,,1993,pp.285. [22] S.Ikenaga,F.Lewis,J.Campos,andL.Davis,Activesuspensioncontrolofgroundvehiclebasedonafull-vehiclemodel,inAmericanControlConference,2000.Proceedingsofthe2000,Vol.6,2000,pp.4019. [23] W.Jones,Easyride:BoseCorp.usesspeakertechnologytogivecarsadaptivesuspension,Spectrum,IEEE42(2005),pp.12. [24] D.KarnoppandD.Margolis,Adaptivesuspensionconceptsforroadvehicles,VehicleSystemDynamics13(1984),pp.145. [25] L.Woods,ComputerOptimizedAdaptiveSuspensionSystemHavingCombinedShockAbsorber/AirSpringUnit;EPPatent0,080,291. [26] B.F.Spencer,S.J.Dyke,M.K.Sain,andJ.D.Carlson,Phenomenologicalmodelofamagnetorheologicaldamper,ASCEJournalofEngineeringMechanics(1996). [27] J.W.Kim,Y.H.Cho,H.Lee,andS.B.Choi,ElectrorheologicalSemi-Aactivedamper:PolyanilinebasedERsystem,JournalofIntelligentMatrialSystemsandStructures13(2002),pp.509. [28] O.M.AnubiandC.Crane,Semi-globalOutputFeedbackAsymptoticTrackingforanUnder-actuatedVariableStiffnessMechanism,inProceedingsofthe13thIFTOMMWorldCongressinMechanismandMachineScience,June,,2011. [29] A.Knaapvander,DesignofaLowPowerAnti-Roll/PitchSystemforaPassengerCar,DelftUniversityofTechnology,1989. 181

PAGE 182

[30] P.J.T.VenhovensandA.C.M.Knaapvander,DelftActiveSuspension(DAS).BackgroundTheoryandPhysicalRealization,SmartVehicles(1995),pp.139. [31] D.Karnopp,M.Crossby,andR.Harwood,VibrationControlUsingSemi-ActiveForceGenerators,JournalofEngineeringforIndustry96(1974),pp.619. [32] S.M.Savaresi,E.Silani,andS.Bittanti,Acceleration-DrivenDamper(ADD):AnOptimalControlAlgorithmforComfort-OrientedSemi-ActiveSuspensions,ASMEJournalofDynamicSystems,MeasurementandControl127(2005),pp.218. [33] S.M.SavaresiandC.Spelta,MixedSky-HookandADD:ApproachingtheFil-teringLimitsofSemi-ActiveSuspension,ASMEJournalofDynamicSystems,MeasurementandControl129(2007),pp.382. [34] K.S.Hong,D.S.Joen,andH.C.Sohn,ANewModelingoftheMacphersonSuspensionSystemanditsOptimalPole-PlacementControl,inProceedingsofthe7thMediterraneanConferenceonControlandAutomation(MED99)Haifa,Israel,1999. [35] M.S.Fallah,NewDynamicModelingandPracticalControlDesignforMacPhersonSuspensionSystem,ConcordiaUniversity,Montreal,Quebec,Canada,2010. [36] O.M.AnubiandC.D.Crane,NonlinearControlofSemi-activeMacPhersonSuspensionSystem,inASME2012InternationalDesignEngineeringTechnicalConferences&ComputersandInformationinEngineeringConference,2012. [37] C.D.C.OlugbengaMosesAnubiandW.Dixon,NonlinearDisturbanceRejectionforSemi-activeMacPhersonSuspensionSystem,inASMEDynamicSystemsandControlConference,Ft.Lauderdale,FL,2012. [38] S.Guo,S.Yang,andC.Pan,DynamicModelingofMagnetorheologicalDamperBehaviors,JournalofIntelligentMatrialSystemsandStructures17(2006),pp.3. [39] H.TanghiradandE.Esmailzadeh,AutomobilePassengerComfortAssuredthroughLQG/LQRActiveSuspension,JournalofVibrationControl4(1998),pp.603. [40] E.Elbeheiry,D.Karnopp,andA.Abdelraaouf,SuboptimalControlDesignofActiveandPassiveSuspensionsBasedonaFullcarModel,VehicleSystemDynamics26(1996),pp.197. [41] F.LinRobustControlDesign:AnOptimalControlApproach,JohnWiley&SonsLtd,WestSussex,England,2007. [42] H.D.T.E.OnoS.HosoeandY.Hayashi,NonlinearH1ControlofActiveSuspen-sion,VehicleSystemDynamicsSupplement25(1996),pp.489. 182

PAGE 183

[43] A.Zin,O.Sename,M.Basset,L.Dugard,andG.Gissinger,Anonlinearvehiclebicyclemodelforsuspensionandhandlingcontrolstudies,inProceedingsoftheIFACConferenceonAdvancesinVehicleControlandSafety(AVCS),Genova,Italy,October,,2004,pp.638. [44] J.H.Taylor,1999,DescribingFunctions.inElectricalEngineeringEncyclopediaJohnWileyandSons,Inc,NewYork. [45] J.Jerz,VariableStiffnessSuspensionSystem;USPatent3,559,976. [46] T.Kobori,M.Takahashi,T.Nasu,N.Niwa,andK.Ogasawara,Seismicresponsecontrolledstructurewithactivevariablestiffnesssystem,EarthquakeEngineeringandStructuralDynamics22(1993),pp.925. [47] I.YounandA.Hac,Semi-activesuspensionwithadaptivecapability,JournalofSoundandVibration180(1995),pp.475. [48] Y.Liu,H.Matsuhisa,andH.Utsuno,Semi-activevibrationisolationsystemwithvariablestiffnessanddampingcontrol,JournalofSoundandVibration313(2008),pp.1628. [49] O.M.Anubi,C.Crane,andS.Ridgeway,DesignandAnalysisofaVariableStiffnessMechanism,inProceedingsIDETC/CIE2010.ASME2010InternationalDesignEngineeringTechnicalConferences&ComputersandInformationinEngineeringConference,2010. [50] F.L.Lewis,D.M.Dawson,andC.T.AbdallahRobotManipulatorControl,TheoryandPractice,2ndMarcelDekker,Inc.,2004. [51] J.BallandJ.Helton,H1controlfornonlinearplants:Connectionswithdifferentialgames,inProceedingsofthe28thConferenceonDecisionandControl,Tampa,Florida,1989,pp.956. [52] J.HeltonandM.JamesExtendingH1ControltoNonlinearSystems,SIAM,1999. [53] P.Soravia,H1ControlforNonlinearSystems:DifferentialandViscositySolutions,SIAMJournalofControlandOptimization34(1996),pp.1071. [54] A.J.SchaftvanderL2-GainandPassivityTechniquesinNonlinearControl,Springer:Berlin,1996. [55] H.KhalilandJ.GrizzleNonlinearsystems,Vol.3,PrenticeHal,lNewJersey,1996. [56] NHTSA,NewPassengerCarFleetAverageCharacterisitcs;. [57] J.Schoukens,R.Pintelon,Y.Rolain,andT.Dobrowiecki,FrequencyResponseFunctionMeasurementsinthePresenceofNonlinearDistortions,Automatica37(2001),pp.939. 183

PAGE 184

[58] A.J.StackandF.J.Doyle,AMeasureforControlRelevantNonlinearity,inAmeri-canControlConferencel,Seattle,1995,pp.2200. [59] I.2631-1InternationalOrganizationofStandardization.MechanicalVibrationandShock-EvaluationofHumanExposuretoWholeBodyVibration.Part1:GeneralRequirement.,Geneva,1997. [60] Y.StarosvetskyandO.Gendelman,Vibrationabsorptioninsystemswithanonlinearenergysink:Nonlineardamping,JournalofSoundandVibration324(2009),pp.916. [61] A.Vakakis,Inducingpassivenonlinearenergysinksinvibratingsystems,TransASME,J.Vib.Acoust.123(2001),pp.324. [62] E.GourdonandC.Lamarque,Nonlinearenergysinkwithuncertainparameters,Journalofcomputationalandnonlineardynamics1(2006),pp.187. [63] X.Jiang,D.M.McFarland,L.A.Bergman,andA.F.Vakakis,Steadystatepassivenonlinearenergypumpingincoupledoscillators:theoreticalandexperimentalresults,NonlinearDynamics33(2003),pp.87. [64] A.F.Vakakis,L.Manevitch,O.Gendelman,andL.Bergman,Dynamicsoflineardiscretesystemsconnectedtolocal,essentiallynon-linearattachments,JournalofSoundandVibration264(2003),pp.559. [65] O.GendelmanandY.Starosvetsky,Quasi-periodicresponseregimesoflinearoscillatorcoupledtononlinearenergysinkunderperiodicforcing,Journalofappliedmechanics74(2007),pp.325. [66] O.Gendelman,Y.Starosvetsky,andM.Feldman,AttractorsofharmonicallyforcedlinearoscillatorwithattachednonlinearenergysinkI:Descriptionofresponseregimes,NonlinearDynamics51(2008),pp.31. [67] Y.StarosvetskyandO.Gendelman,Attractorsofharmonicallyforcedlinearoscillatorwithattachednonlinearenergysink.II:Optimizationofanonlinearvibrationabsorber,NonlinearDynamics51(2008),pp.47. [68] O.Gendelman,L.Manevitch,A.F.Vakakis,andR.M'CLOSKEY,Energypumpinginnonlinearmechanicaloscillators:PartI:DynamicsoftheunderlyingHamiltoniansystems,JournalofAppliedMechanics68(2001),pp.34. [69] O.V.Gendelman,Transitionofenergytoanonlinearlocalizedmodeinahighlyasymmetricsystemoftwooscillators,Nonlineardynamics25(2001),pp.237. [70] O.GendelmanandA.F.Vakakis,Energypumpinginnonlinearmechanicaloscillators:PartII:ResonanceCapture,JournalofAppliedMechanics68(2000),pp.42. 184

PAGE 185

[71] O.M.Anubi,D.R.Patel,andC.D.CraneIII,Anewvariablestiffnesssuspensionsystem:passivecase,MechanicalSciences4(2013),pp.139. [72] J.Smoker,A.Baz,andL.Zheng,VirtualRealitySimulationofActiveCarSuspen-sionSystem,InternationalJournalofVirtualReality8(2009),pp.75. [73] A.AlleyneandJ.Hedrick,Nonlinearadaptivecontrolofactivesuspensions,ControlSystemsTechnology,IEEETransactionson3(1995),pp.94. [74] H.MerrittHydrauliccontrolsystems,WileyNewYork,1967. [75] P.Kokotovic,H.Khali,andJ.O'reillySingularperturbationmethodsincontrol:analysisanddesign,Vol.25,SocietyforIndustrialMathematics,1987. [76] E.Kim,Nonlinearindirectadaptivecontrolofaquartercaractivesuspension,inControlApplications,1996.,Proceedingsofthe1996IEEEInternationalConferenceon,1996,pp.61. [77] L.B.S.H.LaalejZ.QandP.Martynowicz,MRdamperbasedimplementationofnonlineardampingforapitchplanesuspensionsystem,SmartMaterialsandStructures21(2012). [78] M.A.XubinSongandS.C.Sotuhward,ModelingMagnetorheologicalDamperswithApplicationofNonparametricApproach,JournalofIntelligentMaterialSystemsandStructures16(2005),pp.421. [79] E.Guglielmino,T.Sireteanu,C.W.Stammers,G.Ghita,andM.GiucleaSemi-activeSuspensionControl:ImprovedVehicleRideandRoadFriendliness,Springer,2008. [80] Y.K.WenandM.Asce,MethodforRandomVibrationofHystericSystems,JournaloftheEngineeringMechanicsDivision102(1976),pp.249. [81] I.H.ShamesandF.A.CozzarelliElasticandInelasticStressAnalysis,Taylor&Francis,NJ,1997. [82] J.Koo,F.Goncalves,andM.Ahmadian,Acomprehensiveanalysisofthere-sponsetimeofMRdampers,Smartmaterialsandstructures15(2006),p.351. [83] G.J.Forkenbrok,W.Garrot,M.Heitz,andB.C.O'Harra.,Acomprehensiveexperimentalexaminationoftestmaneuversthatmayinduceon-road,untripped,lightvehiclerollover-phaseivofNHTSA'slightvehiclerolloverresearchprogram,,NationalHighwayTrafcSaftetyAdministration,2002. [84] M.Walz,Trendsinthestaticstabilityfactorofpassengercars,lighttrucks,andvans,,2005. [85] D.Sampson,Activerollcontrolofarticulatedheavyvehicles,UniversityofCambridgeUK,2000. 185

PAGE 186

[86] C.CarlsonandJ.Gerdes,Optimalrolloverpreventionwithsteerbywireanddifferentialbraking,inProceedingsofIMECE,2003,pp.16. [87] B.ChenandH.Peng,Differential-braking-basedrolloverpreventionforsportutilityvehicleswithhuman-in-the-loopevaluations,VehicleSystemDynamics36(2001),pp.359. [88] L.Palkovics,A.Semsey,andE.Gerum,Roll-overpreventionsystemforcommer-cialvehiclesadditionalsensorlessfunctionoftheelectronicbrakesystem,VehicleSystemDynamics32(1999),pp.285. [89] V.Tsourapas,D.Piyabongkarn,A.Williams,andR.Rajamani,Newmethodofidentifyingreal-timePredictiveLateralloadTransferRatioforrolloverpreventionsystems,inAmericanControlConference,2009.ACC'09.,2009,pp.439. [90] M.TheriaultGreatMaritimeInventions1833-1950,GooseLaneEditions,2001. [91] S.DeMolinaandS.Deferme,Passiveanti-rollsystem;USPatent6,220,406. [92] P.BockandG.O'rourke,Anti-rollsystemforwheeledvehicles;USPatent5,383,680. [93] J.Kincad,B.Mattila,andT.Ignatius,Semi-activeanti-rollsystem;USPatent6,428,019. [94] I.AGNER,Anti-rollSystem;WOPatent2,004,085,178. [95] M.Nishikawa,Anti-rollsystemforvehicles;USPatent4,345,661. [96] M.Lund,Anti-rollsystemforavehicle;USPatent4,589,678. [97] ,Anti-rollsystemwithtiltlimitation;USPatent5,040,823. [98] M.LundandG.Garrabrant,Anti-rollsystemwithtiltlimitation;USPatent4,966,390. [99] R.JazarVehicleDynamics:TheoryandApplication,Springer,2008. [100] R.RajamaniVehicledynamicsandcontrol,Springer,2011. [101] L.Ray,Nonlinearstateandtireforceestimationforadvancedvehiclecontrol,ControlSystemsTechnology,IEEETransactionson3(1995),pp.117. [102] H.PacejkaandE.Bakker,Themagicformulatyremodel,Vehiclesystemdynamics21(1992),pp.1. [103] M.Oppenheimer,D.Doman,andM.Bolender,Controlallocationforover-actuatedsystems,inControlandAutomation,2006.MED'06.14thMediterraneanConfer-enceon,2006,pp.1. 186

PAGE 187

[104] M.Bodson,Evaluationofoptimizationmethodsforcontrolallocation,inAIAAGuidance,Navigation,andControlConferenceandExhibit,2001. [105] InternationalStandardISO3888-2:PassengerCars-testtrackforaseverelane-changemaneuver-part2:Obstacleavoidance;. [106] K.NamandA.Araposthathis,Amodelreferenceadaptivecontrolschemeforpure-feedbacknonlinearsystems,AutomaticControl,IEEETransactionson33(1988),pp.803. [107] P.IoannouandJ.Sun,Robustadaptivecontrol,(1996). [108] G.KreisselmeierandB.Anderson,Robustmodelreferenceadaptivecontrol,AutomaticControl,IEEETransactionson31(1986),pp.127. [109] B.Badji,E.Fenaux,M.ElBagdouri,andA.Miraoui,NonlinearsingletrackmodelanalysisusingVolterraseriesapproach,VehicleSystemDynamics47(2009),pp.81. [110] S.C.Baslamisli,I.E.Kose,andG.Anlas,Gain-scheduledintegratedactivesteeringanddifferentialcontrolforvehiclehandlingimprovement,VehicleSystemDynamics47(2009),pp.99. [111] G.C.GoodwinandK.S.SinAdaptivelteringpredictionandcontrol,DoverPublications,2009. [112] M.CorlessandG.Leitmann,Continuousstatefeedbackguaranteeinguniformultimateboundednessforuncertaindynamicsystems,AutomaticControl,IEEETransactionson26(1981),pp.1139. [113] O.M.Anubi,D.Patel,andC.D.Crane,PassiveVariableStiffnessSuspensionSystem,ASMEEarlyCareerTechnicalJournal11(2012). [114] T.ButzandStrykOvonModelingandSimulationofRheologicalFluidDevices,TechnischeUniversitatMunchen,Germany,1999PreprintSFB-438-9911. [115] C.ColleteandA.Preumont,Highfrequencyenergytransferinsemi-activesuspen-sion,JournalofSoundandVibration329(2010),pp.4604. [116] N.CiblakandH.Lipkin,SynthesisofStiffnessbySprings,inProceedingsofDETC'98,September,,1998. [117] H.Dugoff,P.Fancher,andL.Segel,Tireperformancecharacteristicsaffectingvehicleresponsetosteeringandbrakingcontrolinputs,,1969. [118] E.Elbeheiry,D.Karnopp,andA.Abdelraaouf,AdvancedGroundVehicleSuspen-sionSystem-aClassiedBiography,VehicleSystemDynamics24(1995),pp.231. [119] H.FRAHM,VibrationsofBodies;USPatent989,958. 187

PAGE 188

[120] G.GentaandL.MorelloTheAutomotiveChassis,Vol.2:SystemDesign,Vol.1,Springer,2009. [121] G.Holler,Antilockbrakingsystembasedrolloverprevention;USPatent6,741,922. [122] J.B.HuntDynamicvibrationabsorbers,MechanicalEngineeringPublications,1979. [123] A.IsidoriandA.Astol,DisturbanceAttenuationandH1ControlviaMeasurementFeedbackinNonlinearSystems,IEEETrans.Automat.Contr37(1992),pp.1283. [124] L.JansenandS.Dyke,SemiactivecontrolstrategiesforMRdampers:compara-tivestudy,JournalofEngineeringMechanics126(2000),pp.795. [125] J.JiandN.Zhang,Suppressionoftheprimaryresonancevibrationsofaforcednonlinearsystemusingadynamicvibrationabsorber,JournalofSoundandVibration329(2010),pp.2044. [126] A.Kalyani,M.Ahuja,A.Kumar,R.Kumar,K.Dhuri,andN.Tambe,SystemsandMethodsProvidingVariableSpringStiffnessforWeightManagementinaVehicle;USPatent20,120,049,479. [127] L.Kitis,B.Wang,andW.Pilkey,Vibrationreductionoverafrequencyrange,JournalofSoundandVibration89(1983),pp.559. [128] C.S.KyleandP.N.Roschke,FuzzyModelingofaMagnetorheologicalDamperusingANFIS,inIEEEFuzzyConference,SanAntonio,Texas,USA,2000. [129] G.Leitmann,Semiactivecontrolforvibrationattenuation,JournalofIntelligentMaterialSystemsandStructures5(1994),pp.841. [130] F.LinRobustControlDesign:AnOptimalControlApproach,JohnWiley&SonsLtd,WestSussex,England,2007. [131] J.LinandJ.Kanellakopoulos,Road-adaptivenonlineardesignofactivesuspen-sions,inAmericanControlConference,1997.Proceedingsofthe1997,Vol.1,1997,pp.714. [132] Y.Liu,H.Matsuhisa,andH.Utsuno,Semi-activevibrationisolationsystemwithvariablestiffnessanddampingcontrol,Journalofsoundandvibration313(2008),pp.16. [133] J.Lu,Afrequency-adaptivemutli-objectivesuspensioncontrolstrategy.,ASMEJournalofDynamicSystems,Measurement,andControl126(2004),pp.700. 188

PAGE 189

[134] S.Masri,Onthestabilityoftheimpactdamper,TransASME,J.Appl.Mech.33(1966),pp.586. [135] D.J.MeadandD.MeadorPassivevibrationcontrol,Wiley,1998. [136] P.Muller,K.Popp,andW.Schiehlen,Covarianceanalysisofnonlinearstochasticguideway-vehicle-systems,inProceedingsoftheSixthIAVSDSymposium,1979,pp.337. [137] S.Natsiavas,Steadystateoscillationsandstabilityofnon-lineardynamicvibrationabsorbers,JournalofSoundandVibration156(1992),pp.227. [138] A.H.NayfehandD.T.MookNonlinearoscillations,Wiley-VCH,2008. [139] J.Ormondroyd,Theoryofthedynamicvibrationabsorber,TransactionoftheASME50(1928),pp.9. [140] A.Pavlov,N.Wouwvande,andH.Nijmeijer,FrequencyresponsefunctionsandBodeplotsfornonlinearconvergentsystems,inProceedingsofthe45thIEEEConferenceonDecision&Control,December,,2006. [141] R.RajamaniandS.Larparisudthi,Oninvariantpointsandthierinuenceonactivevibrationisolation,Mechatronics14(2004),pp.175. [142] R.E.Roberson,Synthesisofanonlineardynamicvibrationabsorber,JournaloftheFranklinInstitute254(1952),pp.205. [143] A.J.Schaftvander,L2-GainAnalysisofNonlinearSystemsandNonlinearStateFeedbackH1,IEEETrans.Automat.Contr37(1992),pp.770. [144] J.Shaw,S.W.Shaw,andA.G.Haddow,Ontheresponseofthenon-linearvibrationabsorber,InternationalJournalofNon-LinearMechanics24(1989),pp.281. [145] C.Spelta,F.Previdi,S.Savaresi,P.Bolzern,M.Cutini,andC.Bisaglia,Anovelcontrolstrategyforsemi-activesuspensionswithvariabledampingandstiffness,inAmericanControlConference(ACC),2010,2010,pp.4582. [146] C.Spelta,F.Previdi,S.Savaresi,P.Bolzern,M.Cutini,C.Bisaglia,andS.Bertinotti,Performanceanalysisofsemi-activesuspensionswithcontrolofvariabledampingandstiffness,VehicleSystemDynamics49(2011). [147] B.F.Spencer,G.Yang,J.D.Carlson,andM.K.Sain,SmartDampersforSeismicProtectionofStructures:AFullScaleStudy,inSecondWorldConferenceonStructuralControl,Kyoto,Japan,1998. [148] Y.StarosvetskyandO.Gendelman,Attractorsofharmonicallyforcedlinearoscillatorwithattachednonlinearenergysink.II:Optimizationofanonlinearvibrationabsorber,NonlinearDynamics51(2008),pp.47. 189

PAGE 190

[149] K.Sung,Y.Han,K.Lim,andS.Choi,Discrete-timefuzzyslidingmodecontrolforavehiclesuspensionsystemfeaturinganelectrorheologicaluiddamper,Smartmaterialsandstructures16(2007),p.798. [150] N.TischlerandA.A.Goldenberg,Stiffnesscontrolforgearedmanipulators,inProceedings2001ICRA.IEEEInternationalConferenceonRoboticsandAutomation,Vol.3,2001,pp.3042. [151] E.Wang,X.Ma,S.Rakheja,andC.Su,Semi-activecontrolofvehiclevibrationwithMR-dampers,inDecisionandControl,2003.Proceedings.42ndIEEEConferenceon,Vol.3,2003,pp.2270. [152] G.Yao,F.Yap,G.Chen,W.Li,andS.Yeo,MRdamperanditsapplicationforsemi-activecontrolofvehiclesuspensionsystem,Mechatronics12(2002),pp.963. [153] J.YaoandC.Song,SimulationandAnalysisofVariableStiffnessDouble-layerVibrationIsolationSystem,JournalofWuhanUniversityofTechnology(Information&ManagementEngineering)32(2010). [154] J.YimandJ.H.Park,NonlinearH1ControlofRoboticManipulator,inIEEEConferenceonSystem,ManandCybernetics,SMC,Tokyo,Japan,October,,1999,pp.960. [155] M.Yokoyama,J.Hedrick,andS.Toyama,Amodelfollowingslidingmodecon-trollerforsemi-activesuspensionsystemswithMRdampers,inAmericanControlConference,2001.Proceedingsofthe2001,Vol.4,2001,pp.2652. [156] M.Zapateiro,F.Pozo,H.Karimi,andN.Luo,SemiactiveControlMethodologiesforSuspensionControlWithMagnetorheologicalDampers,Mechatronics,IEEE/ASMETransactionson(2012),pp.1. [157] L.ZuoandS.Nayfeh,Lowordercontinuous-timeltersforapproximationoftheISO2631-1humanvibrationsensitivityweightings,JournalofSoundandVibration265(2003),pp.459. [158] L.ZuoandS.A.Nayfeh,StructuredH2OptimizationofVehicleSuspensionsBasedonMulti-WheelModels,VehicleSystemDynamics40(2003),pp.351. 190

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BIOGRAPHICALSKETCH OlugbengaMosesAnubireceivedhisB.S(Hons)insystemsengineeringfromtheUniversityofLagos,Nigeriain2006.HethenservedintheNigerianNationalYouthServiceCorp(NYSC)in2007.HeiscurrentlycompletinghisdoctoraldegreeinMechanicalEngineeringattheCenterforIntelligentMachinesandRobotics(CIMAR),attheUniversityofFlorida,Gainesville.Hisresearchinterestsare;VehicleSystemDynamicsandControl,SuspensionDesignandAnalysis,NonlinearControl,RobustControl,OptimalControl,Robotics.HeisamemberoftheAmericanSocietyofMechanicalEngineers(ASME),andtheSocietyofAutomotiveEngineers(SAE) 191