Coordination of coupler-point positions and crank rotations in connection with Roberts' configuration

Coordination of coupler-point positions and crank rotations in

connection with Roberts' configuration

Citation for published version (APA):

Dijksman, E. A. (1969). Coordination of coupler-point positions and crank rotations in connection with Roberts'

configuration. Journal of Engineering for Industry : Transactions of the ASME, 91(feb.), 55-65.

Document status and date:

Published: 01/01/1969

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E. A. DIJKSMAN

Principol Reseorch Officer, Technologicol University, Eindhoven, The Netherlond.

Coordination of

Coup Ier

·Point Positions

and Crank Rotations in Connection With

Roberts' Configuration

In tbis paper It geomelrica/ sa/uI ion is given la the problem of finding four-bar linkages having 5 given couPler-poinl pasitiollS coordinated with 4 given crank angles. It ma)' also be possible to fil/d solulions if one git'en couPler-poinl position together wilh ti given crank Img/e, correspollding to this position, is replaced by Wo given link lenglhs. It lias been found Ihlls fllr Iblll four-bar linkllges, slltisfying the staled èonditions, lire easy la oblaiu if Ibe problem is rell/ted 10 Roberls' law. To get a view of tlle possibilities, Ihe degmeraliolls of co/(/lIlle rircle-poilll curt'es and cognate center-point cun'es, lillked to mch olher by Ibe configllration (CR) of Roberts, are intlestigated. As an áample at tbe end of tbe paper 1I straigbt-/ine mechanism has been designed so Ibal the coupier point moves a/olll: IMs line witb fin approximately uniform tle/ocily.

Introduction

A

~'ItI';(j(TI';N·I'l.r

O('(\[Il'l'illl( prohll'llI in

illdu~(I'illl

1I11"'halli~",i()1I i~ hllVill1( (l1l1'1" of 1I\e(.(uuli"IIIS I'e:wh (·el·t.lIill (lo~i­ I iOIl" 111 "Pl'('ili('" t.iIllPs. 1':xllllliulIl iou of I hilI prohlern Iplld" 10

whnl. i" kllown I~~ 1111' " .... ylll hl'si" of IIlPelllLlli .... ms... 111 t.he

pl'Oh-ICI1I .... Iul,·" hl~I'I', we will ('oulint· olli'selvl's 10 Ihe use of plau:!1' IIl1'l'hallislIIs, wit.h liS few ("Jllil/wtilll( pail' .... a" p;",sihle, T Ill'

re-fol'(', if tlnl,v ow' poinl of t he lIIedulIlisllI to he designed mllsl n""'h l(ivl'lI !loHit.itlu" al kilOWIl vllllleH of I illW, t.his llIeehlLnisllI "houi.! IlI' a fonl'-hltl' linkal(e, Iiere t.he I iml' illC'l'l'lllelit. is J'e(ll'C-senll'd hy t.he eJ'lIl1k I'olatiolls of olie of Iwo ('mllk~, whieh ILl'e rotllt illl( lol' oReilllll.ill/l:) ILI'O 11 I 11 I Ii fixe" pivot. The poillt whose

11101 iUII is 10 hl' spel'ilipd will he IJLkl~1I a.., I he I'ollplel' puilll. E,

whidl is " fixed !'oillt. 'lil I he ('ollpll'l' pillue lnoving wilh the ('Ollpll'l' of I he four-hlll' lillklLl(!!. This (,ollplel' poi"t. R move,.; nlollJ,!; n ('Clllpl!!1' (~III'Ve, I/:Clilll( I hl'olll(h It IlIlmbl'r of l(iVl'l1 pt)~itious, This "lIml,1'1' of po~ilio"s is limiled ltS follow~:

Thp av"ilable dl'l(l'ee" of fl'I'odolll ill tlll~ de,.;il(lI is nim', ~pe­

('iti(':lll,\': Two 1'01' lilt' po,.;il iOIl of ("lI'h ('1'1111'1' of pivot Oll fmlllp,

0111' fol' 11,,· Ipul(t hof ('Iwh 1lI0Villl( lillk or tilt' lillklLl(e, ILlld olie fol' 1·/l.l'h Ilpl'isillJl; Hidp of IllP ("lIlpll'I' I I'i 11.11 I(le of whi('h t,he eOllpl!'1' is t 1U' hZl:·l(~. Ii'urtlwl'lnlll'(', sÎIH'e lL ~illJl:lt\ ('oIHlil iOH i:--i lW(~nMSILI'.V 1,0

'L"'HIII',' 'h:tl, 11 l(iv"11 ptlilll lil'''; 011 Ule ('0111''''1' "'IrvI', rol' ;, I(iv('"

püilJts (Hl, fo),!, J~';IJ 1i'-tJ Hnd J~'h) t,hiK HIPJLllri fivl' (·onditions. IJ('lu'e

olll~' ,l (,l'lIl1k tllIl(lI'''; IIIIL,\' I,,· speeilied, The,.;e nlll(le:; l'ol'l'espolld

I" 1111' SlII','essive I inlt's ill I(cl I i 111( fl'olH one (givell) po~itiou of lilt' ('0111'11'1' pOillt. 1.0 1.lw llt'xL

Tlw J.(·t1l11td.l'ieal Hollilioll of I hl' sttlted pl'Ohlmn is tiet! 'iu with Ihe (:olllil(lIl'11l,ioll «('H) or I! oh{'l'1 s, lUl wiU Hoo" be see,,; 8el' I~ig,

1. The (·tlnfiI(IU'at itlll (·oll"ist. ... or three tl~U'creTiI fom'-hlLl' li"kl\ges, ,)f whi('h I he t.hl'(l(· ('ollpll'l' poillt"

/I:

eoilll'ide in l!!teh positiou of

CH' TIWl'efof'(', p:u'h ('ollplt'1' poilll desel'ihes Lhe id~ntiral

,'ollpl('1' (,III've (\lohel't,,' Inw), Ktlldyilll( ('//, ij, (',1\11 be seen t,hnt the lillk .10:1, t.1l!' ('ollplel' plnlle :I'C' E -whil'll moves wit.h 1,lw

,'ollpll'l' .1

'e'·

'1\1111 l.he liuk ('oC' nll havc Ilw ,.;ame IIlIgleM of I'Otlll iOl\. ThiH follows fl'OIll I he flwt. Ihnt. al uny time A' R is plu'allpl to A,,:1 IIlHIILIso Ihal, ("R iii plLmlielto CoC",

COlllrihlll ... rI hy t"ho De"i~1I \';ngilleel·illl/:.Divi.ion !lntl \H'eHlmteo tlt the ,\lp"h""iHtIlH ('ollfpl'('IWA,' At.lallt.lt, <l .... O"toh"t' 6-9, 1968, of

THF: A"" .. teAN SO"IF:1'Y OIo' J\l\o:CHANICAI. ~;NOlNEElt8, ManU8cdpt.

,.e,,~·i\'l'd at. ASM r,; lIA/U\qlulI'tet's, June 18, 1968. Paper No. 1l8-;\'Iedl~ Hl,

Journalol Engineering lor Industry

Fig. 1

lf Uw fOIlI' are lenl(ths, rnellsllred IIlong the ('onpier C'ln've be-l,wel'lI I Iw(ive kuo"'11 )lositioll~ of t.1l!' eOllpler point R, SIIl'I'I's~ivl'l,v l'OI'l'espolld wilh 11'11' l(ivIJll ('I'lulk I'olltlioll~ ot' tlHl I'I'a"k .1",1 (Ol' equlIlI,v wil h Ilwse or (he ('mak C"O"), t.hen the eOl'l'eH)londing

allgle~ of I'olnlion of the eouplel' trillllgie A 'C'E are known luo, Th!'"e anglt'H combined with the five known po~itkllls of coupl!'r puin!. E, leau to five known posit.ions of the coupier plane represen! ed by ~_I'G' E. Fom positions of 11 moving plane are

already sllffieiell I t.o fix the Ho-ealled centcr-point cw1Je of these )lositions, ~[oreover, for eut'h set of four positiolls, t.hree cellter-point lillrvps lire t1efined hy C/I' These 1I1'e caUed the 8-clll've ('orrespondillg 10 the foUl' PI)."lltlllS of ~ .t'C'E, the P-clll've for Ihe fOllr )losil.ions (jf ~ A,BE !lnd t.he 7'-eurve 1'01'

I,he foUl' )losil.ioilS of ~ U"C' E; see Fig, 2,

Assoeiated wil,h any five given positions of t.he eOllpler plane defined by ~ .-1'G' E, t.here are live center-point curves which all interseet in either 0, 2, or 4 relll center points. These are t.he so-caUeti Burmester points. The fixed pivots Ao nnd Co must be chosen at. olie of t.hllSe (Bnrme81;er) center poinls, This elUl be done in 0, 1, Ol' in 6 ways, depending on the llllmber of real

DUI'ffiester point.s.

Fot· any one sct. of renl cent.er point8 Ao I\nLl Co, it. is I\lways

possible to liud the cOJ'Ml!!ponding elrde points AI' and Cl'.

Now one fOUl'-bar linkago of CR is fully known. The other two tour-bil I' Jinkages of CR are then the Bolutions to our previously

state<! problem. Thereforc the number of IIOlutions is always even aml is not more than 2,6 == 12. which is thc CIIIle if five 8-I!Urves illtersect in four real (Burmestel') center points. lf these live 8-<ml'ves have only two rcal points in common, whieh are l,hclI t.ht' ollly I.wo renl <lentel' points of BurmMt!ll', UIl!

totalllum-bel'

or

tlulUUullN i>! reduced 1.0 2, If all Burmester points are complex, thera il> no real solution to the problem.Therefora 0, 2, or 12 !!olu1.ions exist.

Tba formulat.ioll and elaboraiion or' thc pl'oblem are givell next Îu lw ier.

Coordinating 5 Coupler·Poinl Posltioos

Wllh 4 Crank Angles

[1]

1

I'robillm. Design a Cour-bar Iinkage, given live pOIlitions of thc coupier point E" E2, Ea, E" and E, and the oorresponding crank

allgll~

4:

AIA~A2

=

<{J'2,

4:

AIA~3 - 11'11,

4:

AIA~ • .. 11'1., and

4:

AIAolh

=

ip,~ hetween thet>e positioll!!. SoMIon. Ree Fig. 3,

i Det,ermille thc pOllitioll1! o( the 10 poles (thet>e IU'tl the vir-t,ual conters of rotal ion for live pUoIitiolls of thc plaue): 812• 8');J,

S34, 8", 8 .. , Sa., S~., S'2, 8"", allu 861 with thc cOllditiollS: li;jE, ""

l;ijif:j auu

4:

E,8,jEj ~ <{Jij == ipik - <{Jjk if i F j F k F i and

i,j, and k = 1,2,:3,4, Ol' 5,

2 Draw lhe S·"III've lI't'b belungillg to the !\()uplel' pointR E" Ei" E" allel E. IIIHI t .. the three I\nglcs 1"1'1, 11'14, /tlld 11',. and gOÎlIg

thmllgh Ul(\ ti \lolt·~ S", B,., Su, Su, 1'1", nml SM wit.h Uit' holp (lf

olie uf thtl thrce oppotjite.:pule <lUlIdrilnt.m'al .. S"S •• ,-;.J;'I, SI28~.-SuS", alltl8,~"S .. S .. , .

3 Draw !lex!, thc S-eurve HlIl!4& belollginl( t,t) til(' (0111'

coupler-1 Numbe1'8 in brnckot~ deahcnute lteferellc68 at end of pnper,

i I

:'

\

.

)

point positiolls

E •• Ea.

1$" and

E.

and to the three angles 1P23, 11'24,

and 11'11 and going through the 6 pnles

&a,

8",

Ste,

8$, 8at• alld

8.,. with tbe help of oue of the three opposite-pole quadrilateraIs

8",&.8 •

.843, 8uSwS •• 838,

and 8~.tS24S..,.

4 Choose Ao as one of the real intersectiot1l:l of the two

8-curves whieh do not coineidc with

Ste.

811, and &,. (The lIumber

Fil, 2

I

" ,.' I ,

...

"

...

,

56

I

FEBRUARY 1969 i

,

....

i .,

...

, .... ' 1 _ l 1 l i ... _ _ ' 4'''''''-''-''''''' • ••• , •• # .,>< Fig. 3

Traosactlens of lb. AIME

_{. ,}

/ \

""\

\

/

'''.4

of s\tch ihtt>rHeclious is /1.1, 1Il0;!l 4, 10 wit: 3 X 3 millus the two (illlllgÎlIlLry) i~ol.l'Opic point.,; t,hrough which both curves pa.~s,

U!ulmillllH lh(· t,hl'ce lIlcnt,iollcd point>'! 8", 81.6, alld 8 ... )

r.

J)t'It>!'mÎne point, A,' wit.h lil!' l~ondÎtion,,: ~ .108",1,'

=

'P"/'!. nnd

1.

Ao8 ... 4,' = 'P'''/'2 lUId verify the correetness of thc

e(IIIn.lioll~:

1.

Ao8,.A,'

=

'P ..

12

/l.nd ~ AO<"!16/tl '

=

'PII/2. 1\ Takt- point C.-·--whieh is n vertex of t.he frn.me trinngle

A.H"C,,-II" 0111' of I.he renminilIg ÎIlt.el'l'leetioll8 of the two S-Cllrves.

I·'m l.he (,hoie .. of Uo, 1\1. 1I\(l.~1,

a

real point<'! lire avaihtble.

7 !>ewrmint' pnint. C,' with t.he f'.ondit.iolls: ~ C081~C.' ~

'P • ./'J nnd

1.

('.8,,,(',' = 'P,ll'J aml vel'ify the (\nI"fed,lles~ of t.he

('(!Il1tt iOll":

1.

(.'"""',,C,' 'P"

/:1.

1\11(1

1.

(,'.8",(\' 'P6,/2.

X Dmw Uw pllmllelogmm.1 o!I,I'J,Al'.

!I !':slllhli",h pivot. puinl H. on frame with Lhe conditioll f::,. ,1,'/1;,(;,' , ... f::,.

,1.B.<:0.

10 J)1'h'I'milie poillt. RI WÎUI I,he conditioll f::,. Al' E,C.' ,...., f::,. :1

"1,/<",.

II Thc fom-hllr linku,ge Aought, rOl' is

0

AIl/hB,B.; I.he cor-rt\SllHlltliull f'ollplcr \J'i,uIKle ÎI'! f::,. :lIR,EI.

All HII('l'IllLl ive ,.olul iOIl of I hi" pmblem wilt immt'diat.ely he ["Uilt! if we- go on wil h:

I:! Dmw lhe ptU'l4l1t>logmm O.C/E.C,· /llld the paraUelogmm

n"IJ,/I},H,".

1:\ 'I'he fOlIl'-hlLl'lillklt!(e (j.(,," IJl'

IJ",

togeLher wit.h the eoupler Irinllgl" (',"U," R., UIIlI! il'! Iin IIh.enl!tlive Holut.ioll of the problcm.

Ih~I'(!hy 1 hl' lIH1vililollink C.CI • al!'!n goes t.lJI'(mgh t.he given allgles.

Thl' I \~" (·"l.t;lm\(' "olul,ioli" 1\.1'1' PIIl'I of (1/1' '

Clrcle·Point Curves

'1' .. (.Iwh of lil{! t·hl'tlt) nmtl,l'l'-poilll. 1.ItI'Vo.<, linked with the four pOHitioli" of C /I, Il (IOlTe.~)I01Hlil1g CÎ1'I:/e-point curve ean be indi-(·I\l.e\!. 111 fitis way I,he 81_{-clIIl've belulIgB to the S-curve, the}

pl-I'urv(\ 1.0 tht' 1'-eurve anti t.hc '1'I-eurve to the T-curve of (Jli'

Th!> I.hl'('\' !'ÎI'l'Ie-point curves .'j', Pi, and T' arti related through Cu; H(~(\ jj"ig. 4.

111 Ilw IIl'xl Pl'ohll'nl art allp!iml ion rnllking UR", of the 8 1-curve

will hl' "hnwlI.

Problem. Design fi four-hul' linlmge wiUI 1\ given crank lellgth

A./'

1lII,I 11 givlln frame lellglh .. f~B~, if 4 positiolUl E" E2. Ea, and

Pi, of tltl' ('"upie!' puint /l.nd III~o the correspondillg crank an,les

Journalof Engineering lor Industry

~ A,AoA, ... 'PIl, ~ A,AoAa ... 'Pl', and ~ A,AoA • ... 'PI< be-tween these positions are given.

Solutloft. See Fig. 5.

I Determine the positiollil of the 6 poles 812, Su, Sa., 841 • 8",

alld 8 .. with the condit.ions: S;jE,'" 8i;E; and ~ E.8ijE; ...

'Pij'" 'Pil, 'Pik if i ;1Ii! j ;1Ii! k ;1Ii! i and i,j,k ... 1,2,3, or 4.

\? Determille the image point 8>:)1 of S'3 with regard to the Rt.raight. Iille S,,8'3.

3 Determine the image point 834 • of 8 .. wit.h regard to the

Htraight line 8'88",

4 Determine t,he imnge point S2.1 _{of S,. with regal'd to the}

Htl'aighl, lino 8 • .8, •.

ti Draw Ihe 8'-elll've through the six poles Su, 8 .. ', 8 .. 1,

8 .. , 8", aud S .. • wit.h thc help of one of the three opposite-pole quadrilateraIs 81.8>:)IS .. 18." S,.8,.'8 .. '8", and SI,S".'S'U1841 ,

6 Take point

A,'

in one of the intersectiolIs of ,a circle, with center E, and radius EIA!' "" Ao:!,,, with the SI-curve.

7 Detel'mÎne point A. with the conditions: ~ Ao8aA,' ... 'P2,/'J all d ~ Ao8l.A I'

=

<Pad2 and verify the correctness of t he eqllatillll ~ .408".1 I ' = 'P .. /2.

8 Draw parallelogram AoA,E.AI"

I) Det.ermille the posiliOll8 i12, Aa, and A. of point A with tbe eonditiolls: ~ A,A.A2 ... 'P12, ~ A,Ao:l, "" 'P18, and ~ A,AoA. 'Pu.

10 Determiue the 6 poles 1'''' P .. , Pa., P41 , 1'", and 1' ..

he-longing to thc 4 pOllitiollR AlE" A,E;, AaEs. and A.E. of the eouplet' pln.lte A UE,

11 J)raw the P-f!HI'Ve througb the 6 polcs P" with the help of OIlO of t.ho three opp,>'!ito-pole qlladrihLterah! p,"P'l3I', .• P .. , 1'121' •• 1'.,1'", and P,.PUP,.P".

12 Take point B. M olie of the illterseeLions of a circle, with .

center A. and radi~ A.B., with the P-curve.

13 The sllcoossive positions BOl ( == B.), Bo" Boa, and Bo. of point Bo with regard to the coupier plane A,E, ean be found with the equalit.ies: A AIEIBoo :::: A A.E,Bo; f::,. .1,E1B03

::e

A AaE.Bo; and A AIEIB ..

::e

f::,. A~E2B •.

14 Det,el'mine the center point BI of a circle going throllgh

poiut.~ BOl,

B."

fioo, and Ilcu.

15 The quadrilateral (AoAIB,B.) with the coupier triangle

A,B1EI is a Bolution to the stated problem.

\

\'"

\

\

Fig. 5

Calnale Degeneralians al Center-Point Curves and

Cirtle·Paint Curves by Means af C

n

Proposition 1. '1'" (lll("h [Joint, X Hf tho ('lH I piel' I,I'ÎI/.Ilglo A BE of ('/I, 1111' {'O!!;lIlt\(, p"jllt~ X' !md X' of Uw HlIe(l(~siv(l eouplel' t,l'i-:tll!!;I"," ;I'/~(''' ttn!! fo;n"(tN fil'(l a!lded, whieh LI1\1'e similal' oOllplel'

"UI'V"~ ,,~ the om> Inteed hy point X of Cu. The (lognal,iOIl of point,; X,X', aml X' i~ defilled hy 1110 sirnilfil'Îl,ioH: D. .tBX ~ D. A'X'C' ,~, D. X'U'f"; !lee Fig.!.

I'{)J' tlw slik,· "f IlI'ieflle>!ll, Ih(\ pro"f "f IlI'OliH"it,joll L Willllot he

giv,'n ht'1"Il 141; "'(11t1'1~', Iww(wel', X = foJ yields I,he !lqllltlities

.Y' = X* = /fJ.

Propotoitlon 2, lf D.

:I,U,S, ,-"

D.

,1,'.\','0,'

~ ~

x,'U,'e,'

lIlHI il' .\ " is lt puilll of 1,Iw B'-em'vl', I,hen X I is a !)oillt. of pi and

.\' I ' is a poll! j, uI' Tl IUld HO 011 ln a ('yoHo way,

Proof, 011 tt""oIIHL of pl'lI[loHiUon 1 the p"illl.l'! X,X' aml X' I 1'U.('e ",illlilnl' ('IIlIpll'l' ('IU'veS, '1'hul'!, for Lhe pOKitiulII'I 1,2,a, Iwd ,\ of ('/I, th,' ~ll\lihll·iti(!I'I [J .\,X"X,X, ,"

IJ

XI/X/X.'X.' ...,

o .\'

,'S,"X,,"X ," 111'(\ vItIid. If 0I1n

or

t,IU'Hfl 'luncll'ilnl~I'lll:! is

!lil ill"t'I'ih,>,1 " .. 1)'101:011, i,lwu S .. !U'(' t.hl\r !~II. 1:'\0 if, fOl' eXllmplp,

X,'

iH 1\ ,'ir"I,' Iloint, of D.

A,'e,'R"

UWII

x,

ill Il llÎ!'de point of D. .I,Il,H, ,,"d S,' is It dr"le puilIL of D. foJ,U,"C,'. Thus the IJl'Ilposil ion applieH.

,ft'l'Om prol'""it.ion :! it ('lUI be soon, that C,' of S' I\nd B, of pi

Itl'(\ ,'ogllute "oiub'! in Llm jll~t. nwnt.ioued wny, 'l'hi,.. j .. aillo the !':\s() wii hA, .. I' 1'1 nut!

C,'

of 1" llud, fill!llly, wit.h lt,' of TI and ,I,' of S'.

PropositIon 3. H X,.\", !tud X' of Ulll l'e;<!l,ed.ive couplel' t.ri-Ilngl!'R il Ufo:, .1' Be' nnd EB"C' of C_H !U',1 (.ogllld,e points hy wlty of UUl IIÎmihu'itim, D. .iJlX _ D. A'X,C" _. t:. X "JI"C", t.hen (·V!,,·,\' point. X of t:. ,1 BE Cnll b(1 ubLainod (\'Om X' by a

olle-<lil\! mapping, which is n prodl\ct, of a l'!imilal'Ît)'~trttll>lfol'mation,

lUi inverHion on lt elfde, t.hrollgh H wiLh (:tintel' A, and a trans-fOl'llllit ion, whieh em'!'ospollds tu a roLl\tioll of 180 deg arnmt ('fIlIpier :1/1; IlIHI HO 011 in It eyll\ill way.

58 ;'

fEIIRUARY 1969

Fig. 6

Proof. Tilc ~imilarity tl'am!fol'lnatioll will be chosen in sueh a way, t.hat. il' goes to A and C' to B, Then each point X / of

t:. A

'c'

Jij wi\l he t.ransformed illto a point, Y by turning ahont

1\ ehwIIII puin!. anti a geometrit:al muJt,iplieation throllgh a point (11'pendilljt 1111 I,he llh(lIIeU poin\" CUlIsequelltly D. i1 'X'C' ... D. ,l

r

Il. AM also D. A 'X'C' ~ t:. ABS, Lhe relation D...Il' B ...,..,

D. A /lX applies. 'rhns the poill!.,';! X aud Y are related by the eqllnt.IIHI/'I.TT' ';L\'

=

.Ui"

alld

4:

BA Y ,.. -

4:

BAX; Ree Fig. 6. The Ihst, equatJon repl'esents t,he just meutioned invel'sion 011

Ihe eÏl'('le, I.hl'ough B with center A. From the second equation we aee that t he poillts X and l' are opposlllg points with I'e!lpect

1.0 Ilouplcr A Tl.

Fm' the t.rallsformation of X t.o X* a similar mapping OCelll1il in which B' is t.he ~'enter of the iuversioll-circle through C*. Simi-lal'ly the tl'allsl'm'matioll of X' iuto X' corresponds to a mapping in whillh (." il! the center of the illvel'l\ion-cirele throllgh A', Thel'eful'e Ihe pl'opositiun holds gooIt

Dy maan!! of pl'oposition 3 it is possible to oreate cognate curves: I']very three loci of cognate points X,X', alid X"i are cognate CUl'Ves. It may be clear that generaUy cognate C\lrves are not similar, since the shape of a curve is changed by the in-vel'Hiol\ on a üÎrl.le,

}<'rom propo.~it iOlls 2 and 3 the n\;lxt proposition occurs

immedi-alHly,

P,opolilloll 4, The 8·-curve of CR is similar to a curve, whieh is

the illvcl'IIÎon 011 a circle, throlll/:h Band with center A, of an-ot,h(,r milva, which, in turn, is opposed to the pl-curve with reilp(lI't lu cOllplel' A,B,

PropolHIOII 5. Tha corresponding poles of the SI,PI, alld

TI-l'urveR are eogml.te points u defined by the similaritiea in propoili-'-iou 1.

Hn it. IIIIl~t be flroved Lhat

t:.. J ,'St:(Y '" L::,. A/BIPtt '"'" L::,. Tt2Bt'Ct'}

U "t'S13(lt' '" L::,. A tBIPt3 "'" L::,. T.aB.'CI '

2. J,'S"C.' '" L::,. AtJl,PII ' " L::,. TuR1'Cl'

mul al"o Ihat

è-. Á,'S,atC.' """,L::,. AIB, tP23 I "'" L::,. "utRI'CI'}

[:" :1 ,'S,,'C,' '"'" L::,. A\B1P24 • ""'" L::,. Tt,IBI'CI '

,':. .1,'s.,·e.' "'"

L::,. A,/.I.l' .. 1 "'" L::,. TulR1'CI '

(1)

(2)

Proof, If.\"'

=

8., Uum X' is IL double point of thc I'ollplel'

"III'V(' t,rnl'I,,1 by X', beeallHe X.' "'"

X,' "'"

812 cOl'respondiug t,o

Ihe Jlo~iLi"IL~ .-h'K.'C.' alld AI'X,'C,', }<'rom propOi!itioli 1 it ioHuw:'! theu t1mt POilll, X is abio a double point of the couplel'

(:urve tmcI'd b .... X and, cOllsequent.ly,

XI

== X, '"

Plf ,

There-fore, è-. :!t'S,.CI ' '"'" L::,. AIR/PI2 and analol/:ously each similarity

or sy~tlmt (Il.

Th!' polei'! s",',8 .. 1,8.,I, alld P'.lIl1,PHI,P .. ', aud a,lso T2l!1,T2il,

1' .. • IU'!' ohill.ined f\'Om t,he I'CIlpeeUve poles

&,

82<, 8 ... , and Po, P .. , /'34, Illl(/ 1'~., T .. , 1'" hy brillgillg t,hem back from tha

indi-CIII.('d JlI~~iti()m' 10 po»itioll I of CR' Consequently,

A

.1I'8=sICl'

"'" è-. :1/8 .. (:.' ~ L::,. A2B.P~'1 ""'" è-. AIR.P:lIl· and a.lso L::,. A.JlIPI.1 '- L:,. .-I,U.I'", ""'" è-. 7' •• B1'Ct • ' " L::,. T:lIl·R\'OI' alld!'lo on. Thus

j)l'up",;it,ioll,j h<lld'! good.

lemma A. Th", illVersiull'oll 1\ (lÎ!'do, Ulrtlllgh IJ and with center :I, t.1'Il1I"rorm!l a dl'(,le, nol; going throllgh A, into allother eirde, '1'111' tlll'l'C ('irdl;ll; have the sallle radicnlll.xis; see I<'ig, 7,

Proof. I';anh l!Ill'Ilight lille through origin A intersectH the

11II-Irall~f()rlllell eirde iu t,hepoillts p, and F.. If these are real

p"illl~, lIl!'y will he tmllsrormed illl~) the poiut.~ FI ' and F.', for

whil'h

'1'11\11'1 I,he (lower of point "I wit.h r(IApe(lt to the h'allsformed

"UI'V" iH (J()f1l-1tl!.ut; LheJ'erOl't1 the tl'l!.ll>lrormoo curve is a eil·cle.

'flm inler'l<ectlou" of the untrallsformoo cirde with the invel'8ion-eil',,)e (.1,

:fR)

remain al, the HlI.rrte plane throughout the iuv61'siou,

~) Ilw t,ranllfol'med eÎl't.le ali<1l paHSIl!! throllgh the same

inter-"Il\lt.im.", mullItt' ('OUlIllOIl chlll'd iR lihll l'I\,ijl:t\1 axis of the three

,'i!'('h~ .. ,

Lemma 8. 1'lte ÎllverNion Ol! !l. ch'cle, t.hrough R I\nd with center A, t.mn",rlll'lllR a eil'(.le t.lu'ough A int.) the l1l.dicMl\xis of'l,he t.wo

dreh~ IUIlI (\ollvel'!!l'ly.

JOlrnal O,

Elglle,rllg lor Industry

\,~"""'"

\

...•...

..

~

...

-

...

-

..

LemmCi C. The invel'8ion on Il. eircle, throu.gh Baud with center A, t.rllJl8forml'l a strai,ht line through A into the same Hne,

Lemma D. The invel'8lon on the circle (A,AB) trllJl8forms 8.11

orthogonal byperbola going through A iuto a circul&r curve of the thil'd order, with a double point at A and with perpendicular tangents at A aud converselYi

see

Fig. 8.

Proof. The invel'8io!ll'l of the two perpendieular asymptotic linea of the hyperbola are two circles perpendicular to each other at point A, The perpendicular tangents to the circles at A

(',oincide wit,h Lhe tangentIJ t<l t.he t.I'tl,nsfonned curve at.4., 80 A,' is a double point of febiH curve ;l.IId t,he tangellts to Ihis clll'Ve at this {fouble point are perpendicular to each other, Eaeh straight line through A iutel'8e~t.~ the transformed curve in three pOmts, whieh are t.he twica-counted point A and . .a third point, . whieh is the transfol'med intel'sectiou point, not coinciding with

A, of the straight Hlle with the hyperbola, Therefore, the tl'allAformed curve is of the third order,

'rhe tangent, to the hypel'bola at the origin A isa.lsll (,he sole

I\sympl,otic dir,at,Uou of the trnnsfol'moo curve, Therefol'e, the ' line at infiniLy intel'8euts thii'l curve at one uympt,otic point and at two conjugatoo complex points, The hyperbola may be seeu

llB the locus of illtel'sect.ion points of corresponding samples out of two penci!!! of lines, l<'UI'thermore, the inversion on the circle (.1,AR) tmnsforms eI'Wh pencil of linea, going through a buic point not coinciding with A, iuto a pencil of circles, going thl'ough A and throngh the inverf,oo basic point of the pencil of

lintll!, Thel'efol'e, the transformoo curve is t,he locus or inter-seetiolls of correspolldillg samples out, of two pencils of circles. The isotmpic points of the plane are illtel'8eCtions of this type. COllseqllelltly, the trall8formoo curve is eirclllal', So the lemma holds I/:ood.

P'opolilloll 6. Ir the 8·-cUl've of CB degeneratea illto an

orthol/:-OIud hypet'bola UlI'<JIIgh at leut 4 poles and illto the line at in-finity, f,hen the P·-curve degeneratea int<l a pole curve with a double point at AI. and the TI-curve into a pole curve with a double point at Cl'; and so on iu acyclic way throughout

OB; 800 Fig. 9,

Proof. Not more than 2 poles of the 81_{-curve lie at 1}

00, whieh

is a part of this curve, The 2 poles are OPPOi!Îte polea, lf not,

mfJre than 2 will he at /"" which is in contradiction with the initia! statement, 80 at leut one opposite-pole quadrilateral

('Aln he found of which the vertices are finite, Suppose this one

is

0

81

08 ..

'8 .. 1841 , Sinoe tlte orthOgOlla! hyperbola through the

fiuite poles is a part of 81, this must be ft, parallelogram [4), As

may he deal' the hyperbola is also going through the ftuite circle points ,{I' and Ol"

From propositions :.I /pld 4 alld lemma D it ean he seen that the PI-curve degeneratell into a circwar cprve through RI with two perpendicular taIllents at A" (CoI\iMlquelltly [4) each of the oppOi!ite-pole quadrilateràls PltP'I!lIPN·P41, PtaP .. tp.,'Pal,

and PaaPnlPulpu will he a qnadrilo.teral circumscrihed about a (lÎl'llle, )

Fig. 9

Under Um I.rll.lIHfol'ulI\lion of point,,; of é::. ,U/& into CO(l;llate jloint" uf f::,. U'C' E, point BI' il'l the een tel' of inversion /lot coÎn· eidillK with Cl' whi(Ih is t.he eogilltte point. of the double point

A I of PI, 1.40 in I,hirl ('Ik!e the tl'allJ!fol'lIIatioll allow8 a douhle

point to I'emaill 1\ double poinL Thul:! ,/" ilol a pole curve through

B,' wit.h I Wo pel'p(mdirular tangent,~ nt, Cl" (Col1l.lCquently

o

7' .. 7' .. , '1'2,'7'" is a 'I"adl'ilatt,eral .:Î1'CllIIlM"rihed l\bllllt a circle.) With I·lai>! the propo"ÏI,ioll is proved.

Proposillon 7. Ir S' of Cn Î~ a pole elH'V!l wit,h a double point

1101, (,oineidilljt wilh one of t!te el1d8 of 'itmplel' .4}C,

'J

then also

1" and 'J'I RI'" pole CIIl'VIlt! wil,h a double point not, coÎlllliding

wit.1a Ol\(' of I.he clldH of I he (iol're._pondillll: ,i(lllfllel'l'; 800 Fig. 10. Proof. 111 Ihi~ (il\M1l 1\ double poiut of S' .loflM 1101. Iluincide with

A I' Ol' (', I j t,herefore ilO double poillt (Ioinddes with a fiuite ceuter of inver"ion. 80 each douhle point transforml'l into ltnother one IInd eaeh of t.he throo L~lgllate circle--point curves has a double point, Thl\!I every linrrlll!llOndillK op~it,e-pole qUll(ll'Îlateral

m

('ir"tm~''''l'ihed IIhOIlI. a (,h'('IIl.

60

I

FeBRUARY 1969

Fig. 10

Propos'tlon 8. lf the circle-poillt curve of any coupier triangle

in position 1 of OR degenerates into a circle and a straight lille

going throllgh the center of thil:! circle, thelI, generally, su do the drcle--poillt curves of the other two coupier triangles Îll posi-tion 1 of Oli' Thereby, those turning-points rotating abollt the same fixed pivot of 0 Ii both !ie in pOl!lition 1 either on the CÎrclllar part or 011 the straight-line part of the belollging cil'cle--poinl; curve. Ir the three circle--point curves COlltaÎ1l!1I0 couplet' of OR'

only OM of the three circulatr part:; passes through both elld points of the corresponding coupIer; sec Fig. U,

Proo'. Suppose _{8 '} degenerates into a eircle thl'Ough A,' aud

C,'

alld a ~t,raight line. Then, IInder pmpositiolls 2 and 4 alld lemma

B.

I,he pl.curve degeneratIlt! Înto a circle through A, and a straight line through B, and through the center of this circle. AlldJ under propositions 2 and 4 and lemmas A and C. the TL

curve degenerates into ft, straight Hne through!

B,'

a.nd a eircJe

through Ol" .

Now sup\lOtle 8' cOlll!i"t.8 of ft ,'irde through .4t ' alld a straight

Transacti ••• , tbe ASME

/

Fil. 11

T-_ T,.

/

Jour ..

1

of

Enll~eerinl

for IndustrY

liue through Cl', then PI consists of a circ\e through AI and BI alld a straight Hne, Thc Tl-curve then consists of a ei rele through

BI' and a straight \ine through Cl"

If SI breaks up in to a straight Hue through

ot

I ' al\( I a circle t,hrough C/. t.hen pi break" UI> int,o 1\ eirde t,hrol1gh U, anti a

straight Hne through AI, whereas Tl breaks up Înto a cirde through BI' Md through Cl' and a straight. line thl'ough thc center of this eÎl'cle.

If finally coupier A I'CI' is part of SI, then each coupier of C f/

ÎH part of the cOl'respondillg Ilil'cle-point, curve.

All

J,lI'IM,fll

nro

based 011 the lemmas Md on propositions 2 alld 4. In Fig. 11

one of the just mentioned cases is shown.

PropolItIon 9. If the center-point curve of ally coupier of C f/

degeneratel! into an orthogonal hyperbola, through the 6 poles and through both fixed pivots of the corresponding four-bar linkage, and in10 the !ina at infinity, then this is also the case with the centel'-point curves of tbc other two coupier triangles of C 1/; see Fig. 12.

Proof. Suppose the S-curve degenerates iuto an orthogollal hyperbola, through .10 anti Co anp through the 6 poles, and into the line at infinity.

T\le

SI-curve, which is the circle-point curve conllected' with the corresponding cellter-point curve, wi\l then be broken up inw thc straight !ine AI'CI' alld into a circle through the poles SII,S .. I,S",SU.,S421_{, and SUl. (In this}

regard, each point of the line at illfinity is cOllnected with a point of this circle. Therefore AI' Md Cl', which are COllnected with the cOIl-esponding poiut.s A. and Co of the hyperbola, !ie on the straight !ine which is part of SI.)

Under proposition 8, the pl-curve degenerates Î1lto the straight, line A d~1 and into a (,iI'CIe. Poin t A I of thi8 line is the een ter of

illversioll. Thu8, with the transformation of points of D. AI'C,'E

11110 points of D. A IBIE, t.he eirc1e transforms into allother circle

2 The oonnection between deaenerations of the eircle-point curve

(l,nd the corresponding center-point curve is known [2].

, /

, /

Fig. 12

Fig. 13

1111<1111(' ~tnû!1;hl~ Ene i1l1,(j l\Ilol,her line. Therefore pi degenerates iulo Ihe Itue .'I'/)I Rnd inf,o a eirde through the 6 polas PI2,PJ8 ,-P'l'l',P2.',1',,1, /Iud 1'... Thc P-eurve, whieh iK connected with thc l"-c'llrve. will then dcgenemte into a hyperbola t,hrollgh Aa

and lI. Illld into Ilw line af, infinity. The proof may be continl\ed

in a ('~'di(' wa~' I hI'OUII;hollt, (!//; UlOrefore the propusit,iol) hnlds

gO()(1.

Pl'Opollflon 10. 11' I,h., B-('IlI'Vtl of C_H j\"n~i"I.~ or 11 I!jtraight Iille

t hWlljI;h :1., /11111 .), pol."" HwI

(Ir

a dl'j'lc (Iu'uugh ('0 I\lId :& ptlle,~,

t hPII I' hreak. ... up illt~1 /lil ort,hpgonal hyperbola thl'uugh A 0,

Uil, til u I 4, po1('1' , lIIul int,u the Hue at infillity t,hl'uligh 2 polll!4;

'1' breaks up int .. Il. circ1e throllgh Co and :l polet!, alid int" a straight line through Bo and 4 poles, and su on in a oydi(' way

thl'ollghout CR; see Fig.

la.

Proof. Thtl SI"mu'Vll is relal!!.1 with the 8-mll'Ve, SI being a pole curve through Cl' with a double point, for illstanee, at 8$JI =

811. The isogonal transformatioll conneets every point of the straight line through Ao wit,h one point,; thu8 (,he double point cuincides also wit.h Al '. (The cirele-point curve SI ill position 1

cau he obt.ained from the cent.er-point cHI've 8 by

i.sOfICMllran3-!ormatüm with l'egn.rd to the pole triangle SI,SwS"1 ÎII addition to a l.\II'nillll: of lSO deg about I,he!!ide 81,8".) Under pl'9po!!itioll

6 the ]>1-clll'Ve then consists of an orthogonal hyperbola through

A I, Bh and through the vertices of the paraIlelogram

PltP",I-PüIP .. , and of theline at infinity going through PUl'" and Pa"'.

FlII't,hermore the TI-eurve then win be a pole curve through Ol' with a double point at BI' "" T23 1 ... TI.. Sinee

0

PlSP"P ... P31

is aJ..~0 a parallelogram, the P-eurve oonsists of an orthogonal hyperbola through Ao, BIJt and through 4 polas, and

ot

the line at illfinity t,hl'ough P'IO" and PI.'''.

The shape of 7'1 eorresponds to the degeneration of T illto a cil'c\e through O. and 2 polas, and into a straight line through Bo

and 4 polas. (The double point BI" - T'131 _{- TI, is connected}

again with the straight line going through B. and Ta, T24, Ta,

and 7'3")

Pro,.IIIIo" 11. If (,he 8-eul've of C N degellerates illl~) 1\ 8(.l'l\ighl,

lint' t.hruugh 4 pol"" alld into a dl'de thl'Ough both cOI'I'espouding fixed pivots and through 2 poles, then thi" is aL~o the CMe with

I,he P alld with t.he '1'-curve of OR; !lee Fig. 14.

Proof. The shllpe of the S-eurve illdueas a pole cUrve as the SI-eurve going through Al' alld Ct' with a double point, for in-8tanoo, at S'tIJl - Su. Uuder proposition Î the pi-curve will he

1\ polo curve through AI and BI with a double point at P2.1

=

PH. The P-cllrve, which is the centel'-point curve connected with the pl-(llll'Ve, wiIJ then degenel'ate into a straight line throllgh 4 poles and into a circ\e through the fixed centers 11.0 and Bo and through two polas. The shape of the T-curve is derivable

in a similar way,

PropoII'lon 12. If the S-Illlrve of CR consists of a cil'e1e throngh 4 polas and or the line cOlltaining the fixed link and 2 poles, then this is alRo t,he ellSe with the P and with the T-eIll'V1lS of CR;

seeFig.15.

Proof. The Sllpp(~'!ed !llu~pe of the S-cllrve iuduces a dC/I,'ellel'1t-t.iol! ur the SI_jHII'Ve into a drcle t,hrough 4 poles and into a stmight Iinc thwlIgh :ol po16ll. Here t.he straight liue is e'oll-nected with the fixed liuk AoC.; therefure this line is eollinear

with th ... j'ollpler

,h'Cl'.

Under proposition 8 the pI-curve

I

.. !

\\'1.-"1.

FIg. 14

Fig. 15

Im'uk" lip int .. n f'Î1'1'le I.hl·Olll/:h .. poll'M luul iul.o "he eouplt,l' ti ,Jl,

thl'olll/:h 2 Iloh"" Thon Lh" I'-""I've '''"l11o,~I.otl with 1"

dOI(III1-cmh\' inl.o 1\ dr,'I,' \.hl'olll(h 4 pol"", nlul 1111,0 !I. lilltl, eollinclu' wit.h t.he fix!'!! link A.U., thrnnl/:h 2 Iloles, Th .. Hhll.pe of the T-em'Ve

ill derivable in ft ~imihll' way,

Propolltlon 13. If Uw S-('HI've of 0 l/ brel\k.~ up inl.rI a st I'Itil(h\ lill(' thwlIl(h :ol pol"", 'Llld illl .. 11 ('Irele passing t,hn>llgh A. and C. !mt! 4 poll·s, t ht'1l I he P-elll've is a pule 1'1I1've t.hl'ol1l(h A. wit.h a

clollbln puilIl nt. Ilo, alltl the T-"m've is 1\ pnle mll've thl'ough C.

with a dOllhlt, IlOillt. at U.; ILIld ~o on ill lt I'ydic way I.hrnughout

C ,,; spe I"il(, W,

Proof, The ,,!tape of Ihe S-C'III'V" illdw',lS a degellel'lltiol1 of I.he S'-('III'V!' iI,ttl It "irde I.hl'olll(h 4 pule;; !ind inl.., It sl.r!\ight line

I hl'llllp;h :ol polt.,.., 'I'his I'i",'lc, ill 1'01l1lente,1 with Ihe "in'le thl'OlIgh :\. "lid ('. Il.lld, IIII\I'el'm'(', I""':;(l>' I.hl'ollgh .1,' 1\lId Cl'. UIIdei'

IH'Opt,.-il iOIl X lilt' 1"-C'III'Vt' !,IWII IH'ouk.~ up illio !\ !'Îl'ele t.hnmjl;h

.4., and:J poles, and iuto a straight line through BI and 4 poles. Huiler the same proposition the TI-cUI'Ve consist.'l of a eÎl'de th.·oujI;h Cl' and 2 poles, and of a straight. line through Hl" aud '" pollll<. 'fhen t,he 7'-eurvll, (lOnnet·ted with t,he 7T

1..curve, is a

)l(Jle ('lIl've t.hrough (Ju with a douhltl point at, COl' inst,IUlI'.e, 7' .. =

TH. In thi~ connection the straight line tht'ough BI" is eOllnected with the double point and, in particlllar, BI" with

Ho;

therefore

B. = TH "" 7',..

Similat'ly the P-(mrve iR identical with a pole curve thl'uugh AG and wil.h ft double )l(Jillt at Ro = PH - 1'23.

'tOpoaltlon 14. IC the S-curve of C II consista of a circle thtough

Co and 4 poles, and of a straight !ine throllgh AG and 2 poles, thail t he P-<'lll'Ve cOllsista of a eh'ele throllgh Bo and 4 poles, !tud of a Itl.l'aight HUil t.hrough A 0 and 2 poles, Whel'C8.8 I,he T-curve is

a pole cllrve Ihrough B. and C., with ft. double point not at B.

Ol' Co; and st) 011 in aeyclic way throughout C_It; see Fig, 17.

Proof. Tbe shape of the S-curve induces a degeneration of the 81_{""'llI've inln a cil'de through Cl' and 4 poles, IUld into a straight}

lille t,hl'Ough ,1. tand 2 poles. U nder pl'Oposition 8 the pi-curve bl'eakR lip illio a eil'de thl'ough RI and 4 poles, anti into a st.raight line I.hl'Olll(h A I 8.ml :J polel!. Under the same proposition the

~l'1-<'IlI'Ve break~ up into a circle through Cl

*

and Hl" and 2 poles and illto a straight line throllgh 4 poles.

The shltpe of the pl-cllrve induces a degenet'ation of the

P-(·.III'Ve into It circle thrOllgh B. and 4 poles, and into a straight line thl'Ollgh.1, and 2 poles, Furthermore, t.hc TI-curve induces a degellerat.ioll of thc 7'-curve Înto It pole <,urve throllgh Rh !\nd Co with a double ptlillt at. 7"1 = 7'34 1I0t Iyillg al olie of the fixed

pjVllt~,

'ropo&ltlon 15. Ir thll S-cUl've of CR cunsist>! of a circle throllgh

ij polas and of a Iiue collinoor with tha fixed Huk .4.c~ thelI. t he

P~curve is idelltical with a pole elll'Ve thl'OlIgh B. with 8. double point at Ao, and the T-CIII'Ve is identical with a pole curve throllgh Bo with a double point at Cc; anti so 011 in acyclie way thl'Oughout (J n; see Fig, IR.

Proof. To the given 8-eurve helongs a 81_{-I'Ul'Ve consisting of}

an ol'thognllal hyperbola throllgh

A,', (.\',

lllld ti finite pole:>, and

of tbe Iiue at infillity; see also Fig, 9, Under propositioll ti Ihe P'-cul've wiJl be a pole curve thl'ollgh B, with a double point at

A". It followl!j thell that t.he P-curve be('omes a pole curve lhrollgh B. with a double point at .10, Also under propositioll ti

Ihe shape of the PI-curve let\ds 1.0 the T'-curve through

B,·

with a double point Itt C,', It f,,!lows then, thaI. the ']'-curve

hll.~ t.he shape of !\ pole el\rv~ thrnugh Ru with a double point at

C •.

FiV. l '

'''.17

"

..

..,..

fig. 18

A Strailht-Line Meehanism Wlth ApproxlmateJy

Uniform VeloeUy Along Thai Line

lf t he B'-clII've !l8 weU !IS I,he P-curvil degel1erute illto a circle

unrl It ~tmight Ihw, n ~hlt .. p simplificutiulI will fleeur with regard

10 thc d~i!!;u di,t'nsscd iu thc secUoH, "Ch'cle-Point Curves."

A demonst mtion of t.his occurrcnce wiU be given by the next ('xumJll ...

Problem. I )c~i!l:n a four-har lillkage wil.h a givclI crank leugth

A ~fl nlld a givl'l' lengt h of frame A.H~, if 4 oqllidistallt positions

E"B"E., ll,nd E, of the coupl",r point E In'e givell on a straight line: /<kH~ = E,E; '"

N;/tJ:,

MOI'cover, the magnitudes of crank nngl{'''; <): A ,..1,,:1,

=

<): A.t1.Aa ""

1:

A811.A. are also givell.

SoMIon. H,'(' l"ig, W.

.1 1';"lllhlish \.111' !JoRit.ioll of point 8" with the eo.lIditions

S ..

fr:,

= ,'{"B. IUIlt <): EvS"B, ""

1:

A,Ao/1. .. !p,..

2 llott'!'mÎne poillt ,~" wit h j,he nunditiolls ,~,;Ë~

'"'

S;;E~ and

1:

B,S .. !!.', = a(

1:

A ,A

o.t.) '"

3<p".

:{ 1 )mw the "'I"uight, Hlle EIS".

4 T,tk(· Jloinl, .·tl ' at Itll illtel'!«lciioll of thc straight line E,Sa

with a ('i!'!'le wil h center El nnd ratlius

E1A;! ""

AoAI' (The F<1l'aÎghl lillc 1';,8 .. i" part of the 8I..ellrve.)

64 /

FE8RUARY 1969

5 Determine point Aa' with the conditiollS

8

12AI' == Bt~~47

and

1:

A,'8 .. A.' = !P, ••

a

Take

.-t

G at the interseetion point of the perpendicular

billecLors of thc line segments At'A.' and EaEa. (This last olie coincides wit,h the straight line S,.E."', which is part of the S-eurve.)

7 Draw parallelogl'am A.A IEIA

I'.

8 Determine thc positiolls A.,AII, and A4 of point A with the couditiollil

A.Ao

=

A~-i-; '" A.Ao

=

AIA. and!ptt "" ~ AIA.A. ==

~ A.Ao.4., ""

1:

A8A&>1. '" !Pat.

\! F..stablish thc posit.iollS of the poles P12 and P18 in

accord-anee with the pOlSitiOIlS AtE!> A.E., and AaEa of coupIer plane

A BE.

10 Draw I,he eh-cle through Pa and Pla whieh bas thc

per-pelldinular bj,;eetor of E.E-; as a diameter. (The circle as well

AA the perpendimdar billcctor of

E.Ea

are parts of the P-eurve.) 11 Take Bo as olie of the four illtersections of the P-curve with the cirolc with oontel' Ao and radius AoBo.

12 The successive positiolts BOl = &, Be, Bea, and BOl of point Bo wit,h regard to the coupier phl.ne AlE" ean be found with the equalities

t:.

A,EIB .. ~

t:.

A.E.Bo;

t:.

AIE,Bea ~

Ll AaEaBo; and

t:.

A,EIBOI ~

t:.

AtEtBo.

..

o ...

..

0 ..

....

_I!~-FItI.

l'

13 Take point BI

at

thc center of a circle throup a.,B..,Btt,

Md Bo<.

14 Thc four-bar linkage (AoAIBIBo) together with theooupler triangle A,BIEI representlJ a solution to the presented problem. (Tbe tota.! number of 1I01utions is 2.4 - 8.)

JHrnal

0'

Enlineertel 'ar In.lstr,

Explanatlon. On account of the symmetrica.! positions of the

points Et, E., E., and E. with regard to the perpendicular bisector

of

EDE.

and also of the equa.!ness of tlla angles IPI', IPDI. aud tpu,

the 8-cnrve degencrl\tes into a circle through thc poles

B

12,

B18,

St.,

Md BI4. Md into the line BI.BD through

Eo"'.

The 8'·

CUMl'e, belonging to

8,

then consists of a circle through the poles Bil, 818, BUl, Md 8 .. 1 Md of a straight line 8 •

.&.'

through "cirele

point" El.

If Al' is chosen on the line which is part of 81, theu the p.

curve consists of a circle through 4 poles Pil, Pil' P.., and Pu

Md of the straight Une P •• P'l4 going throu,h At lWd Eo"', This

oan be Been from Figs. 15 or 17.

The event of Fig. 15 occurs if Eo is taken on thc line PI4P";

the other avent OCOUfl! if B. is taken on tbc ~ircle whieh is part of the P-curve,

RIferencIS

Ir 1 FreudellJltein, F., and Sandor, G. N., "Syntheeis of Path Generating Mechl\llism!l by Mealll! of 8. Prograauned Diaital

Com-puter," JOUKNAL 011' El'IGIJl1lEKING roK INDU!I'1'.I\T, TRANS. ASME,

Vol. 80, Series B, Ms,y 19391 pp. ~168.

2 Claussell, V.,

"über

die Mittalpunktkurve und ihre Bond_ fAlIe," dJaaerta.Üon, T.H. Brauneohweig, 1964.

3 Rain. K.t "Punk~~und Winker.uoMnungen an Kurbel

und Bohwinge von Gelenk;(erecken.~" MIl8ChWtlbaulBllfrirb BeU. GsrN~n,dl, Vol. 12, No.. 5, 1944,.pp. 2153-256.

4 DijklIInIlIlt E. A., "H" ontweJpen van stangeIuneohaniemen, ot PoI.lJli6f""nMch ~t ~ At V-oI. 22, ?oio. ~25. 1967.

"