Six-bar cognates of a Stephenson mechanism
Citation for published version (APA):Dijksman, E. A. (1971). Six-bar cognates of a Stephenson mechanism. Journal of Mechanisms, 6(1), 31-57. https://doi.org/10.1016/0022-2569(71)90005-X
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Jnl. Mechanisms Volumes 6, pp. 31-57/Pergamon Press 1971/Printed in Great Britain
Six-Bar Cognates of a Stephenson Mechanism
E. A. Dijksman :"
Received 8 May 1970
Abstract
The paper treats the analogy of Roberts' Law for Stephenson-1 and Stephenson-2 mechanisms. Besides the investigation of cognates producing identical six-bar coupler curves of genus 7, the plural generation of identically moving coupler- planes by alternative six-bar linkages is observed.
Zusammenfassung- Verwandte Sechsgliedrige Gelenkgetriebe der Stephensons- chen Mechanismen: E. A. Dijksman.
Die Arbeit behandelt ein Analogon des Robertsschen Satzes for Stephenson- Getriebe 1. und 2. Art. Neben der Untersuchung yon verwandten Getrieben, welche identische Koppelkurven vom Geschlecht 7 erzeugen, wird auch die mehr- fache Erzeugung der Bewegung einer Koppelebene durch austauschbare Sechs- stabgetriebe betrachtet.
P e ] m ~ l e - PO.9.CTBetlHblC ttlCCTH3BCHHHKH MCXaH~I3MOB CTCI~CI-ICOHa: E. ,,~H~KCMaH.
B pa6ore pa¢CMaTpHBaeTc~I aHa:IOFHR 3aKOHa Po6epTca ~t,la MCXaHH3MOB CTe(~HCOHa I H 2 THR~'I. K p o M e ~tcc.ae,aoaaHna po,.acr~eHHblX MexaHIt3MOB C H,aeHTHqHblMH raK =4a3blaaeMbtMn srtctmt,,,~H waTyHHblMH KpHBblMH ILICCTOFO po~.a, TaK)Ke pacc~,+aTpHBaeTc~ MHOFOHHC~'ICHtIOC o r p a 3 o B a H H e H,.q, CHTH4HbIX I~aTyHHblX n.r[ocKocTe~, a.rlbTCpHaTHBHbtMH WCCTH3BCHHbIMH I.HapHHpHblMH MCXaHH3- MaMH.
Summary
SIX-BAR linkages with the kinematic configuration of the mechanism of Stephenson have been investigated regarding the generation of identical moving coupler-planes and the generation of coupler-curves. The six-bars considered are those in which the fixed link is a binary one. It is shown that the knee-curve may generally be generated by a coinciding turning-joint of three equivalent six-bars of the Stephenson-1 type. More- over, there is a two-fold generation of each of the two moving links attached to this turning-joint. Special cases are observed and an application on the well-known straight- line linkwork of Hart is given.
It is further shown that each of the binary links connecting the two moving triangles of a Stephenson-2 six-bar mechanism may be generated by two different six-bars of that type. The curve generated by a coupler-point attached to such a link will be produced by four different Stephenson-2 mechanisms. The cognates are also investigated in special cases such as the case where the two moving triangles of the Stephenson-2 mechanism are similar. One may then obtain ~'-' different cognates of Watt's form.
In addition, it will be shown that the moving triangle of a Stephenson-2 six-bar, not directly attached to the fixed link, may be generated by three cognates including the
*Senior Research Officer, Eindhoven University of Technology, The Netherlands.
initial one. And the turning joint between this triangle and one of the t~vo binary links referred to a b o v e in a Stephenson-2 mechanism, m a y generally be generated b y f i c e dif- ferent cognates of that type.
Finally, it will be shown that in case this triangle degenerates into a binary link, (at the same time turning a five-sided sub-chain into a four-sided one),
afice-fold
genera- tion of a curve produced by any coupler-point attached to the binary link will occur.1. Introduction
In 1846 Robert S t e p h e n s o n designed a link reversing gear in order to control the mo- tion of a steam engine. At the time, the mechanism was e v e r y w h e r e used as a valve gear for locomotives: Although the actual mechanism of S t e p h e n s o n had very specific dimensions, Burmester[1] in 1888 associated the name o f Stephenson to any link- age of this kind. T h e kinematic configuration of such a linkage, named after Stephen- son, is shown in Fig. 1. It is a six-bar linkage with a five-sided loop and a four sided loop as two
independent
sub-chains.Figure 1. Stephenson's form.
N o w a d a y s , only historical value can be attached to the fact that S t e p h e n s o n made bar 5 the body (frame-work) o f the locomotive. (The mechanism obtained by this choice of frame is now c o m m o n l y called a Stephenson-2 mechanism.) Successively choosing other bars as frame, one obtains no more than three essentially different mechanisms (see Fig. 2). O f these, the one with three pivot centers on the frame has already been investigated by Rischen [2], who found
six
m e c h a n i s m s of this type with the same six- bar curve, generated by point E of the mechanism. All six mechanisms having the same configuration of S t e p h e n s o n and all having three pivot centers on the frame are calledcognates.
T h e n a m e cognate was first introduced by H a r t e n b e r g a n d Denavit in their paper[10, I l] on "'cognate linkages," with r e s p e c t to alternative four-bar m e c h a n i s m s , coupler-points o f which trace identical coupler curves. T h e n a m e was coined in c o n n e c t i o n with the three alternative linkages appearing in the well-knov, n t h e o r e m o f R o b e r t s - C h e b y s h e v . H a r t e n b e r g and Denavit also e x t e n d e d the m e a n i n g of cognation to special forms of six-bar linkages. It is for this reason, that the writer o f the p r e s e n t paper took the liberty to do like wise. e v e n in t h o s e c a s e s w h e r e there is no restriction on the link lengths o f the sixbars of the linkage.
Since Roberts [ 121 discovered the e x i s t e n c e of alternative linkages in the case of four-bars, m a n y authors, including C a y l e y [ 1 3 ] , Schor[14], M e y e r zur C a p e l l e n [ 1 5 ] and de J o n g e [ 1 6 ] have occupied t h e m s e l v e s with this theorem. Each of t h e m has f o u n d a n o t h e r proof o f Roberts" Law. and accordingly the n u m b e r of proofs h a s steadily increased and a situation has been arising that is similar to the one which developed a r o u n d the proposition o f P y t h a g o r a s at o n e time.
A c c o r d i n g to the spirit of t h e s e times, h o w e v e r , (with the ever-growing attention given to the practical use of the c o m p u t e r s and so on) m o s t of t h e s e efforts h a v e been in the range of analytical t r e a t m e n t in contrast to the geometrical way of finding tour-bar c o g n a t e s , as originally practised by Roberts !
It was, m a y b e , for this reason, that n o b o d y paid m u c h attention to a new geometrical proof of R o b e r t s ' Law, as given by H. P f l i e g e r - H a e r t e l [ 1 7 ] during the 2nd World War. A l t h o u g h he was the first to use the
E STEPHENSON-1 MECHANISM Figure 2. E STEPHENSON-3 MECHANISM STEPHENSON-2 MECHANISM
Stephenson-1, -2 and -3 types of mechanism.
principle of stretch-rotation in this matter, he also hesitated to rely on it entirely and therefore added a proof based on a known derivation in complex numbers.
Also, in a paper[ 18] about parallel moving bars a proof of Robert's law, based on the principle of stretch- rotation, has been given.
In the paper presented here, the geometrical principle of stretch-rotation is entirely relied on. and the application of this principle has been extended to six-bar linkages of Stephenson's form.
It may be emphasized, therefore, that in spite of the existence of computers, the quickest way to find the cognates of such forms follows the classic treatment of geometry, although ! am sure that, once we know of their existence, other, analytical, proofs may be found with which the designer will be enabled to calculate the dimensions of the cognates with the help of those same computers.
Finally, it is thought necessary to emphasize a fact which is probably not recognized, that the proofs of the existence of the six-bar cognates of Stephenson's form are also given in this paper by means of the prin- ciple of stretch-rotation: the presentation given here provides not only the way of finding cognates, but also simultaneously the proof, that the given statements are true.
In a d d i t i o n to the i n v e s t i g a t i o n o f R i s c h e n , w e a r e i n t e r e s t e d in t h e n u m b e r o f c o g - n a t e s w i t h r e g a r d to t h e r e m a i n i n g t y p e s , viz. t h e S t e p h e n s o n - 1 a n d t h e S t e p h e n s o n - 2 m e c h a n i s m . F u r t h e r , w e w a n t to k n o w h o w to d e s i g n s u c h a l t e r n a t i v e m e c h a n i s m s . T h u s , f r o m t h e S t e p h e n s o n - 1 m e c h a n i s m , for e x a m p l e , w e i n v e s t i g a t e t h e m u l t i - g e n e r a - t i o n o f t h e c u r v e p r o d u c e d b y a n a r b i t r a r i l y c h o s e n p o i n t E in t h e m o v i n g p l a n e 5. E a c h m e c h a n i s m o f the S t e p h e n s o n - 1 t y p e w h i c h is a b l e to g e n e r a t e t h e i d e n t i c a l s i x - b a r c u r v e , will t h e n b e c a l l e d a c o g n a t e o f t h e initial m e c h a n i s m s . A s i m i l a r a s s u m p t i o n will h o l d f o r c o g n a t e s o f t h e S t e p h e n s o n - 2 t y p e . It is, h o w e v e r , i m p o r t a n t to i n c l u d e in s u c h d e n o m i n a t i o n s o f c o g n a t e s a n i n d i c a t i o n o f t h e link to w h i c h the g e n e r a t i n g p o i n t is g o i n g to be a t t a c h e d ; for, g e n e r a l l y , it is n o t p o s s i b l e to g e n e r a t e t h e s a m e c u r v e b y d i f f e r e n t S t e p h e n s o n - 2 (or -1) m e c h a n i s m s if t h e g e n e r a t i n g p o i n t E is n o t a t t a c h e d to s i m i l a r l y s i t u a t e d links in t h e m e c h a n i s m s . In o r d e r to i n v e s t i g a t e e s s e n t i a l l y d i f f e r e n t c a s e s , it m u s t b e c l e a r t h a t o n l y the p l a n e 5 o f a S t e p h e n s o n - I m e c h a n i s m is o f i n t e r e s t ( T h e i n v e s t i g a t i o n o f c o g n a t e s g e n e r a t i n g t h e i d e n t i c a l s i x - b a r c u r v e p r o d u c e d b y a p o i n t o f p l a n e 4 m a y b e d o n e in a s i m i l a r w a y ) . A S t e p h e n s o n - 2 m e c h a n i s m , h o w e v e r , has t w o l i n k s , viz. t h e links 2 a n d 3, w h i c h p l a y a n i n t e r e s t i n g p a r t in o u r s e a r c h f o r c o g n a t e s .
2. The Two-Fold Generation of the Moving Plane 5 of a Stephenson-1 Mechanism and the Three-Fold Generation of the Knee-Curve*
T h e initial Stephenson-1 mechanism consists of the five-bars- OA,~FDCB,~, r>
F A B C D , the four-bar ~ A o A B B o and the rigid triangles zSAoAF. ABoBC and ekFDE
(see Fig. 3). T h e curve described by point D of the mechanism is called the knee-curve which is an algebraic curve o f order 14 and .genus 7[3]. It will be shown that this curve may be generated b, three different Stephenson-I mechanisms. T h e curve generated
E ;'5" knee - CUFVe
A~
~ C
Figure 3. Initial Stephenson-1 mechanism.
by a point E of the moving plane 5 may be generated by not more than two different Stephenson-I mechanisms. It will also be shown that the two last mentioned cognates generate the entire moving plane 5. According to Fig. 4 the design for such an alterna- tive six-bar mechanism may be obtained in the following way.
(a) F o r m the linkage parallelogramsABCB' andABBoB". (b) T u r n the four-barA.ABVBo about A through the angle.
= ~ BVAB ' = ~ B,BC. and multiply the four bars in size simultaneously fromA by
the factor j~, = B'A/BVA = CB/B"B.
(c) One obtains the four-barU]AoAB'B[~ ~ L3AoAB'Bo.
(d) One now forms the rigid quadrilateral FAAoA(~ and the rigid triangle A B'B'oC which is similar to A BCBo (The corresponding links of the four-bars A~AB'B,'j and
AoABVB,, move at the same angular velocity at any time. T h e opposite sides of the
parallelogram A B ' C B move also at identical angular velocities. T h e r e f o r e , so do the links B'B[~ and B'C. Since they also have a turning-jointB' in c o m m o n , both links be- long to the same moving plane 2' and A B'B(~C is a rigid triangle. T h e same may be said of AAB'B v. By agreement we further state that: If at any time two non-identical moving planes have the same angular velocity, such planes will beat- the same figure, and the non-identity will be indicated by different numbers of primes attached to the figure).
*The name of the c u r v e has been i n t r o d u c e d by R. Miiller in 1895. ( " U e b e r eine g e w i s s e K l a s s e yon i J b e r g e s c h l o s s e n e n M e c h a n i s m e n ' " Z. f Math. u. Phys. Vol, 40 pp, 257-278.1
+(The a u t h o r uses the term "'five-bar'" to mean w h a t is more usually called "'five-sided, or p e n t a g o n a l , l o o p " ; similarly "'four-bar" signifies a "'quadrilateral loop").
35 o"
" E
Figure 4. The two-fold generation of plane 5 of a Stephenson-1 mechanism. (e) N e x t turn the two five-bars © F A B ' C D and © FA~B'oCD a b o u t F through the angle/3 = & AoFA; and multiply the five-bars in size simultaneously by the factorf~ =
FA o / FA ;.
(f) One thus obtains a new six-bar consisting of the five-bars O FA"B"C"D" and
© FAoBgC"D", the four-bar F-IAoA"B"B'~ and the rigid triangles &AoFA", &B~B"C",
and Z~ FD"E. In this six-bar, point B; is a second pivot center on the frame of the alternative mechanism (see also Fig. 5). Like the initial one, the six-bar thus found has one degree of freedom of movement. It has also a configuration similar to a Stephen- son-l mechanism. T h e r e f o r e it may be called a Stephenson-1 cognate. T h e cognate
D"
" URVE
Y
thus found p r o d u c e s the s a m e moving plane 5 as the initial one. ] ' h a t is to say. any generating point of plane 5 will be p r o d u c e d by both m e c h a n i s m s and the generating points of the two m e c h a n i s m s are coincident points at ever,," point or" time. So both points will generate the identical curve.
2.1 General account of the presented design
A linkage m e c h a n i s m consists or several closed kinematic sub-chains or loops. F o r instance, the S t e p h e n s o n - I m e c h a n i s m has two five-bars and one tour-bar.
T h e s e sub-chains, h o w e v e r , m a y be seen as closed v e c t o r polygons, which can be r e p r e s e n t e d by equivalent v e c t o r equations.
Let the v e c t o r equation
a ~ + a ~ + a : : , - v a , ) = O i l )
represent the f o u r - b a r d o A BBo, and
bl + ba + b 4 -c- ba + ao = 0 (2)
represent the five-bar A,~FDCBo (see figs. 4 and 5.) (The o r d e r of these v e c t o r s m a y not be interchanged, otherwise other linkages are represented. F o r instance, the equation al + aa + a._, + ao = 0 r e p r e s e n t s the fotw-barAoAB'Bo). T h e n the equation
(bl - a~) + ba + b , + (ba -- aa) -- aa = 0 (13)
r e p r e s e n t s the second five-bar A FDCB.
It is clear that the last equation is not i n d e p e n d e n t of the first two. T h e r e f o r e , we say that a S t e p h e n s o n - I m e c h a n i s m consists of two independent sub-chains, viz. a four-bar and a five-bar,
If we multiply* equation (1) by the c o n s t a n t c o m p l e x n u m b e r ba/aa and subtract the result from (2), we get
bl -- (ba/aa)al + bs + b 4 - - (ba/aa)a~ + a0(l -- ba/aa) = 0 (4) which is a five-bar of the form
C 1 -~ b 5 ~- b 4 @ c 2 At- c 0 = 0 . (4a)
This r e p r e s e n t s the five-barA;FDCB[~ (see Fig. 4).
Interchanging the s e q u e n c e of the bars in the kinematic chain r e p r e s e n t e d by equa- tion (3), means a p e r m u t a t i o n in the s e q u e n c e of the t e r m s in this equation.
T h e r e f o r e , the equation
(bl - - al) + ba + b 4 - - ae + (ba - - a 3 ) = 0 (5) r e p r e s e n t s the five-barA FDCB'.
Both five-bars r e p r e s e n t e d by the equations (4) and (5) are i n d e p e n d e n t of each other. If we subtract the left-hand side of (5) from (4), we obtain the v e c t o r equation of
the four-bar A ~A B' B0
a l ( 1 - - b J a 3 ) + ( a 3 - - b 3 ) + a ~ ( 1 - b 3 / a z ) + a 0 ( 1 - b J a 3) = 0 . (6) The link
A~Bo
of the five-bar (4) and of the four-bar (6), however, is only atranslating
link. It can be made
afixed
link if we multiply the left-hand sides of (4), (5) and (6) by the complex number bl/(bt - (ba/a3)al), for then the point A; of the linkA~B'o
which is already only translating, transforms itself into the fixed center Ao. And so the cognate Stephenson-1 mechanism will be obtained. The center of similitude of the last multi- plication coincides with turning joint F. Therefore, the transformed moving planes 1 and 5 remain attached to the initial ones.The multiplied five-bar (4), for instance, will then contain the vector bl again, that is the bar
AoF,
of the initial mechanism. Thus, thedyadAoFE
is a common dyad to both cognates.The fact that no other cognates for the generation of the curve produced by point E exist, will be proved by indirect demonstration (reductio ad absundum).
The generating point E is connected with the fixed centerA0 through the
dyadAoFE.
This dyad is represented by the vector sum (bt +d~). It is not possible to find a cognate by interchanging the terms in this vector sum. This is due to the fact that a four-bar link- age of which the sides move p~,rallel to the respective planes 0, 2, 3 and 5, cannot be composed. Likewise, it is prohibited to interchange the moving planes 3 and 4. And lastly, the fixed link 0 may never be interchanged, since, generally, no other moving plane of the initial mechanism moves parallel to the fixed link.
We assume that in a supposed cognate, point E has to be connected with the fixed link through some dyad. Instead of going through all the arising possibilities we just take one dyad and prove that such an arbitrarily chosen combination cannot be effectu- ated. Suppose, then, that the dyad that connects point E of the cognate with the fixed link, is represented by the vector sum (ea + e4). Here, e3 is a vector moving parallel to a fixed line in plane 3, and e4 moves parallei to plane 4. Although the fixed center does not necessarily coincide with A 0 or Bo we may nevertheless indicate a hexagon of which the triad
BoCDE
takes a part.So we may write
e o + e z + e 4 + ( b ~ - - d s ) + b 4 + b 3 = 0
or by interchanging the terms through making use of linkage parallelograms, we get the four-bar
eo + (e3 + bz) + (e4 +
b4)
"I"- (b5 -- ds) = 0.Since it is not possible to design a four-bar with links moving parallel to the planes 0, 3, 4 and 5, we encounter a contradiction. Therefore, the assumption made that the dyad mentioned may be represented by the vector sum (e3 + e4), is not valid.
In this way going through all other combinations we find that, unless E coincides with D, there is only one dyad possible, viz. the dyad bl + d s , the one used in the cognate already found.
In the case that E coincides with turning joint D, we may also consider this point a generating point of the moving plane 4. Since the latter takes a similar position in the Stephenson-I mechanism, we may also interchange the planes 1 and 2 instead of the
planes 3 and 2 as was done for the preceding cognate. We thus obtain a n o t h e r cognate for the k n e e - c u r v e as d e m o n s t r a t e d in Fig. 6. So all in all there are three cognates gener- ating the knee-curve.
0 KNEE-CURVE
/ \
1 \ eo ,' - - ~ f ~ / ' i I
a ~%0B; - ' % 0~
Figure 6. Two, out of three, cognates for the knee-curve. F r o m the design s h o w n in Fig. 4 we derive that
and
0 F A " B " C " D " ~ 0 F A B ' C D :
K] A , , A " B " B[/ ~ ~ A,'~A B ' B,;.
C' F A o B , ; ' C " D " ~ 0 F A ; B D C D .
It also follows that
and
F A A " ~ A FA[IAo ~ z% F D D "
A C " B " B ( / ~ A B o B C ~ ~ A o . 4 A / ,
Similarly, Fig. 6 d e m o n s t r a t e s that A F~A ~A 2 ~ A A, A F.
2.2 S p e c i a l c a s e w h e r e ~ A . A F ~ ~ B o B C
In this case A/~ ~ F and the preceding design collapses. So a n o t h e r a p p r o a c h is a d o p t e d at the point where first the second multiplication could not be brought into effect. T h e design for this case m a y be carried out with the following instructions (see Fig. 7):
(a) F o r m the linkage parallelograms A B C B ' . A B B o B ' . A o F E F " , A o F D F ~ and
B o C D C ' .
(b) T u r n the f o u r - b a r A , F 4 B ' B , about A through the angle c~ = L B ' A B ' = ~ B o B C = L A o A F and multiply the size of the fc, ur-bar geometrically fi'om .4 by the factor
,, 39 '
\'@
F\
F'{
/ / ~ . X ., BoB,,
Figure
7. Special case of the Stephenson-1 mechanism with A AoAF ~ A BoBC.(c) One obtains the four-bar [] F A B'B'o ~ [] A o A BVBo. (d) One forms the rigid triangle B ' B ~ C ~ A B C B o ~ A A F A o . (e) Next, one forms the linkage parallelogram B ~ F D F ' .
(f) Then, turn the five-bar O A o F A D C ' B o about Ao through the angle/3 = ~ D F E = F A A o F '' and multiply the size of the five-bar geometrically from Ao by the factor ft3 = E F I D F = F"Ao/FZ~Ao .
(g) One obtains the five-bar © A o F " D " C " B o' ~ 0 A o F ~ D C 'Bo. (h) Form the rigid triangle A D " F " E ~ ~ E F D .
(i) Make the four-barD AoB[/ B " A " ~ [] F ' DCB~. (j) Form the rigid triangles F " A o A " and C "B'o'B".
The cognate obtained consists of the five-bars © A o F " D " C " B ' ~ , O A " F " D " C " B " , the four-bar [] AoB~;B"A" and the rigid triangles A A o F " A " , A B~C"B" and & D " F " E (see also Fig. 8).
In order to justify the correctness of the above result, it is sufficient to prove that the triangles to be indicated as rigid triangles are rigid indeed. In all cases arising, this may be done by proving that two sides of such a triangle are moving at identical angular velocity.
With regard to the geometric properties of the cognate obtained, one can prove that A A o F " A " ~ ~ A F A o, and likewise that A BoC"B" ~ & BCBo. Thus
A A o F " A " ~ A B~'C " B " .
So we see that for the initial mechanism as well as for the cognate one, the two triangles rotating about a fixed center are similar.
0
Ix, ~ \ s 1
\
/
Figure 8. Special case of the Stephenson-1 mechanism with &A,~AF ~ &BoBC.
We also find that
15 D " F " E ~ ~ E F D ~ ~ B,'~'AoB,,
2.2. I. A p p l i c a t i o n on a s t r a i g h t - l i n e linkwork o f H a r t . An exact straight-line link- work of Hart[4], which is a special case of the mechanism of Kempe[5], which is also called the focal mechanism of Burmester[6], may be seen as a Stephenson-1 mechanism for which A A o A F ~ z5 BoBC. The four-bar A , A B B o in this linkwork (see Fig. 9) is chosen arbitrarily. ~P~H OF 0 PERPENOICULAR [ . . . Lc : o
ro
A o B u/"x, "(~
I q z: F . -~ A\ i \, \ ! \ \ I \ \ / \ ' \ \ I " \ - -F \ k ~ 6 #' = ~- / / ' ~ 0 '/ Do . . . -_ LWe took A0A = 1. AB = 4, BBo = 3 and AoBo = 5. T h e remaining dimensions of the
linkwork were calculated from formulae obtained by Wunderlich[7], and so we got
AoF = 25/9, BoC = 75/9, FD = 20/3 and CD = - 20/9. T h e initial linkwork of H a r t so
consists of the five-bars AoFDCBo, A F D C B and the four-bar AoABBo. T h e knee-curve
generated by point D is in this case an exact straight line perpendicular to the frame- line AoB o. M o r e o v e r , the angle 4_ AoFD =- ~ BoCD as shown by Kempe.
If we apply the design procedure of the preceding section to this linkwork, we obtain the six-bar cognate consisting of the five-bars A o F " D " C " B o , A " F " D " C " B " and the
four-bar Ao A " B" Bo.
It follows from the design that [] AoFDF" and [] BoCDC" are linkage parallelo-
grams. It may be proved that A " B " lies parallel to AB at any time. Further, one finds
that the points D, A and A" are always in line, and the same is true for the points D, B and B". We see that the cognate linkwork is also an exact straight-line linkwork of Hart,
thus c o n n e c t e d with the initial one (see Fig. 9).
In addition, we observe that the cognate linkwork may also be obtained by making the coupler AB of the initial linkwork the fixed link instead of AoBo, by simultaneously
multiplying the entire linkwork by the factor CD/AF and, finally, by turning the link-
work thus obtained about some straight axis in the plane around by rr radians. This proves that the turning joint D of the initial mechanism also m o v e s along a straight line perpendicular to the c o u p l e r A B [7].
3. The 4 Cognates for a Stephenson-2 Mechanism with Respect to a Point E of Plane 2
3. I. The two-fold generation of plane 2 and the 3/4-cognate for that plane
T h e initial Stephenson-2 mechanism consists of the five-bars AoABCCo, AoFDCCo,
the f o u r - b a r N A B D F and the rigid triangles A AoAF, A BCD and A FDE. T h e centers
of pivot on the frame in this mechanism are the points Ao and Co (see Fig. 10). Design instructions for what is called the 3/4-cognate are as follows (see Fig. 11):
(a) F o r m the parallelograms A BDB' and A BCB".
(b) T u r n the five-bar 0 AoA BVCCo about A through the angle a = 4_ B%4B' = ~ C B D
and multiply the five-bar geometrically from A by the factorf~ = B'A/BVA = DB/CB.
(c) One obtains the five-bar OA'oAB'C'C'o ~ O A o A B ' C C o and the rigid triangles A AoA~A and A B ' C ' D .
(d) N e x t , turn the five-bar O FA~C~C'D, together with the four-bar F A B ' D about F
through the angle/3 = 4. A~FAo and multiply both geometrically from F by the factor fa = FA,,/FAo.
(e) One thus obtains the five-bar O FAoC~'C"D" ~ 0 FA~C~C'D and the four-bar [] FA "B"D".
(f) Lastly, form the rigid triangles S AoA"F, S B " C "D", A FD"D and A FD"E.
Co " " C : s :;
/f:'/
i~O " " "~ 'ala 2 - - ', 5 / ,, ~ o 20Figure 11. The 3/4-cognate of a Stephenson-2 mechanism for plane 2.
One recognizes the cognate six-bar, consisting of the five-bars O A , A " B " C " ( ' i / . © A , F D " C " C ~ ( . the four bar [] A " B " D " F and the rigid triangles k, A . A " F . 2X B " C " D ' . LxFD"D and & F D " E .
T h e cognate obtained has the same kinematic configuration as a Stephenson-2 mechanism (see Fig. 12). T h e angular velocities of the links in the cognate can at any time also be o b s e r v e d in the initial mechanism. With the exception of the angular ve- locities of the links 3 and 4, they are distributed in the same way among the links as is the case with the initial mechanism. One finds that only the angular velocities of the links 3 and 4 have been interchanged in the cognate. T h e r e f o r e , the cognate is called a
3 / 4 - c o g n a t e for short.
T h e other thing one observes is that the entire plane 2 of the initial mechanism is generated by the 3/4-cognate. T h u s a two-fold generation of the moving plane 2 exists. A six-bar curve p r o d u c e d by an}, point E of this plane will then be generated by both cognates. T h e same holds for turning joint D of the initial mechanism (see Fig. 12). One notes also that the dyad A ~ F E is always c o m m o n to the initial and to the 3/4-cognate.
T h e geometric properties of the 3/4-cognate may be derived through the following observations (see Fig. 1 I):
~ B DC ~ & B C D AB"D"C". A A A , , A , ; ~ A ' "' F r o m (d) we derive that © F A , , C , ; ' C " D " ~ 0 FA(,C[~C'D and E / F A " B " D ' " -- ~ F A B ' D . c s ci~ 8!i
Further we see that
A FAoAo ~ A F A A " ~ A F D D " and thus A FAA'o ~ A F A " A o .
Moreover.
Thus
Finally:
L (AA[),A"Ao) = [3 and ~_AoAA'o = C~.
L A A o A " = c~ + ~ = ~ CoAoC o'.
A,)Co
Aoc,,
. A ; c ; = r _ , A,~4 ..4;,.4 _ .4o.4A , , c ; =
A;C'o AoCo
" "f ; ' =,4'o.4 A(,A" AoA"
T h e r e f o r e , we conclude thatA A o A A " ~ A A,)CoCI;.
in the special case where A F D E ~ AFA~)A~) we have that E --- D", in which case the generating point E b e c o m e s a turning joint of the 3/4-cognate.
In the special case where A A o A F ~ A C B D , one finds that A[) = F. In that case instruction (d) can no longer be followed. Cognates may then be obtained as in the Sec- tions 3.2.1 and 3.3. I. One finds a total number of oc" cognates in this case.
3.2. T h e 1~2-cognate o f a S t e p h e n s o n - 2 m e c h a n i s m (with E a p o i n t o f p l a n e 2)
The initial mechanism being the same as in the preceding section, the design instruc- tions for what is called the I/2-cognate are (see Fig. 13):
a-
(a) F o r m the parallelograms A o F E F ' , A o F D F v and A F D F ' .
(b) T u r n the five-bar A o F r D C C , ~ about A,~ through the angle c ~ ' = ~ F ' A o F - : = 4. E F D and multiply the five-bar g e o m e t r i c a l l y by the t'actorf,, = F ' A J F V A o = E F / D F . T h u s @ A o F ' D ' C " ' C , ; " ~ O A o F v D C C ~ .
(c) F o r m the rigid triangles A A o C / / ' C o and A D ' F ' E . (d) M a k e ~ A " ' B " ' D ' F ' ~ ~ A B D F ' " .
(e) F o r m the rigid triangles A C " ' B " ' D ' and A A " 'F'A,~.
T h e cognate obtained consists of the five-bars O A , , A " ' B " ' C " ' C , ' / ' . O A o F ' D ' - C"'CI;'. the f o u r - b a r ~ A " ' B " D ' F ' and the rigid triangles A A o A " ' F ' . A B " ' C " ' D ' and the coupler-triangle A D ' F ' E . T h e alternative m e c h a n i s m has the same kinematic configuration as the initial m e c h a n i s m and is therefore a cognate S t e p h e n s o n - 2 m e c h a n - ism. H e r e , only point E of the cognate g e n e r a t e s the identical six-bar curve.
When one o b s e r v e s the distribution of angular velocities in the cognate obtained, one sees that only the angular velocities of the links I and 2 of the initial m e c h a n i s m have interchanged. T h e r e f o r e , the cognate is called a l / 2 - c o g n a t e for short (see Fig. 13). T h e g e o m e t r i c properties derived from the figure are obtained in the following way:
F r o m (b)it follows that
A C o " A o C o ~ A E F D ~ A D ' F ' E . Since ~ A " ' F ' A o = £ A o F A and A " ' F ' = A " ' F ' D'F__~'= F___D_D D ' F ' F ' E = F D E F F'__.EE=F'___ff_E=AoF F ' A o D ' F ' E F F A F ' E E F F A F D E F F A F A we find that A A " ' F ' A , , ~ A A o F A . Since 4 ( D ' C " ' . D C ) = ~ ' and 4_(BD, B " ' D ' ) = c z ' + r r - £ A " ' F ' A o B " ' D ' C " ' = 2,"r- ( £ B D C + /> A o F A ). M o r e o v e r , we have B"'D' B"'D' F'D' CD . A o F = DB A o F C " ' D ' F ' D ' A o F C " ' D ' C D C D F A " H e n c e , i f A B " C " D " ~ A B C D and B" -= A and D ' =- F w e find that
A B " ' D ' C " ' ~ A A a F C v See also Fig. 13.
3.2.1. S p e c i a l case, w h e r e A A , ~ A F ~ A C B D . In this case we find that A0 ~ C ' and so B " ' =- C " ' . So point B " ' of the l / 2 - c o g n a t e describes a circle about the fixed c e n t r e C o . O n e thus obtains a cognate o f a special form, c o n s ~ s t m ~ o t O A o F D C Co . ~JAoA"'C"'C',;', [ ] A " ' C " ' D ' F ' and the c o u p l e r triangle A F ' D ' E (see Fig. 14). Such an alternative m e c h a n i s m m a y be regarded as a special form of Stephenson-2, but also as a special m e c h a n i s m of W a t t ' s form. In addition, we know[8] that for a m e c h a n -
45
A .*0
( ~
C'- B-
Figure 14. The 1/2-cognate of a Stephenson-2 mechanism in case
/k A o A F ~ /k C B D .
Figure 15. Special cognate of mechanism.
ism of Watt's form, the entire moving plane in which point E is fixed, may be generated by ~o2 different mechanisms of Watt's form. See for example the cognate obtained in Fig. 1 5, where point E is made a turning joint between the moving links F ' E and B Y E
(This cognate consists of the four-bars []AoAVCVC70 , [] F ' E B V A v and the rigid tri- angles A A o F ' A v and A A V B V C V ).
The cognate obtained is a special one, because 8 = ,5- D ' F ' E and f8 = F ' E / F ' D ' .
Should other values for 8 and f8 be chosen, one of the ~'-' different cognates should be obtained. I I I I I Co
3.3. The 1/2-3/4 cognate o f a S t e p h e n s o n - 2 m e c h a n i s m ( with E a point o f ptane 2 ) Starting with the same initial mechanism as in the preceding Sections 3.1 and 3.2, the design instructions for the cognate presented here are (see Fig. 16):
(a) Form the linkage parallelograms A o F E F ' , A B D B ' and A B C B " and also the rigid triangle A A B " B ' .
(b) Turn the four-bar ~ B ' D F about A through the angle c~ = ~_B"AB' = & C B D and multiply the four-bar geometrically from A by the factorf~ = B " A / B ' A = C B / D B ,
(c) One obtains the f o u r - b a r ~ , A B ' D Z F ~ ~ [ ] A B ' D F . (d) Form the rigid triangles A o A F ~ and CBVD n.
(e) Form the linkage parallelogram A o F Z D Z U and the rigid triangle F ' A o U .
(f) Turn the five-bar Q A o F D n C C o about A,~ through the angle 5' = & F ' A o F t and muhiply the five-bar geometrically from A0 by the factorf~ = F ' A o / U A o = E F / C ° F .
(g) One obtains the five-bar @ AoF'D~C~Co ~ ~ @ AoFtD~CCo. (h) Form the rigid triangles A A,~CoC/and A D ' F ' E .
Fig 1 6 A Fig 16 B O-:O b E
~ C
b
F_ F ~ F t b Co C s A s B' . -~---~--~. - -' . . . . " ~ ' I - ' ( 1 ",'-I ~-o,7_i
/ 'C'
)s\\
/ B s(i) Make the four-bar [] F'D'BSA" ~ [] F A B D . (j) Form the rigid triangles B ' C ' D ~ andAoA'F'.
The alternative six-bar obtained consists of the five-bars ©AoF'D'C*Co" and © AoASB'C*Co ~, the four-bar [] A ' B ' D ~ F ' and the rigid triangles A BsC*DL A AoA~F ' and the coupler triangle & D*F'E.
The alternative six-bar has a kinematic configuration similar to the initial mechanism and may therefore be called a Stephenson-2 cognate. Here. only point E of the cognate generates the initial six-bar curve produced by the initial Stephenson-2 mechanism.
The angular velocities of the links l and 2 and also those of the links 3 and 4 are interchanged when one compares the distribution of angular velocities in the obtained cognate with the distribution in the initial six-bar. Therefore, the cognate is called a l / 2 - 3 / 4 - c o g n a t e (see Fig. 17).
The geometric properties of this cognate may be obtained from Fig. 16 in the following way. F" A s A o
"} ~
_-7---= " B \©
Co ~/-8
D s C s B sFigure
17. The 1 / 2 - 3 / 4 c o g n a t e of a S t e p h e n s o n - 2 m e c h a n i s m . JM VoL 6. No. I - DF r o m
A " F ' A " F ' D E D'~F ' D z U D F F'A,, F - A o D F
~ . ~ o
F ' A o D F F'Ao D ~ U A F F'A,, FtAo A F F'Ao D F F~Ao A F F z A o F'-'Ao D A F ~ A F A F ~ A F A F A" a n d t h e f a c t t h a t A F Ao ~ ' = ~- A o F Z A . W e d e r i v e t h a t A ' F Ao A A o F ~ A • F r o m t h e s a m e f i g u r e w e a l s o h a v e S i n c e a l s o A F ' A o U ~ ~ Co~A,,C. w i t h ,'y = 2,"r -- ( 6- E F D + 6- D B C ). F ' A o = E F . F D = E F . B D A o U F D FZXD :~ F D B C w e find t h a t a s l o n g a s A E F C ° ~ A F ' A o U A B ° C ° D ° ~ A B C D w i t h D o =-- D a n d B ~ -= F . Hence, DSB ~ I)~F ' D a F t E F FZXAo -- , __ o A B - - D A F t A F F C ° A F D s C ~ D A C A F C D A F F A F D~B ~ A B FAAo B D F a A o F A o M o r e o v e r , T h u s A CoMoCo ~ A E F C ° ( s e e F i g . 1 6 A ) . From Fig. 16 w e s e e t h a t A A F A F ~ z~ B C D ~ A B " D a C w i t h s t e p g o f t h e d e s i g n i n s t r u c t i o n s w e h a v e Ds C "~ E F DZxC -- F C o ,
On the o t h e r hand, one may o b s e r v e that L ( D S B L A B ) = 4- ( F ' D " , F A ) = - y + & ( F t D A, F A ) = - - 7 + 4 - ( A o F A , F A ) and ( D ~ C L A B ) = - - y + L T h u s L B ' C ~D~ = 4- A o F F ' .
And so, finally, we have
A B~C~D ~ ~ A A o F F zx.
E F ' FAo DZXF t FAo F t A o
D S F ' DZXF t D'*F ' F Z X A o F ' A o Since
( D ~ C , C B V ) = - - 7 + ~- ( F AF. F A ).
and 4- E F ' D s = - - ~ £ F~XAoF+ ( 2 z r - 7 ) = ,5- F A o F 'b, we conclude that A E F ' D s ~ A F A o F ' b .
H e r e , the point F 'b is defined by the relationship A A o F A F 'b ~ A A o F t F ' ~ A F C b E (see Fig. 16B).
F o r the particular point E defined by the relation A F D E ~ A B D C , one finds that E = C b . T h e n y = 0 a n d f r = 1. C o n s e q u e n t l y T h u s C o s ~ Co~ C s = C , D s = D ~, F ' = F t. A o F t _ F A D A FZXDZX D B F D D B C B D F C B A " F t C B C D zx C D zx B A D C B A D C D B B A C D - B S D ~ CD" M o r e o v e r A F r E D zx ~ A A o F F zx ~ A B s C D zx.
In this particular case, therefore, the " c o u p l e r - p o i n t " E of the 1/2-3/4 cognate is defined in a way similar to the one in the initial six-bar (see Fig. 18).
3.3.1. S p e c i a l c a s e , w h e r e A A o A F ~ A C B D . In this case we have F zx = A0. T h e r e f o r e , the m e t h o d described in Section 3.3 collapses. T h u s another method has to be developed, of which the design instructions are as follows (see Fig. ! 9):
(a) F o r m the linkage parallelograms A o F E F ' , A B D B ' and A B C B " and also the fixed triangle A A B " B ' .
(b) T u r n the four-bar [] A B ' D F about A through the angle a = ~ B " A B ' = ~ C B D = 4- A o A F , and multiply the four-bar by the factorf~ = B " A / B ' A = C B / D B = A o A / F A .
~S
O
j,p,-0.
a~Figure 18. The 1 / 2 - 3 / 4 c o g n a t e of a S t e p h e n s o n - 2 m e c h a n i s m in case E is b o u n d by: S F D E ~ A B D C .
Ds R~ c • - i
F i g u r e 19. T r a n s f o r m a t i o n of a S t e p h e n s o n - 2 m e c h a n i s m , for w h i c h S A,~AF ~ A C B D , into o n e of z-' W a t t ' s s i x - b a r c o g n a t e s .
(c) One obtains the f o u r - b a r D A B ' D ' ~ A o ~ [] A B ' D F .
(d) F o r m the rigid triangle A C B " D £
(e) T u r n the four-bar [] A,~DaCCo about A0 through an arbitrarily chosen angle y -
D"AoD ~ and multiply the four-bar geometrically with an arbitrarily chosen factor
f~ = D.~Ao/D~Ao"
(f) One obtains the four-bar [] AoD°'C'~C% ~ [] A o D ~ C C o .
(g) F o r m the rigid triangle ~ A o D ~ F ' .
(h) Make the f o u r - b a r D A ~ B ~ D " F ' ~ D A B ' D F .
(i) Finally, form the rigid triangles F 'A*E and B~C*D't
T h e cognate obtained consists of the four-bars [] AoD'~'C'~C,( and C]A~B~D"F ' and the rigid triangles ZX F ' A " E , A B*C.'D ~ and A AoD'~F '.
Since y and alsoJ~, may be chosen arbitrarily, one finds :~ cognates of Watt's form. This result is in agreement with the known* fact [8] that a mechanism o f Watt with point *This fact is only recently known, since the paper concerning the six-bar cognates o f Watt's type has only just been accepted for presentation in the coming Fall 1970 at the A.S.M.E. Conference on Mechanisms, Columbus, Ohio, U.S.A. tt should also be noted that a special case of these cognates has already been in- vestigated by A. H. Soni[9]. He found cognates of W a n ' s type in case one of the four-bar sub-chains of the source mechanism ',,,'as parallelogram-shaped.
E arbitrarily chosen in the plane as shown in Fig. 20, has indeed :~z cognates of this type.
A special cognate of this type will be obtained by making the generating point E at
the same time a turning joint of the mechanism. The resulting mechanism will then be the same as the one obtained in Fig. 15, since the mechanism obtained through this section, will have the same distribution of angular velocities over the links, and the points A0 and E will be identical points for the two resulting mechanisms. (The latter remark connects the result of the section under consideration with that of Section 3.2.1. Therefore, it is of little importance which design procedure one prefers: in the end one obtains the same mechanism.)
oS ~ C s
F
Figure 20. Transformation of a particular Stephenson-2 mechanism, for which
/k AoAF ~ A C B D , into one of the == Watt's six-bar cognates.
3.4. General remarks about the 4 cognates of the Stephenson-2 mechanism with E a
point of plane 2
Including the initial six-bar, we now have 4 cognates o f a Stephenson-2 mechanism. One may prove that the 3/4-cognate of the 1/2-cognate is identical to the 1/2-314 cognate. One may also prove that the 1/2-cognate of the 3/4-cognate turns into the
1/2-3/4 cognate. And, finally, that the 1/2-3/4 cognate of the 1/2-cognate becomes identical to the 3/4-cognate of the initial mechanism. It is clear that all transformations are reversible.
In a similar way as the one of Section 2.1 one may prove that no other cognates exist than the 4 already obtained. In order to shorten the length of the manuscript, all proofs of the facts just mentioned have been omitted and the reader is invited to compose them himself.
4. The Three-Fold Generation of Plane 3 of a Stephenson-2 Mechanism
4.1 The l/2-cognateforplane 3
The initial mechanism is the same as the one of Section 3 and consists of the five- bars © AoABCCo and O AoFDCCo, the four-bar I-1ABDF, the rigid quadrangle B C D E and the rigid triangle A A o A F . The instructions for the design of the 1/2-cognate are successively (see Fig. 21):
(a) Form the parallelograms AoFDF v and A F D F ^.
(b) Turn the four-bar f-IDFAAB about D through the angle a = 4 F ^ D F v =
,~- AFAo and multiply the four-bar geometrically from D by the factor f,, = FVD/F^D = AoF]AF.
5
,
/
\
..:-, z__2, t £ ) 1 0 7 / / . . . . ~,
'
~
C O
Figure 21.
The three-fold generation of plane 3 of a Stephenson-2 mechanism (the 1/2 cognate).(c) One obtains the four-barlN D F V A V B v ~ [] D F A A B .
(d) Finally. one forms the rigid triangles F V F A D . A V A o F ~, B V C D and C D E .
T h e I/2- c o g n a t e obtained consists of the five-bars O AoAVBVCC,~ and O A o F V D C C o .
the four-bar [] A T B ~ ' D F v and the rigid triangles A A T A o F v , / X B V C D and A C D E .
O n e may o b s e r v e that
B V B D ~ Z ~ A o A F ~ 2x F V F A D - A A V A o F 'v.
In the special case where z~AoAF ~ S C B D , o n e finds that B v -= C a n d the cognate has the similar relation S &~A'VF v ~ A B C D .
O n e also notes that the cognate six-bar g e n e r a t e s the entire moving plane 3 and also the curves produced by any point E of that plane.
4.2. T h e 1 / 4 - c o g n a t e f o r p l a n e 3
Starting with the s a m e initial m e c h a n i s m as in the preceding section, one obtains the 1/4-cognate. using the instructions (see Fig. 22):
(a) F o r m the parallelograms A o A B A ~ and F A B A ' .
(b) T u r n ~ B A ' F D about B through / 3 = 4 _ A ' B A ~ = 4 - F A A o and multiply the four-bar byf~ = A i B / A 'B = A o A / F A .
(c) One obtains the f o u r - b a r D BAZXF"D '' ~ [] B A ' F D .
(d) Finally. one forms the rigid triangles A ' B A ±, A,,AZ~F". D B D " and B C D " .
T h e l / 4 - c o g n a t e obtained consists of the five-bar O A o A A B C C o and O A o F " D " C C o ,
the four-bar 3 A ~ B D " F '' and the rigid triangles A , v 4 ± F '', C B D " and l C D " E .
O n e o b s e r v e s that A F V A o A v ~ A F A A , , ~ A ' B A L ~ A A o A £ F " ~ A D B D " .
C o m p a r i n g the l / 4 - c o g n a t e with the 1/2-cognate, one finds that D" =- B v.
In the special case where A A o A F ~ A C B D , one sees that D" == C ~- B 'v and c o n s e q u e n t l y ~ A o A - F " "~ S D B C .
Ao ~
~ Bv-D ''
Co
Figure 22. The three-fold generation of plane 3 of a Stephenson-2 mechanism
(the 1/4 cognate).
By indirect demonstration (reductio ad absurdum) one may easily prove that no other cognates for an arbitrarily chosen point E of plane 3 exist. Thus, including the initial mechanism, there are three Stephenson-2 mechanisms generating the identical curve produced by any point E of the moving plane 3.
5. The Five-Fold Generation of the Curve Produced by Point D of a Stephenson-2 Mechanism
The curve generated by point D of the mechanism is algebraic of order 16 and has the genus 7[3]. Turning joint D of a Stephenson-2 mechanism may be regarded as a special point both of link 2 and of the moving plane 3. Therefore, the design instructions of the two Sections 3 and 4 may be applied.
In case E -- D, however, the I/2-cognate of Section 3 turns into the l/2-cognate of Section 4.
Therefore, we have the five cognates:
(a) the initial six-bar with D a turning joint between the moving planes 2 and 3 (see Fig. 10),
(b) the 1/2-cognate with D being a similar turning joint (see Fig. 21 ), (c) the 3/4-cognate with D merely a specific point of plane 2 (see Fig. 12),
(d) the l/2-3/4-cognate, also with D a specific point of a similarly situated plane in the kinematic configuration (see Fig. 23), and
(e) the 1/4-cognate, where D is merely a specific point of plane 3 (see Fig. 22). There are no other cognates obtainable. With two cognates, point D turns out to be a turning joint. In the remaining three cognates, point D is merely a specific coupler- point attached to some moving link.
The six-bars listed under a, b, c and d are cognates discussed in Section 3. The cognates listed under a, b and e are of the kind treated in Section 4.
cl
_ C ~ < 4 ~ ~ . . . As ~ \---_~
\/
Co
~
-B s6. The Five-Fold Generation of a Coupler-Point E, Attached to Link 3 of a Stephenson-2 Six-Bar, in Case B -= C
T h e m e c h a n i s m under consideration is s h o w n as the initial six-bar in the Figs. 2 4 - 2 7 . It consists of the four bars
AoABCo
andABDF
and the rigid trianglesA,~AF
andBDE.
T h e six-bar c u r v e c o n s i d e r e d is the one p r o d u c e d by the coupler-point E. T h e m e c h a n i s m shown m a y be r e g a r d e d as a S t e p h e n s o n - I six-bar, as a S t e p h e n s o n - 2 m e c h a n i s m or as a Watt m e c h a n i s m . In all cases o b s e r v e d , cognates can be found, but s o m e of them are identical, and for this reason the total n u m b e r of cognates will be restricted to five. If the m e c h a n i s m is t a k e n for a Stephenson-1 m e c h a n i s m , the one cognate obtained in that case will be aI/4-cognate,
also obtainable if the initial mechan- ism is taken for a S t e p h e n s o n - 2 six-bar. As s h o w n in Section 4, three c o g n a t e s m a y be found, as long as the initial m e c h a n i s m is regarded as a S t e p h e n s o n - 2 six-bar. T h e s e are the initial m e c h a n i s m , the 1/4-cognate obtained in Fig. 24. and theI/2-cognate
shown in Fig. 27. T h e l / 4 - c o g n a t e , mentioned before, will also be obtained, if the
. . . : :
Z c-B
O"- B"
Figure 24. The 1/4 cognate for plane 3 if B --- C. (The five-fold generation of the curve produced by point E of the plane).
C ~
v ~ ~ - . - - ~ - " 7 x'.,
Ao Co
Figure 25. Transformation of a particular Stephenson-2 mechanism (with B --- C) into a mechanism of Watt's type. (The 2/3-cognate).
2: ~ . '4 v . / ' F : ' h v_ M ~ 1 - - - \ / A 1 ] x '~ :x ,
A o ~
,",_
,y
c~
"x \ \ \ \ \ \ \ \ \ '\ ,\ " 4 C xFigure 26. 1 / 4 - 2 / 3 cognate in case B -= C. Five-fold generation of the curve.
0
@
F
Figure 27. The 1/2-cognate for plane 3, if B ~ C. (The five-fold generation of the curve produced by point E of that plane).
initial mechanism is looked upon as a special configuration of the Watt mechanism [8]. But besides that, we also find two o t h e r cognates of Watt's type of the initial mechanism. T h e s e are the 2~3-cognate (see Fig. 25) and the 1 / 4 - 2 / 3 c o g n a t e tsee Fig. 26). In designing the Watt's 2/3-cognate of the 1/4-cognate, one obtains the 1/4-2/3 cognate again. And finally, the l/4-cognate taken from the Watt's 2/3-cognate also produces the 1/4-2/3 cognate.
T h e design procedure for the 2/3-cognate will here be given briefly (see Fig. 25): (a) F r a m e the parallelograms F D C D zx and F D E D ; ~ .
(c) Form the rigid and mutually similar triangles F A A * , F D / ~ D * and D * E B * (all similar to ~, D C E ) .
(d) Make the four-bar [] A o A * C ' C * ~ [] A o A C C o .
(e) And finally, form the rigid triangles Ao A * F , A u C o C ~ and A * B ' C * . The mechanism obtained is indicated in Fig. 25 by thick solid lines.
• , ) ,-i
The design instructions for the 1/4-_/.~ cognate are briefly (see Fig. 26): (a) Make the four-bar[] C E F ' A " ~ [] C D F A .
(b) Frame the rigid triangle /~ A A V C . the parallelogram A o A A V A y and the rigid triangle A F V A ' A ' .
(c) Make the four-barf--1 F ' A ~ ' C ~ E ~ ~ ~ F A C D .
(d) Frame the rigid triangles A o A ~ C ~ and A F r E e E , the linkage parallelogram
A oA C A "~ and the rigid triangle/~ A 0A ×A y
(e) Make the f o u r - b a r f - q A o A Y C ~ C o y ~ [ ] A o A ~ C C o .
(f) Frame the rigid triangles A Y C Y F " and A o C o C o y.
The six-bar obtained is shown by Fig. 26 by thick solid lines and has one degree of freedom in movement like the initial mechanism.
There are no other cognates to be obtained. The initial six-bar, and its l/4-cognate only, have coincident turning joints B and C. This is not the case with the three remain- ing cognates. On these grounds, no other cognates are to be found in this particular case.
R e f e r e n c e s
[ll B U RM ESTER l_., Lehrbuch der Kinematik. p. 42 I. Leipzig (1888).
[2] RISCHEN K. A.. Ueber die achtfache Erzeugung der Koppelkurven der zweiten Koppelebene,
Konstruktion 14 (10), 381-385 (1962).
[31 PRIMROSE E . J . F . , F R E U D E N S T E I N F. and ROTH B., Six-bar motion, 11 The Stephenson-I and Stephenson-2 mechanisms. Arch. Ration. Mectz. Anal. 24 ( I ), 42-72 (1967).
[4] H ART H., On some cases of parallel motion, Proc. Lond. Math. Soc. 8. 286-289 (1877). [5] KEM P E A . B., On conjugate tour-piece linkages. Proc. Lond. Math. Soc. 9, 133-147 (1878). [6] BU RM ESTER L.. Die Brennpunktmechanismen, Z. Math. Phys. 38 (4). 193-223, (1893).
[7] W U N D E R L 1 C H W., On Burmester's focal mechanism and Hart's straight-line motion, J. Mechan-
isms 3, 79-86 (1968).
[8] DI.I KSMAN E. A., Six-bar cognates o f Watt's form. presented at the ASME Conference on Mechan- isms at Columbus. Ohio, U.S.A. Paper No. 70-Mech-30 (1970).
[9] SONI A. H., Coupler cognate mechanisms of certain parallelogram forms of Watt's six-link mechan- ism. J. Mechanisms 5,203-215 (1970).
[10] H A R T E N B E R G R.S. and D E N A V I T J . , Cognate linkages, Mach. Des. 31,149-152 (1958).
[11] H A R T E N B E R G R. S. and D E N A V I T J., The Fecund Four-Bar. Trans. Fifth Mechanism Conf. (1958). Machine Design Publ.. Cleveland, Ohio. p. 194-206.
[ 12] ROBERTS S., On three-bar motion in plane space, Proc. Lond. Math. Soc. 7, 14-23 (1876). [ 131 C A Y L E Y A., On three-bar motion, Proc. Lond. Math. Soc. 7, 136-166 (1876).
[14l SCHOR J. B., On the theorem of Roberts-Chebyshev, J. appl. Math. Mech. U.S.S.R. Acad. Sci. 5.323-324(1941).
[151 MEYER Z U R C A P E L L E N W., Bemerkungen z u m S a t z v o n RobertsiJberdiedreifacheErzeugung der Koppelkurve. Konstruktion 8 (7), 268-270 (1956).
[16] DE J O N G E A. E. R., The correlation of hinged four-bar straight-line motion devices by means of Roberts' theorem and a new proof of the latter. Ann. N. Y. A cad. Sci. 84, 75-145 (1960).
[17] P F L I E G E R - H A E R T E L H., Abgewandelte Kurbelgetriebe and der Satz yon Roberts, Reuleaux- Mitteilungen (Getriebetechnik) 12 (4). 197-199 (1944).
[18] DIJKSMAN E. A., How to compose mechanisms with parallel moving bars (with application on a level-luffing jib-crane consisting of a four-bar linkage and exploiting a coupler-point curve). De In-
