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On an unpublished work of Alt
J. Mechanisms Vol. 1, pp. 165-170. Pergamon Press 1966. Printed in Great Britain
ON AN UNPUBLISHED WORK OF ALT Prof. FRANK ERSKINE CROSSLEY Georgia Institute of Technology, Atlanta, Georgia (Received 13 April 1966)
Abstract--The existence of an unpublished paper of Alt is here reported, in which he has made a collection of ten-link plane chains. Comparison of this with a previous paper of the author raises the number of variants to 230. IT had been reported by Hain that, while discussing the number of structural permutations of plane linkaget (kinematic "number" synthesis) with the late Professor Alt in about the year 1953, the latter had mentioned he had ascertained that there were 226 variants of plane chains with ten members and one degree of mobility. The results of this work [1] of Alt however were never published. The author also published a collection of these ten-bar chains, consisting of 222 varieties [2]. While compiling this collection, the greatest problem was found to be, to distinguish whether two arrangements, which might appear unlike, were actually the same or different: this led [3] to a definition of isomorphism between linkages, which will be restated later. Subsequently in the summer of 1964, while on a lecture tour of Germany, and being a guest of Professor Overlach of the Technische Universitaet Berlin (Charlottenburg), it was discovered that Alt's work was still retained in the files of the Lehrstuhl fuer Foerderund Getriebetechnik. This work of Alt consists of a sheaf of thirty-one sheets of beautifully executed drawings; two typical sheets are reproduced here (Figs. 1, 2). It can be seen that Alt used a method similar to that of Franke [4] in order systematically to comb out all the variants. He considered first only the pattern made by all members omitting the binary links: these were arranged in "groups" (I through VII) according to the combination and number of each size of the constituent links (see Table 1): these were then divided into "subgroups" according to their connective patterns. Unlike Franke, it can be seen that Alt did not consider that a connecting path joining the larger bodies with binary chains should have a length of less than one (i.e. one binary member and two joints); Franke on the contrary takes zero (i.e. one joint only) as the minimum length of a path. A comparison of this work with the author's article [2] was thus made possible: this revealed that the author had overlooked one grouping (Alt's sub-group 10 II G) completely, which added three variants to his total, while Alt had missed entirely a sub-group of 10 V; there were also a small number of other omissions on both sides. Alt's collection moreover contains 16 pairs of arrangements, which are isomorphic to one another. The result of the comparison is thus to add eight new forms (Fig. 3) and one correction (Fig. 4) to the author's previous collection [2]. The number of varieties previously thought to exist in each group is therefore amended, and the new totals are given in Table 2. It is felt with some assurance that this is now a complete census, for it agrees with a third independent census made by Davies [5], which is set forth in an accompanying article. 165
166
FRANK ERSKINE CROSSLEY
Die zehngliedrigen zwangl/~ufigen kinematischen Ketten n = 10. e = 13, n 2 = 4 + n 4 + 2 n 5
G r u p p e 1 n2=4, n 3 = 6 . n 4 = 0 , n s = 0 Untergruppen 10 1 G und I0 1 H
Blatt 4
i
IOIG
I0
I
IO I H 2
IOIH3
Fro. I. Sheet 4 from AIt's unpublished work.
On an unpublished work of Alt
167
D i e z e h n g l i e d r i g e n zwangliiufigen k i n e m a t i s c h e n K e t t e n n = 10, e = 13, n 2 = 4 + n 4 + 2 n 5
G r u p p e I I n 2 = 5, n a = 4, n 4 = 1, n 5 = 0
Blatt 10
U n t e r g r u p p e n 10 II F u n d 10 II G
lOaF
r
lOT[G
IO]TG2
~ ~ 1 0
I01TGI
I01"[G3 FIG. 2. Sheet 10 from Alt's unpublished work.
~F5
168
F R A N K ERSKINE CROSSLEY TABLE 1. CONTENTSOF PROFESSOR ALT'S COLLECTION Constituent members
No. of subgroups
No. of variants
Comments (F.E.C.)
14
57
includes 5 isomorphic pairs
5-4-1-0
23
95
includes 8 pairs and one rigid form
!II
6--2-2-0
16
49
includes 2 isomorphic pairs
IV
7-0--3 -0
3
3
Group I
I1
n2--n3--n4--n 5
4
6- 0 - 0
V
6--3-0-1
10
13
VI
7 -1 - I - 1
7
7
VII
8--0-0-2
2
2 Totals
226
includes 1 isomorphic pair
17
TABLE 2. CENSUSOF IO-LINK CHAINS
Group 1
II
Constituent members n2 -- n3 -- n4 -- n5 4
No. of variants 50
6-- 0 - - 0
5 - - 4 - I- 0
95
III
6
57
IV
7- 0 - 3 - - 0
2
2 --0
V
6 3-0-I
VI
7-1--1-
VII
3 15 I
8
8--0--0--2
2 Total
230
T h i s c o u n t is o f c o u r s e d e p e n d e n t o n t h e d e f i n i t i o n o f w h a t c o n s t i t u t e s a u n i q u e a n d distinct arrangement: h e r e t h e c r i t e r i o n is t h a t o f t o p o l o g i c a l i s o m o r p h i s m [3]. T w o p a t t e r n s f o r m a n i s o m o r p h i c p a i r , t h a t is t o say, t h e y a r e e q u i v a l e n t a n d n o t d i f f e r e n t , if, w h e n t h e m e m b e r s o f o n e l i n k a g e a r e g i v e n a set o f n a m e s (say M l, M 2 . . . . . Mlo), it
On an unpublished work of AIt
169
is possible to designate the members of the second linkage with the same set of names, such that the members of the second set are joined by a kinematic pair (joint), whenever but only if their corresponding members of the first set are so joined. When such one-for-one correspondence does not exist, then the two geometrical patterns are different.
GROUP
4-6-0-0 47
48
49
~'~:~-:. [.:~:~.
GROUP
5-4-1-0
~~"":..:':~' ::~:'" A
92
$4
93
9S
FxG. 3. Eight new forms of lO-Link Chains to be added to Crossley's earlier collection.
170
FRANK ERSKINE CROSSLEY
iii i FIG. 4. The corrected form of No. 62 (Group 5 - 4 10-Link Chains [2].
I - 0 ) in the collection of
REFERENCES [1] H. ALT, "Die zehngliedrigen zwangl~ufigen kinematischen Ketten', eine noch nie ver0ffentlichte Sammlung. [2] F. R. E. CROSSLEY,Antriebstechnik 3, 181 (1964). [3] F. R. E. CROSSLEY,Developments in Theoretical and Applied Mechanics (ed. W. A. Shaw), Vol. 2, pp. 467-486. Pergamon Press (1965). [4] F. FRANKE, Ion Aufbau der Getriebe (3rd edition), Band 1. VDI-Verlag (1958). [5] T. H. DAVIESand F. R. E. CROSSLEY,J. Mechanisms 1, 171 (1966). APPENDIX A m o n g the collection o f Professor Alt, the following forms prove to be i s o m o r p h i c pairs, a c c o r d i n g to the a b o v e definition. It is o f course recognized that Alt m a y not have subscribed to this definition.
10IC5=4 10ID5=I 10I J 6 = 5 10IK3=2
10II 10II 101I 10II
F5=3I L5=3 Q3=2 T l=N2
10104=2
1011 10 II 10II 10 1I
T3=2 T 5=4 W2=I X 4=2
There is also one o f A l t ' s linkages, which is rigid, with F = 0 :
This pair appears in Fig. 2.
10IIIE3=2 IOIIIO3=Q2 10VI D I=E1
it is the form l0 II H 6.
ON AN UNPUBLISHED WORK OF ALT Prof. FRANK ERSKINE CROSSLEY Georgia Institute of Technology, Atlanta, Georgia (Received 13 April 1966)
Abstract--The existence of an unpublished paper of Alt is here reported, in which he has made a collection of ten-link plane chains. Comparison of this with a previous paper of the author raises the number of variants to 230. IT had been reported by Hain that, while discussing the number of structural permutations of plane linkaget (kinematic "number" synthesis) with the late Professor Alt in about the year 1953, the latter had mentioned he had ascertained that there were 226 variants of plane chains with ten members and one degree of mobility. The results of this work [1] of Alt however were never published. The author also published a collection of these ten-bar chains, consisting of 222 varieties [2]. While compiling this collection, the greatest problem was found to be, to distinguish whether two arrangements, which might appear unlike, were actually the same or different: this led [3] to a definition of isomorphism between linkages, which will be restated later. Subsequently in the summer of 1964, while on a lecture tour of Germany, and being a guest of Professor Overlach of the Technische Universitaet Berlin (Charlottenburg), it was discovered that Alt's work was still retained in the files of the Lehrstuhl fuer Foerderund Getriebetechnik. This work of Alt consists of a sheaf of thirty-one sheets of beautifully executed drawings; two typical sheets are reproduced here (Figs. 1, 2). It can be seen that Alt used a method similar to that of Franke [4] in order systematically to comb out all the variants. He considered first only the pattern made by all members omitting the binary links: these were arranged in "groups" (I through VII) according to the combination and number of each size of the constituent links (see Table 1): these were then divided into "subgroups" according to their connective patterns. Unlike Franke, it can be seen that Alt did not consider that a connecting path joining the larger bodies with binary chains should have a length of less than one (i.e. one binary member and two joints); Franke on the contrary takes zero (i.e. one joint only) as the minimum length of a path. A comparison of this work with the author's article [2] was thus made possible: this revealed that the author had overlooked one grouping (Alt's sub-group 10 II G) completely, which added three variants to his total, while Alt had missed entirely a sub-group of 10 V; there were also a small number of other omissions on both sides. Alt's collection moreover contains 16 pairs of arrangements, which are isomorphic to one another. The result of the comparison is thus to add eight new forms (Fig. 3) and one correction (Fig. 4) to the author's previous collection [2]. The number of varieties previously thought to exist in each group is therefore amended, and the new totals are given in Table 2. It is felt with some assurance that this is now a complete census, for it agrees with a third independent census made by Davies [5], which is set forth in an accompanying article. 165
166
FRANK ERSKINE CROSSLEY
Die zehngliedrigen zwangl/~ufigen kinematischen Ketten n = 10. e = 13, n 2 = 4 + n 4 + 2 n 5
G r u p p e 1 n2=4, n 3 = 6 . n 4 = 0 , n s = 0 Untergruppen 10 1 G und I0 1 H
Blatt 4
i
IOIG
I0
I
IO I H 2
IOIH3
Fro. I. Sheet 4 from AIt's unpublished work.
On an unpublished work of Alt
167
D i e z e h n g l i e d r i g e n zwangliiufigen k i n e m a t i s c h e n K e t t e n n = 10, e = 13, n 2 = 4 + n 4 + 2 n 5
G r u p p e I I n 2 = 5, n a = 4, n 4 = 1, n 5 = 0
Blatt 10
U n t e r g r u p p e n 10 II F u n d 10 II G
lOaF
r
lOT[G
IO]TG2
~ ~ 1 0
I01TGI
I01"[G3 FIG. 2. Sheet 10 from Alt's unpublished work.
~F5
168
F R A N K ERSKINE CROSSLEY TABLE 1. CONTENTSOF PROFESSOR ALT'S COLLECTION Constituent members
No. of subgroups
No. of variants
Comments (F.E.C.)
14
57
includes 5 isomorphic pairs
5-4-1-0
23
95
includes 8 pairs and one rigid form
!II
6--2-2-0
16
49
includes 2 isomorphic pairs
IV
7-0--3 -0
3
3
Group I
I1
n2--n3--n4--n 5
4
6- 0 - 0
V
6--3-0-1
10
13
VI
7 -1 - I - 1
7
7
VII
8--0-0-2
2
2 Totals
226
includes 1 isomorphic pair
17
TABLE 2. CENSUSOF IO-LINK CHAINS
Group 1
II
Constituent members n2 -- n3 -- n4 -- n5 4
No. of variants 50
6-- 0 - - 0
5 - - 4 - I- 0
95
III
6
57
IV
7- 0 - 3 - - 0
2
2 --0
V
6 3-0-I
VI
7-1--1-
VII
3 15 I
8
8--0--0--2
2 Total
230
T h i s c o u n t is o f c o u r s e d e p e n d e n t o n t h e d e f i n i t i o n o f w h a t c o n s t i t u t e s a u n i q u e a n d distinct arrangement: h e r e t h e c r i t e r i o n is t h a t o f t o p o l o g i c a l i s o m o r p h i s m [3]. T w o p a t t e r n s f o r m a n i s o m o r p h i c p a i r , t h a t is t o say, t h e y a r e e q u i v a l e n t a n d n o t d i f f e r e n t , if, w h e n t h e m e m b e r s o f o n e l i n k a g e a r e g i v e n a set o f n a m e s (say M l, M 2 . . . . . Mlo), it
On an unpublished work of AIt
169
is possible to designate the members of the second linkage with the same set of names, such that the members of the second set are joined by a kinematic pair (joint), whenever but only if their corresponding members of the first set are so joined. When such one-for-one correspondence does not exist, then the two geometrical patterns are different.
GROUP
4-6-0-0 47
48
49
~'~:~-:. [.:~:~.
GROUP
5-4-1-0
~~"":..:':~' ::~:'" A
92
$4
93
9S
FxG. 3. Eight new forms of lO-Link Chains to be added to Crossley's earlier collection.
170
FRANK ERSKINE CROSSLEY
iii i FIG. 4. The corrected form of No. 62 (Group 5 - 4 10-Link Chains [2].
I - 0 ) in the collection of
REFERENCES [1] H. ALT, "Die zehngliedrigen zwangl~ufigen kinematischen Ketten', eine noch nie ver0ffentlichte Sammlung. [2] F. R. E. CROSSLEY,Antriebstechnik 3, 181 (1964). [3] F. R. E. CROSSLEY,Developments in Theoretical and Applied Mechanics (ed. W. A. Shaw), Vol. 2, pp. 467-486. Pergamon Press (1965). [4] F. FRANKE, Ion Aufbau der Getriebe (3rd edition), Band 1. VDI-Verlag (1958). [5] T. H. DAVIESand F. R. E. CROSSLEY,J. Mechanisms 1, 171 (1966). APPENDIX A m o n g the collection o f Professor Alt, the following forms prove to be i s o m o r p h i c pairs, a c c o r d i n g to the a b o v e definition. It is o f course recognized that Alt m a y not have subscribed to this definition.
10IC5=4 10ID5=I 10I J 6 = 5 10IK3=2
10II 10II 101I 10II
F5=3I L5=3 Q3=2 T l=N2
10104=2
1011 10 II 10II 10 1I
T3=2 T 5=4 W2=I X 4=2
There is also one o f A l t ' s linkages, which is rigid, with F = 0 :
This pair appears in Fig. 2.
10IIIE3=2 IOIIIO3=Q2 10VI D I=E1
it is the form l0 II H 6.