solutions-manual-feedback-control-of-dynamic-systems-franklin-5th-edition

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1、Chapter2DynamicModelsProblemsandSolutionsforSection2.11.WritethedifferentialequationsforthemechanicalsystemsshowninFig.2.38.Solution:ThekeyistodrawtheFreeBodyDiagram(FBD)inordertokeepthesignsright.For(a),toidentifythedirectionofthespringforcesontheobject,letx2=0andÞxedandincreasex1from0.Thenthek1sp

2、ringwillbestretchedproducingitsspringforcetotheleftandthek2springwillbecompressedproducingitsspringforcetotheleftalso.Youcanusethesametechniqueonthedamperforcesandtheothermass.(a)m1x¨1=−k1x1−b1xú1−k2(x1−x2)m2x¨2=−k2(x2−x1)−k3(x2−y)−b2xú21112CHAPTER2.DYNAMICMODELSFigure2.38:Mechanicalsystems13m1x¨1

3、=−k1x1−k2(x1−x2)−b1xú1m2x¨2=−k2(x2−x1)−k3x2m1x¨1=−k1x1−k2(x1−x2)−b1(xú1−xú2)m2x¨2=F−k2(x2−x1)−b1(xú2−xú1)2.Writetheequationsofmotionofapendulumconsistingofathin,2-kgstickoflengthlsuspendedfromapivot.Howlongshouldtherodbeinorderfortheperiodtobeexactly2secs?(TheinertiaIofathinstickaboutanendpointis1

4、ml2.Assumeθissmallenoughthatsinθ∼=θ.)3Solution:Let’suseEq.(2.14)M=Iα,14CHAPTER2.DYNAMICMODELS
lO2
mgMomentaboutpointO.lMO=−mg×sinθ=IO¨θ2=1ml2¨θ33g¨θ+sinθ=02lAsweassumedθissmall,3g¨θ+θ=02lThefrequencyonlydependsonthelengthoftherod23gω=2ls2π2lT==2π=2ω3g3gl==1.49m2π215Figure2.39:Doublependulum

5、q(a)Comparetheformulafortheperiod,T=2π2lwiththewellknown3gformulafortheperiodofapointmasshangingwithastringwithqlengthl.T=2πl.g(b)Important!Ingeneral,Eq.(2.14)isvalidonlywhenthereferencepointforthemomentandthemomentofinertiaisthemasscenterofthebody.However,wealsocanusetheformularwithareferencepoin

6、totherthanmasscenterwhenthepointofreferenceisÞxedornotaccelerating,aswasthecasehereforpointO.3.Writetheequationsofmotionforthedouble-pendulumsystemshowninFig.2.39.Assumethedisplacementanglesofthependulumsaresmallenoughtoensurethatthespringisalwayshorizontal.Thependulumrodsaretakentobemassless,ofle

7、ngthl,andthespringsareattached3/4ofthewaydown.Solution:16CHAPTER2.DYNAMICMODELS3l

412kmm33lsin
lsin
1244Ifwewritethemomentequilibriumaboutthepivotpointoftheleftpen-dulemfromthefreebodydiagram,M=−mglsinθ−k3l(sinθ