Four-bar of special forms of Watt's six-bar mechanism : discussion on a paper of A.H. Soni about "coupler cognate mechanisms consisting of linkage parallelograms supported by four-bars"

Four-bar of special forms of Watt's six-bar mechanism :

discussion on a paper of A.H. Soni about "coupler cognate

mechanisms consisting of linkage parallelograms supported

by four-bars"

Citation for published version (APA):

Dijksman, E. A. (1971). Four-bar of special forms of Watt's six-bar mechanism : discussion on a paper of A.H. Soni about "coupler cognate mechanisms consisting of linkage parallelograms supported by four-bars". Journal of Mechanisms, 6(2), 195-202. https://doi.org/10.1016/0022-2569(71)90030-9

DOI:

10.1016/0022-2569(71)90030-9

Document status and date: Published: 01/01/1971

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Four-bar Cognates of Special Forms of Watt's Six-bar

Mechanism

(Discussion on a paper of A. H. Soni about "coupler cognate

mechanisms consisting of linkage parallelograms supported by

four-bars")

E. A. Dijksman*

Received 8 July 1970

Abstract

In a discussion of Soni's paper[l] a geometric way has been proposed in deriving the cognate six-bar mechanisms as found by Soni. Moreover, it is proved that the curves produced by the six-bar mechanisms of the considered type and structure are ordinary four-bar coupler-curves of degree six and genus one.

Zusammenfassung--Verwandte Gelenkvierecke fUr Sonderformen des

Wattschen Sechsstabgetriebes. (Diskussion des Aufsatzes yon A. H. Soni fiber "Verwandte Koppelgetriebe, aufgebaut aus einem dutch Gelenkvierecke gestStzten Parallelogramm"): E. A. Dijksman.

In einer Diskussion der Arbeit von Soni [1] werden rein geometrische Ableitungen der verwandten Sonischen Sechsstabgetriebe erl&utert. Uberdies wird gezeigt, dass die von dem betrachteten Sechsstabgetriebe erzeugten Punktbahnen gewShnliche Koppelkurven sechster Ordnung vom Geschlecht eins sind.

Pe3mMe - - Po~craeHHocrb mapHHpHMX ¢IeTbI~X3BCHHHKOB C qaCTHblMH (~OpMaMH Lt[eCTH3BeHHblX mapHHpHblX MexaHH3MOB (O~:;yX~eHHe CraTbH A. F. COHH O pO2,CTBeHHblX MexaHH3MaX, o6p~L3OBaHHblX npHCOC~HHeHHCM cTpyKTypHoR rpynnb[ K l~aTyuy lZlapHHpHOFO xi~Yrbl~X3BeHHOFO MeXaHa3Ma napamlenorpaMMa): E. ~Hi;IKCMaH.

B o~y)g~l.eHHH CTaTbH COHH O6-b~ICHeH tlHCYO FCOMeTDHqeCKH~ nyTb rlonytleHHS COHHeBCKHX pO~CTBCHHblX mI~t'H3BCHHblX MeXaHH3MOB, o6pa3osaHa~Ix'HpHc0C~HHeHHeM cTpylcrypHO~ rpynnb=

K m a T y ~ y m a p . a p H o r o ~eTUpex3SeHBMKa. K p O M e xoro, yKa3siRaeTC~, ~TO LUaTyHHbl¢ KpHsble, onHCaHHbIe maTyHHhlMH TOLIKaMH St-lmeyrIOM~lHyTr~IX tlaC'THblX MexaHH3MOB, TaK~Ce MOryT 6~-ITb rIocTpoeHbl C rlOMOLU~IO l.tlapHHpHblX ~IOTI~IpeX3BeHHHKOB.

1. Introduction

IN THE Summer edition of this journal a very interesting paper appeared about coupler cognate m e c h a n i s m s written by P r o f e s s o r A. H. Soni. In this p a p e r three cognate six- bars of a particular t y p e o f W a t t ' s six-bar m e c h a n i s m has been found. T h e type con- sidered is the one, which has o n e linkage parallelogram as a kinematic sub-chain in the six-bar (see Fig. I).

T h e m a n n e r in which he derived his cognates is partly geometric and partly algebraic through making use of the t h e o r y on c o m p l e x numbers.

In order to e m p h a s i z e these results, I intend to d e v e l o p in this discussion another, purely geometric, w a y of obtaining the m e n t i o n e d c o g n a t e m e c h a n i s m s of Soni. More- o v e r I intend to p r o v e too that the curves described by the coupler-points of this

*University of Technology, Eindhoven, the Netherlands.

196

particular type of m e c h a n i s m are identical with ordinary four-bar coupler curves of degree six and genus one.

...:'"

::::

: I ... i

So~rce': mee f'~lnism ond four-bor coupler

A curves

Figure 1.

2. The Six-Bar C o g n a t e M e c h a n i s m s of Soni

A r a n d o m l y chosen six-bar m e c h a n i s m with the linkage-parallelogram B C D K ,

as s h o w n ~: in Fig. I, will be termed the s o u r c e m e c h a n i s m in the course of this dis- cussion. (With the exception of Section 3, it remains the s a m e m e c h a n i s m throughout the paper. In fact, h o w e v e r , the s a m e p u r p o s e would be served by any linkage of this kind).

T h e source m e c h a n i s m consists of a four-bar A,~4BBo, the linkage-parallelogram

B C D K , the arbitrarily chosen coupler-triangles A B K . C D F and K D E and, finally, of the rigid triangle BoBC.

T h e curves described by the coupler-points E or F are under discussion. T h a t is to say, we are looking for cognate six-bars of the s a m e type as the source m e c h a n i s m and of which the coupler-points E or F generate the s a m e (identical) c u r v e as generated by point E (or F) of the source mechanism.

2.1 Ttle g e n e r a t i o n o f the curve p r o d u c e d by p o i n t E o f the s o u r c e m e c h a n i s m

Let us first consider the curve g e n e r a t e d by coupler-point E. attached to link K D

of the mechanism. One of the (two) Soni cognate m e c h a n i s m s may then be obtained by following the next design instructions: (See Fig. 2).

(a) F o r m the linkage-parallelograms A , c l B A v. AVBB,,B v and A,w4KA ~. (Note that

,kA~Av,4~" ~ ~ A B K ) .

(b) T u r n the four-bar A,v4rBVBo about A0 o v e r the angle cz = g MrA,v4 v = ~ K A B

and multiply the four-bar geometrically from Ao by the f a c t o r f ~ = ,4~Mo/AVAo = K A / B A (One thus obtains the tour-bar AoA~'B"Bo ~" which is similar to the four- bar A oA r BV Bo).

(c) F r a m e the rigid triangleAoBoBo ~" which is similar to the c o u p l e r - t r i a n g l e A B K .

(d) F o r m the linkage-parallelogram A ~ ' K E K L ( N o t e that AK~'A~B ~" is a rigid triangle. This is due to the fact that A r B ~' and A~'K ~ both have the identical angular velocity o):m and also have the turning-joint A ~ in c o m m o n . )

(e) N e x t , c h o o s e a r a n d o m turning-joint D ~" in the moving plane attached to link

K~'E.

~ . ~ , u r - b a r coupler- curve

... ~'1 ... ...::.,~=

¥ " ' " "

I ~ ~ ° ~ ,." Tu~-i~n, D v randomly

. - , . ~ ~ , - - So

I~..,, ~;:" "'"'~';; Source mechonisrn and

" ~ "2 " first cognate

F i g u r e 2.

(f) Further, form the linkage-parallelogram BVK"DvC v. ( N o t e that AB~'CvB, v is a rigid triangle, since two sides of the triangle at all times have the same angular velocity O~o).

T h e source mechanism is now supplemented with a six-bar cognate mechanism of the same type and structure. T h e obtained cognate is indicated by hard-solid lines in Fig. 2. Like the source mechanism it has one degree of f r e e d o m of movement. Both mechanisms, the cognate and the source mechanism, have one kinematic sub-chain which is a linkage-parallelogram and both have the same coupler-point E at all times coinciding. T h e r e f o r e , both points E generate the identical coupler-curve and the mechanisms may be called cognates.

(One may note that D ~', or otherwise C ~', may be chosen randomly. T h e r e f o r e , there is a doubly infinite n u m b e r of solutions of this kind.)

A second cognate of the source mechanism as found by Soni, may be obtained through the next sequence of design instructions: (See Fig. 3).

(a) F o r m the linkage-parallelogram B o B A B ~, BaAA~A -~, A K E K A and A o A K A A "~. ( N o t e that ~kAoAaA "x is a rigid one, since two sides of this triangle have the same angular velocity o~:~o at any point of time).

A

Turning-joint D ^ arbitrarily ChOsen in plane K^E l .,.... ... . i v .-'"L'Four-bor coupler curve

/ h

198

(b) T u r n the f o u r - b a r A , v 4 a B a B o a b o u t A 0 o v e r the angle 8 = < AXA,~4 ~ and m u l t i p b the f o u r - b a r s i m u l t a n e o u s l y from A, by the factor fa = A ~AJA~Ao (One thus obtains the four-barAoA-XB~Bo ~ which is similar to the f o u r - b a r A , ~ B a B ~ p .

(c) F r a m e the rigid triangle AoBoB,~ ~ which is similar to the triangle ,4,~AaA ,x, and form the rigid triangle A.XBAK "~.

(d) N e x t , c h o o s e a turning-joint D ,~ s o m e w h e r e in the moving plane attached to link KXE.

le) F o r m the linkage-parallelogram B-XK.~D'~C ~ and { f) F r a m e the rigid triangle B,/XB.xC.X

Again the source m e c h a n i s m is s u p p l e m e n t e d by a n o t h e r six-bar cognate mechan- ism of the s a m e structure. T h e obtained c o g n a t e is indicated by the hard-solid lines of Fig. 3. Point E of the cognate six-bar generates a c o u p l e r - c u r v e identical with the c u r v e produced by point E of the source m e c h a n i s m .

N o w , two cognates have been found based on an interchange in the s e q u e n c e of the moving links of the initial four-bar A,~IBBo. This initial four-bar, which is a kinematic sub-chain of the source m e c h a n i s m , has the next s e q u e n c e in the n u m b e r e d links: 0 - 1 - 2 - 3 .

T h e first cognate of Soni is based on the s e q u e n c e 0 - 2 - 3 - 1 and the second cognate p o s s e s s e s the s e q u e n c e 0 - 3 - I - 2.

Clearly, three other possibilities exist, and they will p r o b a b l y c o r r e s p o n d to as m a n y cognates of W a t t ' s type.

In order not to lengthen this discussion I do not intend to broaden the field of possible cognates. I will merely confine m y s e l f to a (geometric) discussion of the presented cognates of Soni.

One interesting point remains to be established, and that is the astonishing fact. that the curve generated by point E of the source m e c h a n i s m m a y also be g e n e r a t e d by an ordinary four-bar mechanism.

T o prove t h i s - r a t h e r u n e x p e c t e d - f a c t . 1 intend to give the relational dimensions of such a special cognate in the next s e q u e n c e of design instructions: (See Fig. 4).

, , . . . . • . . . , . , . . , . . , Four-bar coupler ...- . . . curve ~ . " Ao

E~'

/ 3 '

A'

/

/

- ~ : . ! - g . -

/

/

Translating bar--w

"" K ~

~:~

~.k' /

:..::. ~2

I

~ / ~

~[i:~"

.~,;,..o/72 Four-bar cognate of

B'"

"

source mechanism

(a) T u r n the four-bar AoABBo of the source mechanism about A o v e r the angle o~ = ~ KAB and multiply the four-bar geometrically from A by the f a c t o r f ~ = KA/BA. (One thus obtains the four-bar HA KA' which is similar to A oA BBo.)

(b) F o r m the rigid triangle AAoH which is similar to A A B K and note that link HA'

is a translating bar.

(c) F o r m the linkage-parallelograms BoBKB' and AoHA'A'o.

(d) F r a m e the rigid triangles AoBoA 'o which is similar to the coupler-triangle A B K .

(e) F r a m e also the rigid triangle KDB' which is similar to ABCBo. ( N o t e that the

points E, K, B', D and A' are points rigidly attached to the moving plane 3'.)

(f) T h e coupler-point E of the coupler-triangle A ' B ' E attached to the coupler A ' B '

of the f o u r - b a r A o A ' B ' B o generates the same coupler-curve as produced by point E of the source mechanism.

(g) N o t e that VqAoA'B'Bo ~ ~AoAB'"Bo as long as point B'" is defined through the

linkage-parallelogram ABBoB'". ( T h e center of similitude coincides with the

fixed center Bo and the multiplication factorfB = BK/BA ).

(h) N o t e also that any point of the coupler-plane 3' of the source mechanism will

be generated by the same four-bar A'oA'B'Bo. T h e r e f o r e , there is a cognate

generation of the entire coupler-plane of the source mechanism.

Clearly, two o t h e r four-bar cognates may be indicated through Roberts' Law*. And they too, in turn, give rise to six-bar cognates of Watt's type.

A n y w a y , we now have established the important fact, that four-bar coupler-curves may also be generated by an infinite row of six-bar cognates (of special type) of which the source mechanism is an example.

2.2 The generation of the curve produced by point F of the source mechanism

Let us now consider the curve generated by the coupler-point F, attached to link

CD of the source mechanism. Soni proposes a cognate as shown in Fig. 4. Starting

with the source mechanism again, the geometrical design of such a cognate may be obtained through the next sequence of instructions: (See Fig. 5).

T u r n i n g - j o i n t K : ' a r b i t r a r i l y Source mechanism c h o s e n in plane A ~ B > and first c o g n a t e , ,~D ~ D > / 2 > "o-"

t..::~:~

~

.,~::.:.::~-

Ao,

\

center in the frame

Figure 5.

*Since AA,BoA' o coincides with the fixed triangle in the configuration of Roberts, the other two four-bars

200

(a) F o r m the linkage-parallelograms A B B o B " ' and B,jCFC >.

(b) T u r n the four-bar A , v 4 B ' " B , about Bo o v e r the arbitrarily c h o s e n angle e = < A,~B,~I,~ and multiply the four-bar geometrically from B,, by the arbitraril>

chosen factorJ~ = A,~>Bo/A,~Bo (One thus obtains the four-bar Ao>A>B>B,, which

is similar to the four-barA~AB'"B,,).

(c) F o r m the rigid triangles A,~B,v4,~ > and BOB-"('".

(d) C h o o s e the turning-joint K > arbitrarily in the moving plane attached to the link A>B > (Thus ,A.4 >B>K > is a rigid triangle.)

(e) F o r m the linkage-parallelogram C>B>K>D > and

(f) Finally. form the rigid triangle F C ' D >.

T h e obtained cognate, drawn with hard-solid lines in Fig. 4, has the same structure as the source mechanism. And the coupler-point F describes the identical coupler- curve. ( N o t e that the design of the altermttive m e c h a n i s m gives f r e e d o m of choice of the f r a m e - c e n t e r A,~ and also of the tu rni ng-joi nt K > (or D >).)

Finally. we will show the surprising ['act that the c u r v e generated by point b of

the source m e c h a n i s m may also be g e n e r a t e d by three cognate four-bar linkages. O n e of them m a y be obtained as follows: {See Fig. 6)

O \ \ \ I--..x \ \ # ' o A D

Ko-

2, iii i

....

Foor-bo

t::iP

F o u r - b a r cognate of

source mechanism

Figure 6.

(a) T u r n the four-bar A~v4BB,~ a b o u t B0 o v e r the angle/3o = < CB,,B and multiply

the tour-bar g e o m e t r i c a l l y from B0 by the factor J~,, = CBo/BB,,. {One thus

obtains the four-barA{'Ct"B"Bo which is similar to the initial four-barAoABB,,).

(b) F r a m e the rigid triangle AoB,,A o which is similar to k B B o C and

(c) F o r m also the rigid triangle A " C D . ( N o t e that E]A"CFD forms a rigid quadri-

lateral).

(d) T h e c u r v e p r o d u c e d by point F. m a y also be g e n e r a t e d by the coupler-point t:' of the coupler-triangle A"B"F. attached to the coupler A"B" of the four-bar A d'A "B"Bo.

T h e r e f o r e . here too. the c o n s i d e r e d c u r v e is an ordinary four-bar c o u p l e r - c u r v e of degree six and genus one. T h e other two f o u r - b a r cognates* are to be found with the well-known L a w of R o b e r t s - C h e b y s h e v .

*The first obtained four-bar cognate, solely, has the advantage of generating the entire coupler-plane Z of the source mechanism.

3. The Source Mechanism as Part of the Generalized Pantograph

Finally, one may remark that, since the source mechanism contains a linkage- parallelogram, the mechanism must have something to do with the so-called general- ized pantograph of Kempe-Burmester[3,4]. (See Fig. 7).

This pantograph consists of a linkage-parallelogram with rigid triangles attached to each side of it.

In the design-position of the linkage, the four connected vertices of these triangles form a quadrilateral of which the sides pass through the turning-joints of the parallelo- gram. Moreover, the diagonals of the quadrilateral lie parallel to the sides of the parallelogram in this position.

Since 1888 it is known that, if one of the mentioned vertices should be made a center of pivot on the frame, the remaining three vertices will describe similar curves*. Now suppose, the source mechanism may be brought into the position? where the turning-joints A, B and Bo are in line. (Such a position will be called a design-position.

If there is no such position, one draws the plan with the points An, A, B and B0 in line, similar to Cayley's plan for determining the link-lengths of the four-bar cognates of Roberts). * 0 , 8 ~A,o A' 6ererolized i pontocJroph Figure 7.

In the source mechanism, brought into the design-position, one so recognizes immediately part of the generalized pantograph. Therefore, the source mechanism in this position may be supplemented by the links A~A' and A'~4" in accordance with the pantographic proposition. One may remark that for each position of the source mechan- ism, the quadrilateral AA 'A"Bo will always remain similar to itself+~.

Hence, a possible second design-position of the source mechanism does not furnish a pantographic solution differing from the one obtained with the first design- position.

The additional frame-centers A~ and A~; may be obtained through the similarity:

[~AoA'oAoBo ~ []AA'A"Bo.

As a result of the pantographic addition one recognizes the four-bar A~A"CBo.

Thus turning-joint D (and any coupler-point attached to link CD) generates an ordinary coupler-curve. Since on the other hand the five-bar Ao'A'DA"A~' has two input-rods always rotating at equal angular velocity, any point attached to link A'D or KD will describe ordinary coupler-curves too.

* In the case under consideration, these curves are circles.

t l f the crank BBo o f t h e s o u r c e mechanism may turn the full circle about the frame-center B0, there generally are two such positions.

202

Clearly. the application of the principle of the generalized pantograph u[ &empe- Burmester leads also to the already found cognate tour-bar linkages which generate the same curves. The reader may choose which way of obtaining he prefers.

References

[1] SON I A. H., Coupler Cognate Mechanisms of certain parallelogram forms of Wart's six-link Mechanism

J. Mechanisrns 5, (2), 203-216 (1970).

[2] D I J K S M A N E. A., Six-bar cognates of Watt's form, Trans. ,4SME, J. Engng. Ind. 93B. 183-190 (19701. [3] K E M P E A. B., On Conjugate Four-piece Linkages Proc. Lond. Math. Soc. 9. 146 (1878).

[41 B U R M E S T E R L., Lehrbuch der Kinematik, Erster Band en "die ebene Bewegung": Felix-Verlag p. 595-596 (1888).

[5] BR1CARD R., Lefons de Cin~matique. Gauthiers-Villars et Cie. Editeurs, kivre V, Notes et l~tudes Diverses. (Gdndralisation du pantographe) p. 277-280 ( 19271.

[6] S H A M B U R O V V. A., A new Method of the Synthesis of Pantographs and other transforming Mechan- isms, Trans. o f the Institute o f Science o f Machines, Academy of Sciences USSR Seminar on the Theory