1Chapter8:CompositivemotionofaparticleTheoreticalmechanics2第八章點(diǎn)的合成運(yùn)動(dòng)3§8–1Conceptofcompositivemotionofaparticle§8–2Theoremofcompositionofvelocitiesofa particle§8–3Theoremofcompositionofaccelerations whentheconvectedmotionistranslation§8–4Theoremofcompositionofaccelerations whentheconvectedmotionisrotation LessonforproblemsolvingChapter8:Compositivemotionofaparticle4§8–1點(diǎn)的合成運(yùn)動(dòng)的概念

§8–2點(diǎn)的速度合成定理

§8–3牽連運(yùn)動(dòng)為平動(dòng)時(shí)點(diǎn)的加速度合成定理

§8–4牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)點(diǎn)的加速度合成定理習(xí)題課第八章點(diǎn)的合成運(yùn)動(dòng)5§8-1

conceptofcompositivemotionofaparticle1．Coordinatesystem：

1)Staticcoordinatesystem:AcoordinatesystemfixedtotheearthgroundiscalledStaticcoordinatesystem(SCS).

2)Movingcoordinatesystem:Acoordinatesystemfixedtoamovingobjectrelativetotheearthgroundiscalledmovingcoordinatesystem(MCS).Forexample,arunningcar.

Intheprevioustwochapters,wehaveconsideredthemotionofaparticleorarigidbodywithrespecttoasystemoffixedcoordinateaxes.Inpractice,however,weoftenobservethemotionwithrespecttoamovingbody.Forexample,weobservethemotionofanflyingairplanefromamovingcar,orobservethemotionofraindropsfromatravelingtrain.

Whydoobtaindifferentresultswhenweobservethemotionaobjectondifferentcoordinatesystemsorreferenceobjects?Itisknownthattheremayexistrelationshipbetweentwodifferentobjects.Next,wearegoingtostudytherelationshipbetweenthemotionsofanobjectandthereferenceobject.Firstofall,webeginwiththeintroductionofsomeconcepts.Kinematics6§8-1點(diǎn)的合成運(yùn)動(dòng)的概念

一．坐標(biāo)系：

1.靜坐標(biāo)系：把固結(jié)于地面上的坐標(biāo)系稱為靜坐標(biāo)系,簡(jiǎn)稱靜系。

2.動(dòng)坐標(biāo)系：把固結(jié)于相對(duì)于地面運(yùn)動(dòng)物體上的坐標(biāo)系，稱為動(dòng)坐標(biāo)系，簡(jiǎn)稱動(dòng)系。例如在行駛的汽車。前兩章中我們研究點(diǎn)和剛體的運(yùn)動(dòng)，一般都是以地面為參考體的。然而在實(shí)際問題中，還常常要在相對(duì)于地面運(yùn)動(dòng)著的參考系上觀察和研究物體的運(yùn)動(dòng)。例如，從行駛的汽車上觀看飛機(jī)的運(yùn)動(dòng)等，坐在行駛的火車內(nèi)看下雨的雨點(diǎn)是向后斜落的等。

為什么在不同的坐標(biāo)系或參考體上觀察物體的運(yùn)動(dòng)會(huì)有不同的結(jié)果呢？我們說事物都是相互聯(lián)系著的。下面我們就將研究參考體與觀察物體運(yùn)動(dòng)之間的聯(lián)系。為了便于研究，下面先來介紹有關(guān)的概念。運(yùn)動(dòng)學(xué)73．Threekindsofmotionandtheirvelocitiesandaccelerations

1)Absolutemotion：motionofthemovingpointrelativetotheSCS.

2)Relativemotion：motionofthemovingpointrelativetotheMCS.

Forexample,motionsofapersoninarunningbus

3)Convectedmotion：motionoftheMCSrelativetotheSCS.

Forexample,motionofarunningcarerelativetotheearthground.

Thevelocityandaccelerationofthemovingpointinitsabsolutemotionarecalledabsolutevelocityandabsoluteacceleration

Thevelocityandaccelerationofthemovingpointinitsrelativemotionarecalledrelativevelocityandrelativeacceleration

Thevelocityandaccelerationoftheconvectedpointinitsabsolutemotionarecalledconvectedvelocityandconvectedacceleration

convectedpoint:thepointintheMCScoincidingwiththemovingpointatanyinstantoftime.ImagethatthemovingpointwerefixedtotheMCSandmovedtogetherwiththeMCS.SuchapointintheMCSiscalledconvectedpoint.MotionofaparticleMotionofarigidbody2.Movingpoint：amovingpointunderstudyingKinematics8三．三種運(yùn)動(dòng)及三種速度與三種加速度。

１．絕對(duì)運(yùn)動(dòng)：動(dòng)點(diǎn)對(duì)靜系的運(yùn)動(dòng)。

２．相對(duì)運(yùn)動(dòng)：動(dòng)點(diǎn)對(duì)動(dòng)系的運(yùn)動(dòng)。例如：人在行駛的汽車?yán)镒邉?dòng)。

３．牽連運(yùn)動(dòng)：動(dòng)系相對(duì)于靜系的運(yùn)動(dòng)例如：行駛的汽車相對(duì)于地面的運(yùn)動(dòng)。

絕對(duì)運(yùn)動(dòng)中,動(dòng)點(diǎn)的速度與加速度稱為絕對(duì)速度與絕對(duì)加速度相對(duì)運(yùn)動(dòng)中,動(dòng)點(diǎn)的速度和加速度稱為相對(duì)速度與相對(duì)加速度牽連運(yùn)動(dòng)中,牽連點(diǎn)的速度和加速度稱為牽連速度與牽連加速度牽連點(diǎn)：在任意瞬時(shí)，動(dòng)坐標(biāo)系中與動(dòng)點(diǎn)相重合的點(diǎn)，也就是設(shè)想將該動(dòng)點(diǎn)固結(jié)在動(dòng)坐標(biāo)系上，而隨著動(dòng)坐標(biāo)系一起運(yùn)動(dòng)時(shí)該點(diǎn)叫牽連點(diǎn)。點(diǎn)的運(yùn)動(dòng)剛體的運(yùn)動(dòng)運(yùn)動(dòng)學(xué)二．動(dòng)點(diǎn)：所研究的點(diǎn)（運(yùn)動(dòng)著的點(diǎn)）。9Explanationoftheconceptsbyexamples:

4.Guidelineforselectionofmovingpoint:Ingeneral,thecontactpointbetweenthedriveanddrivenobjectsisusuallyselectedasmovingpoint,whichmovesrelativetobothSCSandMCS.

5．GuidelineforMCSselection:AMCSshouldbeacoordinatesystem(object)onwhichthetrajectoryofthemovingpointisknownorclear.Movingpoint:MCS:SCS：PointAonrodABFixedtocamＯ'FixedtothegroundKinematics10下面舉例說明以上各概念：

四．動(dòng)點(diǎn)的選擇原則：一般選擇主動(dòng)件與從動(dòng)件的連接點(diǎn)，它是對(duì)兩個(gè)坐標(biāo)系都有運(yùn)動(dòng)的點(diǎn)。

五．動(dòng)系的選擇原則：動(dòng)點(diǎn)對(duì)動(dòng)系有相對(duì)運(yùn)動(dòng)，且相對(duì)運(yùn)動(dòng)的軌跡是已知的，或者能直接看出的。運(yùn)動(dòng)學(xué)動(dòng)點(diǎn)：動(dòng)系：靜系：AB桿上A點(diǎn)固結(jié)于凸輪Ｏ'上固結(jié)在地面上11Relativemotion

convectedmotion:Curvilinearmotion(arc)RectilineartranslationAbsolutemotion:RectilinearmotionKinematics12運(yùn)動(dòng)學(xué)相對(duì)運(yùn)動(dòng)：牽連運(yùn)動(dòng)：曲線（圓?。┲本€平動(dòng)絕對(duì)運(yùn)動(dòng)：直線13Absolutevelocity:relativevelocity:convectedvelocity:Kinematics14運(yùn)動(dòng)學(xué)絕對(duì)速度：相對(duì)速度：牽連速度：15Absoluteacceleration:Relativeacceleration:convectedacceleration:Kinematics16絕對(duì)加速度：相對(duì)加速度：牽連加速度：運(yùn)動(dòng)學(xué)17Movingpoint：A（ondisk)MCS：rodO'ASCS：supportAbsolutemotion:curvilinear(circle)Relativemotion:rectilinearmotionconvectedmotion:rotationMovingpoint：A1(onrodO'A1)MCS：circulardiskSCS：supportAbsolutemotion:curvilinear(arc)Relativemotion：curvilinearmotionconvectedmotion：rotationKinematics18動(dòng)點(diǎn)：A（在圓盤上)動(dòng)系：O'A擺桿靜系：機(jī)架絕對(duì)運(yùn)動(dòng)：曲線（圓周）相對(duì)運(yùn)動(dòng)：直線牽連運(yùn)動(dòng)：定軸轉(zhuǎn)動(dòng)運(yùn)動(dòng)學(xué)動(dòng)點(diǎn)：A1（在O'A1

擺桿上)動(dòng)系：圓盤靜系：機(jī)架絕對(duì)運(yùn)動(dòng)：曲線（圓?。┫鄬?duì)運(yùn)動(dòng)：曲線牽連運(yùn)動(dòng)：定軸轉(zhuǎn)動(dòng)19

IfthemovingpointisontheCAMMovingpoint:A(onrodAB)

A（onthecam）MCS：cam

rodABSCS：earthground

earthgroundAbsolutemotion：rectilinear

Circle(reddashedline)Relativemotion:curvilinear(circle)curve（pathunknownconvectedmotion:rotation

Translation[Note]Clearstatethattowhichobjectthemovingpointisattached.ThemovingpointcannotbeattachedtotheMCS,otherwiseitsrelativemotionisrestontheMCS.Kinematics20

若動(dòng)點(diǎn)A在偏心輪上時(shí)動(dòng)點(diǎn)：A(在AB桿上)

A（在偏心輪上）動(dòng)系：偏心輪AB桿靜系：地面地面絕對(duì)運(yùn)動(dòng)：直線圓周（紅色虛線）相對(duì)運(yùn)動(dòng)：圓周（曲線）曲線（未知）牽連運(yùn)動(dòng)：定軸轉(zhuǎn)動(dòng)平動(dòng)[注]要指明動(dòng)點(diǎn)應(yīng)在哪個(gè)物體上，但不能選在動(dòng)系上。運(yùn)動(dòng)學(xué)21§8-2Theoremofcompositionofvelocitiesofaparticle

Therelationshipamongtheabsolute,relativeandconvectedvelocitywillbeestablishedandstatedinthetheoremofcompositionofvelocitiesofaparticle.＝＋Whent

t+△t

AB

A'B'

M

M'EquivalenttoM

M１M′MM'

istheabsolutepathMM'

istheabsolutedisplacementM1M'istherelativepath

M1M'

istherelativedisplacementDividedtheaboveequationby,,givesWhen1.ProofKinematics22§８－２點(diǎn)的速度合成定理

速度合成定理將建立動(dòng)點(diǎn)的絕對(duì)速度，相對(duì)速度和牽連速度之間的關(guān)系。＝＋運(yùn)動(dòng)學(xué)當(dāng)t

t+△t

AB

A'B'

M

M'也可看成M

M１

M′MM'為絕對(duì)軌跡MM'為絕對(duì)位移M1M'為相對(duì)軌跡M1M'為相對(duì)位移將上式兩邊同除以后，時(shí)的極限，得取一．證明23Kinematics24運(yùn)動(dòng)學(xué)25Explanations：

va—absolutevelocityofthemovingpoint;

vr—relativevelocityofthemovingpoint;

ve—convectedvelocityofthemovingpoint, whichisthevelocityofapoint(convected point)ontheMCS;

I)WhenMCSisintranslation,allpointsinithavethesamevelocity. II)WhenMCSrotates,vemustbethevelocityofthepoint intheMCScoincidingwiththemovingpoint.

Conclusion:Atanyinstantoftime,theabsolutevelocityofamovingpointequalstothegeometricsumofitsrelativevelocityandconvectedvelocity.Thisisthetheoremofcompositionofvelocitiesofaparticle.Kinematics26說明：

va—?jiǎng)狱c(diǎn)的絕對(duì)速度；

vr—?jiǎng)狱c(diǎn)的相對(duì)速度；

ve—?jiǎng)狱c(diǎn)的牽連速度，是動(dòng)系上一點(diǎn)(牽連點(diǎn))的速度

I)動(dòng)系作平動(dòng)時(shí)，動(dòng)系上各點(diǎn)速度都相等。

II)動(dòng)系作轉(zhuǎn)動(dòng)時(shí)，ve必須是該瞬時(shí)動(dòng)系上與 動(dòng)點(diǎn)相重合點(diǎn)的速度。

即在任一瞬時(shí)動(dòng)點(diǎn)的絕對(duì)速度等于其牽連速度與相對(duì)速度的矢量和，這就是點(diǎn)的速度合成定理。運(yùn)動(dòng)學(xué)27

Theequationgiveninthetheoremofcompositionofvelocitiesisavectorequationataninstant,whichincludessixcomponentssuchasmagnitudesanddirections.Ifanyfourcomponentsaregiven,theothertwounknownscanbesolvedfromthisequation.2.Examplesoftheoremapplication[Example1]Thesmallvehicleinabridgecanemovesalongitshorizontaltrackwithauniformvelocityvh，andweightAmovesverticallyatavelocityvv

relativetothesmallvehicle.FindtheabsolutevelocityofweightA.Kinematics28

點(diǎn)的速度合成定理是瞬時(shí)矢量式，共包括大小?方向六個(gè)元素，已知任意四個(gè)元素，就能求出其他兩個(gè)。二．應(yīng)用舉例運(yùn)動(dòng)學(xué)[例1]

橋式吊車已知：小車水平運(yùn)行，速度為v平，物塊A相對(duì)小車垂直上升的速度為v

。求物塊A的運(yùn)行速度。29Drawtheparallelogramofvelocitiesshowninthefigure,themagnitudeanddirectionofthevelocityoftheweightAcanbeobtainedas

Solution：movingpoint:weightA

MCS:smallvehicle

SCS:earthgroundRelativemotion:rectilinear;Relativevelocityvr=vv,direction

convectedmotion:translation;convectedvelocityve=vh,

directionAbsolutemotion:curvilinear;Absolutevelocity

va,magnitudeanddirectionareunknown

Employingthetheoremofcompositionofvelocities,yields

Kinematics30運(yùn)動(dòng)學(xué)作出速度平四邊形如圖示，則物塊Ａ的速度大小和方向?yàn)榻猓哼x取動(dòng)點(diǎn):物塊A

動(dòng)系:小車

靜系:地面相對(duì)運(yùn)動(dòng):直線;相對(duì)速度vr=v

方向

牽連運(yùn)動(dòng):平動(dòng);牽連速度ve=v平方向絕對(duì)運(yùn)動(dòng):曲線;絕對(duì)速度va

的大小,方向待求由速度合成定理：31Solution：ChoosepointAonrodOA桿asthemovingpoint，attachtheMCStorodO1B，andusetheearthgroundastheSCS. absolutevelocityva

=r

direction

OA relativevelocityvr

=?direction//O1B convectedvelocityve

=?directionO1B[Example2]Inthemechanismshowninthefigure,OA=r,

,OO1=l.Atthismoment,OA

OO1

Find:angularvelocity

1

ofrodO1B.Accordingtova=vr+

ve,drawtheparallelogramofvelocitiesasshowninthefigure.Kinematics32解：取OA桿上A點(diǎn)為動(dòng)點(diǎn)，擺桿O1B為動(dòng)系，基座為靜系。 絕對(duì)速度va

=r

方向

OA

相對(duì)速度vr

=?方向//O1B

牽連速度ve

=?方向O1B（）運(yùn)動(dòng)學(xué)[例2]

曲柄擺桿機(jī)構(gòu)已知：OA=r,

,OO1=l圖示瞬時(shí)OA

OO1

求：擺桿O1B角速度

1由速度合成定理va=vr+

ve

作出速度平行四邊形如圖示。33Accordingtotheequationva=vr+

ve

，Drawtheparallelogramofvelocitiesasshowninthefigure.Solution:SelectpointAonrodABasthemovingpoint,fixtheMCStothediskandtheSCStothesupport.absolutevelocityva

=?unknown，directin//ABrelativevelocity

vr

=?unknown，direction

CAconvectedvelocityve=OA

=2e,direction

OA（Lookattheanimationonthenextpage）

[Example3]Inthecam-rodmechanism,OC＝e,

,

（constant）,andthemoment,OC

CAandpointO,AandBarecollinear.Find:thevelocityofthedrivenrodAB.

Kinematics34由速度合成定理va=vr+

ve

，作出速度平行四邊形如圖示。解：動(dòng)點(diǎn)取直桿上A點(diǎn)，動(dòng)系固結(jié)于圓盤，

靜系固結(jié)于基座。絕對(duì)速度va

=?待求，方向//AB

相對(duì)速度

vr

=?未知，方向CA

牽連速度ve=OA

=2e,方向

OA（翻頁請(qǐng)看動(dòng)畫）

運(yùn)動(dòng)學(xué)[例3]

圓盤凸輪機(jī)構(gòu)已知：OC＝e,

,

（勻角速度）圖示瞬時(shí),OC

CA

且

O,A,B三點(diǎn)共線。求：從動(dòng)桿AB的速度。35Kinematics36運(yùn)動(dòng)學(xué)37Asevidencedfromtheexamples,thegeneralstepsshouldbefollowedinsolvingproblemsaboutthecompositionofvelocities:

Selectmovingpoint,MCSandSCS;

Analysisoftherelatedthreekindsofmotions;

Analysisofthreekindsofvelocities;Accordingtothetheoremofcompositionofvelocities,

drawtheparallelogramofvelocities;

Findunknownsfromtheparallelogramofvelocities;Properselectionofthemovingpoint,MCSandSCSisthekeypointinsolvingsuchproblems.Kinematics38由上述例題可看出，求解合成運(yùn)動(dòng)的速度問題的一般步驟為：

選取動(dòng)點(diǎn)，動(dòng)系和靜系。

三種運(yùn)動(dòng)的分析。

三種速度的分析。

根據(jù)速度合成定理作出速度平行四邊形。

根據(jù)速度平行四邊形，求出未知量。恰當(dāng)?shù)剡x擇動(dòng)點(diǎn)、動(dòng)系和靜系是求解合成運(yùn)動(dòng)問題的關(guān)鍵。運(yùn)動(dòng)學(xué)39Guidelinesfortheselectionofmovingpoint,MCSandSCSThemovingpoint,MCSandSCSmustbeinthreedifferentrigidbodies.Otherwise,onemotionoutoftheabsolute,relativeandconvectedmotionwillbeabsent(alwaysrestingindeed)，andconsequentlyitwillnotsuitablefortheapplicationofthecompositionofmotions.

TherelativepathofthemovingpointrelativetotheMCSmustbeeasytodetermine.(excepttofindtheconvectedmotionwhentheabsoluteandrelativemotionsaregiven.)Kinematics40動(dòng)點(diǎn)、動(dòng)系和靜系的選擇原則

動(dòng)點(diǎn)、動(dòng)系和靜系必須分別屬于三個(gè)不同的物體，否則絕對(duì)、相對(duì)和牽連運(yùn)動(dòng)中就缺少一種運(yùn)動(dòng)，不能成為合成運(yùn)動(dòng)

動(dòng)點(diǎn)相對(duì)動(dòng)系的相對(duì)運(yùn)動(dòng)軌跡易于直觀判斷（已知絕對(duì)運(yùn)動(dòng)和牽連運(yùn)動(dòng)求解相對(duì)運(yùn)動(dòng)的問題除外）。運(yùn)動(dòng)學(xué)41

Analysis：Sincethecontactpointbetweenthesetwoobjectschangeswithtime，theircontactpointisnotsuitabletobeamovingpoint，otherwisetherelativemotionisverycomplicatedInthiscase,weneedtoselectanotherpointinsteadofthecontactpointasthemovingpointwhichshouldsatisfytheguideline.Kinematics[Example4]

Theradiusofthecamisr,andasshowninthediagram.RodOAisleaningagainstthecam.DeterminetheangularvelocityofrodOA.42

分析：相接觸的兩個(gè)物體的接觸點(diǎn)位置都隨時(shí)間而變化，因此兩物體的接觸點(diǎn)都不宜選為動(dòng)點(diǎn)，否則相對(duì)運(yùn)動(dòng)的分析就會(huì)很困難。這種情況下，需選擇滿足上述兩條原則的非接觸點(diǎn)為動(dòng)點(diǎn)。運(yùn)動(dòng)學(xué)[例４]

已知:凸輪半徑r,圖示時(shí)桿OA靠在凸輪上。求：桿OA的角速度。43Solution:SelectpointCasthemovingpoint,attachtheMCStorodOAandtheSCStothebase.Absolutemotion:rectilinearmotion,Absolutevelocity:Relativemotion:rectilinearmotion,Relativevelocity:convected:rotation,convectedvelocity:Accordingtodrawtheparallelogramofvelocitiesasshowninthefigure.（）Kinematics44解:取凸輪上C點(diǎn)為動(dòng)點(diǎn),

動(dòng)系固結(jié)于OA桿上,

靜系固結(jié)于基座。絕對(duì)運(yùn)動(dòng):直線運(yùn)動(dòng),絕對(duì)速度:相對(duì)運(yùn)動(dòng):直線運(yùn)動(dòng),相對(duì)速度:牽連運(yùn)動(dòng):定軸轉(zhuǎn)動(dòng),牽連速度:

如圖示。根據(jù)速度合成定理做出速度平行四邊形（）運(yùn)動(dòng)學(xué)45§8-3

TheoremofcompositionofaccelerationswhentheconvectedmotionistranslationNotingtheconvectedmotionistranslation，wehaveEmployingtheoremofcompositionofvelocitiesDifferentiatingitwithrespecttot:

ApointmovesalongacurvedpathABattachedtotheMCSO'x'y'z',atthesametime,thepathABalsohastranslatorymotiontogetherwiththeMCSO'x'y'z'relativetotheSCSOxyz.Kinematics46§8-3牽連運(yùn)動(dòng)為平動(dòng)時(shí)點(diǎn)的加速度合成定理由于牽連運(yùn)動(dòng)為平動(dòng)，故由速度合成定理對(duì)t求導(dǎo)：運(yùn)動(dòng)學(xué)設(shè)有一動(dòng)點(diǎn)M按一定規(guī)律沿著固連于動(dòng)系O'x'y'z'的曲線AB運(yùn)動(dòng),而曲線AB同時(shí)又隨同動(dòng)系O'x'y'z'相對(duì)靜系Oxyz平動(dòng)。47(where aretheunitvectorsoftheMCS,andtheirdirectiondonotchangeduetothetranslationoftheMCS.Hence ).moreover —Theoremofcompositionofaccelerationswhentheconvectedmotionistranslationi.e.whentheconvectedmotionistranslation,theabsoluteaccelerationofamovingpointequalstothegeometricsumofitsrelativeandconvectedaccelerations.∴Itsgeneralformis:Kinematics48(其中 為動(dòng)系坐標(biāo)的單位矢量，因?yàn)閯?dòng)系為平動(dòng)，故它們的方向不變，是常矢量，所以 ）運(yùn)動(dòng)學(xué)—牽連運(yùn)動(dòng)為平動(dòng)時(shí)點(diǎn)的加速度合成定理即當(dāng)牽連運(yùn)動(dòng)為平動(dòng)時(shí)，動(dòng)點(diǎn)的絕對(duì)加速度等于牽連加速度與相對(duì)加速度的矢量和?！嘁话闶娇蓪憺椋?9Solution：ChoosepointAonrodABasthemovingpoint，andattachtheMCSontothecam.[Example1]Theradiusofthecamisarealsogiven.

FindthevelocityofrodABwhenj=60o.LookattheanimationKinematics50解：取桿上的A點(diǎn)為動(dòng)點(diǎn)，

動(dòng)系與凸輪固連。運(yùn)動(dòng)學(xué)[例1]

已知：凸輪半徑求：j=60o時(shí),頂桿AB的加速度。請(qǐng)看動(dòng)畫51Absolutevelocity

va=?,direction

AB

；absoluteaccelerationaa=?,directionAB，tobesolved.Relativevelocity

vr

=?,direction

CA;

relativeaccelerationart=?direction

CA ,directionalongCAandpointingtoC.convectedvelocityve=v0,direction→;convectedaccelerationae=a0,direction→.Applingdrawtheparallelogramofvelocitiesshowninthefigureabove

Kinematics52絕對(duì)速度va=?,方向

AB

；絕對(duì)加速度aa=?,方向

AB，待求。相對(duì)速度vr

=?,方向

CA;

相對(duì)加速度art=?方向

CA ,方向沿CA指向C牽連速度ve=v0,方向→;牽連加速度ae=a0,方向→運(yùn)動(dòng)學(xué)由速度合成定理做出速度平行四邊形,如圖示。

53Notingtheconvectedmotionistranslation,wehaveDrawingvectordiagramofaccelerationsshowinthefigure,andprojectingthemtothenormal

n,yieldsAfterarrangement,finallywehave

nKinematics54運(yùn)動(dòng)學(xué)因牽連運(yùn)動(dòng)為平動(dòng)，故有作加速度矢量圖如圖示，將上式投影到法線上，得整理得

n55§8-4

Theoremofcompositionofaccelerationswhentheconvectedmotionisrotation

Inlastsection,wehaveprovedthetheoremofcompositionofaccelerationswhentheconvectedmotionistranslation.Isthistheoremstillvalidwhentheconvectedmotionisrotation?Beforeansweringthisquestion,let’sanalyzeatypicalexample.

AdiskisrotatingaboutafixedaxisＯwithauniformangularvelocity.ApointMmovesalongacircularpathslottedinthediskwithaconstantvr,what’stheabsoluteaccelerationofthepointM?Kinematics56§8-4牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)點(diǎn)的加速度合成定理上一節(jié)我們證明了牽連運(yùn)動(dòng)為平動(dòng)時(shí)的點(diǎn)的加速度合成定理，那么當(dāng)牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)，上述的加速度合成定理是否還適用呢？下面我們來分析一特例。運(yùn)動(dòng)學(xué)設(shè)一圓盤以勻角速度

繞定軸Ｏ順時(shí)針轉(zhuǎn)動(dòng)，盤上圓槽內(nèi)有一點(diǎn)M以大小不變的速度vr

沿槽作圓周運(yùn)動(dòng)，那么M點(diǎn)相對(duì)于靜系的絕對(duì)加速度應(yīng)是多少呢？57Sincetherelativemotionisauniformcircularmotion.(directionisshowninthefigure)ApplyingthetheoremofcompositionofvelocitiesgivesChooseMasthemovingpoint，andtheMCSisattachedtothedisk.Inthiscase,theconvectedmotionofthepointMisauniformrotation.(directionisshowninthefigure)ItisshownthattheabsolutemotionofpointMisalsoauniformcircularmotion,andthereforeItpointstothecenterＯ.Kinematics58相對(duì)運(yùn)動(dòng)為勻速圓周運(yùn)動(dòng)，（方向如圖）由速度合成定理可得出運(yùn)動(dòng)學(xué)選點(diǎn)M為動(dòng)點(diǎn)，動(dòng)系固結(jié)與圓盤上，則M點(diǎn)的牽連運(yùn)動(dòng)為勻速轉(zhuǎn)動(dòng)（方向如圖）即絕對(duì)運(yùn)動(dòng)也為勻速圓周運(yùn)動(dòng)，所以方向指向圓心Ｏ點(diǎn)59itcanfoundfromtheaboveequationthatinadditiontoanextraterm2vrappears.Itisclearthatthe

absoluteaccelerationofamovingpointisnolongerequaltothegeometricsumofitsrelativeaccelerationandconvectedaccelerationwhentheconvectedmotionisrotation.Whatisthecorrectrelationshipamongtheseaccelerations？Whatistheadditionalterm2vrwhydoesitappear?Next,wearegoingtoseektheanswerstothesequestionsandderivethetheoremofcompositionofaccelerationswhentheconvectedmotionisrotation.Kinematics60運(yùn)動(dòng)學(xué)

分析上式： 還多出一項(xiàng)2vr?？梢?，當(dāng)牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)，動(dòng)點(diǎn)的絕對(duì)加速度并不等于牽連加速度和相對(duì)加速度的矢量和。那么他們之間的關(guān)系是什么呢？2vr又是怎樣出現(xiàn)的呢？它是什么呢？下面我們就來討論這些問題，推證牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)點(diǎn)的加速度合成定理。61AnalysisofthreevelocitiesconvectedvelocityRelativevelocityAbsolutevelocity

Atinstantt,thesystemisinpositionＩandatt+Dt,itisinpositionIIItcanbeobservedthatthemagnitudesanddirectionsofbothconvectedandrelativevelocitiesvaryafteranintervalDt.

ArodOArotatesaboutaxisOwithaconstantangularvelocity

，andthesleeveMmovesalongrodOAwithavaryingvelocity.ChoosesleeveMasthemovingpoint,andattachtheMCStorodOAandtheSCStothestaticsupport.Kinematics62運(yùn)動(dòng)學(xué)三種速度分析牽連速度相對(duì)速度絕對(duì)速度

t

瞬時(shí)在位置Ｉt+Dt瞬時(shí)在位置II

可以看出，經(jīng)過Dt時(shí)間間隔，牽連速度和相對(duì)速度的大小和方向都變化了。

設(shè)有已知桿OA在圖示平面內(nèi)以勻

繞軸O轉(zhuǎn)動(dòng)，套筒M（可視為點(diǎn)M）沿直桿作變速運(yùn)動(dòng)。取套筒M為動(dòng)點(diǎn)，動(dòng)系固結(jié)于桿OA上，靜系固結(jié)于機(jī)架。63where--themagnitudechangeoftherelativevelocityinDt,whichisindependentontheconvectedmotion.--在directionchangeoftherelativevelocitycausedbytheconvectedmotioninDt,whichisdependentontheconvectedangularvelocity

.

AnalysisofthevelocitychangeinanintervalDtrelativevelocity： drawthetrianglecomposedofAftercuttingalengthofinthevector

,canbedecomposedintoand.convectedvelocitydrawthetrianglecomposedofAftercuttingalengthofinthevector

,canbedecomposedintoand.Kinematics64運(yùn)動(dòng)學(xué)其中--在Dt內(nèi)相對(duì)速度大小的改變量，它與牽連轉(zhuǎn)動(dòng)無關(guān)。 --在Dt內(nèi)由于牽連轉(zhuǎn)動(dòng)而引起的相對(duì)速度方向的改變 量，與牽連轉(zhuǎn)動(dòng)的

的大小有關(guān)

。

Dt時(shí)間間隔內(nèi)的速度變化分析相對(duì)速度：由 作速度矢量三角形，在矢量上截取長(zhǎng)度后，分解為和

牽連速度：由 作速度矢量三角形，在矢量上截取等于長(zhǎng)后，將分解為和，65where

—representsthedirectionchangeoftheconvectedvelocitycausedbytheconvectedrotationinanintervalDt,whichisindependentontherelativemotion.—representsthemagnitudechangeoftheconvectedvelocitycausedbytherelativemotioninanintervalDt,whichisdependentontherelativevelocity.AccelerationanalysisAccordingtothedefinition,Physicalinterpretations：Thefirstterm:magnitudeKinematics66運(yùn)動(dòng)學(xué)其中:

—表示Dt內(nèi)由于牽連轉(zhuǎn)動(dòng)而引起的牽連速度方向的改變量，與相對(duì)運(yùn)動(dòng)無關(guān)。

—表示Dt內(nèi)動(dòng)點(diǎn)的牽連速度，由于相對(duì)運(yùn)動(dòng)而引起的大小改變量，與相對(duì)速度有關(guān)。加速度分析根據(jù)加速度定義上式中各項(xiàng)的物理意義如下：第一項(xiàng)大小：67Direction:asDt

0andD

0,itwillgraduallypointtopointAalongtherod.Therefore，thistermistheconvectedaccelerationatinstantt.Thethirdterm:magnitude:

representsthechangerateofthemagnitudeof.

Direction:alongtherod.so，thistermistherelativeaccelerationofthemovingpointatinstantt.

Thesecondterm：magnitude:Thistermistheaccelerationresultingfromthemagnitudechangeoftheconvectedvelocityduetotheexistenceofrelativemotion.Theforthterm：magnitude:Thistermrepresentstheaccelerationresultingfromthedirectionchangeoftherelativevelocitycausedbytheconvectedrotation.Kinematics68運(yùn)動(dòng)學(xué)

方向：Dt

0時(shí)，D

0,其方向沿著直桿指向A點(diǎn)。因此，第一項(xiàng)正是t

瞬時(shí)動(dòng)點(diǎn)的牽連加速度。第三項(xiàng)大?。簽閷?duì)應(yīng)于大小改變

方向：總是沿直桿。因此，該項(xiàng)恰是ｔ瞬時(shí)動(dòng)點(diǎn)的相對(duì)加速度。第二項(xiàng)大小：該項(xiàng)為由于相對(duì)運(yùn)動(dòng)的存在而引起牽連速度的大小改變的加速度。第四項(xiàng)大?。哼@一項(xiàng)表明由于牽連轉(zhuǎn)動(dòng)而引起相對(duì)速度方向改變的加速度。69Finally,weobtainthetheoremofcompositionofaccelerationswitharotatingconvectedmotionasfollow

Whentheconvectedmotionisrotation，theabsoluteaccelerationofamovingpointequalstothegeometricsumofitsconvected,relativeandCoriolisaccelerations.Moregeneralform:TheCoriolisaccelerationcanalsobeexpressedasthefollowingvectorform:

Sincethesecondandthefourthtermhavethesamemagnitudeanddirection,theycanbewrittenintoonetermanddenotedby,whichiscalledCoriolisacceleration.Kinematics70運(yùn)動(dòng)學(xué)所以，當(dāng)牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)，加速度合成定理為

當(dāng)牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)，動(dòng)點(diǎn)的絕對(duì)加速度等于它的牽連加速度，相對(duì)加速度和科氏加速度三者的矢量和。一般式

一般情況下

科氏加速度的計(jì)算可以用矢積表示

由于第二項(xiàng)和第四項(xiàng)所表示的加速度分量的大小，方向都相同，可以合并為一項(xiàng)，用表示，稱為科里奧利加速度，簡(jiǎn)稱科氏加速度。71Solution:movingpoint:pointAonrodAB；

MCS:cam;

SCS:earthground.

absolutemotion:rectilinearmotion;absolutevelocity:va=?tobesolved,direction//AB;relativemotion:curvilinear;relativevelocity:vr=?direction

n;convectedmotion:rotation;convectedvelocity:ve=r，direction

OA，。direction：determinedaccordingtotheright-handrule.[Example2]ThecamrotatesuniformlyaroundaxisOatanangularvelocityof

.

Atthemomentshowninthefigure,OA=r

，theradiusofcurvatureatpointAis

,and

isalsogiven.FindthevelocityandaccelerationoftherodABatthismoment.

Kinematics72解:動(dòng)點(diǎn):頂桿上A點(diǎn)；動(dòng)系:凸輪;靜系:地面。絕對(duì)運(yùn)動(dòng):直線;

絕對(duì)速度:va=?待求,方向//AB;

相對(duì)運(yùn)動(dòng):曲線;相對(duì)速度:vr=?方向

n;

牽連運(yùn)動(dòng):定軸轉(zhuǎn)動(dòng);

牽連速度:ve=r，方向

OA，。運(yùn)動(dòng)學(xué)方向：按右手法則確定。[例2]

已知：凸輪機(jī)構(gòu)以勻

繞O軸轉(zhuǎn)動(dòng)，圖示瞬時(shí)OA=r

，A點(diǎn)曲率半徑

,

已知。求：該瞬時(shí)頂桿AB的速度和加速度。73Employinganddrawingtheparallelogramofvelocities,wehaveKinematics74運(yùn)動(dòng)學(xué)根據(jù)速度合成定理做出速度平行四邊形75Applyingthetheoremofcompositionofaccelerationswitharotatingconvectedmotiondrawingthevectordiagramofaccelerationsasshowninthefigure,andprojectingthemtoaxisn,yieldsKinematics76運(yùn)動(dòng)學(xué)由牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)的加速度合成定理作出加速度矢量圖如圖示向n軸投影：77DABCSolution：TheCoriolisaccelerationofpointM1is

anditpointsinsidethescreen.

[Example3]ArectangularplateABCDrotatesaboutaxiszatanconstantangularvelocityof.PointM1andM2movealongthediagonalBDandsideCDrespectively，andtheirvelocitiesrelativetotheplateareandrespectivelyatthismoment.CalculatetheCoriolisaccelerationsatpointM1and

M2anddeterminetheirdirections.

TheCoriolisaccelerationofpointM2isKinematics78DABC解：點(diǎn)M1的科氏加速度 垂直板面向里。

運(yùn)動(dòng)學(xué)

[例3]

矩形板ABCD以勻角速度

繞固定軸z轉(zhuǎn)動(dòng)，點(diǎn)M1和點(diǎn)M2分別沿板的對(duì)角線BD和邊線CD運(yùn)動(dòng)，在圖示位置時(shí)相對(duì)于板的速度分別為和，計(jì)算點(diǎn)M1、

M2的科氏加速度大小,并圖示方向。點(diǎn)M2的科氏加速度79Solution：ApplyingAnddrawingtheparallelogramofvelocitiesgives.direction：sameas[Example4]Inthemechanismshowninthefigure,O1A＝r,

,

and

1aregiven.TakingpointAonrodO1Aasthemovingpoint，andattachingtheMCSontherodO2B，findtheCoriolisaccelerationofpointA.Kinematics80解：根據(jù)做出速度平行四邊形方向：與相同。運(yùn)動(dòng)學(xué)[例4]

曲柄擺桿機(jī)構(gòu)已知：O1A＝r,

,

,

1;取O1A桿上A點(diǎn)為動(dòng)點(diǎn)，動(dòng)系固結(jié)O2B上，試計(jì)算動(dòng)點(diǎn)A的科氏加速度。81Chapter8Lessonforproblemsolving

1．Conceptsandformulae

1).Onepoint,twosystemsandthreemotionsTheabsolutemotionofapointisthecompositionitsrelativeandconvectedmotionsCompositionofmotions

2).Theoremofvelocitycomposition

3.TheoremofaccelerationcompositionwhentheconvectedmotionistranslationwhentheconvectedmotionisrotationKinematics82運(yùn)動(dòng)學(xué)第八章點(diǎn)的合成運(yùn)動(dòng)習(xí)題課一．概念及公式

1.一點(diǎn)、二系、三運(yùn)動(dòng)點(diǎn)的絕對(duì)運(yùn)動(dòng)為點(diǎn)的相對(duì)運(yùn)動(dòng)與牽連運(yùn)動(dòng)的合成．

2.速度合成定理

3.加速度合成定理牽連運(yùn)動(dòng)為平動(dòng)時(shí)牽連運(yùn)動(dòng)為轉(zhuǎn)動(dòng)時(shí)832．Stepsforproblemsolving

1).Choosemovingpoint,MCSandSCS.

2).Analyzethreekindofmotions：absolute,relativeandconvectedmotions.

3).Basedonvelocityanalysis,drawtheparallelogramofvelocitiesandsolveTherelevantunknownssuchasvelocitiesandangularvelocities.

4).Basedonaccelerationanalysis,drawthevectordiagramofaccelerationsandsolvetherelevantunknownssuchasaccelerationsandangularaccelerations.Kinematics84運(yùn)動(dòng)學(xué)二．解題步驟

1.選擇動(dòng)點(diǎn)、動(dòng)系、靜系。

2.分析三種運(yùn)動(dòng)：絕對(duì)運(yùn)動(dòng)、相對(duì)運(yùn)動(dòng)和牽連運(yùn)動(dòng)。

3.作速度分析,畫出速度平行四邊形,求出有關(guān)未知量(速度,

角速度）。

4.作加速度分析，畫出加速度矢量圖，求出有關(guān)的加速度、角加速度未知量。85

2．Someskillsforproblemsolving

1).Properlyselectmovingpoint,MCSandSCSaccordingtotheguideline,whichincludes:

Tofindtherelativevelocitybetweentwopoints,

selectoneofthemasthemovingpoint,andattachatranslatingMCStotheotherone.

Whenapointtravelsonarigidbodywhichitselfalsomoves.Inthiscasethispointhasacomplicatedmotion.Tosolvesuchaproblem,

selectthispointasthemovingpointandattachtheMCStotherigidbody.

Inthetransmissiontworigidbodieswherethereisapointononebodyalwayskeepscontactwithbutmovesrelativetotheotherone,

ForGuidingrod-blockmechanisms,atypical