mechanical narendra

7/27/2019 mechanical narendra

1/119

FTHEPLANEMOTION

umberof conceptsofkinematicswhich,

andofconsiderableusefulnessinmechanism

dfromundergraduatecurricula.

Euler-SavaryEquation

6,radiiofcurvatureofthe pathsofmoving

nceinkinematicanalysisofmechanisms.

nceinsynthesis.Theonlymethoddiscussed

path curvatureisbasedontheknowledgeof

ionofthepointconcerned:p=v2/anor

arly,asfarasmechanismsareconcerned,this

,becausehere,asinany constrainedmotion,

whicharea geometricalpropertyofthesystem,

eactualvelocitiesandaccelerations.The

inthissection,offersa direct,purelygeo-

blem.

dandmovingpolodes,TT/and wm,which

mentoftheplanem;OfandOmare thecentersof

ttheirpoint ofcontacttheinstantvelocity

yP;pnis thepolenormal,whichoriginatesat

7 (seealsoFig.3-22);ptis thepoletangent,the

btainedby turningpnthrough90 intheposi-

e,sense.

,definedbythe rayangle08,measured

andthedistancePS. Inthefollowing,dis-

atedasdirectedlineelements,i.e.,takenas

ositivesensebeingfromP tothemovingpoint.

ayspositive.(Theoverbarwillbeusedto

rquantityisdirected.)Thepositivesense

toa givenrayisobtainedby turningtheposi-

erclockwise.0sistheinstantaneouscenter

S.

Publ

icDomain,Google-digitized

/http://www.h

athitrust.org/access_use#pd-google


2/119

ICSOFPLANEMECHANISMS

lrotationd


3/119

OFTHEPLANEMOTION

nofa length,Eq.(c)becomes

vailable,oreasilyobtainable,detailsofthe

positionofthevelocitypole andtheangular

epolenormal,specifiedby theangle 0,

engthrf,arenotknown.Theobjectiveof the

hedeterminationofthesetwoquantities,to

tionofEq.(10-1).

tion(10-1)showsthatpointssuch asSi,

scribepathswhichdifferin shapeinthevicinity

vation.Itis thereforereasonabletoassume

yapointI, whichpasses,atthegiveninstant,

path.Forsuchapoint,the radiusofcurva-

stanceOi.>P arebothinfinite.Equation

s,therefore,to

PublicDomain,Google-digitized

/http://www.h

athitrust.org/access_

use#pd-google


4/119


venray therereallyexistsone,andonlyone,

elocusof allinflectionpointsina given

em isacircleofdiameterdf, centeredonpn

scircleis knownastheinflectioncircle.

ctioncircleforthe kinematicsystemof

erivationofEq.(10-2)isinteresting.

(cos0,)i,+(aspjn

/,= oand(a/.)n=0,so that

/.)wm2

omittedbecausediisin effectadirectedlength.

thatd,-,as definedbyEq.(3-15),isidentical

directproofofthisfactis givenbelow.

efirstformoftheEuler-Savaryequation

qs.(10-1)and(10-2),thus:

eamenabletographictreatment,itis trans-


/http://www.h


use#pd-google


5/119

OFTHEPLANEMOTION

oftheEuler-Savaryequation.Since(P>


6/119

CSOFPLANEMECHANISMS

onnectI,with Y(line|),and YwithP

lineparalleltoline |,intersectingtheauxil-

awaline paralleltoline||,and obtainOsat

withtherays.

structionis provedbyconsideringtwosets

angles SI,YandSPZ,

e trianglesSOSZandSPY,

meconstructionfora differentrelativedis-

and/,.

icalDeterminationofd,-

centerofcurvature0sof itspathforma

tdiand thedirectionofpncan bedetermined

pointsandthecorrespondingray angleare

onjugatepoints,ondifferentrays,arespecified.

mpleofthis caseisillustratedin Fig.10-6.

intsOm andOfforwhichthe rayangleiszero.

ureofthepath ofOm.SeeSec.3-13.)


/http://www.h


use#pd-google


7/119

FTHEPLANEMOTION

(10-1),....

> Om= /5m,thisrelationbecomes

hetwopolodesareconvex,andtheminusif one

therconcave.Theresultis thusidenticalwith

aphicalsolution,showninFig.10-6,

constructionofFig.10-5in reverseorder.

airsof conjugatepoints,QandOQonrayq

givenincludedray angle6q,=0, 08

(10-1),

edfor0,, andd,-isthenfoundby Eq.(10-1),

oblemmaybesolvedgraphicallyby

einflectionpoints/,andI,,usingthe con-

erseorder, andthenerectingnormalsthrough

verays.Thetwo normalsintersectat/,the


/http://www.h



8/119

AMICSOFPLANEMECHANISMS

atively,theproblemmaybesolved by

escribednext.

ion;CollineationAxis

fersfromthereverseconstructionofFig.10-5

neandpointZare nolongerarbitrary.Here

swiththeline throughthemovingpointsQand

ttheintersectionofQSwithOQ()S.Becauseof

noflineandpoint,the twoinflectionpointsIq

neously.

icationofBobillier'smethodtothe con-

ircleforthecoupler3 ofafour-barmechanism

struction.(QBoA)-Z-P18-Y-(/t,

atZcoincideswiththe instantaneous

ofthedrivinganddrivenlinks.

PZinBobillier'sconstructionisknownas

raysq ands.Thus,ina four-barmechanism,

ativeinstantcentersPisand P24ofthephysi-

ollineationaxisof drivinganddrivenlinks;it is

ineationaxisofthemechanism.

heoremstatesthattheanglebetweenthe

equaltotheanglebetweentheother rayand

tworays, bothanglesbeingmeasuredinthe

ssofthetheoremin analysisandsynthesiswill

ischapter.

oremmaybeverifiedwiththe aidofFig.10-7.

cle

sP:*


/http://www.h



9/119

OFTHEPLANEMOTION225

uareparallel.Similarly,

ction

erminingtheinflectioncircleis somewhat

onthe displacementvelocityoftheinstant

outdifficulty inthemostcomplexsituations.

gularspeedofthe planem,andvpthespeed

erof rotationchangesitspositionalongthe

stantaneouspoletangent.Vp istherefore

locityoftheinstant center.Asshownpre-

the positivepolenormal.Consequently,the

poletangentisobtainedbyturningdi aboutP

ofcom.Conversely,withVPknown,thedirection

ormal,isobtainedby rotatingVPthrough90

ethod,anarbitrarilyassumedvalueis

ocitywm,andthe correspondingdisplace-

tcenter isdeterminedgraphically,withthe

n general,theabsoluteinstantcenterP

oncircle istobe determinedisdefinedbythe

rays.Theauxiliarylinkageconsists,there-

tedto theframeatappropriatepoints,and

blockswhicharefreeto slidealongtheselinks,

butarepin-connectedtogetheratP. Thus

nectionP,intheauxiliarylinkage,is identical

placementvelocity.

theauxiliarylinkageshavebeendrawn

eexplanations.Inpractice,however,the

uton theoriginalplan,aprocedurewhich

ccuracy.

nstructtheinflectioncirclefor amoving

ne pairofconjugatepoints(S and0s),and

ormalarespecified.

emisshownin Fig.10-8o,andtheauxiliary

tterconsists ofthebars, pivotedtothe


/http://www.h



10/119


sq andr,pin-connectedtogetheratP,with

ngentptandr freetoslidealongs. Thepoint

denotedbyX,.

correspondingtothe arbitrarilyassumed

sec,ccw,isthe requiredpole-displacement

yofthelinkageisvs =umX(P> S).

y proportion,andthevelocityofPfollows


/http://www.h



11/119

FTHEPLANEMOTION

nstructionfortheparticularcaseofspecified

hpolodes.

nstructtheinflectioncircleforthecoupler3

mofFig.10-9a.(Twosetsof conjugatepoints

and 04.)

ed)

wstheauxiliarylinkage.Itconsistsof the

tedtotheframeatOzand04,respectively,and

nnectedtogetheratP, withblock5freetoslide

oslide onbar4.Thepoints onbars2and 4,

oted,respectively,byX2andXt.


/http://www.h



12/119


=w3 X(Pis>A)and VB=waX(Pis> 5);

ally,byproportion,andVP followsfrom

nstructtheinflectioncircleforlink3 ofthe

10a).(Thenameofthemechanismderives

t onlink3describesa conchoidalpath.)

ageconsistsoftheextendedslider2, the

e frameat04,andthetwo blocks5and6,

tP,ofwhichblock5isfree toslidealonglink 4,


/http://www.h



13/119

FTHEPLANEMOTION229

Thecoincidentpointsareagaindenotedby

celinks3 and4forma slidingpair,t>4=u3.

04>Xi).Sincelink2 isinrectilinear

(P13-* 4).vpfollowsfrom

nstructtheinflectioncircleforthebar3 of

ig.10-1la.The mechanismconsistsofthe

othecrankat A,andthestationarysurface1

bar3.

ageisshowninFig. 10-116.Itselements

eframeat02, thebar4,pivotedto theframe

cideswiththecenterofcurvatureMofthe

blocks5 and6,pin-connectedtogetheratP,of

dealongbar2,and block6alongbar4. The

eX2(on 2)andX4(on 4).Sincetheaux-

diculartothe link3duringan elemental

dconsequently,vx,=u3X(04* Xi).Also,

followsfrom

nstructtheinflectioncirclefortheternary

howninFig.10-12a.

mplexproblem,andcanonlybesolvedin

shownin Fig.10-126,involvesthedetermina-

ementvelocityofthecenterP2t,which,to

notedbyR.Theauxiliarylinkagefor this

ntricallypivotedbars3and4 andtheblocks7

heframeat 03=P13,andbar 4atOi= P14.

tedtogetheratR,withblock7free toslideon

.The pointsonbars3and 4,coincidentwith

ely,byX3andXi.Theinput velocitiesare

d= ujX(P1t>D). vRfollowsfrom

nstruction,showninFig.10-12c,is con-

tionofthedisplacementvelocityofP1t.The

ageconsistsof thebar10,pivotedtothe

ssingthroughR,thebar6, pivotedtotheframe

nd12,pin-connectedtogetheratP=P15,

PublicDomain,

Google-d

igitized

/http://www.h

athitrust.org/access

_use#pd-g

oogle


14/119



/http://www.h



15/119

FTHEPLANEMOTION


/http://www.h


use#pd-google


16/119


onbar10and block12onbar6. Thepoint

R,is denotedbyXi0>andthepointson

withP,areshownasYIOandy6, respectively.

= w5X(Pi6>E)andvXl,=componentof^v

sfrom

fortheRelativeMotionofTwo MovingPlanes

teredaroundtheconstructionofthe inflec-

theabsolutemotionofa plane.Occasionally

ethecurvaturesofpathstracedby thepoints

otherplane,also inmotion.Theinflection

elativemotionoftwoplanesis constructed

kinematicinversioninwhichtheparticular

herespectivepolenormal

tionary,andotherwisefollowingtheprocedure

section.Itshouldbe notedthat,asshownin

oncirclesinvolved(motionofmrelativeto /,

m)arepolarimagesofeachother, afactwhich

structtheinflectioncircleassociatedwith

3 relativetothecam2 inFig.10-14.

owstheauxiliarylinkagefortheinversionin

onary.Thelinkageis basicallythesameasthe

eExample4of theprevioussectionandshown

ocitiesfor theinversionare

Xt).VPfollowsfrom


/http://www.h


use#pd-google


17/119

FTHEPLANEMOTION

ectionCircletoKinematicAnalysis

providesthemeansfora purelygeometrical

vatures,itbecomespossibletoanalyzecomplex

oursetotheindirectanalyticalandgraphical

re10-15ashowstheskeletonofacertain

echanismwhoseinputcrank2revolvesatcon-

theangularaccelerationoflink4 andthe


/http://www.h


use#pd-google


18/119


t,as shown,thesolutionbymeansof

atherdifficult,andtheproblemisbest solved

oncircle.AfterthepathcurvatureofBhas

raphicallyorbycomputation

mentary.

nin Fig.10-16a,theproblemmaybesolved

nflectioncircleorbythesubstitutionof the


/http://www.h



19/119

FTHEPLANEMOTION

entmechanism

ntlow-complexitymechanismofFig.10-166.

chlesscumbersome.

eneralCase)

which,byrolling ononeanother,reproduce

thecharacteristicsoftheconstraints.The

s investigatedinthefollowingsections.


/http://www.h



20/119


definedbyQS,isconstrainedtomovein a

fthefact thatthecentersofcurvatureOQ

diS, respectively,aremadetotraversethe

The locusofthecenterofcurvatureofa given

ute.)

thefixedpolodeis ageometricalproperty

terminedhereindirectly,i.e.,byconsidering

eristicsofthevelocitypole:

speedofvelocitypole

tofitsdisplacementacceleration(nottobe

onofpointPm,aPm=wm2d,-)


/http://www.h


use#pd-google


21/119

OFTHEPLANEMOTION237

),a.pwillbecomputedwiththe aidofthe

twasusedin determiningVpbyHartmann's

uations

dwm,which,tosimplifythe calculations,will

ce,inthefollowing,du/dr= 0.

othepositiveand negativesensealonga

glestoit,werelaid downinSec.10-1.Since,

he basisofthesubsequentderivations,all

positive,thediacritical overbarwillbeomitted

ereintroducedinthe finalexpressions.

pendicularto0s >S=ps,consists oftwo

omponentof&x,andthe Corioliscomponent

x.)c0r(b)

das follows:

Oa->X.)-(c)

=vg;and

x.

>X,)rsdps+ Os>X,dri]

.(d) maybeexpressedintermsof thegiven

manner:

ointsontherays,

s ->-S')>(Os-* S).]FromFig.10-16,

6, {&)


/http://www.h


use#pd-google


22/119


-> S') S).]With

rs

= P-> S= rs

(/) into(d)yields

sin0,

px.)cariscalculatedasfollows:

^mvpsinS,=Wm2d,- -sin68(h)

nowbe substitutedinto(6),giving

0.(i)

=UmWiKo(j)

6q.7.

. (Z)

to

h(m)yieldsthe finalresult

ndX's aredefinedbyEqs.(i) and(j).

f thefixedpolode,asdeterminedby

ornegative.A positiveresultshowsthat


/http://www.h



23/119

OFTHEPLANEMOTION239

irectionofthepositivepole normal,which

ectioncircleareonoppositesides ofthepole

pecialCase);Hall'sEquation

ssionforthe polodecurvatureinthemost

wthespecialcasewherethe pathcurvaturesof

edinthefollowingbyY andZ,haveinstan-

es;thatis,dpy/dl dpz/dl=0.It isobvious

ythiscondition,theevoluteserand ezmust

ndinginstant;that is,pfr=pfl =0.

herefore,to

f r" ~PYcos0, sin6y

thersimplified,and

asfirstreportedin theliteraturebyA.S. Hall.1

ationarypathcurvatures,thepolodecurva-

din termsofaratio (R(whosephysicalsignifi-

nSec.10-10)andits derivatives,inthefollowing

> Y)

Ydt,

, tan6a

P)dr, tan^

cle andPolodeCurvature,Trans.FifthCon/,on

58,pp.207-231,PurdueUniversity,Lafayette,Ind.


/http://www.h


use#pd-google


24/119


rItan6y=((R - (R2)taney

an0,

+ (P ->F)=(0r->P)


/http://www.h


use#pd-google


25/119

OFTHEPLANEMOTION

ssedin termsoftan6y,andthe latterin

Eq.(q),thenthefollowingexpressionresults:

beobtainedby substituting

essestheradius ofcurvatureinpolar

heFour-barMechanism;CouplerMotion

ththemotionofthe coupler3relativetothe

m =7r3.Itis obviousfromFig.10-19athat


/http://www.h


use#pd-google


26/119


ddirectlyfor thecomputationofthecurvature

rminethepolodecurvaturesforthecoupler

mofFig.10-20.

3in.,04-> B=-2.3in.,P ->A=1.7in.,

oncircleyieldsthefollowingparticulars:

=346 =-14.Hence,byEq.(10-11),

n iscalculatedbyEq.(10-5),withm =3

of P13,withthecentersofcurvaturedenoted

y,areshowninFig. 10-20.


/http://www.h


use#pd-google


27/119

FTHEPLANEMOTION243

ntheFour-barMechanism;RelativeMotionof

DeterminationoftheOutputAngular

f Change

6revealsthat Hall'sequation,withappro-

,maybeusedtodeterminethecurvatureof the

ththemotionofthe outputlink4relativeto the

13)

ncethediscussionis concernedwiththemotion

hepositivesensealong bars1and3 isthe

orrespondingpointsonlink 4,i.e.,fromPto

tively.

ablyalterednotation,isalso applicablein

gs.10-18and10-19&showsthat dtp2=dy

0randOtto Y.Hence

W2

erelativemotionof theoutputandinput

ocityratioof themechanism.((Rmaybe

((R')2+(R(R'

=

fiednotation,yields

0i

rvelocityo>2,from Eq.(10-14)


/http://www.h


use#pd-google


28/119


&i=2(10-17)

)

epositionofthe relativeinstantcenterP-u

ityratio(Rofthe mechanism.

at,inamechanismwitha uniformlyrotating

e angle61betweenthepolenormalandthe

heoutputangularacceleration.

qs.(10-15)and(10-18)yieldsan expres-

etodeterminethe timerateofchangeofthe

on.

maybeusedinthe analysisofagivenmecha-

onisinthe fieldofsynthesis,particularlyin

rivingcrank2ofthe four-barmechanism

constantspeedofu2=2 rad/sec,ccw.Deter-

,and4.

hepoletangent(and consequentlyofthe

dbyBobillier'stheorem,i.e.,bymakingthe

dthepoletangentequalthe angleformedbythe

Bymeasurement,


/http://www.h


use#pd-google


29/119

OFTHEPLANEMOTION245

oncircle maybedeterminedgraphicallyor

ler-Savaryequationintheform

S)- (Os-> 5)[(P -> 5)-d, cos0. ]

->04)[(P24->04)- d.cosO1]

n.

9 in.

nd w4=0.78rad/sec

rad/sec2

),p2= 10.12in.

enterofcurvature0r2of t2liesonthe sideof

othe inflectioncircle.)ByEq.(10-5),with

0.39and(R'=0.92,

ineation-axisTheorem;1Carter-HallCircle2

ropertyonwhich Bobillier'stheoremis

spossessesanumberofother characteristics

ismdesign.

Inafour-barmechanism,inthephasescor-

valuesofthevelocity ratio,thecollineation

coupler.(Inthesephases,ifw2 =const,

atesthat(R'=((R2 (R)tan0i.Hence,

,andconsequently(R'=0, 0i=0also. Thus,

d,thepolenormalcoincides withthefixed

aximumandMinimumVelocitiesand theAccelerations

Trans.ASME,vol.78,no.4,pp.779-787,1956.

alysisandSynthesisUsingCollineation-axisEqua-

,no.6,pp. 1305-1312,1957,withdiscussionbyA.S.


/http://www.h


use#pd-google


30/119


erpendiculartothefixedlink,andbyBobillier's

axisisperpendiculartothecoupler.

rkinematicpropertywhichmaybe

naxisisthe following.Ifafamilyoffour-bar

edonthesamefixedbaseline02 >04,whose

nstantaneousvelocityratio(R,thesamefirst

,andthesamesecondderivative(R"=d2&./d


31/119

FTHEPLANEMOTION247

n,appliedtolink1,yields

24-> 00-tU cos0,]

* sint?J

>04)+disint?i, .

(c)

forlink3,

n

(d) into(6)andrearrangementoftermsyields

?icos#3sin i?3sin(?3 t?i)

------------------

-

(2- - -1+3d, -si

f Eq.(e)issubstitutedfor P24 >BinEq. (a),

dto

-cos#3(J)

r--- -

d,

24)+(P24 ->04)

00

mbinedyield

nd(A),Eq.(/) istransformedto

C

R


/http://www.h


use#pd-google


32/119


6)showthatmechanismswhichhavethesame

,Si',and(R"have,ipso facto,thesameinflection

ecurvature.Hence,forsuchafamilyof

0-20)

lar coordinates,ofacirclehavingthechar-

enEqs.(10-15)and(10-19),then,aftersome

raicoperations,thefollowingexpressionis

toquantitiesdirectlydeterminableby

hemechanismofFigs.10-21and10-22,

9,0i=104.5,anddc= 5.6in.

21)yields(R"= 0.72,whichagreeswiththe

ativeexampleof Sec.10-10.

Curve(GeneralCase)

sedinthischapteris thecircling-pointcurve,

ature,thelocusofall thosepointsonamoving

stantconsidered,havestationarycurvature.

ancein synthesis,andparticularlyinthe

nismswithprolongeddwellrequirementsfor

ecircling-pointcurve,andapplytheEuler-

os0, ]=(P->F )2

nsOY >Y=pyandP >Y=fy,becomes

)

arindicatingdirectedline elementswillbe

s,butreintroducedinthefinal expression.)

stobe instantaneouslystationary,then

cementofthepoleP.


/http://www.h



33/119

FTHEPLANEMOTION249

elds,therefore,

w \ = 2ry it(6)

pm=vP/um,wasderivedinSec.10-1.)

sions(c)and (g)into(b)leadsto

d, - sin0f - r) = 2rysin0(A)

etweenEqs.(h)and(a) yieldstheequationof

olar coordinates,withthevariablesfyand0:

ntsinthephaseconsidered.

eevaluationofEq.(10-22),itis necessary

eousrateofgrowthof thediameterofthe

ase ofthemovingbody;thatis,d(d,)/dl.The

sibleif,asshowninFig.10-17,the pathcurva-

are given.

,relevanttothepresentinvestigation,were

he assumptionum=const:

ponentofthepole-displacementacceleration:

, . .

W

npole-displacementaccelerationandpole-


/http://www.h



34/119


Eqs.(t)and (j)ofSec.10-7.

sconstant,

fthelastthreeequationsyields

10-23,thecircling-pointcurveisa looped

ranchescrossatrightanglesat theinstant

oinfinityin anasymptoticalapproachtoaline

iththe polenormal.0Misobtainedby

minatorinEq. (10-22):


/http://www.h



35/119

FTHEPLANEMOTION251

notpassthroughthe instantcenter.Itsoffset

6g=6m. However,ifthenumeratorand

iatedseparatelywithrespectto#v,thelimiting

0-25)

egeneratesintoacircleandastraightl ine

mevalue.ByEq.(10-22),if d(di)/dl=0,

)reducesto

0 =equationofacircleof diameter2/M,

ndpassingthrough

straightli necoincident

CurvefortheCouplerofa Four-barMechanism

hetwocouplerhingesare obviouslypoints

.Hence,iftheirpolarcoordinatesaresub-

woindependentequationsareobtained,from

TVcanbe calculated.WithMandTVknown,

easilyconstructed.

wthecircling-pointcurveforthecouplerof

fFig.10-24.

hepoletangentand polenormalarefound

dthepolarcoordinatesofA andBaredeter-

substitutedintoEq. (10-22),yield

60

tsonthecurveare locatedasshownindetail


/http://www.h


use#pd-google


36/119


gle6C =125.ByEq.(10-22),

0cos125'

ythedegreetowhichthe actualpathofa

curvemaybeapproximated,inthevicinityof

heosculationcircle.ForthepointC, thetwo

nguishablefromoneanotherforabout70 of

ce,iftwolinks5and6 wereaddedtothe

outputlink6wouldremainsensiblyat restdur-

etweenlinks5 and6whichcoincideswith,

hepath ofCmaybefoundeither graphically,

ectioncircle,orbycalculation,asfollows:


/http://www.h


use#pd-google


37/119

FTHEPLANEMOTION253

n,appliedtoA,yieldsthediameterofthe

A)=(0, ->A)[(PU->A)-di cos.]

n,appliednexttopointC,yieldsthe required

>O -dicostfj

.


/http://www.h


use#pd-google


38/119

ESIS;

thedesigner'sworkis thedevelopment,or

sfortheperformanceofspecifiedoperations.

ntivenessandintuitionplayamajorpart in

arecertainfundamentalproblemsofsynthesis

ationalway.Suchproblemsinclude:

numberofprescribedpositions

tionsofthe outputandinputlinks,i.e.,

prescribedpath,i.e.,pathgeneration

nstantaneousmotioncharacteristics

numberofmethods,bothgraphicaland

oftheseproblems.Theaccentwillbe onthe

useitisthemoredirectof thetwoandyields

uracy,withmuchlesseffort.

utionwill bebasedonthefour-barmecha-

gationofthecharacteristicpropertiesofthis

prisinglyversatile,linkageiscalledfor.

e

nthe followingdiscussionarelistedbelow

therotocenters,aredefinedbyFig. 11-1.

r

ardisplacementofqbetweenphasesmandn

isplacementofsbetweenphasemandn

splacementofrbetweenphasesmand n

ebetweenrands


/http://www.h


use#pd-google


39/119

ESIS;GRAPHICALMETHODS

withrespectto theframe,definingthe

mtophasen

ningtherelativedisplacementoflinksq

nd n

ducedanddefinedasrequired.

ism.Figure11-2showsthethreetypesof

rank-rockermechanism.Therotationofq

,whilescanonlyoscillatethroughalimited

thattheforwardandreturnstrokesof sdonot

onsofq.

uble-crank,ordrag-link,mechanism.The

reunrestricted.Auniformrotationofq

tationofs.

ble-rockermechanism,qandscanonly

nedas inversionsofthesamelinkage,pro-

thsoftheshortestandlongest linksislessthan

shof'srule):


/http://www.h


use#pd-google


40/119


rankcrank-rockermechanism

amedrag-linkmechanism

ouplerdouble-rockermechanism

anism,derivedfromGrashof'slinkage,the

eterevolutionwhiletherockersmovethrough

arlinkagedoesnotsatisfyGrashof'scriterion,

gementoflinksis used,onlydouble-rocker

ngcouplersresult.

riencehasshownthatmechanismswith

helimitsofGrashof'srulearenot satisfactory

ioncycle,linksassumerelativepositions

centerpositions),withconsequentchatterand

es.Moreover,sincesuchlinkagesrequirethe

emachiningtolerances,ifundesirableeffectson

cteristicsaretobeavoided,theyaremore


/http://www.h



41/119


nlinkageswithamoresubstantialinequality

he designstage,totheeventualperformance

smissionanglen oritssupplementn'.Alt,1

eof thisparameterasaqualityindex forthe

commendedaminimumvalueof40 forlow-

eedapplications.Otherkinematicianshave

owever,itisextremelydoubtfulwhethersuch

accepteduncritically.Clearly,inlow-speed

ansferwillbe"best"if n=90 andimpossible

gswinkelundseineBedeutungfuerdasKonstruieren

erkslattstechnik,vol.26,no.4,p. 61,1932.


/http://www.h


use#pd-google


42/119


bviousthatthe effectofinertiaforcesin

annotbeaccountedforbyasimplegeometrical

ggestedbyAlt maybeusedsafelyasa

theinitialdesignina givensituation.A

sclosethesuitabilityorsatisfactorinessofthe

tline)

nd,forinstance,thatthe inertiaeffects

reductionofn,whichmaybe quitedesirable

achesits extremevaluesinamotioncycle

ollinearwiththefixedlink p.Thecorre-

tedby1and 3.Akinematicallyoptimum


/http://www.h



43/119


issionangles,isonein which^min=^Vn.The

for prescribedvaluesofMminand^13isshowni n

,thedrivenlink,ofarbitraryl ength

sesSiand s3,andthelinesTI andr3aredrawn

ns,determinedbythespecifiedtransmission

gthsrx=BiAix=B$A3Xarethen trieduntil

tainedwhichpassesthroughO,.Therequired

themidpointofA3Ai.Inthe alternative

-46,abundleofrays isdrawnthroughO,,

A* andliner3in pointsA*.Thesolution

ointsA* =AiandA$= A3,forwhich

mplishedbyplottingthelengthsBiAfversus

sBtA*.Theintersectionofthelocusobtained

nedrawnthroughB3at45 totherayr3, clearly

nismasQuick-returnMotion.1Figure11-5

chanisminthetwoextremepositionsofthe

rvedthatthecrankangle


44/119


hesisofa crank-rockermechanismfora

rnaction,i.e.,forprescribedvaluesof


45/119

ESIS)GRAPHICALMETHODS261

s. Since,byEq.(6),OqB2 OqBt=2q,

ectionofthepivotcirclewithahyperbolaof

2andBi(Fig.11-66).

. Since,byEq.(a),OqB2+OqBt=2r,

ectionofthepivotcirclewithanellipse ofmajor

BA(Fig.11-6c).

positionfa.Oq islocatedattheintersection

hercircle,of radiusMOwhosecenterMis

ctorofO,B2sothat thecentralangle

snotcompletelyfree;an inspectionof

g.1l-6arevealsimmediatelythat:

nsidepivotcircleandfa< (90 -^J

outsidepivot circleandfa>(90 -^J

#24 180 canberealizedbymeansofa

ftheinitialcrankpositionfa =90 ^24/2.

pivotcirclepassesthroughOandanyother

ationforthepivot Oq.Since,inthiscase,the


/http://www.h


use#pd-google


46/119


nsisinfinite,additionaldesignrequirements

sconditions1to 4above.

180 and


47/119


edinmanyways,e.g.,byattachingratAand

xedpivotsOqandO, maybelocatedany-

tors[n^]i2and[na]u ofAiAzandBiB2,respec-

four-barlinkagesisevidentlyinfinite.

alsobe broughtaboutbyasimplerotation

oint ofintersectionof[nA]uand[ns]i2.This

he rotocenterofthebody,orplane,r forthe

-r2.Inthefollowing,sincethereisno possi-

xrwill bedropped,andtherotocenterwill

AninspectionofFig. 11-9revealsthat,as

2R^B2arecongruent,theamountofrotation

tualangulardisplacement[tpr]uofthebody.

propertiescanalsobededucedbyinspection:

theguidinglinks (cranksorrockers)qands


/http://www.h


use#pd-google


48/119


s,capableofguidingtheircouplerr through

ndequalanglesatthe rotocenter[Rr]u=R^;

sareequaltohalfthe angulardisplacementofr:

,andeffectivelengths(betweenhinges)of

mechanismsalsosubtendequalanglesatRi2:


/http://www.h


use#pd-google


49/119


cablemechanismsisinfinite,itis possible

mutuallycompatible,requirementstobesatis-

n.Examplesofsuchadditionalcriteria,the

hisfour,arethefollowing:

hefixedpivotsOq andO,androtationsof

s, [


50/119


underconsideration,pointsOqand D

ivotsof afour-barmechanismwhosecoupler

therequiredpointC,respectively.The

plerislocatedat theintersectionofthenormal

12.Thepositionof Ciisobtainedbymaking

uidinglinkssubtendequalangles atthe

dm,wereattachedtotheslider-crank

shedlines,adwellmechanismwouldresult,

hitlinkm,wouldremainvirtually atrest

dfromAIthroughA2 andA3toAt.

ghThreeDistinctPositions

uidingabodyr, definedbythelinesegment

dpositionswouldbebymeansof twolinks,

to theframeatthecentersof circlesdrawn,

A2,A3andBi,B2,B3.

erthanAandB maybeusedequallywell

beroffour-barmechanismscapableofguiding

ribeddistinctphasesbecomesinfinite.This

nofadditionalrestrictionsonthe design.


/http://www.h



51/119

ESIS;GRAPHICALMETHODS267

ationsofOq andO,arespecifiedinadditionto

ecorrespondingmovinghingesCandD is

kinematicinversioninwhichthebodyitselfis

e threephases,forinstance,AiBi,andthe

erelativepositionsofO,andO3are determined

2B2Og,A2B2O,andA3B3Oq,A3B3O,tothe

=A42B20,,etc.)

hemovinghingesin phase1arethen

e circlesdrawn,respectively,throughOq,011,

ngle

m =ABinthreedistinctphasesand the

RU,RM,andRu.

therotocentertriangleyieldsanumberof

orems.

terTriangle.InFig.11-136,the bodyhas

ethepointsCand D,judiciouslyplacedsothat,

cideswithR^and,inpositions 1and2,D

eextendedlineRuRi3coincideswith[nc]i2

edlineRuR^coincideswith[nc]isand [rio^s,

attheanglesof therotocentertriangleare

dingbodyrotations.

eterminationoftherotationalsense,the

mmended:

montobothrotocenters

mmontobothrotocenters

montobothrotocenters

maybetransferredfromposition1to posi-

rclockwiserotation


52/119



/http://www.h


use#pd-google


53/119

HESIS;GRAPHICALMETHODS

gle,for example,AIaboutside1,yieldingthe

ctionsof Acabouttheother twosidesof

d correlatedpositions.Thecorrectnessof

elf-evident:

sefulinitself,

mportanttheorem.

the Heights.Thecorrelatedpositionsof

alpointHccoincideswith theorthocenter(the

ofthe rotocentertriangle,arelocatedonthe

edasfollows.

R^XR^,inFig.11-15,

ngleH2R^


/http://www.h


use#pd-google


54/119


rtoR^R^andRitRnis perpendiculartoXR23,

rtoRuR andR^Ruisperpendiculartofli2Z,

ngle

ngleRuZH3,


/http://www.h


use#pd-google


55/119


ntRuH^subtendsequalanglesp, atH2,

hat allsixpointslie ononeandthesamecircle.

nthis respectmaybeseenfromFig.11-156.

nts PcwhosecorrelatedpositionPIison

theimagecircle1,i.e.,the reflectionofthe

tside1.Similarly,imagecircle2isthe locus

latedphasePz liesonthecircumscribedcircle,

ocusofallcardinalpoints Pcwhosecorrelated

e circumscribedcircle.However,thethree

epointin common,viz.,theorthocenterHc.

sofaPointin aStraightLine.Ifthree

ointarecollinear,theyare alsolocatedonthe

es,andthelinecontainingthempassesthrough

ollinearcorrelatedpositionsofthepoint A

ocentertriangle.Hcislocatedatthe inter-


/http://www.h


use#pd-google


56/119


siderationoftheanglesaroundthe vertexAi

ndiculartoR^RnandRiJIeis perpendicular

sequalanglesatAiand Hc,itfollowsthat

sRi2,Ru,andHc,that is,imagecircle1.

armannerthatA2 liesonimagecircle2,and

at,asindicatedbytheconstruction,Hc

.

>u+o-=360

^23

7=180

onthesamecircle,R^Aisubtendsequal

equently,

,andA3 lieononecircle,

sofaLinethrougha Point.Ifthethree

ectatagivenstationarypointS, thenli,l2,

,throughHI,H2,andH3,and Sislocatedonthe

ntertriangle.

tions ofZ,dennedbythecorrelatedposi-

niquelydeterminetherotocentertriangle.

sectionofthenormalbisectors[n^]i2and

eenomittedsoasnotto clutterupthedrawing

onlines.)

sect,respectively,theangleliSls=


57/119

HESIS;GRAPHICALMETHODS273


/http://www.h


use#pd-google


58/119


nsequently,

rto6i sandR^Sis perpendicularto623,

thesamecircleasRu,Ru,and#23.

slythatHI, H2,andH3arecorrelatedposi-

hereforethe directionsHiS,HzS,andH3S

nsofaline. Thatthesedirectionsareidentical

tively,followsfromthefactthat,asshownin

rtriangleandagivenpoint Sonthecircum-

thecorrelatedlinepositions.[Theconstruc-

dproceedasfollows.Therotocentertriangle

enseofthe rotation^13.SincetherayRi3S

ions1and3,lines Ziandl3are inclinedtoitat

dN3 arecorrelatedpositionsofthepointN.

ngthe angleZi


59/119

le


/http://www.h



60/119PublicDomain,Google-digitized

/http://www.h



61/119


ditscircumcircleareconstructedinthe

of thecorrelatedpositionsofAandof asuit-

coupler.TheorthocenterHcislocatedatthe

s,anditsposition checkedbymeansofthe

2,andH3are located,respectively,atthe

mcirclewiththeextendedaltitudesHM^,HcRu,

din aconvenientpositiononthecircum-

einphase1 ismadeparalleltoHiM. (An

curacyoftheworkingistheconditionthatO,,

ionof thethreecorrelatedpositionsofthe

cumcircleandthattheextendedlinesAiBi,

passthroughHi, HI,andH3,respectively.)

ghFourDistinctPositions;Burmester'sCurve

ody,or plane,throughfourdistinctposi-

agesisrathercomplex.Itssolutionentailsthe

ne,ofpointswhosefour correlatedpositions

oints(of whichthereisaninfinitenumber)

inghinges,andthecentersofthecorresponding

ocusofthe fixedcentersisknownasBur-

ter-pointcurve.

r'sCurve.1Figure11-20showsabodyr,

ctpositions,togetherwith thesixassociated

RM,Rzt,andRat,whicharelocatedatthe

entifiednormalbisectors[HA]and[UB\.(The

andR-uhavenot beenshownbecausethey

bsequentdiscussion.)

cognitionofthegeometricalrelations,point

vein acircularpath.Thusallnormal

hthepointOA.Theradius 0^Amovesfrom

oughanangle5i2;fromposition2 toposition3

tion3 toposition4through534.Hence,

eangleincludedby thebisectors[n^]i4and

gasami,GraphicalLinkageAnalysis,MachineDesign,

1953.


/http://www.h



62/119


bythebisectors[uaJuand[nA]23,

ces#14^34andR12Rnsubtendequalangles

hown,ina similarmanner,thatotherpairs

osubtendequal(or supplementary)anglesat

le11-1.

eanalysisis that,ifapoint Omistoserve

ntainingfourcorrelatedpositionsof some

thecommonvertexoftwotriangleswhich

ndhavecorrespondingrotocenterdistances

ructionshownin Fig.11-21yieldstwopoints

eseconditions.Theradiiofthe twocircles

maybe determinedeithergraphicallyor,alter-

lideruleto theratioofthe basedistances

dingvalues.Arepetitionof theconstruction

s (andconsequentlydifferentapexanglese)

Thecurvepassesthroughthesix rotocenters


/http://www.h



63/119


BMESTER'SCURVE


/http://www.h



64/119

ICSOFPLANKMECHANISMS

becomeclearlater,andthroughthesixpoints

ondingrotocenterlines,asheree =0.Thus

urveareknownimmediately.

urmester'scurvehasbeenselectedasaloca-

ecorrespondingmovingpointislocatedby

version,asshowninFig.11-22.(Figure

uationasFig.11-20. O,hasbeenplacedarbi-

t.)Thebody risassumedtobe stationary,

fO,are determinedbyreferringthetriangles

O,tothebaseline AiBi.Thecouplerhinge

tedatthecenterof thecirclethroughO,,

epresentexample,becauseO.hasbeenselected

tocenter,twoofitsrelativepositionscoincide.

sible todrawacirclethroughthreepoints,it

ocentersmustbepointsofBurmester'scurve.)

emofhavingto designalinkageforthe

body,thedesignershouldcheckfirst whether

ofthe availablerotocentersorpointsofinter-

ticablemechanisms.If,forsomereason,no

eobtainedinthismanner(e.g.,the mecha-

tetransmissionanglesormayhavetopass,

anin-linepositionofall links,whichnormally

ion),thenBurmester'scurvemustbedevel-

her locationsofthefixedpivotsinvestigated.

bodyr(Fig.11-23)is hingedatAto the

totheframeat Oq.ThepointBis tomove


/http://www.h


use#pd-google


65/119


onsBi,B2,B3,andB^A secondguiding

ablefour-barlinkage.

Bknown,thecorrelatedpositionsAi,

ocatedandthefourrotocentersRu,Ru,RM,

thesecenters,partofBurmester'scurveis

ectedarbitrarilyaslocationforthe fixedpivot

tionsofO,aredeterminedbykinematic

iBi0J1,etc.SinceO,isentirelyarbitrary,

vepositions,O}= O,,O]1,O}11,andO}v.

e,inwhosecenterCiislocated.

n;GeneralDiscussion

ntfacetofkinematicsynthesisisthe devel-

ators,devicesusedinmechanicalcontrolsystems

inwhichthedisplacementoftheoutputlink

e input.


/http://www.h


use#pd-google


66/119


edmechanisms,adefiniterelationshipexists

tsofthedrivinganddrivenlinkswhichis gov-

rtionssuggeststhepossibilityof usingbar

orfunctiongeneration.Theideaisattractive

msofferanumberofconstructionalandopera-

aseofmanufacturewithinclosetolerances,excel-

otsandsliders,and absenceofabruptchanges

nsmoothandquietoperation.However,two

sttheir useasfunctiongenerators,viz.,their

efinite numberoflinks,toreproduceagiven

alexactitudeoverafinite rangeandtheextreme

mechanismswithmorethanfourmembers.

ringproblemsrequirerigorouslyexactanswers.

barlinkageandits sliderderivativespossess

oallow,ingeneral,the findingofasolutionin

ructural"erroriskeptwithinaprescribed

nctiongenerationbymeansofbarandslider

pproximate.Moreover,becauseofthephys-

outputdisplacementranges,onlyalimited

anbe mechanized.

tofunctiongenerationarepossible.

tion.Thegeneratedandspecifiedfunctions

nctpoints.Betweenthese"precision"points,

buttheerror iskeptwithinpermissiblelimits.

alprocedures.)

neration.Thespecifiedandgeneratedfunc-

erivativesatonepoint.The approximation

npoints,but withinamuchshorterrange.

alyticalprocedures.)

saandb.

therprecisionpointsnor precisionderiva-

eerror isconfinedtoanacceptabletolerance

erange.Nomographicprocedures(notdealt

"methods,andexperimentalmethodsfall

on.Thefirststepinthesynthesisof afunc-

rsionofthe givenfunction

gtheoutputand inputdisplacements,e.g.,

s,


/http://www.h


use#pd-google


67/119


nsformationequations

ors,indegrees perunitofcorrespondingvariable

utlink positionscorresponding,respec-

andy

aybeofthe openorthecrossedtype.The

sgenerallyyieldsthemorecompactmechanism.

the factthattheinputand outputdisplace-

e.Consequently,ifacrossedlinkageis desired,

chosenasfollows:

ction (ydecreasingwithincreasingx)and

n

aFour-barMechanism.Thecharacteristic

inkingthepositionsof theoutputandinput

tions(11-1)and(11-2)showthatthe

signparametersinvolvedinthemechanization

specifiedrangeisseven:

ple,q/p,r/p,s/p

andk$(or rangesA


68/119


erivativetothree,etc.)However,sincethe

osynthesizeamechanismfora seven-point

ve,twoparameters,atleast,areassumedin

ndinglyfewerpointsofagreement.(Ina

thefollowingparametercombinationscould

ctors,onescalefactorandoneinitial position,

deratio.) Shouldtheresultantmechanism

,byexceedingthepermissibleerrormargin,

tbemade,basedondifferentassumptions.

orationalmethodsforchoosingtheparameters,

erbe guidedbypastexperienceorsimplyrely

bmayserveas aroughguideforthe start:

^/A


69/119


mainstationaryinits phaseSi=O.B1,and

2is referredtoitby meansoftherotation

edin Fig.11-25c.BecauseofthisrotationB2

dAitoAu.TherotocenterRuis thenlocated

ormalbisectorsof 0,OJ1andA1Au,anditwill


/http://www.h


use#pd-google


70/119


torofOqOl* passesthroughO,andmakesan

of centersOqO,.

256andc revealsthatRumaybelocated

raysdrawn,respectively,throughOqand

i2/2totheline ofcentersOqOasshown

-26showsafour-barmechanisminthree

ethereforethreerotocentersassociated

vetoq,namely,R12,Ru)andRu-

3 aredeterminedasinthe two-position

terminationofthelocationofR23relative

uiresfurther discussion.Theintersectionof

ely,at 023/2and ^23/2tothelineof

appropriatepivotpointsdefinestherelative

2-3in phase2oithemechanism.Conse-

edby Rnm.Itspositionreferredtoqh

nedby rotatingtherayOqR23(2)through 4>u-

iwouldbe obtainedbyrotatingtheray


/http://www.h


use#pd-google


71/119


beextendedtotheconstructionofrelative

threepositions.

der-crankMechanisms.Figure11-27

therelativerotocentersinmechanismswhich

tructionisself-explanatory.InFig.11-276,

nreferredto q%.Inordertorefer ittoSi,

oveda distance lu.

gnaslider-crankmechanismforthegenera-

x,withprecisionpointsatx =1,2,3, and

0.47712,and0.60206.Theindependent

ntedbythehorizontaldisplacementofthe

dthedependentvariableybya counterclock-


/http://www.h


use#pd-google


72/119


equirementofafour-pointapproximation,

maybechosenarbitrarily.Themostcon-

ethetwo scalefactorsandtheinitialposition

v ingA l=k1A x=A x

y,givingA0=fc* Ay=100Ay

in., lu=3in.

u=6012'

itionmechanisminvolvestheconstruction

chrequiresthedeterminationoftworelated

ccordancewithTable11-1forexample,R^Ru

,thesecentershavebeenreferredto qt.

ter'scurveisa theoreticallypossiblelocation

eAi, thechoiceoftheinitialposition isarbi-

Ai locatedonthecrankcircle,theconstruc-

ofthesecondcouplerhingeBproceedsas

edtobe stationaryinitsphases1,and the


/http://www.h


use#pd-google


73/119


oundby makingA2>A11= lu,

t>AIV=ln; Biisthecenterof the

11,andAIV.

sibleto synthesize,bythismethod,a

tapproximation.Thechoiceoftheinitial

uldnolongerbearbitrary.Thepoint would

onoftwo Burmester'scurves,developed,

ntsetsoffourpositions.Ofcourse,thesolution

n;ReductionofPointPositions1

Thethree-positionproblemistrivial,and

meansforexplainingtheprincipleofthe

s.

nthespecifiedandgeneratedfunctionsis

s,four designparametersarearbitrary.

ceisthatofthe twoscalefactorsk^,andk+,of

ofthelinkratiop/q. (Theactualdimensions

hesolution,exceptfortheover-allsize ofthe

ptionsleadtothevaluesoffa2,fa3,^12,and

edbythe design.

etriebelehre,"pp.326-341,HermannSchroedelVerlag,

en,Point-positionReduction,Trans.FifthCon/,on

1-193,PurdueUniversity,Lafayette,Ind.


/http://www.h


use#pd-google


74/119


the positionofthecouplerhingeB,the

and3are referredtotheoutputlinks inits

ofthecirclethroughthethree relativeposi-

edas follows:

edthrough 0i2.

edthrough 0i3.

reductionofpointpositionsisthefact that

wacirclethroughthreepoints.In thefour-

tions,thedesignparametersaresochosenthat

hepoint

mberof

ducedto

this

parameters

d fc^(or

erto

elative

derto

relative

einitial

kmust

le0,+i

djusted

entersOqOs

+i,asshownin Fig.11-30

ativesignof 0,n+iisdisregarded.)

chanizethefunctiony=log xinthe

cisionpointsat x1 1,y1=0; X2=4,

=0.8451;andx4= 10,2/4=1.0000.

yassumptions,k$=10 perunitofx

y, thelinkdisplacementsarecalculatedas

30 i3=60 0i4=90

=-368'0= -5042'0=-60

hosenthatoneoftheinput angles,here0i2,

Oandthecorrespondinglinkratiop/qis


/http://www.h


use#pd-google


75/119


ermined,theremainderofthesolution

eresultantmechanismisshowninFig.11-31.

precisionpointsexceedthe permissible

ativesolutions,obtainedbybisectingeither

veto beinvestigated.Otherchangeswhich

redistributionof precisionpointswithinthe

fthe assumedscalefactors.

onlythreedistinct relativepositionsofA

>i andthedistributionoftheprecision points

oinputdisplacementsm,narebisectedbythe

oftheprecisionpositionsis nolongercom-

n,onlyonedesign parameter,viz.,fc$may

ngementinwhichthe crankpositions1,4

ydisposedwithrespecttoOqO,.Theproblem

nofalinkratio p/qsuchthattheoutput-

enbythelinkage,is equaltothatobtainedby

onvaluesofthe specifiedfunction,where

nt_ A^j_ Ay.-

entA^0Ay0


/http://www.h


use#pd-google


76/119


e input-displacementratioR^,,similarly

ent_ A,_ Az,-,...^

entA


77/119


a trial-and-errorprocedure.Asmall

tween1 and3istried graphically,andthe

andA^0aremeasured.Thusthe approxi-

nestablished.Itsexactpositionis foundby

cal(R* versusp/qcurve.Ithas thefol-

ueof1.

p/q= 1.

Gfyisslightlysmallerthan (R^;outsidethis

osesaserious restrictiononthetypeof

mechanizedtoa five-pointapproximationby

point positions:

-r-1andconsequently-~> -r2-

besuchthatthe slopeofthe"innerchord"is

terchord,"asshowninFig.11-32c.

hodofreductionofpoint positionshasthe

er,therestrictionsplacedonthe designby

blecrankarrangementsoftenprecludethe

tion.In general,therelative-rotocenter

pectsforthesuccessfuldevelopmentofa useful

chanizethefunctiony= sin* xwithinthe

ecisionpointsat xx0,y10; x* =2230',

3=0.5000;x4=6730', yi=0.8535;and

entralsymmetry"ofthegivenfunction,

gementmaybeexpectedtoresultin agood

designis choseninwhichtheangles 4>nand

eofpivotsOqOs,as showninFig.11-33.The

dat1 perdegreeofx.Therequired ratioffl*

eeventuallyyieldsp/q=1.85,giving

ment.


/http://www.h



78/119


ofy

put rotationsknown,therelativepositions

usualmanner.A1andAvcoincidewithA1,

dcoincidentpair,andA1u occupiesthe

ethethreepoint positionshappentobe

smresults.(B1,thesecondcouplerhingein

rightanglesto thelineA1AuAUI.)

eneratedFunctiony=sin's


/http://www.h


use#pd-google


79/119


unctionsarecomparedin Fig.11-34,and

ownin Table11-2.

dnegativeerrorsmaybeequalizedbya

r"range.Should theresultantmaximum

entofthe outputrange)beconsideredexces-

gns,basedondifferentvaluesofthe scale

ei nvestigated.

;OverlayMethod1

aphicaltrial-and-errorprocedurewhich

egeneratedoutputhasneitherprecisionpoints

commonwiththespecified function,but

gMechanismsandLinkages,"pp.147-154,McGraw-

ewYork,1948.


/http://www.h


use#pd-google


80/119


thinanarrowtolerancebandthroughoutthe

ortwodesign parametersarechosenarbi-

onesarevaried inasystematicmanneruntila

uracyisobtained.

pton.Thechosenparametersarek$,k*,and

.Thedesignprocedureisasfollows:

ontobemechanizedissuitablysubdivided,

gularincrementsm,nand\f/m,narecomputed.

ufficientlylargescale,ofthesuccessive

k,andwith theassumedlengthofthecoupler

plercirclesisdrawn,centeredinthesuccessive

(Fig.ll-35a).


/http://www.h


use#pd-google


81/119


e,on transparentpaper,ofthesuccessive

In addition,markedonthisoverlay,

circles,centeredinO,,whoseradiirepresent

putlink (Fig.11-356).

thefirstlayoutandmovedacrossit until

hthecouplercircles1,2,3,etc.,pass through

onecircleofthe overlay.(Intryingto

toverlaycircleswill bescannedinturn.It

polate.)

und,asshownin Fig.11-36,thelengthsof

utlinkandtheinitial positionoftheinput

ereadoff thedrawing.

ng,noacceptablesolutioncanbefound,

allyassumedparametersmustbechanged,

ed.

einthesign ofk+willhavethe effectof

overlay.It shouldbenotedthatthesame

urningtheoverlayfacedown.

n.Thetwochosenparametersarefc^,and

wn,itis notpossibletoconstructaspecific


/http://www.h


use#pd-google


82/119


unction.However,iftherangeAj/ ofthe

dividedintonintervalswhichincreasein a

ndanoverlayisconstructedofzangularspaces

hesameratioas thefunctionintervals,then

pacesmaybeusedtorepresentthe ncorre-

eoutputrangeA^.Thusthis typeofoverlay

aybeused inthesolutionof anyproblem.

t,differentsectorsofthe overlaywillbe

ay,IntervalRatiofffc=y/2

edfromthereferenceline)


/http://www.h


use#pd-google


83/119


taforthe constructionofageneraloverlay,

ectedintervalratio(R,- =-\/2,whichdoubles

e11-37showsaportionof thisoverlay.

ewi tisadvantageoustohavetheinput

yaspossible.Forthis reason,inthecaseof

slope,asshownin Fig.11-38a,Ayisdivided

lis atthelowerendof therange,andthe

hosenforafunctionwithdecreasingslope,as

softhe outputintervalsasfractionsofthe

sinto4,6, and8spaces.Thevaluesare

R, -=-\/2.

hanizethefunctiony=logiox,between

therangedividedintosixintervals.

ndthecoupler-crankratioareassumed.

=60, givingk$=A0/Ax=^-= 6f per

wingvaluesforthe outputintervals(Ay=1,

698,yu= 0.4262

95,yn =1.0000

,andm,n=k^cm,n,

0xn=9.000


/http://www.h



84/119


tudesBasedon(Ri=y/2


/http://www.h



85/119


ationincrossedandopen mechanisms,makesit

moresuited totheformerarrangement.

dontwoprecisionpointscumderivatives,

rily,correspondingtotheextremesofthe

ativesatanumberofselectedvaluesof the

esolutionproceedsasfollows:

dfc^andthe linkratioq/pareassumed.

respondingtothe prescribedorselectedn

culatedandmarkedoffonp[Eq.(10-14)]:

Og>O,

chosenarbitrarilyonthecrankcircle.


86/119

CSOFPLANEMECHANISMS

nsAfPi andA+Pnaredrawnin,and the

usteduntil BfandB+lie onthesamecircular

le\f/^nis comparedwiththerequired

caseatthe firstattempt,thetwoangles

iterated

untilthe

ained.

nreached,

ositions

ponding

esame

wnin

edureis

ratiosq/p

olutionis

reached

torsare

yof scan-

atio,which,

only10to 15min,thisapproachleadsto there-

verlaymethodwith a"free"outputscale.


/http://www.h



87/119


hanizethefunctiony=log x,between

precisionpointsatx=1 and10andprecision

7,and10.

1.Assume

^=^ perunitofx

=-90 perunitofy

culatedbyEqs.(11-6)and (11-7),where

746p,h=0.594p

butnot averysatisfactoryone)obtainable

orsisshownin Fig.11-42.Theparticulars


/http://www.h


use#pd-google


88/119


rones'sandNelson'sMotionAtlas1

hesisofmechanismsforthe productionof

ossessingcertaindesirablecharacteristics,or

gha numberofprescribedpositions.

urvesinthe designofsingle-dwellmechanisms

ouslyinFigs.11-11and 11-18.Otheruses

onsofengineindicators,specialindexingdevices,

dapplicationsassociatedwithagricultural,

ngmachinery.

diusDC

ratD1i radiusDC

onAtlascontainssome7,000couplercurves,

callyoperatedfour-barmechanismwhoselink

tematicmanner.Ineachsetting,thecoupler

mberoftracingpoints,arrangedina grid

oducedbymechanicalmeans,thepagesof

a familyofcouplercurvesisfoundwhose

hedesiredgeneralshape.Correspondingto

arateoverlayofthegivencurveis thencon-

son,"AnalysisoftheFour-barLinkage,"JohnWiley

1.


/http://www.h


use#pd-google


89/119

ESIS;GRAPHICALMETHODS30")

axesof bothareequal,andtheir shapesare

ocedureislaborious,butoftenyields a

eratethespecifiedcurvewithinanacceptable

anaidin thedesignofdwelllinkages.For

ofFig.11-43,witharise-dwell-fall-dwell

eru,wasdevelopedin thismanner.Itsbasic

obtainedasa resultofamethodicalsearchof

a couplercurvewhichhadtwoappropriately

engthwiththesamecurvature.Thelinkage

andr/p=s/p= %,andasfaras canbe

drawinginstruments,thepathofthecoupler

= 90 andBC/p=1.6/3,satisfiesthe

eductionofPointPositions1(FixedPivot

er)

generation,themethodofreductionofpoint

eapproach,ofsomewhatlimitedscope,to

on.Thebasisofthe methodwillagainbe

trivialthree-pointconstruction,andthe

endedtothe morecomplexproblemsinvolving

positionsofa point.

ig.11-44,thecouplerpointCis required

prescribedpositionsCi,C2,and3.Ifno

bemet,the problemadmitsasixfoldinfinity

ofthefree choiceofthefollowingdesign

(2d.o.f.),directionofthebaseline and

gthOqAofthe crank(1d.o.f.),andthe

(1d.o.f.).

foundbymeansofkinematicinversion,i.e.,

tionaryinits phaseTIandmovingtheframe.

arefound,therefore,bymaking

d0J"=A^3C30.

ofthecircumcircleofthetriangleO,O}1O}11.

ismsformorethan threeprescribedpoint

signparametersaresochosenthatsome rela-

e,leavingthreedistinctpositionsthroughwhich

etriebelehre,"pp.326-341,HermannSohroederVerlag,


/http://www.h


use#pd-google


90/119


dertoachievetherequiredreductionof

votO,of thedrivenlinkismadeto coincide

softhe coupler,forexample,[Rr]u,asshown

ositionof thesepointsonthenormal

elyarbitrary.)Asa consequenceofthis

metricallydisposedwithrespecttothe

e

,A

O,Oy=

;because

and

ry,butonceassumed,determinesthe

Oqon thebaselineis alsoarbitrary,but

ranklengthOqA.Theintermediatepositions

don thecrankcircle[(C2A2=CiAi),etc.],

fO,constructedinthe usualmanner.Because

th[Rr]u,0,IVcoincideswithO}= O,,andthe

sitionsis reducedtothree.

infinityof solutions(4),asfollows:

o.f.),directionofbaselineandpositionof

ebetweenCandA(1d.o.f.).

scaseO,ismadeto coincidewithtwo

sly.Itisthereforelocatedattheintersection

oneof thefollowingcombinations(onepoint


/http://www.h


use#pd-google


91/119


r):

e O,Oqisarbitrary.

madetocoincidewith[Rr]nand[Rr]3t-

=^16/2and$.A3O,Oq=^34/2.PointsA\

respectiverayswithanarbitrarilyassumed

attheintersectionofthebase linewiththe

smarkedonthe crankcircle,andtherelative

cted.Asaconsequenceoftheparticular

swithO]=O,,O}vcoincideswith0J11,and

inctposition.

utionshasbeen reducedtoatwofoldinfinity

fthebaseline(1 d.o.f.)anddistancebetween


/http://www.h


use#pd-google


92/119


ally,thecouplerpointisnotmerely

numberofdiscretepositions,buttotracea

taindesignpositionsarechosen.It ispossible

nstructiontosixpoints,providedthe sixth

nthe samecircle,centeredat0asthe

n.Thusthethreerotocenters[Rr]u,[Rrhs,

oincidewithOasshownin Fig.11-47.

sthatof thedirectionofthebaseline.

a3havebeendrawn,asuitablelengthCA+is

thepointsAf,A$,and A^arelocatedon

ndthecenter0+oftheir circumcirclecon-

he casewiththefirstrandomattempt,0+

nbaseline,theconstructionisreiteratedwith

dthelocusofO+developed.Theintersection

lineyieldsthetrue positionofOq.Next,the


/http://www.h


use#pd-google


93/119


erminedbyinterpolation,theactualpositions

ys,andtherelativepositionsofO,constructed.

,,0,vwith0.11,0JV with0.m,andBiislocated

mcircle.

lutionshasbeenreducedto asingleinfinity,

oiceofthebase-linedirection.Thusit would

rtherandspecifyaseventhdesignposition.

ina systematicsearchforaparticularbase

awingacirclethroughfour distinctrelative

eductionofPointPositions(MovingHinge

er)

tions,accomplishedbyplacingthefixed

centersofthecoupler,hascertainlimitations.


/http://www.h


use#pd-google


94/119


d itselftothesynthesisof couplercurves

ofdouble-dwelllinkageswithextendeddwell

sareovercometoalargeextentbythe

ichthe movinghingeB,ratherthanthe

esamepositionas thecouplerrotocenter.

oachesareavailabletothedesigner;should

red result,theothercouldbesuccessful.

re11-48showsfourpositions,Ci,C2,C3,

raversedbythecouplerpointC.Asshownin

problemhasafourfoldinfinityof solutions.

equiredreductionof pointpositions,the

coincideintwopositionsofthecouplerwith

nter,forexample,[Rr}u,whosepositiononthe

tirelyarbitrary.Alsoarbitraryis theposition


/http://www.h


use#pd-google


95/119


auseofthecoincidenceoffi1|4with [Rr]u

nd BzAzaresymmetricalwithrespectto

inedtoit atanangleXu/2,whereXH isthe

ecouplerbetweenpositions1and 4;thatis,

tobe choseniseitherthelength CAorthe

rmediatepositionsofA andBcannow be

z CiAi,whichdeterminesthepositionof

the positionofB2is foundwithA2B2=AiBi,t

edpivotO,islocated atthecenterofthe

B3.

olutionoftheproblem,thefollowingkine-

noted(Fig.11-48).In theinversion,in

stationaryinthephaseri,the fourrelative

ecenteredat AI;furthermore,therays

aresymmetricalinrelationtothe lineBi,tAi

/2.

movinghingeBismadeto occupytwo

ntwiththerespectiverotocenters.InFig.

.RJis,andBs,4 with[/2r]s4.Thepointpairs

thesequenceinwhich thepointpositionsare

closesthe other.Consequently,thefollowing

rsarepossible:

Rr]^;[Rr]16 [Rr]!H;

[Rr]34

irectionofthelineBi,6Oq arearbitrary,but

neisno longerso,aswillbe seenfromthe

locatedinanarbitrary positionontheray

, andOvqarethepositions occupiedbyOqin

coupleriskeptstationaryinits phaserI.

atrandom,thesepointswill notsatisfythe

ttheylieonone andthesamecircle.To

aysSi,60J andBi,sO mustbemirrorimages,

iAi,oftheraysBi,6O\ andBi,6O\11;thatis,


/http://www.h


use#pd-google


96/119



/http://www.h


use#pd-google


97/119


thisdiagram,therayB3,tOqisinclinedto

tanangle

edto thenormalbisector[ncjisat

gandB3itOqmakeequalangleswith thecorre-

gleincludedbythem(vertexOq)is equalto

sectors(vertex^15.34); i.e.,

^15-34.83,4

tsOq,B1i6B3,4,and#15.34(the pointof

bisectors[nc]isand[nc]n)lie ontheperiphery

enotedbyk ]5.34,isthe locusofallpossible

catedatthe intersectionofthechosen

16.3t,thepositionsofOqrelative toriare

edat thecenterofthecirclethrough

distanceACarethusdetermined.Next,

arefound,andthemechanismiscompleted

rof thecirclethroughJ5i,6B2,and#3,4.

itionof AIonthecrankcircle;A2,82 =AiBi

ethepositionof B2.)

boveconstructionmaybeextendedtosix

mhasasingle infinityofsolutions,dueto

engthCB.The couplerhingeBismadeto

ositionsBi,e=[Rr]u,#2,5= [Rrlu,and

catedatthecenterofthecorrespondingcircum-

ytheconditionthatit mustliesimultaneously

ci6-34.

explanatory,showstheapplicationofthe

ofadouble-dwellmechanism.Inpractice,the

kerarmmay havetocorrespondtodefinite

le.Therequiredcorrelationmaybeobtained

, orfollower,ofadrag-linkmechanism,

ouble-dwelllinkage.Thecoordinationofthe


/http://www.h


use#pd-google


98/119


otatingprimarydriver(determinedbythe

epositionsqiq3andqtq^ (whichdefinethe

hedbymeansofBurmester'scurve.

oberts'sTheorem1

ecurvegeneratedby agivencouplerpoint

anbe reproducedbytwootherfour-bar

ththefirst.Itis ofpracticalimportance

nerconcernedwith thesynthesisofamechanism

rticularcurvetwoadditionalalternatives,

o bemorefavorablethantheoriginaldesign

glesorspacerequirements.

Motionin PlaneSpace,Proc.LondonMath.Soc.,no.7,

enbergandJ. Denavit,TheFecundFour-bar,Trans.

s,pp.194-206,1958,PurdueUniversity,Lafayette,Ind.


/http://www.h


use#pd-google


99/119


maryfour-barmechanism,p=OgO,,

,towhichadditionallinksareattached,

DCE,suchthatOqDandDCforma parallelo-

dADCEis similartoAABC.

CFG,such thatO,FandFCformaparallelo-

ACFGissimilar toAABC.

areadded,whichformaparallelogram

hanismhas13lowerpairs(two eachatOq,O,,

B,D,E,F,G, andH)andis thereforeconstrained:

2l=3(9)-2(13)=1

hroughoutthemotionofthis compound

nsstationary,sothatthelinkagecouldbe

eframewithoutaffectingitsmobility.If

mechanismmaybesplitupinto threeseparate

showninFig.11-52,eachofwhichwillgivethe

C.

thepoint0


100/119


os(+a)+-sc os(^+a-180)

(+a)sc os(^+a)

sc os^)c osa

sin^)sina]

sc os^)sina

n*)c osa]

ateralOqABO,,

os^=p

n^=0


/http://www.h


use#pd-google


101/119


ly provethat0


102/119


otheconclusionthatthefollowingcombi-

nismsarepossible:

hanismsandonedouble-rockermechanism

isms

echanisms

aybeappliedalsotoslider-crankmechanisms.

s,whilenotingthatO,is atinfinity,itwill be

natemechanismsvanishes.Thusthereexist

hanismswhichwillproducethesamecoupler

nsionofthetheoremisoflittle practical


/http://www.h



103/119

ESIS;

eanalyticalapproachtosynthesisisits

tunately,thissuperiorityovergraphical

etbya numberofadversefeatures.For

rogressivephysicalimagesofthemechanism

nofdesirablemodificationsduringthedevelop-

lt,obtainedafterlengthyand laborious

etobecompletelyimpracticable.Moreover,

impossible,toassessintelligentlytheeffects

thedesign,themethodoftendegeneratesinto

ocedure.Itsfullpotentialitiescanbe

ofahigh-speeddigitalcomputer.

n;Freudenstein'sEquation1

echanismareregardedasdirectedlineele-

1,then

c os4>(a)

in(b)

qs.(a) and(b)yieldstherelation

os^ 2pqcos 2qscos(0 \p)(c)

onofthesideratios

+61,cos\p=cos ( 4d(12-2)

edbyFreudenstein,formsthebasisof a

mateSynthesisofFour-barLinkages,Trans.ASME,

1955.


/http://www.h



104/119


achtosynthesis.Itisadaptabletoboththe

ion-derivativemethodsoffunctiongeneration.

edfromthediscussionof Chap.11.In

atedandspecifiedfunctionsagreein anumber

s,andeverywhereelseanunavoidablestructural

osemagnitudedependsonthespacingofthe

tureofthe function.(Thelowestmaximum

acinginwhichereachesthesamevalue between

tsandbetweenthe limitsoftherangeand the

)Inthesecondmethod,thegeneratedandideal

tandhave anumberofderivativesincommon.

edesignofafunctiongenerator,thegiven

rtedintothefunction^=/(


105/119

hods321

n;Precision-pointApproximation

r,precisionpointswillbe designatedby

gewillbedenotedbythe newsymbols

y,butneednot,coincidewiththefirst precision

echosenparametersareft*,ft*,fa,and

ecomputedbymeansofEq. (12-3),as

i,7i,etc.

esintoEq.(12-2) yieldsthreeequationsfrom

maybe calculated:

fa.=cos(fa \p1)

a>=cos(fa fa)

a,=cos (fa fa,)

parametersassumedareft*,ft*,andfa,

requations:

(fa+fa,1)=cos[(fa fa,1)-fa]

(fa+fa,2)=cos[(fa fa.i)-fa]

(&+ fa,3)=cos[(fa fa,3) fa]

+faA) =cos[(fa-fa,i) -fa]

sideratios yieldsacubicequationintanfa,

+m3tanfa+ml=0

ntsm,thereaderis referredtoFreudenstein's

fa known,thesideratiosarefoundas inthe

n.

Thearbitrarychoiceofft* and ft* leadsto

,cos(fa +faA)=cos[(fa,1- faA)+fa-fa]

,cos(fa +fa,i)=cos[(fa,i- fa,i)+fa-fa]

R,cos(fa +fa,3)=cos[(fa,3- fa,3)+fa-fa]

(R,cos(fa+ ^,-,4)=cos[(faA-faA)+ fa-fa]

,cos(fa +fa,s)=cos[(fa,s- fa.t)+fa-fa]

sideratios yields,afterextremelytediousand


/http://www.h


use#pd-google


106/119


ns,thefollowingexpressions,whichmustbe

WJ)

tf-4F[F'3]

bic expressionsintan^t-,andF[,F2, andF'3

an#,-.(For detailsoftheseexpressions,the

enstein'soriginalpaper.)After 0,-and\pfhave

atiosaredeterminedas inthethree-point

onofaparticularproblem,basedonthe use

ke10 to12weeks.

;Precision-derivativeApproximation

iminarystepin themechanizationofa

ethodofprecisionderivatives,itis necessary

tocorrespondingderivativesofthe function

^dx

bedvaluesofthese derivativesintothebasic

tionswhichdeterminetheparticularlinkage.

equationsareobtainedbysuccessive

2).

os\p=cos ( iA)(12-2)

(1-


107/119

METHODS323

+Vsin}}

)2c oa(0-tf )(12-9)

+W''"-(^')31sin ^}

(1 -^)3]sin(pand thesideratiosaredeterminedfromthe

btainedduringthereductionprocess.

errepresentationofvectors(Fig.12-2),

havedevelopedaveryelegantapproachtothe

Sandor,Synthesisof Path-generatingMechanismsby

gitalComputer,Trans.ASME,vol.81B,no.2,pp.

orandF.Freudenstein,KinematicSynthesisofPath-

yMeansoftheIBM650Computer,IBM650Program


/http://www.h


use#pd-google


108/119


hanismsforthepurposeofgeneratingapath

rypoints,with prescribedcrankrotations.

stillmanageablewiththe aidofadesk

ntsynthesisdemandstheuseofa high-speed

procedurehasbeenprogrammedtosuitthe

whichautomaticallydeterminesallexisting

2linkages),selectsoneonthe basisofa

omputesthecoordinatesofthetracingpoint

rotation.

definedas

mumandmaximumskeletaldimensionsoflinkage

eencouplerhingesand

drivinglink

yofanunrestrictedcrankrotation,thequality

favorof linkagespossessingthischaracter-

ratherthan2w.

rMechanismsforSpecifiedInstantaneous

mponents1

y,andaccelerationdiagramsofthefour-bar

12-3aredefinedbythe vectorequations:

+s=q +r

A+VBA

&A+&BA

ntocomponentsinthedirectionofthe xand

ionscanbereplacedbysix algebraicequations:

esisofFour-barMechanismsbytheMethodof Com-

chanics,vol.24,no.4,pp.22-24,1957.


/http://www.h



109/119

METHODS

3variables,viz.:

of thedrivinglink,itsangularvelocityW3,

a,

thecoupler,itsangularvelocity,,and

p

tions,problemsofsynthesisinwhichseven

cribedmaybesolvedeither directlyorbythe

ch.

efollowingdesignparametersarespecified:

onwanda.


/http://www.h



110/119


t outlinedbelowwouldbeadoptedifthe data

inkswereinterchanged.)

biningEq.(12-12)withEq.(12-15),and

),thefollowingequationsareobtainedin

eexpressedintermsofthe unknownangular

bedquantities:

17)arecombinedto eliminatear,giving

qxry)+ur2(rx2+rv2)

y rysx)(e)

(d)intoEq.(e)yields aquadraticequation

sequaltouq,giving infinitelylonglinksq

,whichrepresentstheactualangularvelocity

theexpression

mechanismshallsatisfythefollowing

secaq=0

4.625in.w,=5 rad/sec


/http://www.h



111/119

ods327

66,090=0

sec

1.77in.q =1.92in.

r= 7. 22in.

followingparametersarespecified:

,andaq

arctan >uanda,

outlinedinthefollowingwouldbeused if

ddrivenlinkswereinterchanged.)

sethe gradualeliminationoftheunknown

ldanequationofthe fourthdegreeinqx,thus

ofthegradual-approachmethodofsolution.

andthecorrespondingvaluesofwr andtan

,usingthegeneralprocedureoutlinedin Illus-

pis reiteratedanumberoftimes,andthe

gainstqx ortabulated.Thefinalresultis

edure.

mechanismshallsatisfythefollowing

= 10rad/ secaq= 0


112/119

esultsforNumericalExample2


/http://www.h


use#pd-google


113/119

METHODS329

ecaq 0

secar=15rad/sec2

a, =10rad/sec2

=sx+p (a)

=sy(b)

y= 5sy(c)

rx= 5sz(d)

4rx+15ry =25sz+10sv (e)

4ry- 15rx=25sv-10sz (/)

mbining(b) with(c),and(6)with(/):

lOs^(A)

cedtofiveequations,Eqs.(a), (d),(e),(g),

eunknownquantitiesqx,rx,ry,sx, andsy.

ombining(g)with (e),and(g)with(h):

t)

to fourequations,Eqs.(a),(d),(i),and(j),

uantitiesqx,rx,ry,andsy.

minatedis ry.Equation(i)combinedwith

nextbycombining(k)with(a),and(k) with(d):

m)

nd(m)resultsin

hanismare:

.240

5


/http://www.h


use#pd-google


114/119


rMechanismsforPrescribedExtremeValues

theDrivenLink; MethodofComponents1

(Fig.12-3)the drivingcrankqisassumedto

ularvelocityw,.Since thedrivenlinks

tion,co,isvariableboth inmagnitudeandin

linkagewhichwouldgiveprescribedvalues

umw,withaspecified constantwg.

>g =const,Eqs.(12-16)and(12-17)reduce

w,2+sya,(12-16a)

xar=s^Wj2 sxa,(12-17o)

seof therequirementthatw.=maxor min,

equationsarefurtherreducedto

12-166)

=svw,2(12-176)

ration,velocity,andreducedacceleration

ectionsof thelinksmaybeexpressedin terms

follows:

wgo>, o>.[(o>3 wr)2(w, Wr)2+ay*],.-.

'^\2I21(b)

!-o>r)2 +ar2]

,r)2+af2](d)

+ar-'

+


115/119

ods

+(w, wr)2w,2wr2

(w,-wr)2+ar2]

)2+ar2

scorrespondingto maximumw,are

1,andthose referringtominimumw,bythe

emofthreeequationsis obtained:

l2Wrl2

wrl)2+ari2]

2w,22Wr22

(w,2 Wr2)2+ar22]

wrl)2+ari2]

)2(w,2 0)r2)2+ar22]

enEqs.(g)and(i) yields

,2 wr2)2

O,22_ (Wg W.2)2(w,l Wrl)2Wrl2,..

accomplishedbyeliminatingar22between

ningtheresultantequationwithEq.(J)] gives

, W,2)2[(w,l Wrl)2 (w9 Wrl)2]wrl2

rl)2[wri2 (w, wrl)2]

Wrl)2[wrl2 (w,l Wrl)2]}= T(Jk)

ndsideofEq.(k) issimilartothe left-hand

es1 and2areinterchanged.Equation(k)

andwr2.Since itcontainstwounknownquanti-

atthe problemhasaninfinitenumberof

nga definiteresultistoassumea valuefor

elimitssetbythe followingcriteria:

3Wrl w,l)]0

s,derivedintheAppendixatthe endofthis


/http://www.h



116/119


ondtimederivative

edriven linkattheinstantwhen thelatter

es.

tedinto Eq.(A;),thecubicequationis

eroots, theonewhichsatisfiescondition

valuesforari2 andari2hasphysicalmeaning.

if eitheruT1orwr2is takenaszero,the

toaquadratic,andthe calculationsaresimpli-

ebornein mind.Inthecaseofa crank-

dw,2haveoppositesense:

-linkmechanism,wiandw,2havethesame

signacrank-rockermechanismtothe

d/sec,w,2=8rad/sec,p= 10in.

ay betakenaszerosincethis value

.

4.11rad/sec

sfycondition(12-21)andisdiscarded.The

sconditionandyields positivevaluesfor

= 197andar22=3,740.

edintoEqs.(a) to(/),yielding:

towi:

q =3.41in.

.s =5.69in.

r=9.74in.

tow,2:

.

.


/http://www.h



117/119

hods

nFig.12-5.The correctnessoftheresult

ofFreudenstein'stheorem,accordingtowhich,

e,thecollineationaxisisperpendiculartothe

signadrag-linkmechanismforthefollow-

rad/sec,u>,2=5rad/sec,p=10 in.

er tosatisfycondition(12-21),wri>

rthercalculationsrevealthat,ifonlyinteger

hechoiceofsuitablevaluesofur1 islimitedto

c,sincevaluesoutsidethisrangegivenegative

ctedvalueofwri=15 rad/sec,Eq.(k)reduces

o!r2- 3,125=0

real root,wr2=12.5rad/sec,whichsatisfies

dsan2=9,490and ar22=703.

smare:

tow,i:


/http://www.h



118/119


owl2:

nFig.12-6.

quationa,=w,[w,(3wr uq) &j,(3wr w,)]

ombiningEqs.(12-16a)and(12-17a):

x2+r2) =w,2^* +rysy)+a,(rxsy r^)(l)

ryandrxsx+rysyare recognized,respectively,

and r-s, andtheterm(r^ r^x)asthe

maybe writteninthefollowingform:

wr2r2=w,2rscos (


119/119

METHODS

hevelocitiesVAandVBin thedirectionAB

n Fig.12-7,

mechanical narendra

Documents

Transcript of mechanical narendra