CHAPTERONETHE GEOMETRYOF MOTION1.IINTRODUCTIONThe theory ofmechani sms and machi nes i s an appl i ed sci ence whi chi s usedtounderstand therel ati onshi ps between thegeometryandmoti onsofthepartsofamachi neormechani smandtheforceswhi chproducethesemoti ons. Thesubj ect, andthereforethi sbook,natural l ydi vi desi tsel fi ntothree parts. Chapters Ito 5 are concerned wi thki nemati cs, the anal ysi s ofthemoti ons ofmachi ne parts. Thi s l ays the groundwork forChaps.6 tol l ,wherewestudymethodsofdesi gnofmechani sms andmachi necomponents.Fi nal l y, i n Chaps.12 to17,we take up the study ofki neti cs, the ti me-varyi ngforcesi nmachi nes andtheresul ti ng dynami cphenomena whi chmustbeconsi dered i n thei r desi gn.Asshowni nFi g.l -1,thedesi gn ofamodernmachi nei softenverycompl ex. Inthe desi gn ofa new engi ne,forexampl e, the automoti ve engi neermust deal withmanyinterrelated questions. Whatis the relationship betweenthemotionofthepistonand themotionofthecrankshaft?Whatwillbe thesl i di ng vel oci ti es and the l oads at the l ubri cated surfaces, and what l ubri cantsare avai l abl e forthe purpose? Howmuch heat wi l l be generated,and how wi l lthe engi ne be cool ed? What are the synchroni zati on and controlrequi rements,and howwi l l theybe met? Whatwi l l be the cost tothe consumer, bothfori ni ti alpurchase and forconti nued operati on and mai ntenance?What materi al sandmanufacturi ng methods wi l l beused? Whatwi l l bethefuel economy,noi se, andexhaust emi ssi ons; wi l l theymeetl egalrequi rements? Al thoughal l these and manyotheri mportantquesti ons mustbeanswered before thedesi gn can be compl eted, obvi ousl y not al lcan be addressedi n a book ofthi s2THEORYOFMACHI NESANDMECHANI SMS.rr*h ".,i-i::* .Fi gurel -fAFi geefl oati ngcranewi thl emni scateboomconfi gurati on(8.V.Machi nefabri ekFigee, Haarlem,Holland)si ze. Just as peopl e wi thdi verse ski l l s must be brought together to produce anadequate desi gn, so toomanybranches ofsci ence must be brought tobear.Thi sbookbri ngs together materi alwhi chfal l si ntothe sci ence ofmechani csas i t rel ates tothe desi gn ofmechani smsand machi nes.I-2ANALYSISANDSYNTHESISTherearetwocompl etel ydi fferentaspects ofthestudyofmechani calsystems, desi gn and anal ysi s. Theconcept embodi ed i ntheword"desi gn"mi ght be more properl y termedsynthess,the process ofcontri vi ng a schemeoramethodofaccompl i shi ng agi venpurpose. Desi gni stheprocess ofprescri bi ng thesi zes, shapes, materi al composi ti ons, andarrangements ofparts so that the resul ti ng machi ne wi l l perform the prescri bed task.Al thoughtherearemanyphases i nthedesi gn process whi chcanbeapproached i nawel l -ordered, sci enti fi c manner, the overal lprocess i s byi tsverynature as much ofan art as a sci ence.Itcal l s fori magi nati on, i ntui ti on,creati vi ty, j udgment, andexperi ence. Therol eofsci encei nthedesi gnriekanar.i cs:aln t tneofofTHE GEoMETRYop prortoN 3processi s merel y to provi de tool s to be used by the desi gnersas they practi cethei r art.Iti s i nthe process ofeval uati ng the vari ous i nteracti ng al ternati ves thatdesigners findneed fora large collectionofmathematical and scientific tools.These tool s,whenproperl yappl i ed, canprovi demoreaccurate andmorerel i abl e i nformati on foruse i n j udgi ng a desi gn than one can achi eve throughi ntui ti onoresti mati on. Thustheycanbeoftremendous hel pi ndeci di ngbetween al ternati ves. However,sci enti fi c tool scannotmakedeci si ons fordesi gners; theyhaveeveryri ghttoexertthei ri magi nati onandcreati veabi l i ti es, even tothe extent ofoverrul i ng the mathemati calpredi cti ons.Probabl ythel argest col l ecti onofsci enti fi c methodsatthedesi gner' sdi sposal fal l i nto the category cal l ed anal yss.These are the techni ques whi chal l ow the desi gnerto cri ti cal l y exami ne an al ready exi sti ng or proposed desi gni norder to j udge i tssui tabi l i ty forthetask. Thusanal ysi s, i ni tsel f, i s notacreati ve sci ence but one ofeval uati on and rati ng ofthi ngs al ready concei ved.Weshoul d al waysbear i nmi ndthatal though mostofoureffortmaybe spentonanal ysi s, t he r ealgoali ssynt hesi s, t he desi gn of amachi ne orsyst em.Anal ysi si ssi mpl yat ool . I t i s, however , avi t al t ool andwi l li nevi tabl y be used as one step i n the desi gn process.I-3 THE SCIENCEOF MECHANICSThat branch ofsci enti fi c anal ysi s whi chdeal s wi thmoti ons, ti me, and forcesi scal fedmechani cs andi smadeupoftwoparts,stati cs anddynami cs.Statcs deal s wi ththeanal ysi s ofstati onarysystems, i .e., thosei nwhi chti mei snotafactor,and dynami csdeal s wi thsystems whi chchange wi tht i me.Asshown i nFi g.l -2,dynami cs i s al so made up oftwomaj or di sci pl i nes,fi rst recogni zed as separateenti ti es byEul er i n1775:tThei nvesti gati onofthemoti onofari gi dbodymaybeconveni entl yseparated i ntotwopar t s,t he one geomet r i cal ,t he ot her mechani cal .I nt he f i r stpar t ,t he t r ansf er enceof t hebodyfroma gi ven posi ti on toany other posi ti on must be i nvesti gated wi thoutrespect tothecauses of t hemot i on.andmust ber epr esent ed byanal yt i calf or mul ae,whi chwi l l denet he posi t i on of each poi ntof t he body.Thi si nvest i gat i onwi l l t her ef or e be r ef er abl e sol el yt o geomet r y,or r at hert ost er eot omy.Iti scl earthatbytheseparati on ofthi spartofthequesti onfromtheother,whi chbel ongs properl ytoMechani cs, thedetermi nati onofthemoti onfromdynami cal pri nci pl eswi l l be made much easi er thani fthe twoparts wereundertaken conj oi ntl y.Thesetwoaspects ofdynami cswerel aterrecogni zed asthedi sti nctsci ences ofki nematcs (fromtheGreekwordknema, meani ng moti on)andki netcs,and dealwi thmoti on and the forces produci ng i t, respecti vel y.fNoui comment. Acad.Petrop., vol .20,1775; al so i n"Theori amotuscorporum,"1790.Thet r ansl at i oni s byWi l l i s,"Pr i nci pl es of Mechani sm, " 2nd ed. ,p.vi i i , 1870.bei t s, f l ,gn4THEORYOFMACHI NESANDMECHANI SMSFigurel-2Thei ni ti al probl emi nthedesi gn ofamechani calsystem i sthereforeunderstandi ng i tski nemati cs. Knemati csi s thestudy ofmoti on, qui te apartfromtheforceswhichproducethatmotion.Moreparticularly,kinematicsisthestudyofposi ti on,di spl acement,rotati on,speed,vel oci ty,andac-cel erati on. Thestudy, say, ofpl anetary ororbi talmoti on i s al so a probl em i nkinematics, butinthisbookweshall concentrate ourattentiononkinematicprobl ems whi chari se i n the desi gn ofmechani calsystems.Thus, the ki nema-tics ofmachines and mechanisms is the focusofthe nextseveral chapters ofthi sbook.Stati cs andki neti cs, however,are al so vi tal parts ofacompl etedesi gn anal ysi s, and they are covered as wel l i n l ater chapters.Itshould becarefullynotedintheabovequotationthatEulerbased hisseparati on ofdynami cs i ntoki nemati cs andki neti cs ontheassumpti on thattheyshoulddealwithrgdbodies. Itisthisveryimportantassumption thatallowsthe twotobe treated separately. Forflexiblebodies, theshapes ofthebodi es themsel ves,and therefore thei r moti ons, depend on the forces exertedonthem.Inthi ssi tuati on, thestudyofforceandmoti onmusttakepl acesi mul taneousl y, thussi gni fi cantl y i ncreasi ng thecompl exi tyoftheanal ysi s.Fortunately,althoughallrealmachine partsare flexibletosome degree,machinesareusuallydesigned fromrelativelyrigidmaterials,keepingpartdeflections toaminimum.Therefore,itiscommonpracticetoassumethatdeflectionsarenegligibleandpartsarerigidwhenanalyzingamachine' skinematicperformance,and then,afterthedynamicanalysis whenloads areknown, todesi gn the parts so that thi s assumpti on i s j usti fi ed.1.4TERMINOLOGY,DEFINITIONS,ANDASSUMPTIONSReuleauxtdefinesamachine*asa" combnatonofresistantbodessoarrangedthatbythermeans themechancal forces ofnaturecanbecom-tMuchofthemateri al ofthi ssecti oni sbasedondefi ni ti onsori gi nal l ysetdownbyF.Reul eaux (1829-1905),a Germanki nemati ci an whose workmarked the begi nni ng ofa systemati ctreatment ofki nemati cs. Foraddi ti onalreadi ng see A.B.W.Kennedy,"Reul eaux' Ki nemati csofMachi nery,"Macmi l l an,London,1876; republ i shed byDover,NewYork,1963.lThereappears tobe noagreementatallontheproperdefinitionofa machine.Ina footnoteReul eaux gi ves17 defi ni ti ons, and hi s transl ator gi ves 7 more and di scusses the whol eprobl emi ndetai l .THEGEOMETRYOFMOTION5pelledtodoworkaccompanedbycertaindetermnatemotions."Healsodefines amechanismasan" assemblage ofresistantbodies,connected bymouable joints, to form aclosedknematicchanwithonelink fixed andhauing the purpose oftransformingmotion."Some lightcan be shed onthese definitions bycontrasting themwiththetermstructure.Astructureisalsoacombinationofresistant(rigid) bodiesconnectedby j oi nts, but i ts purpose i s not to do work or to transform moti on. Astructure (such as a truss) i s i ntended to be ri gi d. It can perhapsbe moved frompl ace topl ace and i s movabl e i nthi ssense ofthe word;however, i thas nointernalmobility,norelatiue motionsbetween itsvariousmembers, whereasboth machi nes and mechani smsdo. Indeed the whol e purpose ofa machi ne ormechanism istoutilizethese relativeinternalmotionsintransmittingpowerortransformingmotion.Amachi nei sanar r angementof par t sf or doi ngwor k, adevi ce f orapplyingpowerorchanging itsdirection.Itdiffersfromamechanism initspurpose. Ina machi ne, terms such as force, torque, work,and power descri bethe predomi nant concepts. Ina mechani sm,though i tmay transmi t power orforce, thepredominantidea inthemindofthedesigner is one ofachieving adesi r ed mot i on. Ther ei sadi r ectanal ogy bet ween t het er ms st r uct ur e,mechani sm, and machi ne, and the three branchesofmechani csshown i nFi g.l - 2. Thet er m" st r uct ur e" i s t ost at i cs as t he t er m" mechani sm" i s t oki nemati csas the term "machi ne"i s to ki neti cs.Weshall use the wordlinktodesignate a machine partora component ofamechani sm. Asdi scussedi ntheprevi ous secti on, al i nki sassumedtobecompl et el y r i gi d. Machi ne component s whi ch donot f i t t hi s assumpt i onofrigidity,such as springs, usually have noeffect onthekinematics ofadevicebut doplaya roleinsupplying forces. Such members are notcalled links;theyar e usual l y i gnor eddur i ngki nemat i c anal ysi s,andt hei r f or ce ef f ect s ar ei nt r oduced dur i ngdynami c anal ysi s. Somet i mes,as wi t habel tor chai n,amachi ne member maypossessone- way r i gi di t y; such amemberwoul dbeconsi dereda l i nk when i n tensi on but not under compressi on.Thelinksofa mechanism must be connected together insome manner inordertotransmitmotionfromthedriuer,orinputlink,tothe follower, oroutput l i nk.These connecti ons, j oi nts between the l i nks, are cal l ed knemati cpairs(orjustpairs) because each joint consists ofa pairofmatingsurfaces,twoelements,one mating surface or element being a part ofeach of the joinedlinks.Thuswecanalso define alinkas thered connection between twoormoreelements ofdifferentkinematicpairs.Stated explicitly,the assumption ofrigidityis that there can be no relativemotion(changeindistance)betweentwoarbitrarilychosenpointsonthesame link.Inparticular,therelativepositionsofpairingelementsonanygivenlinkdonotchange. Inotherwords,thepurposeofalinkistoholdaconstant spatial relationship between the elements ofits pairs.Asa resultoftheassumption ofrigidity,manyoftheintricatedetails oftheactualpartshapes areunimportantwhenstudyingthekinematicsofamachine ormechanism. Forthisreason itis commonpractice todrawhighly6TgpoRYoFMACHI NESANDMECHANI SMSsi mpl i ed schemati c di agrams, whi chcontai n i mportant features ofthe shapeofeachl i nk,suchastherel ati vel ocati onsofpai rel ements, butwhi chcompl etel y subdue the realgeometry ofthe manufactured parts. Thesl i der-crankmechani smofthei nternal combusti onengi ne,forexampl e,can be si mpl i fi ed to the schemati c di agram shown i n Fi g. l -4bfor purposes ofanal ysi s. Suchsi mpl i fi ed schemati cs areagreat hel psi nce theyel i mi nateconfusi ng factorswhi chdonotaffect theanal ysi s; such di agrams are usedextensi vel y throughoutthi stext.However,these schemati cs al sohavethedrawback ofbeari ng l i ttl e resembl anceto physi calhardware. Asa resul t theymay gi ve the i mpressi on that they represent onl yacademi c constructs ratherthanreal machi nery. Weshoul d al waysbeari nmi ndthatthese si mpl i fi eddiagrams are intended tocarry onlythe minimumnecessary informationso asnottoconfusethei ssuewi thal l theuni mportantdetai l (forki nemati cpurposes)orcompl exi ty ofthe true machi ne parts.When several l i nks are movabl y connectedtogether by j oi nts, they are sai dto forma ki nemati c chai n. Li nks contai ni ng onl y two pai r-el ementconnecti onsare cal l ed bi nary l i nks; those havi ng three are cal l ed ternary l i nks, and so on. Ifevery l i nki nthe chai n i s connectedto at l east two other l i nks, the chai n formsone ormorecl osed l oops andi s cal l ed acl osed ki nemati c chai n; i fnot,thechai n i s referred toas open. When no di sti ncti on i s made the chai n i s assumedcl osed.I f t hechai n consi st sent i r el y of bi nar yl i nks,i t i s si mpl e- cl osed;compound-cl osedchai ns, however, i ncl ude otherthanbi naryl i nksandthusform more than a si ngl e cl osedl oop.Recal l i ng Reul eaux'defi ni ti on ofa mechani sm,we see that i ti s necessarytohaveaclosedkinematicchainwithone lnk fixed. Whenwesaythatonel i nki s fi xed, wemean that i ti s chosen as a frameofreference foral lotherl i nks; i .e., that the moti ons ofal lother poi nts on the l i nkage wi l l be measuredwi threspect tothi sl i nk,thoughtofas bei ng fi xed. Thi sl i nki napracti calmachi ne usual l y takes the formofa stati onary pl atform or base (or a housi ngrigidlyattachedtosuchabase) andiscalledthe frame orbase link.Thequesti on ofwhether thi s reference framei s trul ystati onary (i nthesense ofbeing an inertial reference frame)is immaterialinthe study ofkinematics butbecomes i mportanti nthei nvesti gati on ofki neti cs, whereforcesaretobeconsi dered. Inei thercase, onceaframememberi sdesi gnated (andothercondi ti ons are met),theki nemati c chai n becomes amechani sm andas thedri veri s moved through vari ous posi ti ons, cal l ed phases,al lother l i nks havewel l -defi ned moti ons wi threspect tothe chosen frameofreference. Weusethetermknematicchaintospecifyaparticulararrangementoflinksandjoints whenitis notclear whichlinkis tobe treated as the frame.Whentheframe l i nki s speci fi ed,the ki nemati c chai n i s cal l ed a mechani sm.Inorder fora mechani sm tobe useful , the moti ons between l i nks cannotbe completelyarbitrary;theytoomustbe constrained toproducethe properrel ati ve moti ons, those chosen bythedesi gner fortheparti cul ar task tobeperformed. These desired relative motions are obtained by a proper choice of thenumber ofl i nks and theki nds of j oi nts used toconnect them.THEGEOMETRYOFMOTI ON7Thus weare l ed tothe concept that, i naddi ti on tothe di stancesbetweensuccessi ve j oi nts, the nature ofthe j oi nts themsel vesand the rel ati ve moti onswhi ch they permi t are essenti al i n determi ni ng the ki nemati cs ofa mechani sm.Forthi s reason i ti s i mportant tol ookmore cl osel y at the nature of j oi nts i ngeneralterms, and i n parti cul ar at severalofthe more common types.The control l i ng factor whi ch determi nesthe rel ati ve moti ons al l owed by agi ven j oi nt i stheshapes ofthemati ng surfaces orel ements. Eachtypeofj oi nt has i tsowncharacteri sti c shapes fortheel ements, and each al l owsagi ventypeofmoti on,whi chi sdetermi ned bythepossi bl e waysi nwhi chthese el ementalsurfaces can move wi threspect toeach other. Forexampl e,t he pi n j oi nti nFi g.l - 3ahas cyl i ndr i c el ement sand,assumi ngt hatt he l i nkscannotsl i deaxi al l y,these surfaces onl ypermi trel ati verotati onal moti on.Thus a pi n j oi nt al l ows the twoconnectedl i nks toexperrencerel atrve rotati onabout thepi ncenter. So too,other j oi nts each have thei rowncharacteri sti cel ement shapesand rel ati ve moti ons. These shapesrestri ct the total l y arbi trarymoti onoftwounconnected l i nks tosome prescri bed type ofrel ati ve moti onand formthe constrai ni ngcondi ti onsor constrai ntson the mechani sm' smoti on.Itshoul d bepoi ntedoutthattheel ementshapes mayoftenbesubtl ydi sgui sed and di ffi cul t torecogni ze. Forexampl e, a pi nj oi nt mi ght i ncl ude aneedl e beari ng so that twomati ng surfaces, as such, are notdi sti ngui shabl e.) ^ 0) r ot .l a sFi gurel -3The si x l owerpai rs: (a) revol ut eor pi n, (b) pri sm, (c) hel i cal , (d) cyl i ndri c, ()spheri c,and (/) pl anar.k )AOf\\,8THEORYOFMACHI NESANDMECHANI SMSNeverthel ess, i fthemoti ons ofthei ndi vi dualrol l ers are notofi nterest, themoti ons al l owed bythe j oi nts are equi val ent andthepai rs are ofthesamegeneri c type. Thusthecri teri onfordi sti ngui shi ng di fferent pai rtypes i stherel ati vemoti onswhi chtheypermi tandnotnecessari l y theshapes oftheel ements, thoughthesemayprovi devi tal cl ues. Thedi ameterofthepi nused (or other di mensi on data) i s al so ofnomore i mportance than the exactsi zes and shapes ofthe connected l i nks. Asstated previ ousl y, theki nemati cfuncti on ofa l i nki s tohol d a fi xed geometri calrel ati onshi p between the pai rel ements. Ina si mi l ar w&y, the onl yki nemati c functi on ofa j oi nt orpai r i s todetermi ne the rel ati ve moti on between the connected l i nks. Al l other featuresaredetermi nedforotherreasonsandareuni mportanti nthestudyofki nemat i cs.Whena kinematicproblemis formulated,itis necessary torecognize thetype ofrel ati ve moti on permi tted i n each ofthe pai rs and to assi gnto i tsomevari abl e parameter(s)formeasuri ng orcal cul ati ng themoti on. There wi l l beas many ofthese parameters as there are degrees offreedom ofthe j oi nt i nquesti on, and they are referred to as the paroarabl es.Thus the pai r vari abl eofa pi nned j oi nt wi l l be a si ngl e angl e measuredbetween reference l i nes fi xedi ntheadj acent l i nks,whi l easpheri c pai rwi l l have three pai rvari abl es (al langl es)tospeci fy i ts three-di mensi onal rotati on.Ki nemati cpai rs weredi vi dedbyReul eaux i ntohi gher pai rsand l owerpai rs, thel atter category consi sti ng ofsi xprescri bed types tobedi scussednext. Hedi sti ngui shedbetween the categori esbynoti ng that the l ower pai rs,such as the pi nj oi nt, have surface contact between the pai r el ements, whi l ehi gher pai rs, such as the connecti on between a cam and i ts fol l ower, have l i neorpoi ntcontact between theel ementalsurfaces. However,asnoted i nthecase ofa needl e beari ng, thi s cri teri on maybe mi sl eadi ng. Weshoul d ratherl ookfordi sti ngui shi ng featuresi ntherel ati vemoti on(s)whi chthe j oi ntal l ows.The si x l ower pai rs are i l l ustrated i nFi g. l -3. Tabl e1-l l i sts the names ofthel owerpai rs and thesymbol s empl oyed byHartenberg and Denavi ttforthem, together wi ththe number ofdegreesoffreedom and the pai r vari abl esforeach ofthe si x.Theturningpairorreuolute(Fig.l-3a)permitsonlyrelativerotationandhence has one degree offreedom. Thi spai ri s oftenreferred toas a pi nj oi nt .Thepri smati cpai r(Fi g. l -3b)permi tsonl yarel ati vesl i di ngmoti onandthereforei softencal l edasl i di ng j oi nt. Ital sohasasi ngl e degree offreedom.+ R.S.HartenbergandJ.Denavi t,"Ki nemati cSynthesi s ofLi nkages,"McGraw-Hi l l ,NewYor k, 1964. Thi s book i s a cl assi con ki nemat i csand t he t i t l e i s mi sl eadi ng; a consi der abl eamountof mat er i al ont hehi st or yof ki nemat i cs,ki nemat i ct heor y. andki nemat i canal ysi s i sal soi ncl uded.THEGEOMETRYOFMOTION9Table1-lThelower PairsPai rPairSymbol vari abl eDegrees ofRelativefreedommotionRevol uteRPri smPScrewSCyl i nderCSphereGFlatFL0AsA0 orASA0 andAs^0,A, LltAx, Ay,A0Thescrew pairorhelical par(Fig.l-3c)hasonlyonedegree offreedombecause the sliding and rotational motions are related bythe helix angle ofthe thread. Thus the pai r vari abl e may be chosen as ei ther AsorA0butnot both. Notethat the screw pair reverts to a revolute ifthe helix angle ismade zeroand toa prismaticpairifthehelixangle ismade 90"'Thecylindric pair(Fig.l-3d)permitsbothangularrotationandanin-dependentslidingmotion.Thusthecylindric pairhastwodegrees offreedom.The gtobularorspheric pair(Fig.l-3e) is a ball-and-socket joint. Ithas threedegrees offreedom,a rotationabout each ofthe coordinate axes'The fl at pai rorpl anarpai r(Fi g.l -3f)i ssel dom,i fever,foundi nmechani sms i n i ts undi sgui sedform.Ithas three degreesoffreedom.Al lother j oi nt types are cal l ed hi gher pai rs. Exampl es are mati ng gearteeth,a wheel rolling on a rail, a ball rolling on a flat surface, and a cam contacting itsroller follower.Since there are an infinite number ofhigher pairs, a systematcaccounti ng ofthemi snotareal i sti cobj ecti ve.Weshal l treateachasaseparate situation as itarises.Amongthehigher pairs there is asubcategory knownaswrapping pairs.Exampl es are the connecti on between a bel t and pul l ey, between a chai n andsprocket,orbetweenaropeandadrum.Ineach case oneofthelinkshasone-way ri gi di tY.Inthetreatment ofvari ous j oi nt types, whetherl owerorhi gher pai rs,there i s another i mportant l i mi ti ng assumpti on. Throughout the book, wewi l lassume that the actualj oi nt, as manufactured, can be reasonabl y representedbyamathematical abstractionhaving perfectgeometry. Thatis, whenarealmachine joint is assumed tobe a spheric pair, forexample, itis also assumedthat there i s no"pl ay"orcl earance between the j oi nt el ements and that anydevi ati ons i n the i pheri calgeometry ofthe el ements i s negl i gi bl e.When a pi nj oi nt i s treated as a revol ute, i t i s assumedthat no axi almoti on can take pl ace;i fi ti snecessary tostudy thesmal laxi al moti ons resul ti ng fromcl earancesbetweenthereal el ements, the j oi nt mustbetreatedascyl i ndri c,thusallowingforthe axialmotion.2J3Ci rcul arLi nearHel i calCyl i ndri cSpheri cPlanarI OTHEORYoFMACHI NESANDMECHANI SMSTheterm"mechanism," as defined earlier, can refertoa widevarietyofdevi ces, i ncl udi ng bothhi gher andl owerpai rs. Therei samorerestri cti veterm,however, whi chreferstothose mechani sms havi ng onl yl owerpai rs;such amechani sm i s cal l ed al nkage. Al i nkage, then, i s connected onl ybythe l ower pai rs ofFi g. l -3.I-5PLANAR,SPHERICAL,ANDSPATIALMECHANISMSMechani sms may be categori zed i nseveraldi fferent ways toemphasi ze thei rsi mi l ari ti esanddi fferences. Onesuchgroupi ngdi vi desmechani sms i ntoplanar, sphercal, and spatal categories. Allthree groups have many things incommon;thecri teri onwhi chdi sti ngui shes thegroups,however,i stobefoundi n the characteri sti csofthe moti ons ofthe l i nks.Apl anar mechani sni s one i n whi ch al lparti cl es descri be pl ane curves i nspace and al lthese curves l i e i nparal l elpl anes; i .e., the l ociofal lpoi nts arepl ane curves paral l eltoasi ngl e commonpl ane. Thi scharacteri sti c makes i tpossi bl e torepresent the l ocus ofany chosen poi nt ofa pl anar mechani sm i ni tstruesi zeandshape onasi ngl e drawi ngorfi gure. Themoti ontrans-formati onofanysuchmechani sm i scal l edcopl anar.Thepl anefour-barl i nkage, thepl atecamandfol l ower,andthesl i der-crank mechani sm arefami l i ar exampl es ofpl anar mechani sms.The vast maj ori ty ofmechani smsi nuse today are pl anar.Planarmechanisms utilizingonlylowerpairsarecalled planarlinkages,'theymayi ncl ude onl yrevol ute andpri smati c pai rs. Al thoughapl anar pai rmi ghttheoreti cal l y be i ncl uded, thi s woul di mpose noconstrai nt and thus beequi val ent toan openi ng i ntheki nemati c chai n. Pl anar moti onal so requi resthat al lrevol ute axes be normaltothe pl ane ofmoti on,and that al lpri smati cpai r axesbe paral l elto the pl ane.Aspheri calmechansm i s one i nwhi cheach l i nkhas some poi ntwhi chremai ns stati onary as the l i nkage moves and i n whi ch the stati onary poi nts ofal l l i nksl i eatacommonl ocati on;i .e., thel ocusofeach poi nti sacurvecontai ned i n a spheri calsurface, and the spheri calsurfaces defi ned byseveralarbi trari l ychosen poi nts are al l concentrc. Themoti ons ofal l parti cl es cantherefore be compl etel y descri bed bythei r radi alproj ecti ons, or"shadows,"onthesurface ofasphere wi thproperl ychosen center. Hooke' suni versalj oi nt i s perhaps the most fami l i ar exampl e ofa spheri calmechani sm.Spheri call nkages areconsti tuted enti rel yofrevol utepai rs. Aspheri cpai r woul d produce no addi ti onalconstrai nts and woul dthus be equi val ent toan openi ng i nthe chai n, whi l eal lother l owerpai rs have nonspheri c moti on.Inspheri c l i nkages, the axes ofal lrevol ute pai rs must i ntersect at a poi nt.Spati al mechani sl r?s,ontheotherhand, i ncl udero restri cti ons ontherel ati ve moti ons ofthe parti cl es.The moti on transformati on i s not necessari l ycopl anar, normust i tbe concentri c. Aspati almechani sm may have parti cl eswi thl oci ofdoubl e curvature. Anyl i nkage whi chcontai ns ascrew pai r, forTHEGEOMETRYOFMOTI ONI Iexampl e, i s a spati almechani sm, si nce the rel ati ve moti on wi thi na screw pai ri s hel i cal .Thus, theoverwhel mi ngl yl arge category ofpl anar mechani sms and thecategory ofspheri calmechani sms are onl yspeci alcases, orsubsets,oftheal l -i ncl usi ve category spati almechani sms. Theyoccuras aconsequence ofspeci algeometry i n the parti cul ar ori entati ons ofthei r pai r axes.Ifpl anarandspheri calmechani sms areonl yspeci alcases ofspati almechani sms,whyi s i tdesi rabl e toi denti fythem separatel y?Because oftheparti cul argeometri ccondi ti onswhi chi denti fythesetypes,manysi mpl i -fi cati ons are possi bl e i nthei r desi gn and anal ysi s. Aspoi nted out earl i er, i ti spossi bl e to observe the moti ons ofal lparti cl es ofa pl anar mechani sm i n truesi ze andshape fromasi ngl e di recti on. Inotherwords,al l moti ons canberepresentedgraphi cal l y i nasi ngl e vi ew.Thus, graphi caltechni ques are wel lsui tedtothei rsol uti on.Si ncespati al mechani sms donotal l havethi sfortunate geometry, vi sual i zati on becomes moredi ffi cul t and morepowerfultechni ques must be devel oped forthei r anal ysi s.Si nce the vast maj ori tyofmechani smsi n use today are pl anar, one mi ghtquesti on the need ofthe more compl i cated mathemati caltechni ques used forspati almechani sms. Thereareanumberofreasons whymorepowerfulmethods are ofval ue even though the si mpl er graphi caltechni ques have beenmastered.Theyprovi de new, al ternati ve methods whi chwi l l sol ve the probl ems i n adi fferentway.Thustheyprovi deameans ofchecki ngresul ts. Certai nprobl ems bythei r nature mayal so be more amenabl e toone method thananother.Methodswhichareanalyticalinnaturearebettersuitedtosolutionbycal cul ator ordi gi talcomputer than graphi caltechni ques.Even though the maj ori tyofusefulmechani smsare pl anar and wel lsui tedtographicalsolution,thefewremainingmustalsobeanalyzed,andtechni ques shoul d be knownforanal yzi ng them.One reason thatpl anar l i nkages are so commoni s thatgood methods ofanal ysi s forthe more generalspati all i nkages have not been avai l abl e unti lqui te recentl y. Wi thoutmethods forthei r anal ysi s,thei r desi gn and use hasnotbeen common,eventhoughtheymaybei nherentl ybettersui ted i ncertai n appl i cati ons.5.Wewi l l di scover thatspati all i nkages are muchmore commoni npracti cethan thei r formal descri pti on i ndi cates.Consi derafour-barl i nkage. Ithasfourl i nksconnected byfourpi nswhose axes are paral l el . Thi s"paral l el i sm"i s a mathemati calhypothesi s; i t i snot areal i ty. Theaxes as produced i nashop-i nanyshop, nomatter howgood-wi l l onl y be approxi matel y paral l el . Ifthey are far out ofparal l el , therewi l l bebi ndi ngi nnouncertai n terms,andthemechani sm wi l l onl ymovebecausethe "ri gi d"l i nks fl ex and twi st, produci ng l oads i n the beari ngs.Ifthel .11J .4.12THEORYoFMACHI NESANDMECHANI SMSaxes are nearl y paral l el , the mechani sm operates becauseofthe l oosenessofthe runni ng fi ts ofthe beari ngs orfl exi bi l i tyofthe l i nks. Acommon wayofcompensati ng forsmal l nonparal l el i sm i stoconnectthel i nkswi thsel f-al i gni ng beari ngs, actual l yspheri calj oi nts al l owi ngthree-di mensi onalrota-ti on. Such a "pl anar"l i nkage i s thus a l ow-grade spati all i nkage.1-6 MOBILITYOneof thefirstconcernsin eitherthedesignor theanalysisof a mechanismisthe number ofdegreesoffreedom, al so cal l ed the mobl ty ofthe devi ce. Themobi l i tytofamechani sm i sthenumberofi nputparameters (usual l y pai rvari abl es)whi ch must be i ndependentl ycontrol l ed i n order to bri ng the devi cei ntoaparti cul ar posi ti on. Ignori ng forthemoment certai n excepti ons tobementi onedl ater,i ti spossi bl e todetermi ne themobi l i tyofamechani smdi rectl yfromacountofthenumber ofl i nksand thenumber and types ofj oi nts whi ch i ti ncl udes.Todevel opthi srel ati onshi p, consi der thatbeforetheyareconnectedtogether, each l i nk ofa pl anar mechani sm has three degreesoffreedom whenmovi ngrel ati ve tothefi xedl i nk.Notcounti ng thefi xedl i nk,therefore, ann-l i nkpl anar mechani sm has 3(n - l )degrees offreedombefore anyofthej oi nts are connected. Connecti ng a j oi nt whi chhas one degree offreedom,such as a revol ute pai r, has the effect ofprovi di ng twoconstrai nts betweenthe connected l i nks. Ifa twodegree-of-freedompai r i s connected, i t provi desoneconstraint.Whentheconstraintsforall joints aresubtractedfromthetotalfreedoms ofthe unconnected l i nks, we fi nd the resul ti ng mobi l i tyoftheconnected mechani sm.When we use i r to denote the number ofsi ngl e-degree-of-freedompairsand iz forthenumberoftwo-degree-of-freedompairs,theresul ti ng mobi l i tymofa pl anar n-l i nkmechani sm i s gi ven bym: 3( n- l ) - 2i , - i zWri tten i n thi s form, Eq.(l -l )i s cal l edthe Kutzbachcri teri onfor theofa pl anarmechani sm.Its appl i cati oni s shown forseveral si mpl e( 1- 1)mobi l i tycasesi nFi g. 1-4.Ifthe Kutzbachcri teri on yi el ds m)0,the mechani sm has mdegreesoffreedom. Ifm : l ,the mechani sm can be dri ven bya si ngl e i nputmoti on. Ifffi:2, then twoseparatei nput moti ons are necessarytoproduce constrai nedmoti on forthe mechani sm; such a case i s shown i nFi g. l -4d.IftheKutzbachcri teri onyi el dsffi= 0,asi nFi g.l -4a,moti oni si m-tThe Germanliteraturedistinguishesbetweenmouabilifyandmobility.Movabilityincludesthe si x degrees offreedom ofthe devi ce as a whol e, as though the ground l i nkwere not fi xed, andthus appl i es toa ki nemati cchai n. Mobi l i tynegl ects these and consi ders onl ythe i nternalrel ati vemoti ons, thus appl yi ng toa mechani sm. TheEngl i sh l i terature sel dom recogni zes thi s di sti ncti on,and the terms are used somewhat i nterchangeabl y.THEGEOMETRYoFMOTION13n = 3 , / r= 3i r =o, m=o( c )n =4 , j ,=4 ,i 2 = O, m= 1( b)n = 4 , j , = 4 ,i 2 = 0 , m= 1( c )n = 5 , / l= S,i r = O, m= 2@\Figure l-4Applicationsof the Kutzbach mobility criterion.possibleand the mechanismforms a structure.Ifthe criterion gives m :- |orless, then there are redundant constraintsinthechain and itformsa-staticalJv-jndetermiate-struclure.-Exar4oles-are-shown--in-qge.-l--5.-Note--inthese examplesthat when three links are joined byasinglepin, two jointsmust be counted;such a connectionis treatedas two separatebut corrcentricpai rs.Figurel-6showsexamplesofKutzbach'scriterionappliedtomechanismswith two-degree-of-freedom joints. Particularattentionshouldbepaid to the contact (pair) betweenthe wheel and the fixed link inFig. l-6b.Here it was assumedthat slippingis possiblebetweenthe links. Ifthis contactincludedgearteeth or if friction was high enoughto preventslipping,the jointwould be counted as aone-degree-of-freedompair, since only one relativemotion would be possiblebetweenthe links.n = 5 , i t = 6 ,i z = O, m= 0( o )n =6 , / r=B,i z = 0 , m= - 1( )Figure 1-5 Applicationsof the Kutzbach criterion to structures.14THEORYOFMACHI NESANDMECHANI SMSn - - 3 , i , = 2 ,i r = 1 , m=1( a ), = 5 , i tj z = o , m( a )n = 4 , j r = 3j r - - 1 , m= 2( b )Figure 1-6Someti mes theKutzbachcri teri onwi l l gi veani ncorrectresul t. Noti cethatFig.l-7 arepresents astructureandthatthecriterionproperlypredictsm: 0. However , i f l i nk5i sar r anged asi nFi g. l - 7b, t her esul t i sadoubl e-paral l el ogram l i nkagewi thamobi l i tyofIeventhoughEq.(1-t)i ndi cates thati ti sastructure. Theactualmobi l i tyofIresul ts onl yi ftheparal l el ogram geometry i sachi eved. Si nce i nthedevel opment oftheKutz-bach cri teri on no consi derati on was gi ven tothe l engths ofthe l i nks orotherdi mensi onalproperti es, i ti s not surpri si ng that excepti ons to the cri teri on canbe foundforparti cul ar cases wi thequall i nkl engths, paral l ell i nks, orotherspeci algeometri c features.Eventhough the cri teri on has excepti ons, i tremai ns usefulbecause i ti sso easi l y appl i ed. Toavoi d excepti ons i t woul d be necessaryto i ncl ude al lthedi mensi onalproperti es ofthemechani sm. Theresul ti ng cri teri onwoul dbeverycompl exandwoul dbeusel ess attheearl ystages ofdesi gnwhendi mensi ons may not be known.Anearl i er mobi l i tycri teri on named after Grbl er appl i es tomechani smswi thonl ysi ngl e-degree-of-freedomj oi nts wheretheoveral l mobi l i tyofthemechani smi s uni t y.Put t i nE i z: 0 and m: 1i nt oEq.( 1- l ) ,we f i nd Gr bl er ' s, = 5 , i ti 2 = o , m( b )Figure l-7THEGEOMETRYOFMOTI ON15cri teron forpl anar mechani smswi thconstrai ned moti on3n- 2i t - 4: 0 (t-2)From thi s wecan see, forexampl e, that a pl anar mechani sm wi tha mobi l i tyofIand onl ysi ngl e-degree-of-freedom j oi nts cannot have an odd number ofl i nks.Al so, wecanf i ndt hesi mpl estpossi bl e mechani sm of t hi st ype; byassumi ng al l bi nar yl i nkswef i ndn : j t : 4.Thi sshows whyt hef our - barl i nkage, Fi g.l -4c, and the sl i der-crank mechani sm, Fi g.l -4b, are so commoni n appl i cati on.BoththeKutzbachcri teri on,Eq.(1-l ),andtheGrbl ercri teri on,Eq.(l -2), werederi ved forthe case ofpl anar mechani sms. Ifsi mi l ar cri teri a aredevel oped forspati almechani sms,wemust recal lthat each unconnected l i nkhas si x degreesoffreedom and each revol ute pai r, forexampl e, provi des fi veconstrai nts. Si mi l ar arguments then l ead tothe three-di mensi onal formoftheKutzbach cri teri on.m : 6( n-1)-5j '-4i r - 3h- 2j o-j 'and the Grbl ercri teri on6n- 5i ' - 7: 0( l - 3)( l - 4)The si mpl est formofa spati almechani smt wi thal lsi ngl e-freedompai rs and amobi l i t y of I i s t her ef or e, f l :j t : 7.1.7 KINEMATICINVERSIONInSec. l -4wenoted that every mechani sm has a fi xed l i nkcal l ed the frame.Unti l aframel i nkhasbeenchosen, aconnected setofl i nksi scal l edaki nemati c chai n. Whendi fferentl i nksarechosen as theframeforagi venkinematic chain, the relatu motions between the various links are not alteredbutthei rabsol ute moti ons (those measured wi threspect totheframel i nk)may be changed drasti cal l y. The process ofchoosi ng di fferent l i nks ofa chai nforthe frameis knownas knematc inuersion.Inanr-l i nkki nemati cchai n, choosi ng each l i nki nturnastheframeyi el ds r di sti nct ki nemati c i nversi ons ofthechai n, ndi fferent mechani sms.Asan exampl e, the four-l i nksl i der-crank chai n ofFi g.1-8 has fourdi fferenti nver si ons.Fi gurel -9ashows thebasi c sl i der-crank mechani sm, as foundi nmosti nternal combusti onengi nes today.Li nk4,thepi ston,i sdri venbytheexpandi ng gases and formsthe i nput;l i nk2, the crank, i s the dri ven output.The frame i s the cyl i nder bl ock, l i nkl .Byreversi ng the rol es ofthe i nput andoutput, thi s same mechani sm can be used as a compressor.tNotethatal l pl anarmechani sms are excepti onstothespati al -mobi l i tycri teri a.Theyhavespeci algeometri ccharacteri sti cs i nthatal l revol uteaxesareparal l el andperpendi cul artothepl ane ofmoti onand al l pri smaxes l i ei nthe pl ane ofmoti on.16 rneonyoFMACHI NES AND MECHANI SMSFigurel-8 Fourinversionsof theslider-crankmechanism.Fi gure l -8bshows the same ki nemati c chai n; however, i ti s nowi nvertedand l i nk2i sstati onary. Li nkl ,formerl ytheframe,nowrotates about therevol ute atA.Thi si nversi on ofthe sl i der-crank mechani sm was used as thebasi s ofthe rotaryengi ne foundi n earl y ai rcraft.Another i nversi on ofthe same sl i der-crank chai n i s shown i nFi g. l -8c;i thas l i nk3, formerl ytheconnecti ng rod,as theframel i nk.Thi smechani smwasusedtodri vethewheel sofearl ysteam l ocomoti ves,l i nk2bei ngawheel .Thefourthandfi nal i nversi on ofthesl i der-crank chai n has thepi ston,link4, stationary. Althoughit is not foundin engines, by rotating the figure 90'cl ockwi se thi s mechani sm can be recogni zed as part ofa garden water pump.Itwi l l be noted i n the fi gure that the pri smati c pai r connecti ng l i nksIand 4 i sal so i nverted; i .e. the "i nsi de"and "outsi de"el ements ofthe pai r have beenr ever sed.1.8GRASHOF' SLAWAveryi mportant consi derati on when desi gni ng a mechani sm tobe dri ven byamotor,obvi ousl y,i stoensure thatthei nputcrankcanmake acompl eterevol uti on. Mechani sms i nwhi chno l i nkmakes a compl ete revol uti on woul dnotbeusefuli nsuch appl i cati ons. Forthefour-barl i nkage, there i saverysi mpl e test ofwhether thi s i s the case.ulTHEGEOMETRYOFMOTION17Grashof's lawstatesthat for a planar four-bar linkage,the sum oftheshortest andlongest linklengths cannot be greater thanthesum oftheremainngtwo lnk lengthsifthereis to be continuousrelatiuerotaton betweentwo members.This is illustratedin Fig. l-9, wherethe longestlink haslength/,the shortestl i nk hasl engths, and the othertwo l i nks havel engthspandq.l nthi s notati on,Grashof' sl awstatesthat one ofthe l i nks, i nparti cul artheshortestl i nk, wi l lrotate conti nuousl yrel ati veto the other three l i nks i fandonl y i fs * / _^ 0;POSITIONANDDISPLACEMENT47magi nar yX S-vT-IIj Rvit _l al( )t' i gure2-15 Correl ati onof pl anarvectorsand compl exnumbers.:rrstworectangul arcomponentsofmagni tudesR' : Rc o s d RY: Rs i n 0, ii t hR : V(RTJ (Rryo IR-------]B :t an-t #(2-22)(2-23)\r' rtthatwehavemade thearbitrarychoicehere ofaccepting thepositive.quare rootforthe magnitude Rwhen calculating frornthe components ofR.Therefore wemust be carefultointerpretthesigns ofR' and Rvindividually* hen deciding upon the quadrant of0. Notethat 0 s defined as the angle f rom:he positiue xaxis tothe positiue end ofoector R,measured aboutthe orign,l the uector, ands postiue when measured counterclockwse.Exampl e 2-2ExpressthevectorsA:10/30" andB:8/-15' i nrectangul arnotati ontandfi nd thei rsum.SolurtoNThevectorsareshowninFig.2-16 andareA : 10cos30oi + 10sin30" i : A.ee+ S.OOiB :8cos( - 15) I+ 8si n( - 15)i : l . l l - 2. 07ic : A. - I i::Xi::;:t. (5oo-207)iThe magni tudeof the resul tanti s found from Eq. (2-23)c: VT69TTE: 16. 6as i s the angl ed :t an t#= l o. l oThe fi nalresul t i npl anar notati oni sC: 16. 6/ 10. 1' Ans.iMany cal cul ators are equi pped to perform pol ar-rectangul ar and rectangul ar-pol ar conversi ons: : r ect l v.48THeonYOFMACHI NESANDMECHANI SMSI'IB: 8 1 - t 5 " Figure2-16 Example 2-2.Anotherwayoftreati ngtwo-di mensi onalvectorprobl emsanal yti cal l ymakes use ofcompl exal gebra. Al thoughcompl ex numbers are notvectors,they can be used to representvectors i n a pl ane by choosi ng an ori gi n and realand i magi nary axes. Intwo-di mensi onalki nemati cs probl ems, these axes canconveni ent l y be chosen coi nci dentwi t ht he xr yraxes of t he absol ut e coor -di nat e syst em.Asshowni nFi g.2-15b, thel ocati on ofanypoi nti nthepl ane canbespeci fi ed ei ther byi tsabsol ute-posi ti on vectororbyi tscorrespondi ng realand i magi nary coordi natesR: R' +j Rtwhere the operator ji s defi ned as the uni t i magi nary number(2-24)j : { - l ( 2- 25\The realuseful nessofcompl ex numbers i n pl anar anal ysi s stems fromtheease wi thwhi chtheycanbeswi tched topol arform.Empl oyi ngcompl exrectangul ar notati on forthe vector R, we can wri teR: R/ - 0- :R cos 0 + j R si n 0Butusi ng the wel l -knownEul er equati on fromtri gonometry,etj o : cos d r- jsi n 0wecan also writeRincomplex polarformasR : Rj dwhere the magni tude and di recti on ofthe vector appear expl i ci tl y. Aswe wi l lsee i n the next twochapters, expressi on ofa vector i n thi s formi s especi al l yusefulwhen di fferenti ati on i s requi red.Some fami l i ari tywi thusefulmani pul ati on techni ques forvectors wri tteni ncompl expol arformscanbegai ned bysol vi ngthefourcases ofthel oop-cl osure equati on agai n. Wri ti ngEq.(2-16) i ncompl expol arform,wehaveCei or : Aei o. c+Bei ot (2-2e)In case I the two unknowns are C and 06.We begi nthe sol uti on by separati ngtherealand i magi nary parts of the equati on. By substi tuti ngEul er' s equati on (2-27)(2-26)(2-27)(2-28)POSITIONANDDISPLACEMENT49\r e obtainC( cos0c + j s i n0c) : A( c os0 +i si nda) * B( c os 0n+i si n0r )@)On equatingthe real terms and the imaginaryterms separatelywe obtain two:eal equationscorrespondingto the horizontaland vertical componentsof the:rro-di mensi onal vectorequati onCcos0c: A cos0a* Bcos gs ( b)C si n0c: A si n0a*B si n0a(c)B'squari ngand addi ngthesetwo equati ons06 i s el i mi natedand a sol uti oni s: oundf or CC: f + B+ 2AB cos(0 - 0) (2-30)The posi ti ve squareroot was* trul dyi el d a negati vesol uti on: oundf romchosen arbi trari l y;thenegati ve square rootforCwithdc differing by180' . The angle is- , Asi n 0A+ Bsi n 0a0 -- tan (2-3t)(2-32)A cos 0A+ B cos 0s' r herethesi gnsofthenumeratoranddenomi natormustbeconsi dered-eparatel y i n determi ni ng the proper quadrantt of0c. Onl y a si ngl e sol uti on i s:ound forcase l ,as previ ousl y i l l ustrated i nFi g. 2-8.Forcase 2athetwounknownsofEq.(2-29) are thetwomagnitudes A.:ndB.Thegraphi cal sol uti ontothi scasewasshowni nFi g.2-9.one-r)nveni ent wayofsol vi ng i ncompl ex pol ar formi s tofi rst di vi deEq. (2-29). ; o- \, , " r rCei (qc-ei l: A+Bri (q 0)( d)' -' trmp&ri ng thi s equati on wi thFi g.2-17,wesee that di vi si on bythe compl exrol arformofauntvectorei qo hastheeftectofrotati ngthereal and:rnagi naryaxes bythe angl e ga such that the realaxi s l i es al ong the vector A.\\' e cannowuseEuler' sequation(2-27) toseparate therealandimaginary, omponent sCc os( 0c - 0 ) : A+B c os ( ds-0ik )C sin (0, - 0o) : Bsin (0n - 0)().:nd wenotethatthevectorA,nowreal ,hasbeenel i mi natedfromonecquati on. The sol uti on forBi s easi l y found:B: Cs i n( c - 0n) -si n ( 0s -0 ). Cal cul ators ofdi fferentbrandsvarysomewhati nthetreatmentoftheuni tsandquadrant:.rngl es.Each ofus mustbecome fami l i arwi ththe characteri sti cs ofhi s orher owncal cul ator.50r nponyoFMACHI NESANDMECHANTsMSFi gur e2- 17 Rot at i onofaxesbydi vi si onof compl expol ar equat i onby e 0^. ( a) Or i gi nal axes;( )rotated axes.The sol uti on forthe other unknown magni tudeAi sfound i ncompl etel yanalogousfashion.Dividing Eq. (2-29)by eiq"alignsthe real axis along vectorB. The equationis then separatedinto real and imaginaryparts and yieldsA=Cs i n( c - 0 ) -si n( 0 - 0s)(2-33)Asbefore, case 2ayi el ds a uni que sol uti on.Thegraphi cal sol uti ontocase2bwasshowni nFi g.2-10. Thetwounknownsare Aand dB.Webegin byaligning the real axis along vectorAandseparating realandimaginarypartsasincase 2a.Thesolutionsarethenobtai ned di rectl y fromEqs. () and (/)0s: 0t + si n- rCsi n( ?c -0i'B ' - (2-34)A : C cos (06'- 0) - Bcos (0s - 0)(2-35)Wenote that the arc sine term is double-valued and therefore case 2bhas twodi sti nct sol uti ons , A,0n and A' ,0' s.Case2chasthetwoangl es0and0sasunknowns.Thegraphi calsofuti on wasshowni nFi g.2-l l .Inthi scase weal i gn thereal axi sal ongvectorCQ :4t i t 0o- 0c) *Bei ( eB- gc)Usi ngEul er' sequati on toseparate components and then rearrangi ng terms,weobtainA cos( 0o-0c) : C- Bcos( 0s -0c)A si n(0^-0c):-B si n(0" - 0c)Squari ngboth equati onsand addi ng,resul tsi n@)( h)(r)Az : C2+ 82 - zBCcos (ds - 0c)asthel awofcosi nes forthevector0s : 0c+ cos- 'C2 +B1-- A20t : Lc- rcos- C2+. 42 , -822CAPOSI TI ONANDDI SPLACEMENT 5l(i)tri angl e. ItcanbeWerecogni ze thi srol ved for0aPutti ng Conthe other si de ofEq.(ft) before squari ng and addi ng.rnotherformofthe l aw ofcosi nes. fromwhi ch(2-36)resul tsi n(2-37)fhepl us-or-mi nus si gns i nthese twoequati ons arearemi nder thatthearc t r si nes ar eeach doubl e- val uedandt her ef or e0a and0, q each havet wo.trl uti ons. These twopai rs ofangl escan be pai red natural l y together as 0n, 0n.rnd 0' o,0' undertherestri cti on ofE,q. (i )above. Thuscase 2chas twoJi st i nct sol ut i ons. as shown i n Fi e.2- 11.r-9 THE CHACE SOLUTIONSTO PLANAR VECTOR EQUATIONS' \ s wesaw i nt hel astsect i on,t heal gebr a i nvol ved i nsol vi ng even si mpl e:"' l anarvector equati ons can become cumbersome.Chace was the fi rst totake.rdvantageofthebrevi ty ofvector notati on i nobtai ni ng expl i ci t cl osed-form.trl uti onS toboth two- and three-di mensi onal vector equati ons.t Inthi s secti on* c' wi l lstudy hi s sol uti onsfor pl anar equati onsi n terms of the four casesof the.()op-cl osureequati on.We agai n recal lEq.(2-16),the typi calpl anar vector equati on. Interms ofrl asni tudes and uni t vectorsi t can be wri ttenCC: AA+BB (2-38)rnd i t may contai n two unknowns consi sti ngof two magni tudes,two di recti ons,, r rone magni t udeand one di r ect i on.Case 1i sthesi tuati on wherethemasni tude anddi recti on ofthesame\ cctor, say C and , formthe twounknows. The method ofsol uti on forthi s.i rsewas shown i n Exampl e 2-2. The general form of the sol uti on i sc: ( A. i +n. i l i +t a + B.j [ ( 2- 3e)Incase 2athemagni tudes oftwodi fferentvectors:,say Aand B,areunknown. TheChace approach forthi s caseconsi stsi nel i mi nati ng one ofthernknowns by taki ng the dot product ofevery vector wi th a new vector choseni M. A. Chace,VectorAnal ysi sof Li nkages,J. Eng.Ind.,ser.B, vol .55,no.3, pp.289-291.\ueust1963.52THEORYOFMACHI NESANDMECHANI SMSso that one of the unknownsis eliminated.We can eliminatethe vector B bytaking the dot product of every term of the equationwith B x .c. ( Bx :A. 1nx) +r n. ox)@)Thus, si ncen x fr i s perpendi cul arto , n . O x : 0; henceA=9' (?' I )A. ( Bx k )Ina similarmanner, weobtainthe unknownmagnitude BB=9' (+' i )B. ( AXK)Forcase2b,theunknownsarethemagnitudeofonevectorandthedi recti on ofanother, sayAand.Webegi n thesol uti on byel i mi nati ng AfromEq. (2-38)C. ( xt - 88. 1xt r )Now,fromthe definitionofthe dotproductoftwovectorsP' Q: PQc os ' fwe note thatBB' ( x k : Bc os f ( c )where { i s the angl ebetweenthe vectors and 1 x t).Thusc os t ' : . t x f t l@)The vectots and x are perpendicularto eachother; hencewe are free tochooseanothercoordi natesystemi havi ngthedi recti onsi: x Iandp : f.In thi s referencesystem,the unknownuni t vector can be wri tten as: c os ( Ax [ ) +s i n@Ifwe nowsubsti tute Eq. (d)i nto (b)and sol ve forcos , we obtai n,c' ( x[ COS 4 l : - -BThen si n@: +V1- goEA:- t C. t x) l 'Substi tuti ngEqs. U) and (g) i nto () and mul ti pl yi ngboth si desby the knownmagni tudeB gi vesB : [c .( x tl Xx k + Y6' -s '6tl ' (2-42)Toobtai n thevector Awemay wi sh touse Eq.(2-38)di rectl y andperform the vector subtraction.Alternatively,ifwe substituteEq. (2-4D andrearrange,we obtai nA:c- t c. t xl l t x +- t / B'- t c. ( xr l l ' @)(2-40)(2-4r)( b)( e)U)(e) fB,I-t--_BPOSITIONANDDISPLACEMENT53The first two terms of this equationcan be simplifiedas shown in Fig. 2-18a.The x tdirection islocated 90o clockwise fromthedirection. ThemagnitudeC .t,i x il is the-projection of C in the x direction.Therefore,when tC ' (A x k)XA x k)issubtracted fromC,theresultisavectorofmagni tudeC . i n the di recti on.Wi th thi s substi tuti on,Eq. (ft) becomesn : [c . +| n' - tc . t x rl l ' ](2-43)Fi nal l y,i ncase2ctheunknownsarethedi recti onsoftwodi fferentvectors, say Aand B. Thi s case i s i l l ustrated i n Fi g.2-18b, where the vector Cand the twomagnitudes Aand Bare given. Theproblemis solved byfindingthepointsofintersectionoftwocirclesofradiiAandB.Webeginbydefining ^ anewcoordinatesystemirAwhoseaxesaredirectedsothatI : C x kand : C, as shown i nthe fi gure. Ifthe coordi nates ofone ofthepoints ofintersection intheipcoordinatesystem are designated as aandu,t henA: ui +r pandB- - ai +( C- u) f i ( t )The equati on ofthe ci rcl e ofradi us Ai su2+u2: A2The circleofradius Bhas the equationu2+( u- C) ' : 82r)[u2+u2- 2Cu+C2: BzSubtracting Eq.(k)from(j)and solving foruyieldsA2 - 8 2 + C2( c. A) A(i)( k)( / )2Cp ^ /- - A /Fi gure2-18l a) ( b )54THEonYoFMACHI NESANDMECHANI SMSSubstitutingthis into Eq. (i) and solving for u givesl l :.-rrl wThefinalstepistosubstitute thesevaluesofuanduintoreplace iandpaccording totheirdefinitions. Theresults are( m)Eqs. (i )and to(2-44)(2-4s)2.IOALGEBRAICPOSITIONANALYSISOFPLANARI,INKAGESThi ssecti on i l l ustrates severalal gebrai c methods ofapproach tothe posi ti onanal ysi sofpl anarmechani sms. Thethreemai nadvantages ofal gebrai cmet hods over t hegr aphi calappr oach of Sec. 2- 7ar e ( 1) t hei ncr easedaccuracywhi chcanbeachi eved, (2)thefactthattheyaresui tabl eforcomputer orcal cul ator eval uati on, and (3) the factthat once the formofthesol uti on has been found, i tcan be eval uated forany set ofl i nk di mensi ons orfordifferentpositions withoutstarting overagain. Themain disadvantage, aswewillsee, is thatthenature ofthe equations maylead totedious algebraicmanipulations infinding the formofthe solution.Letus returntotheanal ysi s ofthe sl i der-crank mechani sm ofFi g. 2-12,whichwassolvedgraphicallyinSec. 2-7.One commonwayofformulatingthis problem algebraically is tonotice fromthe figure that the verticalpositionofpoi nt Bcan be rel ated tothe l ength and angl e ofl i nk2orofl i nk3. ThusA: + r lo, - ( A' ?- P: +c' ?) ' ? rxp -t -pc=en - +JA'R6si n 0z:- RcB si n 0rRnosin0r :- ffisn :the geometry ofFi g.2-l 2awesee thatso thatAlsofrom( a )( b)( c)Rc : RCoS0zIRcncos 03whichcan be rearranged toreadRc -R CoS0z: Rcscos 03@)Next,squari ng and addi ng Eqs. (a)and (d), we el i mi nate the unknown03R - ZRcRa cos 02*R2B: R2cr( e)Thisequationcanbesolvedfortheunknownangle0z as afunctionofthesl i der posi ti on R6:POSITIONANDDISPLACEMENT550 2 : COSSubsti tuti ngthi s sol uti oni nto Eq.i trr the other unknownangl e030 r : COSIRt c+ RLr - R' uo(2-47)2RCRCB\l though transcendental ,these are cl osed-form sol uti ons that can be qui ckl y.,' r al uated foranyset ofdi mensi onalparameters atanyposi ti on R.ofthe. l i der .In the more usualappl i cati onsofthe sl i der-crank mechani sm,the angl e of:he crank 0zi s gi ven and the angl e ofthe connecti ng rod03and the posi ti on of: he sl i derR. ar e t obe f ound.Thi s pr obl em can be sol ved byr ecal l i ng t hat. l f l Ccos, :- / 1I J :6. i e have, f rom Eq.(b),, RL +R2ro-R2r.u2RcRBt(d)gi ves an equati on(2-46)whi ch can be sol ved(2-48).rhere theposi ti ve square rootwas chosen tocorrespond toFi g. 2-l 2a;theregati vesquare rootdesi gnates adi fferentassembl y ofthel i nkswi ththeIrston to the l eft ofpoi nt A.FromEqs. (c)and (2-48),the posi ti on ofpoi ntCRc : Rs aCoS0r +f m(2-49\Wemayaskourselvesinstartingthealgebraicanalysishowwewill:ecognize the "proper"equations fromthe figure; howwe willknowwhere to.rrok orwhenwehaveenoughequati ons. Oneoftheadvantages ofthe-' r)mplex algebra approach ofSec. 2-8 is thatitguides us inthedevelopment,' fthese i ni ti al equati ons. Referri ngagai n toFi g.2-12a, wecanwri tethe"oop-cl osure equati on i n compl ex pol ar formRc : RBsez*Rueiot(f)' ' r' here xri stakenasthereal axi s.Usi ngEul er' sformul a(2-27), wecan.eparate therealandimaginarytermsoftheaboveequation.Thetwo:' quati onsresul ti ng are preci sel y those deri ved fromthe fi gure as Eqs. (c) and,t ) .Whetherthese equations are foundfromthe figure directlyorbythe useofthe compl ex pol ar l oop-cl osure equati on, the sol uti on process can proceedas above, using whatevermanipulations are required tosolve these equationssimultaneously.Withthecomplex-algebraapproach,however,itisoftenpossible torecognizetheoriginalloop-closureequationasoneofthefourstandard cases andthustowritedownthesolutionimmediatelyfromthosederived inSec. 2-8. Equations (2-46) and (2-47),forexample, result directlybyr _cos 0r : ;l - !Rr, - R?r.osi nr 03^cs56 "r geonY OFMACHI NESANDMECHANI SMSthe form ofEq. (/)as case2c and substitutingthe proper symbols into thestandardsolution,Eqs. (2-36)and (2-37).Similarly, Eqs. (2-48)and (2-49)areexampl esof case2b andcoul d be found di rectl yfrom Eqs.(2-34)and (2-35).Tosol ve the sameprobl em byusi ng the Chaceapproach,we start bywri ti ng the l oop-cl osureequati onfrom Fi e.2-l 2aR.. : Rao"o*Rss G)Wi th 02gi ven,the unknownsi n thi s equati onare the magni tudeRc and thedi recti onR.r. The sol uti oncorrespondstocase2b and i s found by maki ngappropri atesubsti tuti onsi n Eqs. (2-42)and(2-43)Rc" :- [Rr . (R. x t )](n. x L) + fRtrr- [Rao . (R. x X)J'.Rc : [Rao . .+ fntr- [Rs . (R. x [)]' l RcExampl e 2- 3Uset he Chace equ: r t i onst of i ndt heposi t i on of t hesl i derofRHr:25 mm. Rcs = 7- 5mm,and 01: 150".Sol ur t oNPut t i ng t he gi ven i nf or mat i on i n vect orf or mwe haveRs r= 25( cos1, 50) i + 25( si n t - sO j: - 21. 7i+12. 5iRt ' i: 7- 5 R, ':Not e t hat, . x:- j .t t r ensubst i t ut i ngi nt o Eq.( 2- - 51) gi vesR, - = {r-2r. 7 + n. 5i l.i* VmiFi e. 2- 12wi t h= - 50. 2i mmAns.Theanal ysi s ofthefour-barl i nkage i sacl assi c probl emwi thsol uti ondati ng backoveracentury. Thegraphi calsol uti on wasshowni nFi gs. 2-13and2-14. Thesame probl emi spresented heretoi l l ustratetheal gebrai csol uti on techni ques further. The notati on used here i s defi ned i nFi g. 2-19.Notefromthefi gurethatsi sthedi agonaldi stance BD.Thel awofcosi nes can be wri ttenforthe tri angl e BADand agai n forthe trangl e BCD.Interms ofthe l i nkl engths and angl es defi ned i n the fi gure, wehave(2-s0)(2-st)( h)(r)forthe transmi ssi onwrittenagain forthe(i)( k)and tlt are both lessr : Vr t , +y :t cos-'ri+l -s't \ t awhere the pl us-or-mi nus si gns refer tothe twosol uti onsangl e yand y' ,r espect i vel y. The l aw of cosi nescan besame twotriangles tofindthe angles $ and ,lt: cos- l#1 , 1 1dr : cos r r +-'l-r/ . fSwhere i t i s noted fromthe fi gure that the magni tudesof @POSITIONANDDISPLACEMENT57than 180"and that r/i s al ways posi ti vewhi l e si n { i s ofthe same si gn as.in d2.From thesewe find the unknown angles03 and 0a0: 180o- +, b( 2- 52)0t : 0-7Q- 53)orplus sign again signifiesthe twoclosures, 0n and 0'4,u' herethe mi nusrespect i vel y.To sol ve ther ectorS : f - f 2The tri angl e BCDthen gi ves the vector equati onS : - f i n here the twodi recti onsi and t vra unknown. Thi s i s case 2c..ofuti ons are gi venby Eqs.(2-44)and(2-45).Substi tuti nggi vessame problemusing theChace approach, wefirstform the( / )( m)andthe(2-s4)(2-ss)( n)The upper set ofsi gnsgi ves the sol uti on forthe crossed l i nkage; the l ower settherefore appl i es to the open l i nkage ofFi g. 2-19.Sol vi ngthesame probl emusi ng compl exal gebra, wecoul dadaptthe.tandard sol uti on ofcase 2cas done above. However,i twi l l i l l ustrate someusefulmani pul ati on techni ques i fwedonot.Westart bywri ti ngthel oop-.' l osure equati on i ncompl ex pol ar form.Usi ng the notati on ofFi g. 2-19, wehavef2ei ot * rygi ot : ft*fagi oat\ \.-. , ) . .- \ . -/' \ a1 1 0 ,r' -,r''ir*\,4(al ' U't i sure2-19( bl58rnponyoFMACHINES ANDMECHANISMswhere xr i s chosenas the realaxi s. Usi ng Eul er' s formul a,we separatethereal and imaginaryparts of the equationl ' 2CoS02* r y cos 03 :r t *f aCos0a @' l12si n 02+ 4 si n 03 : rasi fr0a @)whereangles fu and0a arethetwounknowns.Next,werearrangetheseequati ons toi sol ate the0r termsI ' 3COS 0t : f COS 0a -12COS 02*f1r-sin 0t:rt sin 0 - r2 sin 02and square and add the twoequationsr 1: r i +r ! +r l+ 2r 1r acoS0-2r t r 2coS 02-2r 2r acos( g -0) ( q)thus el i mi nati ng the unknown03.Wecan combi ne a number ofthe knownquanti ti es ofthi s equati on andreduce i ts compl exi ty bynoti ng fromthe fi gure that. s ' : f f-f 2c os02sY :- f z si n 0z_t r \ * r 1-r 1- r l + 2r 1rcos 02(r)( s)(2-s6)maki ng(r)anglein ai denti ti es(2-s7)( w)7 : COS2ryrawhere thi s l ast equati on i s equi val ent toEqs. (/r) and () above. Afterthese substi tuti ons and rearrangi ng,Eq. (q)reduces tos' cos0a*sYsi n da-/ 3coS f* r o:gWhen deal i ng wi thboth si ne and cosi ne ofthe same unknownsi ngl e equati on, i ti s someti mes hel pfultosubsti tute the hal f-angl efromtrigonometry2t an( ql z)I+ t an2(nl z)Substi tuti ngthesei nto Eq. (/), cl eari ngfracti ons,and rearrangi ngterms,weobtain a quadraticequation( r o-r t cos 7 -s' )t an' ! +2sv t un! * ( r - l 3cos 7 * s' ) :g ( u)L Lfrom whi ch we obtai ntwo sol uti ons. 04_- s ) ' +V , \t anl :- - ' "'r q-t t c os y - s ''( u)|-t anr( nl 2\cos 4 : | +ta1fi srn 4 :whenwe substi tute fromEqs. (r),(s) and (2-56), thi s reduces to, ^_ 0o - Jr t+ r . . Vl-co* 7l af r^r - hc os 7- s *ThereforeHavi ng sol ved the basi c four-bar l i nkage, wenowseek an: he posi t i on of t he coupl erpoi ntP.Fr om Fi e. 2- 19,i n compl ex.re writeRP: RPe' uu :r 2er o2* r 51( %* " t0: 2t an- lr zsi n 0zT r si n 7t - r t * 12COs 0z-h COSyPOSITIONANDDISPLACEMENT59(2-58)(2-61)(2-62)Thesol uti onfortheotherunknown,theangl e0t,canbefoundbyt;ompl etel yanal ogous procedure. Isol ati ng the04 terms ofEqs. (o)and (p).efore squari ng andaddi ngel i mi nates 0 andl eaves aquadrati c equati on.rhi ch can be sol ved for0...The sol uti on i s0t : 2t ant -rz sin 02 -r ra sin 7(2-se)expressi onforpol ar notati on\ *\-/ 2CoS0z- r cos 7(2-60)\\' e recogni ze thi sascaseIsi nce Rpand06 are-thetwounknowns.The.,rl uti ons can be found di rectl y byappl yi ng Eqs. (2-30) and (2-31)Rp: fA. - r r: Si t l0: l r: si n(0:- cr)H^ : t ? f ' t - -r :coS0z* r s cos ( 9 * ct )\otthatboththese equati ons gi vedoubl e val ues comi ng fromthedoubl e' .rl ues ford and correspondi ng to the twocl osures ofthe l i nkage.Exampfe 2-4Cal cul ate andpl otthecoupl ercurveofafour-barl i nkage wi ththefol l owi ngpr opor t i ons t r : 200mm, r : =l 00r l l n, r : 2- 50mm, r : 300 I r , r ' s :l 50mm, anda:- 45".Not at i on i s as def i ned i nFi e.2- 19.Sol ut l oNFor each cr ank angl e 02, t he t r ansmi ssi onangl e 7 i s eval uat ed f r omEq.( 2- 56) .Next , Eq.( 2- 59)i s appl i ed t ogi ve0 .Fi nal l y,t he coupl erpoi ntposi t i on i s cal cul at edf r omEqs.( 2- 61)and( 2- 62) . The sol ut i ons f or t he f i r stsever al cr ank angl esar e gi ven i n Tabl e 2- 1.Thef ul l coupl ercur vei sshowni nFi g. 2- 20.Not et hat onl yoneof t het wosol ut i ons i scal cul at edand pl ot t ed.lable 2-1 Calculation of the coupler curve forExample 2-4' ' ' .degy, deg03, degRp.mm06, degRi , mmRi , mm, r . ( ). , r . 1 )l , l . ( ): r t . { ): ,l . 0i , r . 0' 'r . 0- , r( ). rI{ ) r{ )18.218. 92r . 023. 91 1 L t - +3 1 . 335. 239.243. 146.940. 136. 933.73 1 . 530. s30.73 1 . 833. s35. 838. 4110. 52r 299.423287. 824577.525069.224862. 924158.423055.221853.02045r . 81991621361861392041362131312t3126207123t9612r182r2016611914911860THEORYoFMACHI NESANDMECHANI SMSFigure 2-20 Plotofcoupler curveof Exampl e2-4.Beforel eavi ng thesubj ect ofthe four-barl i nkage, l etus consi der agai nEq.(2-56), whi chdefi nes thetransmi ssi on angl e. Asthecrankangl e02 i svaried, the extremes ofthe transmission angle ycan be foundbydifferentiat-ingEq,(2-56) withrespectto02 andsettingtheresultequaltozero.Thisshows that the extremes occur at0t:0and 0z:180"and are gi ven byr i + r 1-( r 1*r ) 2=c os 7=r 1+r i -U, -r=)2(2-63)2ryra 2ryraTheabove, ofcourse, assumes thatthei nputcranki scapabl e ofmaki ng acompl ete rotati on.Ifi ti snotaGrashofchai n(Sec.l -8)ori snotofthecrank-rockerordouble-cranktype,thecrankwillbelimitedtoarangeofval ues for02.Thecal cul ati ons wi l l then cause troubl e outsi de ofthi s range;themagnitude oftheargument ofthearc cosine ofEq.(2-56) willbe greaterthan unityand a real solution willnotbe foundfory. The limitson this rangeare gi ven byr1+ rl -(ry* r)2< c o s 0 r sr l + r l - ( r r -r ) '(2-64)2r2 2r 1r 22.IIDISPLACEMENTOF A MOVING POINTWehave concerned oursel ves so farwi thonl ya si ngl e i nstantaneousposi ti onofa poi nt, but si nce we wi sh to study moti on, we must be concerned wi ththerel ati onshi p between a successi onofposi ti ons.InFig.2-21 aparticle,originallyatpointP,ismovingalongthepathshown and, some ti mel ater, arri ves at theposi ti on P' .Thedi spl acement ofthe pointARpduring the timeintervalis defined as the net change n postonARp: Rb- Rp(2-6s)POSI TI ONANDDI SPLACEMENT6IFi gure 2-21Di spl acement ofa movi ngpoi nt.Displacement isavectorquantityhaving themagnitude anddirectionofther ector frompoi nt Ptopoi nt P' .Itisimportanttonotethatthedisplacement ARpisthenetchange inposition anddoes notdepend ontheparticularpathtakenbetween pointsP.tnd P' .Itsmagni tude i snotnecessari l y equaltothe l ength ofthe path (theJi stance travel ed), and di recti on i snotnecessari l y al ong thetangent tothepath, al though boththese aretruewhenthedi spl acement i si nfi ni tesi mal l y.mal l .Knowl edgeofthepathactual l ytravel edfromPtoP' i snotevennecessary tofindthedisplacementvectoraslongastheinitialandfinalposi ti ons are known.2.I2DISPLACEMENT DIFFER-ENCEBETWEEN TWO POINTSIn thi s secti on we consi der the di fference i n the di spl acementsoftwomovi ngpoints. Inparticularweareconcerned withthecase wherethetwomovingpoints are both particles ofthe same rigid body. Thesituation is shown inFig.:-22, whererigidbody2 moves froman initialpositiondefined byxzyzzztoal .rterposi ti on defi ned byxLy' zzz.FromEq.(2-6), thepositiondifference between thetwopointsPand Q.' ' f body2 atthe initialinstant isRPQ: Ro- Ro\fterthe displacement ofbody2, the twopoints are located at:hat timethe positiondifference isRbo : R' P - R' o @)Duringthetimeintervalofthemovementthetwopointshaveundergone:ndi vi dualdi spl acementsofARpand ARq, respecti vel y.Asthename implies,the dsplacement difference between thetwopointsr.defined asthenetdifferencebetweentheirrespectivedisplacements and( a)P' and Q' . At\