intro to statics and dynamics - a. ruina, r. pratap

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1、SIntroductiontoTATICSDandYNAMICSMsFskˆˆıˆFF21N1N2AndyRuinaandRudraPratapPre-printforOxfordUniversityPress,January2002SummaryofMechanics0)Thelawsofmechanicsapplytoanycollectionofmaterialor‘body.’Thisbodycouldbetheoverallsystemofstudyoranypartofit.Intheequationsbelow,theforcesandmom

2、entsarethosethatshowonafreebodydiagram.Interactingbodiescauseequalandoppositeforcesandmomentsoneachother.I)LinearMomentumBalance(LMB)/ForceBalance˙EquationofMotionFi=LThetotalforceonabodyisequal(I)toitsrateofchangeoflinearmomentum.t2Impulse-momentumFi·dt=LNetimpulseisequaltot

3、hechangein(Ia)(integratingintime)t1momentum.˙ConservationofmomentumL=0⇒Whenthereisnonetforcethelinear(Ib)(ifFi=0)L=L2−L1=0momentumdoesnotchange.Statics˙Fi=0Iftheinertialtermsarezerothe(Ic)(ifLisnegligible)netforceonsystemiszero.II)AngularMomentumBalance(AMB)/MomentBalan

4、ce˙EquationofmotionMC=HCThesumofmomentsisequaltothe(II)rateofchangeofangularmomentum.t2Impulse-momentum(angular)MCdt=HCThenetangularimpulseisequalto(IIa)(integratingintime)t1thechangeinangularmomentum.˙ConservationofangularmomentumHC=0⇒Ifthereisnonetmomentaboutpoint(

5、IIb)(ifMC=0)HC=HC2−HC1=0CthentheangularmomentumaboutpointCdoesnotchange.Statics˙MC=0Iftheinertialtermsarezerothenthe(IIc)(ifHCisnegligible)totalmomentonthesystemiszero.III)PowerBalance(1stlawofthermodynamics)EquationofmotionQ˙+P=E˙K+E˙P+E˙intHeatflowplusmechanicalpower(III)in

6、toasystemisequaltoitschangeE˙inenergy(kinetic+potential+internal).t2t2forfinitetimeQdt˙+Pdt=EThenetenergyflowgoinginisequal(IIIa)t1t1tothenetchangeinenergy.ConservationofEnergyE˙=0⇒Ifnoenergyflowsintoasystem,(IIIb)(ifQ˙=P=0)E=E2−E1=0thenitsenergydoesnotchange.StaticsQ˙+P=E˙P+E˙intIf

7、thereisnochangeofkineticenergy(IIIc)(ifE˙Kisnegligible)thenthechangeofpotentialandinternalenergyisduetomechanicalworkandheatflow.PureMechanics(ifheatflowanddissipationP=E˙K+E˙PInasystemwellmodeledaspurely(IIId)arenegligible)mechanicalthechangeofkineticandpotentialenergyisduetomechanic

8、alwork.SomeD