Six-bar cognates of Watt's form

Six-bar cognates of Watt's form

Citation for published version (APA):

Dijksman, E. A. (1971). Six-bar cognates of Watt's form. Journal of Engineering for Industry : Transactions of the

ASME, 93(february), 183-190.

Document status and date:

Published: 01/01/1971

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E. A. DIJKSMAN

Six -Bar Cognates of. Watt's F

orm

SenIor Re.earch Office",

TedmoIaglc:al Unl.,enlty,

Eindhoven, the Netherland.

TM paper describes

the

equivalent of Roberl.s'

ww

fo,

WaU-l

medwnisms.

It

is

shuwn that two fI,oatint. links of

the

mechanism

may

be in turn generateil by more than

OM

mechanism of this type .. AnotMr inwistigation rlrl/etlls that tM

WaU-l

six-ba;

(coupler-) curve of deg,ee 14

wiU

be producedby an inftnite nunwer of cog'fl4w of ehe

sameform.

.

IntroductloB

SU'ICE ltoherts UP derived biB theorem onthe triple generation of ooupler-curves of planar four-bar linkages, many authors, including Cayley (2), Schor (3), Meyer zur Capellen (4), and Richard de Jonge (5) have occupied themselves with tbis theorem. Each of them has found another proof of Roberts' theorem. And sinèe then many applications and theoretical oonnectione have been found.

Robert8 derived his theorem geometrically by ths

use

of Syl-vester's "plagiograph" (6). In fact the theorem can he seen BB a special.application of this plagingraph and one may even wonder why Bylvester did not see tbis. The întention of the present paper is

to

find how many BU-bar cognates are to he found, that is

-to

say how many plan ar six-bar mechanisms of the

same

type generate the same BU-bar cu",,, described by ooupler-poînts at-tached

to similar

positioned links in tbe mechanisms •

. Clearly, the

degree

arld

gen.us

of such a

curve

depends on tbe type of mechanism and on the choice of the coupler-plane in wbich tbe coupler-point bas been established. It is also important

to

side-traclt mechanisms with specific dimensions. Otherwille, more generations of the curve than generally acoounted for may be

poaaible. .

To demoll8trate tbiB, we state thaI, there is an

infinite

many-. fOldnees in generating four-bar ooupler-curves with six-bar

mechanisms. TbiB can he done, ror example, by making usa of a special CBBe of the mechanism of Kampe (7) (aIso called foool mechanism of Burme8ter

18,

9) (see Fig.l), or by making useof

a

mechanism wbich consists of two connected plagiographs of Rylvester (see the Figa. 2 ao.d 3). .

Burmester showed thaI, aftil· point F of what is known BB a f~

1 Numbera in braeket8 designate Relerenoes at end ol paper.

Contributed by tbs Design Enaineerina Diviaion and presented at the.MechanismB Conference, Columbus, Ohio. November 2-4. 1970.

ofTe AMIIBlOÄXSooIIlTY Ol' MBOIlAmOÄL ENOINllIIRS. Manuscript reoeived at ASME Headquarters, July '1:1, 1970. Paper No. 7()" Mech-30.

Jouroal

0'

EBIiB .... IBI 'or ladlstry .

Fig. 1 Meehanhm of Kem,.

10.

Fig. 2 Two

_liKt"

platfolrapht of lyIv ....

CUI"IIe' may he oonnected witb each of the links

of

a four-bar

link-&ge AoABBo without a.lrecting the movabllity of the m,ecbanism (see Fig. 1). With eaeh point F on the ~ocal curve, tbe ooupler-curve, described by a coupler-point of tbe ooupler..plane attaehed

to AB,

may he generated by two BÎX-bar mechanisms eaeh

oonsist-ing of two piled-up four-bars. (Both six-bars are then 1,0 be seen as degenerated types of Watt's or of Stephenson's fonn.)

The other approach is bBBed on the possibillty of oonnecting any point F -not neceesarily lying on the focal curve of AoABBo-with each link

oe

the four-bar also viith·no eft"ect on the initia! movement of the mechanism. For any point F tbiB may he done • Each point F

or

this CIll'VS is _na two opposite sides

or

the

lour-bar linkage under tbs lI8Jne a.ngle

or

mon;

'1g.3 T_ connect_ pIagIo ... of SyIv •• t., . WATT-' mecIIanI<om '1g.4 _ '1g.5 -) WATT -2 mee"'ni.m

in two ways. Ea.ch way leads to a mecharusm which consiste of two connected plagiograpbs of Sylveeter (see the Fip. 2 and 3 and also the Appendix 'lor further explanation). The firBt mech-anism, according1y, is shown in Fig. 2 where

!J.AFBo,..., !J.AoFeB.

,..., ••. etc. Fig. 3 shows the second meehanism where

!J.AtFB ,...,

!J.AeF.Bo ,..., ...

etc. And 10 ea.ch mechanism makes it possible to form. two six-bare in order to generate the four-bar coupier CUrve of degree six and genus one. It is thus ~ar that at least 2· ..,1

ways are possible to reproduce a four-bar curve with IÎX-bar mecbuÏsms. (These also are to he regarded as degenerated types of Watt's or of Stephenson's form.)

Other methods of doing sometbing similar are sbown by Ha.rtenberg and Denavit (101 and by Roth (111.

Therefore, we must confine ourselves to the general case only, that is to say, we do not consider six-bar curves identioal to (or degenerated into) four-bar ones.

AB is known, there

are

two forms ofsix..bar linkages with one

degre8 of freedom as shoWli in Fig. 4. And for each form. any

link may he taken as the frame. Bo, from Watt's form we

ob-t.ain wo ditterent types of mechanisms (see Fig. 5). Stephen-son's form. leads to three different types (see Fig. 6). Sinoe we coniine ourse1ves to moving lit;iks of which the points generate

cu'rves

of a higher degree than six, we may rule outthe Watt-2 mechanlsm. In the remaining mechanisms some links of

in-terest Pmy a simila.r part in the mechanism. All in all, tbere-fore, six l~ remain of interest in our investigation. The cognates of a Watt-l, a Stephenson-l, or a Stephenson-2 mech-anism have not been inveetigated IlO far. Only the cognates of a

184 /

,nRUAIn 1971

~

9

~--/

STEPHENSO"-. mKII.ni .... fig. 6

fig. 7 Sovrc ... ecllanllm

Stephenson-3 mechanism have been investigated _ by Risch~

(12, 11). - He found sixl _{six-bar linkages of this type---iIÎX}

. cognala of Stephenson's form-all of

tIiem

~rating

the BmM -six-bar curve described by a point E of the mechanism.

The object of the present paper is to analyze -the possibllities of the Watt-l mechanism. That is to say, we

want

to know how many cognates of tbia type of Watt's form. generate the same CUrve by a couple .... point attached to link

KD

or to

OD,

and af ter tbat also how to design such alternatives.

Tbe TWI 'lid Geleratlln

ef

Ibe MlVln, Plale 5 aId,

Ibe-"1/

a

COlnate"

Fig. 7 shows a randomly chosen WGU-{~. It con-siste of two fou .... bar Iinkages, viz.,

AoABB.

and

BODK,

'luid

of _ tbe rigid triangles

ABK, BoOB,

and

KDE,

the latter being a couple .... triangle attaohed to the floating link KD. The moving

Jinks in the mechàniam are numbered from 1 to 5. The link

KD,

hearing the number 5, bas one point .K moving along a four-bar ... couple .... curve of degree six and génus one. Accordbig to

Prim-raas,

Freudenstein, and Roth (13) binge-pivot D of the li1ech-anism descl'ibes an algtJmJtie CUf'I/6

ol

t:mler 14 Gnd lIaua 6.They

named it the W au-1 ~ c:unle. Point E attached to t.he float-ing link

KD

will descrihe what they termed a ~ ~ CUf'I/6 of t:mler 18 and IlImUB 6. Clea.rly, the end-points K and D of the ftoating link

KD

describe curves, to he &een as depnerated

six-bar couple .... curves. .

• Aotually, RJsohen mentloned the nlllX/.her mailt, but two of them are eight-ba.rs and therel'ore not copatell with -the same number ofJinka.

. TransacUaas If Ibe ASME

j,.., .

. . ~'

.<:

~: ,

Fig. 8 shows the design for a.n alternative Watt--l six-bar, pro-ducing the identiCál moving plane 5 and thus

a.Jso

the identica.I six-bar coUp161-UUl'Veli generated by the points E a.nd D. Start--ing from the init1<ü mechanism shown in the Figs. 7 and 8,' the design for the alternative six-bar will be M fo11ows:

(a) Form the linkage parallelogra.ms ABBoB", BvBKB', and

KBCBv. (Note that the tria.ngl~ B" BoB' a.nd KB'BV may he

made rigid Without afl'ecting tbe movabiUty of the mechanism.) (b) Turn the four-bar AoAB" Bo about Bo over the bed a.ngIe

1:

B"

BoB' -

1:

ABK and multiply the four-bar simulta.neously fromBo bythefactorB'Bo/B"Bo'" KB/AB. ' (c) One obtains the four-bar Ao'A'B'Bo Bimilar to AoAB"Bo. (Notethat AB'A'K -- ilBKA -- ilBoA.'Ao.)

(d) Form the

rliid

triangles B'A'K and BoA'oAo, (This may be done without afl'ooting the movability: In' the case of

ilB'A 'K, for insta.noe, the lihka B~A', B" A, BvB, and B'K all

move at a.ny time at the a.ngular ve10city of link 3.

B' A'

and

B'K &lso have turning-joint

B'

in common. And 80 both Iink:s

B' A'

and'

B' K

,beioOI to the same moving plane 3'. Therefore

ilB'A'K is arigid triangle.)

(,) Turn the four-bar KBv CD àbout K

over

the fixed angle

1:

BV KB' ..

1:

CBBo a.nd multiplythe four--bar geometrica.Ily from KbythefaotorB'K/BVK

=-

BoB/CB.

(f) One thus obtains tbefour-bar KB'C'D' BimilartoKBvCD. (g) Frame the rigid and similar tria.ngleè BoC'B' and D'KD.

(Both tria.ngles &re similar toilBoBC.) ,

(A) As we

see

now, the initia! Watt--l six-bar is supplemented by a.nother one, consisting of the frame-Iink Ao' B., of the four -bars A'oA'B'Bo and B'C'D'K, the rigid triangles A'B'K and

Bo(1'B', and therigidquadrilateralKD'DE.

The 8.lteroative six-bar meohanism is of thesame type M the initia! one a.nd hM &lso oné 'degree of freedom of movement. The corresponding links KD' a.nd Kb produce the identica.I mov-ing plane. Only the joints Bo and K are the same M in tbe initial

six-bar. '

Comparing both six-bars, we note a permutation in the se-quence of the angular velooities of the links of the two mach-anisms: ,the two four-bars, viz., 0-1-2-3 and 2-3-4-5 &re trans-formed into 0-1'-3'-2' and 3'-2'-4'-5. Clearly, the angular velooities of the links 2 and 3, only, have been interohanged.

A~ arbitrarily chosen point E in tbe moving plane 5 of

the

initial six",bar, generating a six-bar coupler--ourve, will naturally &lso be generated by point E if attached to the moving pla.ne 5 of the alternative six-bar. Thus the two six-bars are ~

of the same type. (In the subsequent sections a oognate ob-taioed in this way will he oalled a "'/. ctl(lnat," forshort.)

We now make the MSUmption that the angular velooities 00-oorring in all cognates to he found are permuted only. 'rhat is to say that every a.ngular velooity appears in every cognate at the same time. It is then

waar

that not aH permutations &re per-missible: Suppose, for 8Xample, that the angular velooities of the links 3 and 4 are interchanged, we then get, let us say, the four-bar arrangements 0-1'-2'-4' and 2'-4'-3'-5' for a cognate six-bar. But the a.ngular velocities corresponding to the first

Journalof ER.lllerlng fat IllIastry

'lil. 9 SI ... "h ... wlth Ira ... roef

four-bar in this oognate do not fit, The angular velocities of

a.

four-bar may he permuted' only. Thus suoh a supposed per-mutation is not permissible; einoe a permutation must be a per-missible one withineach four-bar. ' It follows, then, that the interchanging of 2 and 3, M waS done, is permissible. The angular veloeitîes of '4 and 5 may &lso he interohanged. And, natura!ly, a oombination of the two permutations is &lso per-missible. In this way one ca.n even tryto ~oreoMt the ,number of

cognates of 8ÏX-bars.

Special

ea...

SIx .... Mechaahm W1th Tranllallall Rocl (CollO =:!!! 0). 8inoe the four-bar BeDK, ,whieh is a chain in, the six-bar' shown in Fig. 7, now'hM at least tbree Iink:s that move at the angular velooity of a corresponding link of the fOUr-bar

AoABBo, which is a.nother-chain in the six-bar, it is olear that the fourth link must &lso move correswndingly. Thus CoI40 \!iiiI

Cols,.

Therefore KD 0' and CD!!!E I'. Starting from the initia! four-bar AoABBo and coupler-tria.ngle ilABK, shown in Fig. 9, the remaining dimensions of the six-bar are to be determined M follows:

(a) Form the linkage parallelogram

BoAoAF.

(b) Turn tbe four-bar BAF Bo about B over the turning-angle

1:

ABK, and multiply geometrica.Ily from.B by the factor BK/ BA; (Thusfour-bar BKDCis formed and OBKDC,..; OBAFB .. )

(c) Form the rigid tria.ngle AB.;BC,' whioh is similar

to

ilABK, (d) Since KD, AF, and AoBo always move at the same angu-lar ve1ooity, link: KD must be tra1l3ltztiJl(/, that is to say: Each point of that plane (5)desc~bes a sbifted coupler-ourve

identi-oally equal to th" coupler-curve generated by the coupler-point K.

Slnoe two six-bars are found to generate moving pla.ne

5,

such is

&lso the

case

with plane Oio On the other hand, one may

a.Jso

design a six-bar using turning':'joint A instead of B

aà

the center of similitude. Thus one obtains two more generations of tbe same parallel moving plane 0'. LBstly, one may usa the third orank-point out of the known conftguration of Roberts M tbe center of similitude and 80 add two more solutions to the problem.

All in all, 6 six-bars are to be lound generating the same parallel moving plane 0'. The six machanisms mentioned bere were &Iready found by K. Dsin {141 using another wày of approach,

, ,

• A.ny permutation of angular ve100itiea in & four-bar is obta.ined by llimple interohaDgingthe sequenoe of tbe linb in the four-bar.

fEBRUARY 1971

/185

"

Flg.10

'/a

cOlln"

TWD SII-Bar CDlnates WUb Intarcbanled Anllliar MDHon

of LInks 4 and 5

('/6

COlnates fDr Sbort)

The initial mechanism from whieh we start the design is shown in the Figs. 7 and 10. lnstead of interehanging the angular motions of links 2 and 3, as done in the preeedÎng section, we now propose to interchange the ansular motions of links 4 and 5. This can he done by interchanging the linkt! 4 and 5. The design instructions are (see Fig. 10):

(a) Form the linkage paraJlelograms CDED" and CDKDv.

(N ote that the triangles DV CD" and KDE are identicoJ and rigid.) (b), Turn the four-bar CDvKB ahout C over-the angle

1:

DVCD" ..

1:

KDE and multiply the four-bar geometricoJly fromCbythefaetorCD"/CDV ... DE/DK.

(e) One thus obtains the four-bar DCD"K'B" '" DCDVKB.

(Note that AK'D"E '" AEDK '" AB"CB.)

(d) Frame the rigid triangles B"CB, B"CBo, and K'D"E.

(e) Turn the four-bar A&ABB. about

Bo

over the angle ~

Bil

BoB

and multiply the four-bar geometricoJly from B. by the

factor B"Bo/BBo. '

(j) One thus obtains the four-bar A." A"B"Bo '" OA&ABBo.

(Note that

AAoB..Ao" '"

ABBoB".)

(/1) Since B"A", BliK', and ABAK all rotate at the same angular velocity, and the first two have a common joint, it is clea.r that AB"K'A 11 is a rigid triangle. This triangle bea.rs the

numeral2" in accordance with the agreement made in the Appen-dÎll: ahout links moving at the same angular velooity. Thus

(A) Frame the rigid triangle B" K' A" and likewise

AAoB&Ao".

(i) It should he noted now that the initioJ Watt-l abc-bar bas been supplemented by another one, likewise coDsisting of two four-bars, viz., A."A "B"B. alId CD"K'B", the rigid 'triangles

A"B"K' and BoCB", the frame-link A."Bo and the

coupler-tri-angleK'D"E.

Both sÎll:-bars are of the same type, and the simila.r triangles

KDE and BD"K' oJways have a coinciding point B which is attached to the corresponding fioating links of the two abc-bars. Thus bath sÎll:-bars are cognates. Since only the angular veloei-ties of the links 4 and 5 are interchanged, the newly found sÎll:-bar will he called a "'/. cognale." (The four-bar arrangemente 0-1-2-3 anil 2-3-4-5 of the initial su-bar have been transformed intoO-l"-2"-3 and 2"-3-5'-4".)

One may ramark that the turning-jointe

Bo

and, C are common to ~th u-bars, viz., to the initioJ one and to the

'Ia

cognate.

18& /

fEBRUARY 1971 Fig. 11 I I I

I@'

I I I .' I 1 1 11 I I

FIg.12 Jourc.llleehon'.m ond

'Ir'l.

cognot.

,'-n ,~

I

A COloale Wltb loimbalied AOillar Moiions of tbe Links

4 anilSand WUb Those ollbe Llnksbnd 3(a

'/a·'/,

CDloale

for Sbort)

"

The initial mechanism fromwhich we start the design is now to he seen in the I<'igs. 7, 11, and 12. The design instructions aré as follows (see Fig.H):

(a) Make thefour-bar KBvK'Esimilartothefour-bar KBCD

(Point K then coincides with the center of similitude).

(b)' Frame the rigid triangle' KBBv. (Note thst AKBBT ,...., tlKDE.)

(c) Form the linkage paraJlelogram BT B!JoB"'.

(cl) Frame the rigid triangle AB'" BTK'.

(e) Make the four-bar Iinkage DK'B"'C'D'" '" ,OK'BTKE

(K' coincides with the center of similitude.) , (j) }'rame the rigid trlangles ED"'K' and BoB"'C'. (Note that tlED"'K' '" ABvB'''K'.) ,

(ti) Form the linkage paroJlelogram ABBoB and note that tlBÀ

BoB'"

is rigid. ' ,

(h) Make the four-bar linkage

DBoB'"

A '11 Ao ,,, similar to the

four-bar DBoBÀ AA.. ,(Point Bo is the center of similitude.) (i) Frame therigid trianglesB&A&A"'oandA"'B"'K'. (Note that AB&A&A.'" ,....,

ABoB

A

B"'.)

(j) Consider the newly found su-bar cognate, consisting of the four-bars Ao'''A'"B'''Bo and K'B"'C'D"', the rigid trie.ngles

AIII_{B'"K' and BoB"'C', the frame-link Ae"'B. and the}

coupier-triangleED"'K'.

The coupler-point E of the initial sÎlI:-bar and of the cognate descrlbe iqenticoJ six-bar coupler-ourves, since the pointe E

attached to the corresponding fioating links of tne mechanisms may eoincide at any moment (see Fig. 12;) ,

The cognate is coJled a

'/r_4/1

cognate since the angular motions of the links 4 and 5 and also those of the links 2 and 3 have been interchanged.

Transactlons ot lhe ASME

'"

.

.. -J('

Fig. 14 ' " cOlnata ond

l/ro'"

cOllIat.

na

Four SII·Bar Cllnates of tb. Watt·l Mecbanism Wltb

Point E Atlacb.d to Link

51umaH,

.

So lar, four cognates have been found including the initial sÎX-bar. These four cognates all produce the same curve described by a point E atta.ched to a floating link of which the end-points

&re connected with the fmme through a dyad. And one now has to investigate if more cognates exist or not. All permissibie permutatiOIl8 of the angular velocities have already beeit taken into account. So that leaves the pOssibility ol designing new ones with the permissibi"" permutations applied to those already found.

It will he proven, however, that such a possibility does not occur. In p&rticular, one finds that the process of finding

1/.,

'Ia,

or I/r'/a cognates is reversibie. (See Figs. 8, 10, and 12.) Further, one finds that the '/a cognate Qf the

lla

cognate is identical to the

I/r';'

cognate ofthe initialsix-bar. (See Fig. 13.) And the

lla

cognate of the '/a cognate is &lso identical to the

'/.-'/&

cognate.

(See Fig. 14.) Lastly, the

'/r' /

a cognate of ·t,he

I/,

cognate turll8 intothe'lacognateagain. (SeeFig.15.) ,

For the sake of completeness, all existing traositioll8 between the four six-bar cognates are shown in Fig. 16. The lour dift'erent cognates are suceessively:

Ao

4.

B

Bo

A.' A' B' B. Aa" A" B"

Bo

'Aa'" A"'B'''Bo C D K C' D' K C D" K' C' D"'K' E' E E E

Jourlal of Engl ••• rlll lor ladustry

initisl six-bar, I/I cognate,

'Ia

cognate, I/r'/, cognate.

FIII. 15

1/.

COllIat. and ' " cOInato

Fla. 16 blstlal lran ... allan. of .bt-llar COlli"'''

From this it could be seen that all lour ol them have the bed center Bo and &lso the generating point E in common. Besides, there are not more tban two dift'erent and corresponding turning-joints C (viz. C and C') and &lso nat more than tv/o different points K (viz. K and K'). In another symbolic notation the lour cognates are: ' 0-1 -2 -3 -4 -5 0-1' -3'-2'-4'-5 0-1" - 2" - 3 ,... 5' -

4"

0- 1'" - 3" - 2' - 5" ...; 4" initisl 8ÎX-bar '/. dognate

'I.

cognate

1/.-'/,

cognate. '

If the lour cognates

are

presented in one configuration, t)1e .symbolic notation shows that in such a configuration (CD for short) not more than three different planes bearing the numersl 2, 3, 4, or 5 exist. One notes &lso that there are 'two cognates in which the genera.ting point E is atta.ched to plane 5, and there

&re also two cognates with E attached to plane 4". Thus thore

are two ways of genera.ting a sbc-bar collpler-curve of order 16 with the coupler-point E attached to a floating link having tbe

angular velocity of plane 4. Proposltlon

In, ge.n.eral, aurre are fIOt

more aaan

4

~r cognGÛI8 of tM

Walt-l ''1fP6 (wilh tM generatin,g point E attached 10 tM ftootîng link which

ia

ccmn,ected 10

tM frame

ehiotcgh ~yadB only) OBlong OB tM jive

occurring an.gu1a.r l16locitieB wiI.A rll8p6Ct 10 !he fm1M Me ge.n.erally

different.. (The proof of this proposition wiII be delivered in th~

Appendix.)

FIg. 17 Soure. m.han ... (2)

,~---_..;.~_

0 "'l,. .... ,

7

~' I ' I-A<' .1 ft /~I I ' I /1 / " .. ' / 1 I / ',~ Ni'. I~ i W I I 1

,..,tI

c

1 1 1 K'

F".

18 Soure. m.chanllm anel coanat. wIth arIIUrartly cho_

turnlno-...

",

".

n.

Inlnlt. Man,lolliness In Prolluclnl a WaH-l

SII-

aar

Coupler-Curve ol

Oegr.14

(anll Genus 5)

AB atated by Primrose, Freudenstein, and Roth [13] the Bix-bar curve produeed by a coupler-point E attached to the floating link CD is an algebraie CUT1/6

of

order 14 and genw 6. In Fig. 17 lIuch a curve is drawn only styJistically. Tbe actual curve may he generated by cognates too. But as long as E remaill8 at-tached toa floating link which is IIUpported by a triad and a single 'IY.ld, it eeems not possible to find a cognate with a changed _dis-tribution of angwar velocities over the links. On the other hand, it is here no longer possible to prove that two six-bar cognates with the sáme distribution of angular velocities are identical. The one thing one can prove is that the oorresponding dyads

BoCE

and, let U8say, Bo'C'E areidentical in both cognates. (SeeFig.18.) .

CoDSequently, the numher of cognates with the same distri-bution of angular velocities among the links must he infinite. And the same number applies to the manyfoldness in reproduclng the

,ntir,

moving plane 4. Starting from the initial mechanism IIhown in Fig. 17, the design of such cognates will.he carrïed out as follows (aee Fig. 18):

(a) Turn the fOUI'>-bar linkage .CDKB about Cover the

a.rbi-era.rily

choaen angle

a ..

~ DoCD and mwtipJythe four-bar geometrically from C by the arbitrarily choaen factor

I .. '"

DtC/ DO. Then the four-bar DCDoK'B' is formed which is similar to the initial four-bar DCDKB. (Note that D'CD and B'CB are similar triangles and may he made rigid.) .

(b) Frame the rigid triangles BoCB' and DoCE.

(c) Make the four-bar linkage [}Ao'A' B'Bo similar to the four-bar DAeABBo. (Point Bo then .is the center of aimilitude: Note that AAo'BeAo '"'" A.B'BoB.)

(d) Frame the rigid triangIe A'B'K', (This 'may he done

188/

PEBRUARY. 1971

F". 19 COgn .... wfth ".

==

E

aince K'B', BoA', and AABK move at the sameángwarvelocity, and the first two have a oommon joint W.)

(e) Conaider the supplemented cognate, conaisting of the four-bars A.'A'B'B. and CDoK'B',

the

rigid triangles A'B'K' and

BoB'C, the frame-Iink Ao'Be and tlIe coupler-triangle D'CE.

(I) Both the initia! and the cognate have the same dyad

BoCE. They aIso have the same distribution of angular veloci-ties among their reapective links, since two links have never been

interchanged in this design.

(/I) Sinèe there are

two

degrees óf ft:eedomin design (viz.,

a

and

I .. ),

they may be usêd, lor instance, in order to choose the

loca.tion of the bed center A.'.

Or,

as will he aeen later on, they may he used in letting the turning-joint Do coincide with the

generating point E. '.

I~ is c1ear anyway thfl,t there are .. I waya of ge~ating. the

curve described by point

E

of plane ,4. This is true even tor all

pointII of that plane at the same time. . Fig. 19 demonatrates the special ca'3e where the points Do and E

have been made identica!. CoDSequently,

a

= ~ DOE and/ .. = CE/CD. One linde the cognate Bix·b8.r AoIA"Bt:J.BoCEK'.

Here, the turning-joint E of the cognate generatee the same curve as point E of the initia! Bix-bar.' In doing 80, point E aIso turna into a (special) point of plane 5' and therefore, one may make the

lla

cognate of this mechanism. (The f/,.cognate does not come ..

--;'Fo;

point B ol plans 4 for which I!J)BC '"'" AB&C the désicn ooIlapses, for then Bt:J. 5ii B ..

Transaellens ol

t •• ASME.

"I

"

,

under diseWlllion, llinee side,DE of the parallelogram CDED"

have zero length. Bomething like it may be sw about the

'/'r-

t

_I,

eognate.)

The

'Ia

eognate for tbis ease is shown in Fig. 20. It oonsists of the four-hars AoIAIR'Bo ll.lld _B'C'D'K', of the rigid triangles B'C'B~ and A'R'K', of the frame-link

Ao'Bo

and of the eoupler-triangle K'D'E. One may observe tbat here

too,

point E seems

00 be

a

special point of plane iV, mee /lK'D'E '" /lB'C'Bo.

Although point E of tbis eognate is attaehed 00 the floating link 5/,

the degree and genllB of the generated curve are those of the curve produood by point E of the initial IIÎX-bar. Tberefore Ilo point E

attaehed 00' the floating link KD of the meebanism of Fig. 7 for which /lKDE '" !J..BC.8o describes an algebraic curve of degree 14

in:stead of 16.

It remains 00 be seen how the initial six-bar may be directly trlloDSformed inoo the J.ast...foimd cognate. Tbis is demonBtrated

in Fig. 21. Starting from the initial IIÎX-bar of Fig. 17,

ihe

design instructions will be I!.B folloWB (see Fig. 21):

(a) Make the four-bar BoCD"C' lIimilar 00 tbe' four-bar

BCDK. (Point C ooincides with the center of similitude. Note that llD"CD,...., /lBoCB.)

(b) Form thelinkagepar~C'D"ED', C'D,"ClJA,ll.lld ABBoB". (Note that ED"C and D'C'DA are identical and rigid triangles. Note aJso that /lB"C'

Bo

ia a rigid one.) "

(c) Make the four-bar B'C'D'K' ,similar 00 the four-bar

BoC'DAc. (point C' ooincides with the center of similitude. Notethat /lED"C S!! IlD'C'DA,...., /lB'C'.8o,...., M'D'E.)

(d) Frame the rigid and similar triangles B'C'Bo and K'D'E.

(8) Make the four-bar BoB'A'Ao' similar to the four-bar

BoB" AA.. (The fixed center

Bo

ooincides with the center, of llimilitude. Note that /lB"BoB' """ /lAoBoAo'.)

(f) Frame the rigid triangles AoBoAo' and A ' B' K'.

(g) Consider the supplemented eognate, oonsisting of the four-hars Ao'A'B'Bo and B'C'D'K', the rigid triangles A'B'K'

ll.lld B'C'

Bo,

the frame-link Ao'

Bo

ll.lld the ooupler-triangle

K'D' E which is similar 00 /lB'C'

.80.

The oognate obtained ia identical 00 the one obtained in Fig. 20. Tbis may be proven by repeatedly ullÏng the lemma formulated in the Appendix.

It is obBerved, finally, that allsix-bar cognates of Watt's form, special or not, have the hed center

.80

in oommon ll.lld all have the same ooordination betweeu crank rotations about the other

fixed center (Ao, A.', Ao", Al" or Ao') and the ooupler-point poIIÏtions of point E.

RefereICIS

" 1 Roberts, 8., "On Thrt&-Bar Motion in PI&lle 8pace," Proc6ed-inf/lIOltAtl 'Lo1ll1lm MathMMtiMJ&cifllll. Vol. 7,1876, pp.14-23.

2 Cayley, A., "On Thrt&-Bar Motion," Proceetlift(JB ol &he ~M~&cielIl, Vol. 7,1876, pp.136-166.

/ 3 Schor, J. B., "On the Theorem of Roberts-Chebyschev,"

J~ ol Applild MathMMtictt al'ld M8C'hanictt, USSR Academy of Soience, Vol. 6, No. 2, 1941, pp. 328-324.

. 4 Meyer sur Capellen, W., "Bemerkungen lIum Satll von Roberts

JO.I.I

ol Elglae.rlll'or Ilduslry

~---L---~D

Fil. 22 D_onslndlon of propolltlon 1

nber die dreifache Erzeugung der Koppelkurve," K~. Vol. 8.

No. 7,1966. pp. 268-270. ' ' " ,

ti Riohard de Jonae, A. E., "The Corre1ation of Hinged Four-Bar Straight-:(.,ine Motion Devices by Meana of Roberts' Theorem anda New Proo! of the Latter," A~

'I

TM NtfI.D York A-um1l ol

8cittnu, Vol. 84, No. 3,1960, pp. 75-145.

6 SyJ.vester, J. J., "Hiatory of tbe Plagiograph," NtJluf:e, Vol .. 12, 1 8 7 6 , p p . 2 1 4 . - 2 1 6 , '

7 Kempe, A. B., "On Conjupte Four-Piece Linkaps,"

Pro-C'tledift(Jllol tM 'Lo1ll1lm Math~ &cieI.1I, VoL 9, 1878, pp. 133-147. 8 Burmester, L., "Die BrennpllIlktmecbanismen," ZeitscMi/t

(ar Matlurmatlk un4 Ph'/l8ÏTlii Vol. 38, No. 4, 1893, pp. 193-223. 9 Wunderlloh, W., "On Bu:rin'eeter's Foeal MeoheÏ8m a.nd

B'a!:t's Straigh~Line Motion," JtJm'7I4l 0/ M~. Vol. 3, 1968.

pp.79-86. .

10 Hartenberg, R. S., and Denavit, J., "Cognate Link:aaes," Mach.iMDHigrI, Vol. 31, No. 8,1968. pp.149-:152.

11 Roth, B., "On tbe Multiple Oenaration of Coupler-Curves," JOt1l1lfAL 011' ENGDml!IlUNG POll INDt1I'l'RT, TBANB. ASME, Series B,

Vol.87,No.2,1966,pp.177-183. ' , ,

12 RiBohen, K. A:, "'V'ber die aohtfaeh, Erzeugung der Koppel-kurven der SWIIÎten Koppelebene," KtmBtrull:tVm, Vól; 14, No. 10, 1962, pp. 381-386.

13 Primroae, E. J. F .• Freuderllltein, F., Roth, R, "Six-Bar Motion. I. The Watt MeohanisJn," ArckiH lor Rational Mil-c1umicttal'ld AnalllN, Vol. 24, No. I, 1967, pp. 22-41.

, 14 Hain, lt., "Erzeugung von Parallel-Kbppelbewegungen 'mit Anwendungen in der Landteehnik," Gru'fllllGgan der Londted&nik,

Vol. 14, No. 20, 1964, pp. 68-68.

APPENDIX

Throughout the paper the next proposition, which tand!! to be Ilo

fundamental one, hl!.B been used on quite a numbei' of occaSions. The proposition is demonstrated in Fig. 22 and may be formu-lated I!.B follows:

PropolitIon 1

11 at

an'll

time tlDO

8imilar

lour-bGr linkages

ABCD mul A I B'C' D'

have a common.

jDim (B B'),

aM

iJ

at

leaat two

pairs of

corre-B'ponding sides are mutuall'll mDIIing

at

,~ Borne angular .ll6locity

uritJt, respect

to

l1Dme

lronn.e-link,

then èorrBaponding sides witA 0

common.

w.mi1lfl"iDim (B) larm two sides ol a rigid triangle, OM (otAer) COl"'f'e6pcmding sides move

at

equal angular fI6l'ocily.

In tbis proposition one defines two lIimilar four-bll.l' linkages as those formed by llimilar qua.drilaterals. And oorresponding si des, being in di1ierent four-bar linkages, are those occupying the same positions in the two lIimilar quadrilaters.1s.

In the kinematic oonflgura.tion of Fig. 22-which, incidentally, ia only a part of some mechaniam-one notes

that

BAA I ll.lld

BCC' are similar and rigid triangles. Tbe oorresponding llides '

A'D' and AD move at any time at the same

angular

velocity with respect 00· some frame-link. This applies with equal force 00 the sides C'D'and CD.

It should he noted aJso thatthe two oonnected pl8giographs of Sylvester, as shown in the Figs. 2 and 3 are bl!.Bed on the propo-llition. (Fig. 2, for f'xample, eonsists of the linkage of Fig. 22 extended with two linkage parallelograms.)

In the 'figures shown here, if at any time two moving planes have the same angular velocities, such planes baar the same identifying numeral. Tbe ditferenee ia indieated by primes attached 00 the numeral.

FIII. 23 Indirect dam_straflon 10. proposIlIon 2

In order to prove the propoeition at the end of the section ",The Four Six-Bar Cognates etc." it suffices to prove the next prop08ition as bas been pointed out in that section.

Proposhlon 2

If a definite distribution of angular ~eWcil.ieB OlIeT IÀIl links of a Wall-I Biz-bar is presçribed, and if IÀIl generating point (E) is '

attached to IÀ/lftoating link (KD) with Ilnd-points linked to lÀeframe IÀrougAdyads only, 7IOoth.er cognates ezist, generaUy. (See Fig. 7.) The proof of this prop08ition willbe delivered by indirect demonstration. Therefore, supp08e there are two different cognates with at any moment the same distributlon of angular velocities. Supp08e the considered cognate of which there are two, is theone of Fig. 7. In each cognate appe&l'S the four-bars 0--1-2-3 and 2:-3-4-5, or let us say, 0'-1'-2'-3' and 2'-3'-4'-5'.

It is clear that both cognates have at any time a common gen-erating point E. Thus plane 5'

==

5 and the triade B.r-C-D-E

and Bo'-C'-D'-E are to be indicated' in a general configuration showing both cognates in one figure. Clearly, !lDD'E is rigid, and part of the configuration as shown in Fig. 23 may be supple-mented by the linkage parallelograms C'D'DDv and BoBo'C'C", and by the rigid triangles CDDv and B.CC". The figure shows

190 /

FEBRUARY 1971

Fig. 24 Damon .... atIon 10 lemma

the four-bar C'C" CDV or in another notation O'-3-4-5v . How-ever, this is in contradiction to the fact that a four-bar linkage of which the sides move parallel to thwespective planes 0, 3, 4, and 5 cannot be compoeed. (Otherwise w.o should be identica1 to zero.) Thus the suppoeition

Bo'

~ Bo has been fa1se. There-fore, both points Bo and points E are identica1.Then, because of the lemma demon,strated at the end of the Appendix, the triads BoCDE and BoC'D'E are to be identica1 also. The same holds for the triade B.BKE and BoB'K' E. Lastly, applying the lemma to the triads BoAoAB and Mo'A'B, one finde that both cog-nates, with the same distribution of angular velocities, are iden-tica1. That proves the propoeition.

Lemma

If twotriads ABCD and AB'C'D are connected in their end-points and lÀe corresponding link8 have identical angular lIelocil.iés

(WIO ~ w.o ~ Wao ~ WIO), then tkll triads ar6 identical. (See'

Fig. 24.) Proof

Complete the figure with the rigid triangles ABB' and DCC' and with the linkage parallelogram B'BCC". Clearly, !lB'C'C" turns out to be rigid. And by different reasoR8 80 does !lCC'C". Thus W20

==

WIO which is in contradiction to the s~pposition that

w.o jid WI.· Thus C'

==

C" or C"

==

C. And af ter some more

reasoning of this kind, one finds that the lemma holde.

.'. i" ~