mechanical narendra
TRANSCRIPT
-
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
-
7/27/2019 mechanical narendra
2/119
ICSOFPLANEMECHANISMS
lrotationd
-
7/27/2019 mechanical narendra
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
-
7/27/2019 mechanical narendra
4/119
ICSOFPLANEMECHANISMS
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-
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
5/119
OFTHEPLANEMOTION
oftheEuler-Savaryequation.Since(P>
-
7/27/2019 mechanical narendra
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.)
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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:*
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
10/119
ICSOFPLANEMECHANISMS
sq andr,pin-connectedtogetheratP,with
ngentptandr freetoslidealongs. Thepoint
denotedbyX,.
correspondingtothe arbitrarilyassumed
sec,ccw,isthe requiredpole-displacement
yofthelinkageisvs =umX(P> S).
y proportion,andthevelocityofPfollows
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
12/119
ICSOFPLANEMECHANISMS
=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,
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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
-
7/27/2019 mechanical narendra
14/119
AMICSOFPLANEMECHANISMS
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
15/119
FTHEPLANEMOTION
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
16/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
17/119
FTHEPLANEMOTION
ectionCircletoKinematicAnalysis
providesthemeansfora purelygeometrical
vatures,itbecomespossibletoanalyzecomplex
oursetotheindirectanalyticalandgraphical
re10-15ashowstheskeletonofacertain
echanismwhoseinputcrank2revolvesatcon-
theangularaccelerationoflink4 andthe
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
18/119
ICSOFPLANEMECHANISMS
t,as shown,thesolutionbymeansof
atherdifficult,andtheproblemisbest solved
oncircle.AfterthepathcurvatureofBhas
raphicallyorbycomputation
mentary.
nin Fig.10-16a,theproblemmaybesolved
nflectioncircleorbythesubstitutionof the
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
19/119
FTHEPLANEMOTION
entmechanism
ntlow-complexitymechanismofFig.10-166.
chlesscumbersome.
eneralCase)
which,byrolling ononeanother,reproduce
thecharacteristicsoftheconstraints.The
s investigatedinthefollowingsections.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
20/119
ICSOFPLANEMECHANISMS
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,-)
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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, {&)
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
22/119
AMICSOFPLANEMECHANISMS
-> 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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
24/119
AMICSOFPLANEMECHANISMS
rItan6y=((R - (R2)taney
an0,
+ (P ->F)=(0r->P)
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
26/119
ICSOFPLANEMECHANISMS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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)
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
28/119
AMICSOFPLANEMECHANISMS
&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,
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
30/119
ICSOFPLANEMECHANISMS
erpendiculartothefixedlink,andbyBobillier's
axisisperpendiculartothecoupler.
rkinematicpropertywhichmaybe
naxisisthe following.Ifafamilyoffour-bar
edonthesamefixedbaseline02 >04,whose
nstantaneousvelocityratio(R,thesamefirst
,andthesamesecondderivative(R"=d2&./d
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
32/119
AMICSOFPLANEMECHANISMS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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-
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
34/119
ICSOFPLANEMECHANISMS
Eqs.(t)and (j)ofSec.10-7.
sconstant,
fthelastthreeequationsyields
10-23,thecircling-pointcurveisa looped
ranchescrossatrightanglesat theinstant
oinfinityin anasymptoticalapproachtoaline
iththe polenormal.0Misobtainedby
minatorinEq. (10-22):
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
36/119
AMICSOFPLANEMECHANISMS
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:
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
37/119
FTHEPLANEMOTION253
n,appliedtoA,yieldsthediameterofthe
A)=(0, ->A)[(PU->A)-di cos.]
n,appliednexttopointC,yieldsthe required
>O -dicostfj
.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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):
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
40/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
41/119
ESIS;GRAPHICALMETHODS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
42/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
43/119
ESIS;GRAPHICALMETHODS
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
-
7/27/2019 mechanical narendra
44/119
ICSOFPLANEMECHANISMS
hesisofa crank-rockermechanismfora
rnaction,i.e.,forprescribedvaluesof
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
46/119
ICSOFPLANEMECHANISMS
nsisinfinite,additionaldesignrequirements
sconditions1to 4above.
180 and
-
7/27/2019 mechanical narendra
47/119
ESIS;GRAPHICALMETHODS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
48/119
AMICSOFPLANEMECHANISMS
s,capableofguidingtheircouplerr through
ndequalanglesatthe rotocenter[Rr]u=R^;
sareequaltohalfthe angulardisplacementofr:
,andeffectivelengths(betweenhinges)of
mechanismsalsosubtendequalanglesatRi2:
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
49/119
ESIS;GRAPHICALMETHODS
cablemechanismsisinfinite,itis possible
mutuallycompatible,requirementstobesatis-
n.Examplesofsuchadditionalcriteria,the
hisfour,arethefollowing:
hefixedpivotsOq andO,androtationsof
s, [
-
7/27/2019 mechanical narendra
50/119
ICSOFPLANEMECHANISMS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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
-
7/27/2019 mechanical narendra
52/119
ICSOFPLANEMECHANISMS
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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^
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
54/119
AMICSOFPLANEMECHANISMS
rtoR^R^andRitRnis perpendiculartoXR23,
rtoRuR andR^Ruisperpendiculartofli2Z,
ngle
ngleRuZH3,
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
55/119
ESIS;GRAPHICALMETHODS
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-
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
56/119
AMICSOFPLANEMECHANISMS
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=
-
7/27/2019 mechanical narendra
57/119
HESIS;GRAPHICALMETHODS273
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
58/119
AMICSOFPLANEMECHANISMS
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
-
7/27/2019 mechanical narendra
59/119
le
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
60/119PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
61/119
ESIS;GRAPHICALMETHODS277
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
62/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
63/119
ESIS;GRAPHICALMETHODS
BMESTER'SCURVE
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
65/119
ESIS;GRAPHICALMETHODS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
66/119
AMICSOFPLANEMECHANISMS
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,
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
67/119
HESIS;GRAPHICALMETHODS
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
-
7/27/2019 mechanical narendra
68/119
AMICSOFPLANEMECHANISMS
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
-
7/27/2019 mechanical narendra
69/119
ESIS;GRAPHICALMETHODS
mainstationaryinits phaseSi=O.B1,and
2is referredtoitby meansoftherotation
edin Fig.11-25c.BecauseofthisrotationB2
dAitoAu.TherotocenterRuis thenlocated
ormalbisectorsof 0,OJ1andA1Au,anditwill
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
70/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
71/119
ESIS;GRAPHICALMETHODS
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-
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
72/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
73/119
ESIS;GRAPHICALMETHODS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
74/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
75/119
HESIS;GRAPHICALMETHODS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
76/119
AMICSOFPLANEMECHANISMS
e input-displacementratioR^,,similarly
ent_ A,_ Az,-,...^
entA
-
7/27/2019 mechanical narendra
77/119
HESIS;GRAPHICALMETHODS293
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
78/119
ICSOFPLANEMECHANISMS
ofy
put rotationsknown,therelativepositions
usualmanner.A1andAvcoincidewithA1,
dcoincidentpair,andA1u occupiesthe
ethethreepoint positionshappentobe
smresults.(B1,thesecondcouplerhingein
rightanglesto thelineA1AuAUI.)
eneratedFunctiony=sin's
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
79/119
HESIS;GRAPHICALMETHODS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
80/119
ICSOFPLANEMECHANISMS
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).
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
81/119
ESIS;GRAPHICALMETHODS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
82/119
ICSOFPLANEMECHANISMS
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)
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
83/119
ESIS;GRAPHICALMETHODS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
84/119
AMICSOFPLANEMECHANISMS
tudesBasedon(Ri=y/2
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
85/119
ESIS;GRAPHICALMETHODS
ationincrossedandopen mechanisms,makesit
moresuited totheformerarrangement.
dontwoprecisionpointscumderivatives,
rily,correspondingtotheextremesofthe
ativesatanumberofselectedvaluesof the
esolutionproceedsasfollows:
dfc^andthe linkratioq/pareassumed.
respondingtothe prescribedorselectedn
culatedandmarkedoffonp[Eq.(10-14)]:
Og>O,
chosenarbitrarilyonthecrankcircle.
-
7/27/2019 mechanical narendra
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
87/119
ESIS;GRAPHICALMETHODS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
88/119
ICSOFPLANEMECHANISMS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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,
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
90/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
91/119
ESIS;GRAPHICALMETHODS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
92/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
93/119
ESIS;GRAPHICALMETHODS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
94/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
95/119
HESIS;GRAPHICALMETHODS311
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,
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
96/119
AMICSOFPLANEMECHANISMS
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
97/119
ESIS;GRAPHICALMETHODS313
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
98/119
ICSOFPLANEMECHANISMS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
99/119
ESIS;GRAPHICALMETHODS
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
-
7/27/2019 mechanical narendra
100/119
ICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
101/119
ESIS;GRAPHICALMETHODS
ly provethat0
-
7/27/2019 mechanical narendra
102/119
AMICSOFPLANEMECHANISMS
otheconclusionthatthefollowingcombi-
nismsarepossible:
hanismsandonedouble-rockermechanism
isms
echanisms
aybeappliedalsotoslider-crankmechanisms.
s,whilenotingthatO,is atinfinity,itwill be
natemechanismsvanishes.Thusthereexist
hanismswhichwillproducethesamecoupler
nsionofthetheoremisoflittle practical
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
104/119
ICSOFPLANEMECHANISMS
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^=/(
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
106/119
AMICSOFPLANEMECHANISMS
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-
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
108/119
AMICSOFPLANEMECHANISMS
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.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
109/119
METHODS
3variables,viz.:
of thedrivinglink,itsangularvelocityW3,
a,
thecoupler,itsangularvelocity,,and
p
tions,problemsofsynthesisinwhichseven
cribedmaybesolvedeither directlyorbythe
ch.
efollowingdesignparametersarespecified:
onwanda.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
110/119
AMICSOFPLANEMECHANISMS
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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
-
7/27/2019 mechanical narendra
112/119
esultsforNumericalExample2
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_
use#pd-google
-
7/27/2019 mechanical narendra
114/119
AMICSOFPLANEMECHANISMS
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-'
+
-
7/27/2019 mechanical narendra
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
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
116/119
AMICSOFPLANEMECHANISMS
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:
.
.
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
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:
PublicDomain,Google-digitized
/http://www.h
athitrust.org/access_use#pd-google
-
7/27/2019 mechanical narendra
118/119
AMICSOFPLANEMECHANISMS
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 (
-
7/27/2019 mechanical narendra
119/119
METHODS
hevelocitiesVAandVBin thedirectionAB
n Fig.12-7,