Fall 2008 1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 5Lecture 5Modeling of mechanical systemsModeling of mechanical systemsFall 2008 2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanicalelectromechanicalBlock diagramsBlock diagramsLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••NyquistNyquistDesign specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratoriesFall 2008 3TimeTime--invariant & timeinvariant & time--varyingvaryingA system is called A system is called timetime--invariant (timeinvariant (time--varying) varying) if system parameters do not (do) change in time.if system parameters do not (do) change in time.Example: Example: MxMx’’’’(t(t)=)=f(tf(t) & ) & M(t)xM(t)x’’’’(t(t)=)=f(tf(t))For timeFor time--invariant systems:invariant systems:This course deals with timeThis course deals with time--invariant systems.invariant systems.SysSysTime shiftTime shiftTime shiftTime shiftFall 2008 4NewtonNewton’’s laws of motions laws of motion11ststlaw: law: A particle remains at rest or continues to move in a A particle remains at rest or continues to move in a straight line with a constant velocity if there is no straight line with a constant velocity if there is no unbalancing force acting on it.unbalancing force acting on it.22ndndlaw:law:: translational: translational: rotational: rotational33rdrdlaw: law: For every action has an equal and opposite reactionFor every action has an equal and opposite reactionFall 2008 5Translational mechanical elements:Translational mechanical elements:(constitutive equations)(constitutive equations)MassMassDamperDamperSpringSpringMf(tf(t))x(tx(t))f(tf(t))xx11(t)(t)KKxx22(t)(t)f(tf(t))f(tf(t))xx11(t)(t)BBxx22(t)(t)f(tf(t))Fall 2008 6MassMass--springspring--damper systemdamper systemMx(tx(t))KKBBFall 2008 7Free body diagramFree body diagramNewtonNewton’’s law: F=mas law: F=maMKKBBDirection of actual force will be automatically determined by the relative values!Fall 2008 8MassMass--springspring--damper systemdamper systemEquation of motionEquation of motionBy By Laplace transform Laplace transform (with zero initial conditions),(with zero initial conditions),Mx(tx(t))KKBB(2(2ndndorder system)order system)Fall 2008 9Gravity?Gravity?At rest, At rest, y coordinate:y coordinate:x coordinate:x coordinate:KKMKKMFall 2008 10Automobile suspension systemAutomobile suspension systemM2f(tf(t))xx22(t)(t)KK11BBKK22M1xx11(t)(t)automobileautomobilesuspensionsuspensionwheelwheeltiretireFall 2008 11Automobile suspension systemAutomobile suspension systemLaplace transform with zero ICsLaplace transform with zero ICsG2 G1G3FFXX22XX11Block diagramBlock diagramFall 2008 12Rotational mechanical elementsRotational mechanical elements(constitutive equations)(constitutive equations)Moment of inertiaMoment of inertiaFrictionFrictionRotational springRotational springJKKBBtorquetorquerotation anglerotation angleFall 2008 13TorsionalTorsionalpendulum system Ex.2.12pendulum system Ex.2.12JKKBBfriction between friction between bob and airbob and airFall 2008 14TorsionalTorsionalpendulum systempendulum systemEquation of MotionEquation of MotionBy By Laplace transform Laplace transform (with zero ICs),(with zero ICs),JKKBBfriction between friction between bob and airbob and air(2(2ndndorder system)order system)Fall 2008 15ExampleExampleByByNewtonNewton’’s laws lawBy By Laplace transform Laplace transform (with zero ICs),(with zero ICs),Fall 2008 16Example (contExample (cont’’d)d)From second equation:From second equation:From first equation:From first equation:G2Block diagramBlock diagramG1(2(2ndndorder system)order system)(4(4ththorder system)order system)Fall 2008 17Rigid satellite Ex. 2.13Rigid satellite Ex. 2.13ThrustorThrustorDouble Double integratorintegrator••BroadcastingBroadcasting••Weather forecastWeather forecast••CommunicationCommunication••GPS, etc.GPS, etc.Fall 2008 18Summary & ExercisesSummary & ExercisesModeling of mechanical systemsModeling of mechanical systemsTranslationalTranslationalRotationalRotationalNext, block diagrams.Next, block diagrams.ExercisesExercisesRead Sections 2.5, 2.6. Read Sections 2.5, 2.6. Derive equations for the automobile suspension Derive equations for the automobile suspension problem.problem.Fall 2008 19Exercises (Franklin et al.)Exercises (Franklin et al.)Quarter car modelQuarter car model: Obtain a transfer function : Obtain a transfer function from from R(sR(s) to ) to Y(sY(s).).Road surfaceRoad
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