Solution Manual For System Dynamics, 4th Edition

Preview (16 of 221 Pages)

100%

Solution Manual For System Dynamics, 4th Edition - Page 1 preview image

Loading page ...

ContentsPrefacevChapter2TheLaplaceTransform1Chapter3MechanicalSystems12Chapter4 Transfer-FunctionApproachtoModelingDynamicSystems~~24Chapter5State-SpaceApproachtoModelingDynamicSystems43Chapter6ElectricalSystemsandElectromechanicalSystems89Chapter7FluidSystemsandThermalSystems111Chapter8Time-DomainAnalysisofDynamicSystems125Chapter9Frequency-DomainAnalysisofDynamicSystems152Chapter10Time-DomaimAnalysisandDesignofControlSystems176Chapter11Frequency-DomainAnalysisandDesignofControlSystems198iii

Solution Manual For System Dynamics, 4th Edition - Page 2 preview image

Loading page ...

DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

Solution Manual For System Dynamics, 4th Edition - Page 3 preview image

Loading page ...

SolutionstoBProblemsB-2-1.£(t)=0t<0=te2tt>0Notethat_1PA=ReferringtoEquation(2-2),weobtainFs)=J[£(6)]=of,[bet]=—1(s+2)?B-2-2.(a)£,(6)=0t<0=3sin(5t+45°)tt=0Notethat3sin(5t+45%)=3sin5tcos45°+3cos5tsin45°=Fsin5t+fcos5tSowehaveF(s)=JIE(6)]=->—_-53_s1&1Z2+522s2+52=3s+5JZ52425(b)f(t)=0t<0=0.03(1-cos2t)t>0Fo(s)=J,[£5(6))=0.032-0.035___0.12ss2+22g(s2+4)1|

Solution Manual For System Dynamics, 4th Edition - Page 4 preview image

Loading page ...

B-2-3.£(t)=0t<0=t%@goNotethatLez)=2s3ReferringtoEquation(2-2),weobtainFs)=of)[£(t)]=J)[b%eat]=—2—(s+a)3B-2-4.£(t)=0t<0=cos2¢)tcos3tt>0Notingthatcos2ctcos3t=(cos5wt+coset)wehaveF(s)=LG[£(6)]=JlslcosSut+cosa/t)]_1s.sN(s?+13w?)s2\2vw?2+w?(2+25whE?+wh)B-2-5.Thefunctionf(t)canbewrittenas£(t)=(¢-a)1(t-a)TheLaplacetransformof£(t)is"asF(s)=§[£(6)]=f[(6-a)L(t-a)]=&—sB-2-6.£(t)=c1(t-a)-c1(t-b)TheLaplacetransformoff(t)isF(s)=c£95_oceS_c(gas_obs)sss2+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 5 preview image

Loading page ...

B-2-7.Thefunctionf(t)canbewrittenas£6)=29-223ac-2)+2:51c-a)aa?5aSotheLaplacetransformoff(t)becomesF(s)=0[e(e))=40-1_125L~(a/5)s,2.51asa:Ssasa2s=-L(10-12.5e7(a/5)s435o-as)alsAsaapproacheszero,thelimitingvalueofF(s)becomesasfollows:—(a/5_,limF(s)=lim-10-12.5¢(a/5)s42.5easa=0a=>0as2(10-12.5e"(a/5)s42.5~25)=mSSa—+02als=lim2:5se(@/b)s_356casa02asd——yn(2.5e7(a/5)s_55cas)=1ipROa—0daER2a:0.5se~(a/5)s42.5ge=@s=lim>a=0--=0.5s+2.5s_28_g22B-2-8.Thefunctionf(t)canbewrittenasfe)=-24¢-241(¢Ep:Np(t-a)1(t-a)a3a22a3SotheLaplacetransformoff(t)becomes.241241Yas_24eeFs)=—3"7-77s©a3§22a(1agas=)=T3\2Ts23|+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 6 preview image

Loading page ...

ThelimitingvalueofF(s)asaapproacheszerois--kas_g-asLinF(s)=lin2A=ase#8-e8)B€a—0a0ads=24(1-ase~'#as—e=as)=14mg—Q- -a—>02ads?-%asas?—24(-se+===e~%3s+geS)=1im————a>03a2s?_dgoe735+2%eas4gas)=lim92a->03alsS4s,8_-kas,85(-S)~%as_ge-asa]>.+e+565)ese=p———a->02asd-%as-as-asda(de-ase-4e7)=lim———————a0dd,da2I_“25eS_5eTyas5-¢BS,gge708=i———a—=+01=-25-s+4s=sB-2-9.f(ee)=1imf(t)=limsF(s)t->00s->0=lims5(s+2-5x2_jp0s(s+1)12-10.::()B-2-10£(0+)=1im £(t)=lims2s+2=0t—0+S»>o0s(s+1)(s+3)4+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 7 preview image

Loading page ...

B-2-11.Define,y=xThen.y(0+)=x(0+)Theinitialvalueofycanbeobtainedbyuseoftheinitialvaluetheoremasfollows:y(0+)=limsY(s)S200Since.¥(s)=JIy(£)]=Jf,[X(t)]=sX(s)~x(O+)weobtainy(0+)=lims¥(s)=lims[sX(s)~-x(0+)]s>0®S->00=lim[s2X(s)-sx(0+)]SUcdB-2-12.Notethatpd[+0]=sF(s)~£(0)dtPAEww)=s28(s)-s£(0)-£(0)dt?Definea(t)=2ze)dt?ThenBey=p=~g(05EsXL|90)=ss)-a0)=s[s2r(s)-s£(0)-£(0)]-£(0)=s38(s)-s2£(0)-s£(0)—£(0)B-2-13.aa.=s+512(a)Fs)“gins+3sF1573where+5=_4aEH,LE=5*5=-2-_azx=)15

Solution Manual For System Dynamics, 4th Edition - Page 8 preview image

Loading page ...

Fy(s)canthusbewrittenas2__1Ble)=g3T7543andtheinverseLaplacetransformofFy(s)isf(t)=2etet(b)P(e)mera)of,"2fs2s(s+1)(s+2)'ss+1s+2where=3(s+4)=3x4_5UT+2)2s=0a=3(s+4)=_3xX3_92"s(s+2)(-1)x1s=-1a=3(s+4)=3x2_g43s(s+1)(-2)(-1)s=-2F(s)canthusbewrittenasF(s)=-6__-9,_32ss+1s+2andtheinverseLaplacetransformofF(8)is£,(t)=6-9et+3eltB-2-14.6s+363(a)F(s)=——5==2+—5—1<2s52TheinverseLaplacetransformofF(s)isf(t)=6+3tA)6+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 9 preview image

Loading page ...

(b)abbSSPESELEEL.TH2(s+1)(s+2)3(s+2)where55+2=5+2_4a=—0712+2(s)s=-1_5s+2_-10+2_by=|2+1°s=-2b=(2H)_5(s+1)-(5s+2“3\s+121dss=-2(s+1)s=-2_5(-1)-(-10+2)_512F(s)canthusbewrittenas-.83F(s)==,8,22s+1l(s+2)2s+2andtheinverseLaplacetransformofF,(s)is£,(t)=-3etspre?+32B-2-15.252+45+5_25F=E&22720=2+<=©)=eDs+17s(s+1)—2+-2+55_,__3,5s+1ss+1s+1sTheinverseLaplacetransformofF(s)isfe)=28()-3eT+57|T=StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 10 preview image

Loading page ...

B-2-16.2F(s)_-St+25+d424s2ss2TheinverseLaplacetransformofF(s)is£6)=&(t)+2+4tB-2-17.ro).CCev1-as2+2s+10(s+1)2+32=..s8*131(s+1)2+32(s+1)2+323Hence£(t)=etcos3t-+e-tsin3tB-2-18.Fs)=-82*t2s+5_.a,b,cs2(s+1)2ss+1wherea=_82+25+5-5s+1s=0p={25+2)(s+1)-(s2+25+5)22-5.3(s+1)21s=02—c=-_S2+25+5=1-2+5_s21s=-1Hence5=34F(s)=—&5+—+———(=)s2ss+1TheinverseLaplacetransformofF(s)is£(t)=5t-3+4et8+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 11 preview image

Loading page ...

B-2-19.210bcSs+aFs)=—="—=—2—4——— ———(s+1)2(s+4)(s+1)2s+1s+4wherea=_25+10--2+10__8s+433s=-1p=-2(s+4)-(25+10)-6-8_-2==>=(s+4)239s=-1c=_25+10_=8+10_2(s+1)299s=-4Hence8rr2r,2_1F(s)3(s+1)29s+19s+4TheinverseLaplacetransformofF(s)isF(t)=8tet-2ot42gat399B-2-20.1111Fe)=t=(hz)s2(s?+w?)s252+w?2Jw?TheinverseLaplacetransformofF(s)isf(t)=1(t-1sinct)w?wB-2-21.—=Cc_e-as)_-B_o-asF(s)52(1-e7%)Ss©a>o0TheinverseLaplacetransformofF(s)isf(t)=ct-c(t-a)l(t-a)-b1(t-a)9I~StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 12 preview image

Loading page ...

B-2-22.¥+ax=o0,x(0)=5,%(0)=0TheLaplacetransformofthegivendifferentialequationis[s2ZxX(s)-sx(0)~%(0)]+4X(s)=0Substitutionoftheinitialconditionsintothislastequationgives(s2+4)X(s)=5sSolvingforX(s),weobtain5sX(s)=—==2—s2+4TheinverseLaplacetransformofX(s)isx(t)=5cos2tThisisthesolutionofthegivendifferentialequation.B-2-23.¥+wZx=t,x(0)=0,x(0)=0TheLaplacetransformofthisdifferentialequationis$2x(s)+ew2x(s)=—L-n2sSolvingthisequationforX(s),weobtainXs)=—Lof111s2(s2+w?22+w2)W2TheinverseLaplacetransformofX(s)isx(t)=—1—(¢-2singt)wy?WnnThisisthesolutionofthegivendifferentialequation.B-2-24.X+2%+x=1,x(0)=0,x(0)=2TheLaplacetransformofthisdifferentialequationis2[s%K(s)-sx(0)-%(0)]+2[sX(s)-x(0)]+X(s)=110|T=StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 13 preview image

Loading page ...

Substitutionoftheinitialconditionsintothisequationgives2[s2(s)-21+2[sK(s)]+X(s)=L-or:(252+2s+1)X(s)=4+SolvingthislastequationforX(s),wegetX(s)=—&XLs(2s?+2s+1)[ENSEEO—262+25+1(2s?+25+1)_20.5=(s+0.5)2+0.25sls+0.5)%+0.25]_4x0.541__(s+0.5)+0.5(s+0.52+0525(s+0.52+0.5TheinverseLaplacetransformofX(s)givesx(t)=4e705gin0.5t+1-e705cos0.5-e705sin0.5¢=1+3¢e05gin0.5t-0-5cos0.5B-2-25.X+x=sin3t,x(0)=0,*(0)=0TheLaplacetransformofthisdifferentialequationis3s2(s)+X(s)==—>—s?+32SolvingthisequationforX(s),wegetXs)=—u33113(2+1)(s2+9)8s2+18&240TheinverseLaplacetransformofX(s)givesx(t)=2sint-+sin3t11i+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 14 preview image

Loading page ...

B-3-1.J=%mR?=%x100x0.52=12.5kgm?B-3-2.AssumethatthebodyofknownmomentofinertiaJisturnedthroughasmallangle6abouttheverticalaxisandthenreleased.Theequationofmotionfortheoscillationis§=-Jk8wherekisthetorsionalspringconstantofthestring.Thisequationcanbewrittenas+X0-00or-2+=8ow,8=0wherekWw=[=nJoTheperiodTgofthisoscillationisny=2K.2()[oNV%Next,weattacharotatingbodyofunknownmomentofinertiaJandmeasuretheperiodTofoscillation.TheequationfortheperiodTisT=2m(2)[x_JByeliminatingtheunknowntorsionalspringconstantkfromEquations(1)and(2),weobtain23,2mfFToTHence2TJ=—0,TheunknownmomentofinertiaJcanthereforebedeterminedbymeasuringtheperiodofoscillationTandsubstitutingitintoEquation(3).12|T=StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 15 preview image

Loading page ...

B-3-3.Definetheverticaldisplacementoftheballasx(t)withx(0)=0.Thepositivedirectionisdownward.Theequationofmotionforthesystemis.mx=mgwithinitialconditionsx(0)=Omandx(0)=0m/s.SowehaveX=gx=gt+x(0)=gtx=%gt?+x(0)=%gt?xXAssumethatatt=ttheballreachestheground.Then100m100=%x9.807t;2fromwhichweobtain=4.516s5Theballreachesthegroundin4.516s.Thevelocityoftheballwhenithitsthegroundis%(4.516)=9.807x4.516=44.288m/sB-3-4.DefinethetorqueappliedtotheflywheelasT.TheequationofmotionforthesystemisgB=1, 80)=0 6(0)=0fromwhichweobtain6=-L[:]7tBysubstitutingnumericalvaluesintothisequation,wehave=_T20x6.28=5pX5ThusT=1256N-mB-3-5.Jo=-1(T=brakingtorque)Integratingthisequation,6=-Le+8(0),6(0)=100rad/sSubstitutingthegivennumericalvalues,20=--Lx15+100J13v+StudyXY

Solution Manual For System Dynamics, 4th Edition - Page 16 preview image

Loading page ...

SolvingforT/J,weobtainT_T=5.33Hence,thedecelerationgivenbythebrakeis5.33rad/s2.Thetotalanglerotatedin15-secondperiodisobtainedfromTt?,45a(t)=--+6(0)t+6(0),68(0)=0,8(0)=100asfollows;28(15)=-5.33x£-+100x15=900radB-3-6.TheequationsforthesystemareFk(x-y)#£;WFk(x-y)=ky}¥lxEliminatingyfromthetwoequationsgivesk.Kk,k.Fek,(2=12p11xKktk,|kytk,EE12Theequivalentspringconstanteqisthenobtainedas=eqTI,LTk,k2Next,considerthefigureshownbelow.NotethatAABDandACBEaresimilar.SowehaveEmADBDwhichcanberewrittenasAkeg.=H_axl2OB-OC—5—xyPTTus=G+Loon22OB+0A—orTED0EkB0C(0B+%OA)=OA(OB-%OC)Solvingforoc,weobtain14Vv+StudyXY

Preview Mode

This document has 221 pages. Sign in to access the full document!

Report

Study Now!

Document Details

Related Documents

A system is a group of parts that work together to

Solution Manual For Thermal Radiation Heat Transfer, 6th Edition

Solution Manual For Thermodynamics: An Interactive Approach, 1st Edition

Solution Manual for Introduction to Robotics: Mechanics and Control, 4th Edition

Solution Manual For Applied Strength Of Materials, 5th Edition

Solution Manual for Machine Elements in Mechanical Design, 6th Edition

Solution Manual For Theory Of Vibrations With Applications, 5th Edition

Solution Manual For Design Of Machine Elements, 8th Edition

Mechanical Vibrations: Theory and Applications 1st Edition Solution Manual

Solution Manual for Fluid Mechanics for Engineers, 2017 Edition

Company

Explore

Study Tools