Engineering Basic Mechanics Ⅱ Dynamics 工程基础力学 Ⅱ 动力学 课件 Chapter 8 Theorem of Kinetic Energy

Chapter8

TheoremofKineticEnergyMainContents§8.1Workdonebyforces§8.2Kineticenergyofaparticleandasystemofparticles§8.3Principleofworkandkineticenergy§8.4Power·powerequation·efficiencyofamachine§8.5Conservativeforcesfield·potentialenergy·thelawofconservationofmechanicalenergy§8.6Integratedapplicationexamplesofgeneraltheorem

Therearemanyformsofmotioninnature,theseformsofmotionareessentiallydifferentfromeachother,butinterdependentandinterrelated,andundercertainconditions,betransformedintoeachother.Forexample:mechanicalmotioncanbetransformedintoelectricity,heat,sound,light,magnetism,etc,conversely,electricity,heat,sound,light,magnetismcanbetransformedintomechanicalmotion.Mechanicalenergy:Theenergyofmechanicalmotionofbodyiscalledmechanicalenergy,itincludeskineticenergy,potentialenergy.Varioustransformationofformsofmotioncontacteachotherbytheenergy.Energyisameasurementofvariousformsofmotion.Inthischapterwestudythecontactsofkineticenergy,potentialenergyandworkdonebyforcesandpower,powerequationetc.PrincipleofworkandkineticenergyPrincipleofworkandkineticenergy6.TheconditionsoftheconservationofmechanicalenergyKeypointsinthischapter:2.Thelawoftheconservationofmechanicalenergy1.Principleofworkandkineticenergy3.Calculationofworkdonebyforces4.Calculationofkineticenergyofasystemofparticles5.CalculationofpotentialenergyofaparticleandasystemofparticlesTheforceactingonanobjectcanchangethestateofmotionoftheobject,andthisstatechangeisnotonlyrelatedtothemagnitudeanddirectionoftheforce,butalsorelatedtotheprocessofforceaction:

※Thisprocesscanbemeasuredintermsoftime,forexample,theimpulseofaforcedescribesthecumulativeeffectofaforceontime.

※Theworkofaforce,ontheotherhand,representsthecumulativeeffectoftheforceonspace,thatis,theeffectoftheforceonanobjectoveradistance.1.Workofaconstantforce§8.1workdonebyforcesistheanglebetweenforceandthedirectionoflineardisplacement.Theworkisascalarquantity,

intheInternationalSystemofUnits,theunitofworkis.2.Workofavariableforce

InthefixedsystemOxyzThentheworkdonebytheforce

onthisdrdisplacementis§8.1workdonebyforcesWhen，，forcedoesnegativework;When，，forcedoespositivework;Itcanbeseenthat：When，，forcedoesnotwork.1.Theworkofaforceisascalarquantity,dristhesmalldisplacementoftheparticleinthedirectionoftheforce;2.δw=Fxdx+Fydy+Fzdz,ingeneral,notnecessarilythefulldifferentialofafunction,sowriteδW,ratherthandW;§8.1workdonebyforcesLetactonaparticleM,anditsresultantforceisProveIftheforceFactingonthepointMmovesalongthecurvefrompointM1topointM2,theworkdonebyFduringthisdistanceisTheorem:Theworkdonebytheresultantforceactingonaparticleoveracertaindistanceisequaltothealgebraicsumoftheworkdonebythecomponentforcesoverthesamedistance.§8.1workdonebyforcesThentheelementaryworkoftheresultantforceisTheworkdonebytheresultantforceonis§8.1workdonebyforces(a)Workofaweight§8.1workdonebyforcesWorkdonebygravityisonlyrelevantwiththeheightdifferenceofthebeginningandendpositionofparticlemotion,irrelevantwiththeshapeoftrajectory.3.WorkdonebycommonforcesWhenthepositionoftheparticleiselevated(z2>z1),gravitydoesnegativework.Whenthepositionoftheparticleislowered(z2<z1),gravitydoespositivework.Forasystemofparticles:Letthemassofparticlebeandtheheightdifferencebetweenthebeginningandendofthemotionbe,thetotalworkofgravityactingonthesystemofparticlesis§8.1workdonebyforces§8.1workdonebyforces(b)WorkofaspringforceTheobjectissubjectedtoanelasticforceanddoesusethetrajectoryofthepointAasshowninthebluecurveA1A2.Intheelasticlimitofaspring,themagnitudeoftheelasticforceisproportionaltoitsdeformationδThedirectionoftheforcealwayspointstothenaturalpositionwhenitisnotdeformed.Theproportionalityfactorkiscalledthespringstiffnessfactor(orstiffnessfactor).IntheInternationalSystemofUnits,theunitofkisN/morN/mm.ThepointAmovesfromA1toA2andtheworkoftheelasticforceis:WiththepointOastheorigin,thevectordiameterofpointAisanditslengthis.Lettheunitvectoralongthevectordirectionbe,andthenaturallengthofthespringbe.ThentheelasticforceWhenthespringisextended,,theforceisintheoppositedirectionto.Whenthespringiscompressed,,theforceisinthesamedirectionas.§8.1workdonebyforcesItcanbeseenthat：1.theworkdonebytheelasticforceoveracertaindistanceisequaltohalfoftheproductofthestiffnessofthespringandthesquareddifferencebetweenitsbeginningandendpositionsofdeformation;Orwriteas2.theworkoftheelasticforceisonlyrelatedtothestartingdeformationandtheenddeformationofthespring,andnottothetrajectory(path)ofthemotionofthemass,anditselementaryworkcanbewrittenasthefulldifferentialofafunction;3.thedeformationofthespringisrelativetothenaturalstateofthespring(i.e.theoriginallengthofthespring);Whereisthedeformationofthespringatthebeginningandendposition§8.1workdonebyforcesLettherigidbodyrotatearoundafixedaxisOz,andtheforceactsontherigidbodyatpointA.ThevectordiameterofpointAwithrespecttopointOis,andRistheverticaldistancefromthepointAtothez-axis.§8.1workdonebyforces(c)Workofforceactingonarigidbodyinfixed-axisrotation（

RigidbodyrotatesaroundOzaxisinafixedaxis）ThenaturalcoordinatesystemisestablishedoverthepointA,andistheunitpositivevectorofthenaturalcoordinatesystem,respectively.§8.1workdonebyforcesBecauseisequaltomomentofforceabouttherotatingaxisz,thusTheworkdonebytheforceofarigidbodyintherotationprocessfromtheangletoisIfarigidbodysubjectstoacouple,theworkofthecouplestillcomputesthroughtheaboveequation,

ismomentofthecoupleaboutrotatingaxisz,isalsoequaltotheprojectionofmomentvectorMofthecoupleinz-axis.TheelementaryworkofforceisWhentherigidbodyrotatesaroundthefixedaxis,therelationshipbetweentheangleofrotationφ

andthearclengthS

is§8.1workdonebyforces(d)Workofforcesactingonarigidbodyinplanemotion

Workofforcesactingonarigidbodyinplanemotionisequaltothesumofworkdonebytheforceandcouplewhichforcesystemsimplifiedtothemasscenter.(e)WorkdonebygravitationalforceThedifferentialworkis

§8.1workdonebyforcesisthegravitationalforceactingontheparticlem,

isradiusvectoroftheparticlem.

Workdonebygravitationalforceisindependentofthepath,dependsonlytheinitialandfinalposition,thedifferentialworkisthetotaldifferentialofafunction.ThefrictionforceactingatthecontactpointBisthestaticslidingfrictionthatpreventstherelativeslidingbetweenthetwoobjects,thenthedifferentialworkof

is(f)WorkofforceactingontheinstantaneouscenterofvelocityArigidbodyrollswithoutslidingalongafixedsurface.Whenarigidbodymakespurerollingalongfixedsurface,workoffictionactingoncontactpointiszero.Generallyspeakingthedifferentialworkofforceactingoninstantaneouscenterofvelocityiszero.§8.1workdonebyforcesbecausepointBisinstantaneouscenterofvelocityoftherigidbodyThusThus

Thekineticenergyofamaterialbodyistheenergyduetothemotion,thekineticenergyisameasureofthemechanicalmotion.1.KineticenergyofaparticleAssumingthemassofaparticleis

,thevelocityis

,thenthekineticenergyoftheparticleisKineticenergyisascalarquantity,isalwayspositive.IntheInternationalSystemofUnitstheunitofkineticenergyis.2.KineticenergyofasystemofparticlesThearithmeticsumofthekineticenergiesofalloftheparticlesiscalledthekineticenergyofthesystemofparticles,thatisRigidbodyisasystemofparticlescomposedofnumerousparticles,themotionofrigidbodyisdecomposedintotranslationwiththecenterofmassandrotationwithrespecttothecenterofmass,accordingtothiscalculatingthekineticenergyofsomeproblemsismoreconvenient：§8.2kineticenergyofaparticleandasystemofparticles(a)KineticenergyofrigidbodyintranslationalmotionWhere

ismassoftherigidbody.(b)Kineticenergyofrigidbodyinrotationalmotionaboutafixed-axisTherigidbodyrotateswithangularvelocityaroundthez-axisfixedaxis,showninfigure§8.2kineticenergyofaparticleandasystemofparticles

istheverticaldistanceofparticletotherotatingaxis.Hencekineticenergyofrigidbodyinrotationalmotionaboutafixed-axisisthevelocityofanyparticleofmass

isWhenrigidbodymaketranslation,velocitiesofeachparticlearethesame,isequaltothevelocityofthecenterofmass,therefore,kineticenergyofthetranslationofarigidbodyis(c)KineticenergyofrigidbodyinplanemotionTaketheplaneinwhichthecenterofmassCoftherigidbodyislocatedasshowninthefigure.ThisinstantaneousmotionoftherigidbodycanbeseenasaninstantaneousfixedaxisrotationaroundthepointPaxis,sothekineticenergyoftherigidbodyinplanemotionisAtdifferentinstantaneoustimes,therigidbodyhasdifferentpointsasinstantaneouscenters,soitisinconvenienttocalculatethekineticenergyusingtheaboveequation.§8.2kineticenergyofaparticleandasystemofparticlesLetthepointPinthegraphbetheinstantaneouscenterofvelocityatacertaininstant,andωbetheangularvelocityofrotationoftheplanegraph.whereJPistherotationalinertiaoftherigidbodywithrespecttotheinstantaneousaxis.Thekineticenergyofarigidbodyinplanemotionisequaltothesumofthekineticenergyoftranslationwiththecenterofmassandthekineticenergyofrotationaroundthecenterofmass.§8.2kineticenergyofaparticleandasystemofparticlesIfCisthecenterofmassofarigidbody,accordingtotheparallelaxistheoremforcalculatingtheinertiaofrotationwehavewheremisthemassoftherigidbody,d=CP,andJCistherotationalinertiaoftherigidbodywithrespecttothecenterofmassaxis.Takingintoaccounttheformulatocalculatethekineticenergy,wegetBecauseof,itfollowsthat§8.3principleofworkandkineticenergy1.PrincipleofworkandkineticenergyofaparticleMultiplyingbothsidesoftheequationby

,

weobtainSince

,theaboveequationcanbewrittenasorThisisthedifferentialformofthekineticenergytheoremoftheparticle:thedifferentialofthekineticenergyoftheparticleisequaltotheelementaryworkoftheforce(resultantforce)actingontheparticle.VectorformofdifferentialequationofparticlemotionThatisthedifferentialformoftheprincipleofworkandkineticenergyofaparticle.§8.3principleofworkandkineticenergy

whereW1,2representstheworkdonebytheforce(resultantforce)actingontheparticleinthepathfrompointM1topointM2.Thisistheintegralformofthekineticenergytheoremoftheparticle:theincrementofthekineticenergyoftheparticleinacertaindisplacementisequaltotheworkdonebytheforce(resultantforce)actingontheparticleinthesamedisplacement.Thistheoremappliesonlytotheinertialsystem,vanddrarethevelocityanddisplacementrelativetotheinertialsystem.Ifthesystemcontainsnparticles,therewillbenequationssimilartotheaboveequation.Addingalloftheseequations,weobtain§8.3principleofworkandkineticenergy2.principleofworkandkineticenergyofasystemofparticleoristhedifferentialformoftheprincipleofworkandkineticenergyofasystemofparticlesAnymasspointinthesystemofmasseswithmassandvelocity,accordingtothedifferentialformofthekineticenergytheoremofmasses,haswhereisthekineticenergyofthemasssystem,expressedinT,sotheaboveequationcanbewrittenasIntegratingtheaboveequation,wegetTheintegralformoftheprincipleofworkandkineticenergyofasystemofparticles:Thechangeinkineticenergyofasystemofmassesduringacertainperiodofmotion,atthebeginningandattheend,isequaltothesumoftheworkdonebyalltheforcesactingonthesystemofmassesduringthisperiod.Thedifferentialformoftheprincipleofworkandkineticenergyofasystemofparticles：Theincrementofthekineticenergyofthemasssystemisequaltothesumoftheelementaryworkdonebyalltheforcesactingonthemasssystem.§8.3principleofworkandkineticenergyIntheaboveequation,T1andT2arethekineticenergiesofthemasssystematthebeginningandendofacertainperiodofmotion,respectively.Example

8-1AsshowninFigure,ahomogeneousplateweightQisplacedontwohomogeneouscylindricalrollers,eachweightisQ/2andradiusisr.IfahorizontalforcePisappliedtotheplate,andlettherollershavenorelativeslidingwithrespecttothegroundandtheplate,determinetheaccelerationoftheplate.§8.3principleofworkandkineticenergy

§8.3principleofworkandkineticenergyExample

8-1Thekineticenergyofsystem:

§8.3principleofworkandkineticenergyExample

8-1θOMDCExample8-2Windlassisshowninfigure.DrumsubjectedtotheconstantcoupleMwillpullthecylinderalongtheslope.TheradiusofdrumisR1,themassism1,themassdistributeintherim;theradiusofcylinderisR2,themassism2,massuniformlydistributes.Assumingtheangleofslopeisθ,thecylinderonlyrollswithoutslipping.Thesystembegantomovefromthestatic,determinethevelocityandaccelerationwhenthecenterCofcylinderpassedthroughthedistances.§8.3principleofworkandkineticenergyExample

8-2θOMDCω2ω1Workdonebyactiveforceiscalculatedasfollows:KineticenergyofthesystemofparticlesiscalculatedasfollowsWhereJ1,JCarerespectivelythemassmomentofinertiaofdrumaboutthecentralaxisO,andthemassmomentofinertiaofcylinderabouttheaxisthoughmasscenterC:ω1andω2arerespectivelyangularvelocityofdrumandcylinder,thatis§8.3principleofworkandkineticenergyExample

8-2（1）HenceApplyingtheprincipleofworkandkineticenergySubstitutingintoaboveequation,weobtain§8.3principleofworkandkineticenergyθOMDCω2ω1Example

8-2Inprocessofmotionofthesystem，velocityanddistanceSarefunctionsoftime,determinethefirstderivativeofthetwosideofequation（1）abouttime,weobtainTheaccelerationofcylindercenterCis§8.3principleofworkandkineticenergyθOMDCω2ω11.PowerIntheInternationalSystemofUnits,

unitofpowerisW,§8.4powerpowerequationefficiencyofamachineWorkdonebyaforceinaunitoftimeiscalledpower,indicatingPPowerofanactingforce:Powerofthemomentofaforce：2.PowerequationTakingthedifferentialformofprincipleofworkandkineticenergyofsystemofparticles,

dividingbothsideby,

weobtainThatispowerequation§8.4powerpowerequationefficiencyofamachine<A>drivingforce：imposingtothemachinefromtheoutside,theforceofdrivingmachinework,intheprocessofmachinework,theseforcesdopositivework,forexamplemotortorque,theliquidpressureinthehydraulictransmissionetc.<B>usefulresistance(productiveresistance):forexamplethecuttingforcewhenmachineworks,theimpactresistanceofworkpieceactingonmachinewhenpunchprocessing,loadofcrane,etc.Theseforcesconsumeenergy,donegativework,buttheyareessential.<C>uselessresistance(detrimentalresistance):forexamplethefrictionalresistancebetweenthecontactsurfaceswhenmachineisrunning,airresistanceetc.theseforcesconsumeenergyinvain,donegativework.§8.4powerpowerequationefficiencyofamachineInmechanicalengineering,weusuallyuseanotherformofpowerequation,becausemachineisatworktoinputcertainwork(power,energy),atthesametimetoovercomeacertainresistance,thentoconsumeoroutputpartofthework(powerenergy),hence,inmechanicalengineering,weanalyzeforcebythefollowingway.Thus,

thedifferentialformofprincipleofworkandkineticenergycanbewrittenasδWuseless：AbsolutevalueofdifferentialworkofuselessresistanceDividingbothsideby,

weobtainThatispowerequationofmachine.AboveequationcanalsobewrittenasItsphysicalmeaningis：inputpowerofthemachine（drivepower）consumeinthreeparts：therequiredpowerofmakingmachinerunning,therequiredpowerofovercomingusefulresistance,therequiredpowerofovercominguselessresistance.δWInput：AbsolutevalueofdifferentialworkofdrivingforceδWuseful：AbsolutevalueofdifferentialworkofusefulresistancedT=

δWInput-

δWuseful-δWuselessdT/dt=

PInput-

Puseful-PuselessPInput

=dT/dt-Puseful-Puseless§8.4powerpowerequationefficiencyofamachineAnalysis：inastartingstate：

inabrakingstate：

inasteadystate:3.EfficiencyofamachineTheefficiencyofthemachineTheefficiencyofthemachineshowthattheeffectiveutilizationdegreeofmachineaboutinputpower,itisoneofthemostimportantindicesforjudgingthequalityofamachine.Ingeneral,PInput

<Puseful+PuselessPInput

=Puseful+PuselessPInput

>Puseful+PuselessTheusefulpower=Puseful+

dT/dt§8.4powerpowerequationefficiencyofamachine§8.5conservativeforcesfieldpotentialenergy(1)forcefield:

ifinaregionofspaceanyparticleexperiencesaforceofacertainmagnitudeanddirectiondependingonposition,theregionofspaceiscalledtobea

forcefield.1.Conservativeforcesfield(3)gravityfield:

particleinanypositionofearth’ssurfacesubjectedtoacertaingravity,

theearth’ssurfacespaceiscalledgravityfield.(2)conservativeforcefield:

ifinaforcefieldtheworkdonebytheforceactingonamovingparticledependsonlyontheinitialandthefinalpositionoftheparticleanddoesnotdependonthetrackshape，theforcefieldiscalledtobeaconservativeforcefield.Gravityfield,

gravitationalforcesfield,

elasticfieldareconservativeforcefield.(4)gravitationalforcesfield:

spacearoundthesun.(5)elasticfield:

spacearoundthespring.(6)conservativeforces:

theforcesofparticleactinginconservativeforcesfieldiscalled

conservativeforces.Characteristicsofconservativeforces:（1）conservativeforcesactingonaparticleisonlysingle-valuedcontinuousfunctionofparticlecoordinate.（2）workdonebyconservativeforcedependsonlyontheinitialandthefinalpositionoftheactingpointanddoesnotdependontrackshape.2.PotentialfunctionTheworkdonebyitonacertaindistanceM1M2is§8.5conservativeforcesfieldpotentialenergyInthepotentialfield,establishafixedcoordinatesystemOxyz,thepotentialforceofaparticleinthepotentialfieldisso§8.5conservativeforcesfieldpotentialenergySincetheintegralisindependentofthepath,Fxdx+Fydy+FzdzisthefulldifferentiationofafunctionU,CallU=U(x,y,z)asthepotentialfunctionoftheforcefield,obviouslyThatis,thepartialderivativeofthepotentialfunctionwithrespecttothecoordinatesisequaltotheprojectionofthepotentialforceonthesamecoordinateaxis,soUisalsocalledastheforcefunction,with3.Potentialenergy§8.5conservativeforcesfieldpotentialenergyPotentialenergy：inconservativeforcefield,particlemovesfrompointMtooptionalpoint,theworkdonebyconservativeforceiscalledthepotentialenergyoftheparticleatpointMrelativetopoint,itisdenotedbyV.Zeropointofthepotentialenergy:inasameconservativeforcefield,differentposition,potentialenergyisdifferent，inordertocomparethepotentialenergyofeachpoint,whenwecalculatethepotentialenergyofeachposition,thepotentialenergyatthepointisdefinedaszero,thepointiscalledzeropointofthepotentialenergy.WhentheparticlemovesfromM1toM2,theworkdonebythepotentialforceiswhichisequaltotheincrementofthepotentialfunction.§8.5conservativeforcesfieldpotentialenergyCalculateseveralcommonpotentialenergy：(a)potentialenergyingravityfieldIngravityfield,verticalaxisisz-axis,thepointiszeropointofpotentialenergy.Thepotentialenergyofparticleatpointisequaltotheworkdonebythegravityfromto,thatis(b)potentialenergyinelasticforcefieldAssumingoneendofthespringisfixed，theotherendisconnectedwithbody,stiffnesscoefficientofspringis.Thepointofdeformationiszeropointofpotentialenergy，thenthepotentialenergyofthepointofdeformationisIftakingthenaturalpositionofspringaszeropointofpotentialenergy,then,thus(c)potentialenergyingravitationalforcefieldTheparticleofmasssubjectstothegravitationalforceofbodyofmass,showninfigurePointiszeropointofpotentialenergy,thenpotentialenergyofparticleatpointOisWherefisgravitationalconstant,

isunitvectorofparticleinradiusvectordirection;thefigureshows,

istheincrementoflengthofradiusvector,assumingisradiusvectorofzeropointofpotentialenergy,thenwehave§8.5conservativeforcesfieldpotentialenergy4.ThelawofconservationofmechanicalenergyMechanicalenergy：thealgebraicsumofkineticenergyandpotentialenergyofasystemofparticlesinatransient.Assumingkineticenergiesofasystemofparticlesintheinitialandfinaltransientofmotionprocessarerespectivelyand,workdonebyforcesinthisprocessis,

accordingtoprincipleofworkandkineticenergywehaveIfinmotionofsystem,onlyconservativeforcedoeswork,thenworkdonebyconservativeforcecanbecalculatedbypotentialenergy,thatisAftertranspositionweobtainAboveequationismathematicalexpressionofthelawofconservationofmechanicalenergy.Suchsystemofparticlesiscalledconservativesystem.Thelawofconservationofmechanicalenergy:themechanicalenergyofasystemofmassesremainsconstantwhenitismovingonlyundertheactionofapotentialforce.§8.5

thelawofconservationofmechanicalenergyIfasystemofparticlesissubjectedtonon-conservativeforces,thesystemiscallednon-conservativesystem,mechanicalenergyofnon-conservativesystemisnotconserved.Assumingworkdonebyconservativeforcesis,workdonebynon-conservativeforcesis,byprincipleofworkandkineticenergywehaveSince

,thenorWhenasystemofparticlesissubjectedtofrictionresistance,

isnegativework,mechanicalenergyofthesystemofparticlesreducedintheprocessofmotioniscalledmechanicalenergydissipation;Fromtheenergypointofview,nomatterwhatthesystem,thetotalenergyisconstant,theincreaseordecreaseofmechanicalenergyonlyshowsthatintheprocessofmechanicalenergyandotherformsofenergy(suchasthermalenergy,electricalenergy,etc.)haveamutualtransformation.whenasystemofparticlesissubjectedtoactiveforceofnon-conservativeforce,ifispositivework,thenmechanicalenergyofthesystemofparticlesincreasedintheprocessofmotion,atthistimesysteminputenergyfromoutside.§8.5

thelawofconservationofmechanicalenergyConditionsofconservationofmechanicalenergy：1.whenasystemofparticlesisonlysubjectedtoconservativeforce,mechanicalenergyisconservative.2.ifasystemofparticlesissubjectedtonon-conservativeforce,mechanicalenergyisgenerallynon-conservative,butwhenthesumofworkdonebyallnon-conservativeforcesactingonthesystemofparticlesiszero（ordoesnotwork,mechanicalenergyofthesystemisconservative.Forexample,rigidbodysystemwithidealconstraintsmovesinconservativeforcefield,sinceallnon-conservativeforcesdonotwork,orthesumofworkiszero,atthistimemechanicalenergyofrigidbodysystemisconservative.Whenweanswerquestionsandanalyzetheforce,wemustpayattentiontowhichareconservativeforces,whicharenon-conservativeforces,whetherconditionsofconservationofmechanicalenergyofthesystemissatisfied.§8.5

thelawofconservationofmechanicalenergyLetacylinderofmassmandradiusrrollwithoutslidinginalargecirculargrooveofradiusRasshowninthefigure.Ifthefrictionofrollingisnott