CramX Logo
RegisterLog in
Solution Manual For System Dynamics, 4th Edition - Document preview page 1

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

Document preview content for Solution Manual For System Dynamics, 4th Edition

Solution Manual For System Dynamics, 4th Edition

Solution Manual For System Dynamics, 4th Edition offers textbook solutions that are easy to follow, helping you ace your assignments.

Eva Reed
Contributor
4.7
0
12 months ago
Preview (16 of 221 Pages)
100%
Log in to unlock
Page 1 of 16
Solution Manual For System Dynamics, 4th Edition - Page 1 preview imageContentsPrefacevChapter2TheLaplaceTransform1Chapter3MechanicalSystems12Chapter4 Transfer-FunctionApproachtoModelingDynamicSystems~~24Chapter5State-SpaceApproachtoModelingDynamicSystems43Chapter6ElectricalSystemsandElectromechanicalSystems89Chapter7FluidSystemsandThermalSystems111Chapter8Time-DomainAnalysisofDynamicSystems125Chapter9Frequency-DomainAnalysisofDynamicSystems152Chapter10Time-DomaimAnalysisandDesignofControlSystems176Chapter11Frequency-DomainAnalysisandDesignofControlSystems198iii
Page 2 of 16
Solution Manual For System Dynamics, 4th Edition - Page 2 preview imageDownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com
Page 3 of 16
Solution Manual For System Dynamics, 4th Edition - Page 3 preview imageSolutionstoBProblemsB-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|
Page 4 of 16
Solution Manual For System Dynamics, 4th Edition - Page 4 preview imageB-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
Page 5 of 16
Solution Manual For System Dynamics, 4th Edition - Page 5 preview imageB-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
Page 6 of 16
Solution Manual For System Dynamics, 4th Edition - Page 6 preview imageThelimitingvalueofF(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
Page 7 of 16
Solution Manual For System Dynamics, 4th Edition - Page 7 preview imageB-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
Page 8 of 16
Solution Manual For System Dynamics, 4th Edition - Page 8 preview imageFy(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
Page 9 of 16
Solution Manual For System Dynamics, 4th Edition - Page 9 preview image(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
Page 10 of 16
Solution Manual For System Dynamics, 4th Edition - Page 10 preview imageB-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
Page 11 of 16
Solution Manual For System Dynamics, 4th Edition - Page 11 preview imageB-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
Page 12 of 16
Solution Manual For System Dynamics, 4th Edition - Page 12 preview imageB-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
Page 13 of 16
Solution Manual For System Dynamics, 4th Edition - Page 13 preview imageSubstitutionoftheinitialconditionsintothisequationgives2[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
Page 14 of 16
Solution Manual For System Dynamics, 4th Edition - Page 14 preview imageB-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
Page 15 of 16
Solution Manual For System Dynamics, 4th Edition - Page 15 preview imageB-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
Page 16 of 16
Solution Manual For System Dynamics, 4th Edition - Page 16 preview imageSolvingforT/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!

Study Now!

X-Copilot AI
Unlimited Access
Secure Payment
Instant Access
24/7 Support
Document Chat

Document Details

Subject
Mechanical Engineering
Textbook
System Dynamics by Katsuhiko Ogata

Related Documents

View all
Solved Assignments

A system is a group of parts that work together to

Textbook Guides

Solution Manual For Thermal Radiation Heat Transfer, 6th Edition

4.6
00
Textbook Guides

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

5.0
00
Textbook Guides

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

4.2
00
Textbook Guides

Solution Manual For Applied Strength Of Materials, 5th Edition

4.2
00
Textbook Guides

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

4.7
00
Textbook Guides

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

4.0
00
Textbook Guides

Solution Manual For Design Of Machine Elements, 8th Edition

4.5
00
Textbook Guides

Mechanical Vibrations: Theory and Applications 1st Edition Solution Manual

4.8
00
Textbook Guides

Solution Manual for Fluid Mechanics for Engineers, 2017 Edition

4.4
00