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Systemics of Assur groups with multiple joints
PII:
Mech. Mach. Theory Vol. 33, No. 8, pp. 1127±1133, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00117-1
SYSTEMICS OF ASSUR GROUPS WITH MULTIPLE JOINTS CHU JINKUI and CAO WEIQING Department of Mechanical Engineering, Xi'an University of Technology, Xi'an 710048, People's Republic of China AbstractÐIn this paper, the composition principle of kinematic chain (K.C.) with multiple joints (M.J.) is ®rstly discussed, and a factor of M.J. about kinematic chain with M.J. is de®ned, so that the structural analysis and synthesis of the K.C. with M.J. could be simpli®ed. The 6-bar and 8-bar Assur groups with M.J. are also investigated. All the independent types of the Assur groups with M.J. are obtained. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
In general, multiple joints (M.J.) may exist in dierent structures of mechanisms, especially in epicyclic gear trains and gear-link mechanisms, but up to now, reports on the synthesis of mechanisms with M.J. are rather few. In refs [1±3], which are circulated world-wide, this topic has somewhat been dealt with. Theoretically speaking, relevant solutions are far from satisfactory. Therefore, further research on the synthesis problem about mechanisms with M.J. are necessary. The second author of this article had got all structures for up to 8-bar Assur groups with simple joints only, as shown in ref. [4]. Based on various theories mentioned in ref. [4], the formation principle of M.J. and the synthesis of Assur groups with M.J. are developed here. Lemma 1 In a simple-joint kinematic chain (K.C.), if there is a link with joint elements more than 2, and all the distances among these joint elements are shrunk to zero, then one M.J. is formed and this link is eliminated. An M.J. formatted by a link with k + 1 joint elements is equivalent to k simple joints. The number of M.J. composed of k simple joints would be denoted by Pk(k e 2). As shown in Fig. 1(a), there are three joint elements on link 4, they would be shrunk into one M.J., so k = 2, P2=1. In Fig. 1(b), there are four joint elements on link 5, they would be shrunk into one M.J., so k = 3, P3=1. In the K.C. with Pk(k = 2, 3, 4, . . .) M.J., if one of these M.J. is replaced by one link with k + 1 joint elements, the modi®ed K.C. can be obtained, the number of links, the number of kinematic pairs and the number of mobility here in the modi®ed K.C. increase by one. The reverse procedure is also true. Lemma 2 On basis of Lemma 1, in a closed K.C. with M.J., one formula can be obtained as below: X ini 2P ÿ ; 1 i2
where i is the number of joint elements of a link, P is the total number of kinematic pairs of single and multiple joints in the K.C., ni is the number of links with i joint elements in the K.C., is called the M.J. factor of the K.C., and we can have the following relation: X k ÿ 1Pk : 2 k2
Proving that according to Lemma 1, replacing all the Pk(k = 2, 3, 4, . . .) M.J. in the K.C. by links with K + 1 joint elements, a modi®ed K.C., which only has simple joints, can be obtained here. The number of links and kinematic pairs increase by Dn, DP, respectively, and we have 1127
1128
C. Jinkui and C. Weiqing
Fig. 1. Composition of M.J.
Dn X
X
Dnk1
k2
Pk DP
k2
Dn DP:
3
Based on graph theory, in the modi®ed K.C., let nj be the number of the links with j joint-elements, q be the number of kinematic pairs, we can get X jnj 2q j2
X j2
jnj
X
ini
i2
X k 1Dnk1 k2
2q 2 P
X
!
Pk :
k2
Rearranging the above formulas using formula (3) we obtain X X X ini ÿ 2P 2 Pk ÿ k 1Pk i2
k2
k2
X ÿ k ÿ 1Pk ÿ: k2
Therefore,
X
ini 2P ÿ :
i2
For example, in Fig. 2, the K.C. in Fig. 2(a) is the modi®ed K.C. of the K.C. in Fig. 2(b±d).
SYNTHESIS OF INTERNAL CHAIN OF ASSUR GROUPS
Among the Assur groups, except those containing only the ``suspension link'' [4], all their internal chains are closed chains. Suppose the number of links in the internal closed chain is n', the number of the pairs is P', the number of mobility is w' and M.J. factor in the internal chain is I, then n 02 n 03 n 0i n 0
4
2n 02 3n 03 in 0i 2p 0 ÿ I :
5
On the basis of formulas (4) and (5), and 3n' ÿ 2P ' = w', we get
Systemics of Assur groups with multiple joints
1129
Fig. 2. M.J. factor and modi®ed K.C.
n 02 n 04 2n 05 3n 06
i ÿ 3n 0i w 0 I :
6
According to Equation (6), the maximum value of I can be obtained I max n 02 max ÿ w 0 n 0 ÿ w 0 :
7
On the basis of formulas (4) and (6), the distributions of the number of links in the internal chain may be obtained. As shwon in Fig. 2(a), the distributions of the number of the link, ni00, in the modi®ed K.C. can be obtained by using the following transformations, and this would be true for k e 3. n300 n 03 I
ni00 n 0i
i 6 3:
8
According to the distributions of the number of links ni00(i>2), the contract mappings of the modi®ed K.C. can be drawn, and distributions of n200 on the dierent edges of the correspondent contract mapping can be speci®ed also, so the graphs of modi®ed K.C. are obtained. In order to avoid appearance of the overconstraint subchain, the distributions of n200 should satisfy the following formula: X ni00 4 nc00 2 L ÿ 1 9 i3
where L is the number of loops of the subchain in the contract mapping; nc00 is the number of the links that should be shrinked for forming M.J. in the subchain. Lastly, converting I links with three joint elements into M.J. in the modi®ed K.C., the structural graphs of the internal chain with M.J. can be obtained. SYNTHESIS OF THE EXTERNAL CHAIN
Let the M.J. factor of the Assur group be , since M.J. may appear in both the internal and external chain, we have I E
10
where I is the M.J. factor in the internal chain, E is the M.J. factor in the external chain. According to the structural properties of the Assur group, there are two types of external chains. The ®rst one is called the k1 type, which is a ``suspension link'' [4], and consists of two pairs at both ends and one constraint. The second one is the k2 type, and consists of a simple external pair with two constraints. There are two pairs on a suspension link, one pair is called the external pair, another is called the internal pair. The external pair can be connected to the ground link, input link or the link of the adjacent Assur group. The internal pair is directly connected to the internal chain. Hence, we have: E E1 E2
11
where nE1 is the number of M.J. formed by connecting the internal pair of the suspension link
1130
C. Jinkui and C. Weiqing
to the internal chain, nE2 is the number of M.J. formed by connecting the external pair to each other. Let the number of the external pair be r, the number of external pairs with one and two constraints be k1 and k2, respectively, the number of links in the Assur group be n, the mobility of the internal chain be w', on the basis of the structural properties of the Assur group [4], we have r k1 k2
12
n 1r2 2
13
n n 0 k1
14
w 0 k1 2k2
15
E1 k1
16
r E2 : 2
17
and in an external chain, we get
In order to avoid the appearance of the decomposable Assur group and the overconstrainted subchain, the synthesis rules stated below about external chains [4] are also suitable for the synthesis of an Assur group with M.J. In the internal chain of an Assur group, the number of the external constrains is w 002s , any subchain with binary links that can be receptible should satisfy the following equations: n 02s ÿ 1 w 002s 2s n 02s 1
18
n 0s ÿ 2 n 02s 1
19
where n 02s is the number of links of the subchain, n 0s is the number of links in a loop to which the subchain belongs.
Example Synthesizing an 8-bar Assur group. Given: r = 3, k2=3, k1=0, I=2, E=0. Since k1=0, n = n' = 8, k2=3, we have w' = 6. From formulas (4) and (6), we get n 02 n 03 n 0i 8 n 02 n 04 2n 05 i ÿ 3n 0i 6 2: So one distribution is obtained below: n 02 8, n 03 n 04 0: On the basis of formula (8), the distributions about modi®ed K.C. would be: n 002 8, n 003 n 0 I 2,
n 004 n 005 0:
Based on the distribution, the contract mapping of the modi®ed K.C. [in Fig. 3(a)] can be drawn. Since n 002 8, according to formula (9), three distributions (0, 4, 4), (3, 1, 4), (3, 2, 3) of binary links are obtained. The distribution (0, 4, 4) is eliminated, since it can produce a decomposition subchain. So, two kinds of the modi®ed K.C. are obtained [Fig. 3(b)]. The internal chains are obtained by shrinking two links with three joint elements of the modi®ed K.C. [Fig. 3(c)]. Figure 3(d) shows the synthesis of the external chains, which are obtained by connecting three k2 type external pairs to the internal chain on the basis of formulas (18) and (19).
Systemics of Assur groups with multiple joints
1131
Fig. 3. I=2, k2=3.
Example Synthesizing an 8-bar Assur group. Given: = 6, r = 4, k1=4, k2=0. Since k1=4, n' = 8 ÿ 4 = 4, so there are only four bars in the internal chain [Fig. 4(a)], and I=0. Based on formula (16), we have E1=4 [Fig. 4(b)]. Owing to r = 4, on the basis of formula (17), we have E2=2. The structural graph of an Assur group is shown in Fig. 4(c). STRUCTURAL SYNTHESIS OF 6-BAR ASSUR GROUP
From formulas (4), (6), (14) and (15), some formulas about a 6-bar Assur group could be developed as below: n 02 n 03 n 0i 6 ÿ k1
n 02 n 04 2n 05 i ÿ 3n 0i I k1 2k2 : When nI=0, the valid value of one set of parameters k1 and k2 about the internal chain are listed below: 1. k1=1, k2=2, the internal chain is a pentagon. As a result of synthesis of the external chain, two independent graphs [Fig. 5(c) and (d)] are obtained. 2. k1=2, k2=1, the internal chain is a quadrilateral. As a result of synthesis of the external chain, six independent structural graphs [Fig. 5(e)±(j)] are obtained. 3. k1=4, k2=0, the internal chain is a single-open-chain. As a result of synthesis of the external chain, two independent structural graphs are obtained [Fig. 5(k), Fig. 5(l)]. When I=1, there is only one group of the valid value k1=0, k2=2. Firstly the modi®ed internal K.C. is synthesized, and then the internal chain can be obtained from the modi®ed K.C. As a result of the synthesis of the external chain, the two independent structural graphs are obtained [Fig. 5(a) and (b)]. Summing up the above results, 12 types of independent structural graphs are obtained (Fig. 5, Table 1). STRUCTURAL SYNTHESIS OF 8-BAR ASSUR GROUPS
From the synthesis of the 8-bar Assur group with M.J., since the minimum value of the mobility in the internal chain is 4, from formula (7), when n' = n = 8, we get I max 4: On the basis of formulas (4) and (6), seven kinds of contract mappings of the modi®ed internal
Fig. 4. I=0, k1=4.
1132
C. Jinkui and C. Weiqing Table 1.
Fig. 5. Structural graphs of 6-bar Assur groups with M.J.
Table 2.
Systemics of Assur groups with multiple joints
1133
chains are obtained. From these mappings, 25 types of the modi®ed internal chains could be developed. From these modi®ed chains, 83 types of the internal chains are deduced. Among them 50 types of the internal chains with I=1, 2, 3, 4 are shown in Table 2, other 33 types of the internal chains with I=0, had already been discussed in references [4, 5]. Based on the 83 internal chains, the synthesis of the external chains are developed according to formulas (18) and (19). Thus, all the independent structural graphs of 8-bar Assur groups, 872, could be obtained. CONCLUSION
The concept of Assur groups developed by the Soviet kinematician provides an idea of building blocks for the structures of mechanisms. The author had once solved the problem of enumeration and classi®cation for simple joints of 8-bar Assur groups resulting in a number of structures of 173 [4]. The number of structures for ten-bar linkages with simple joints and one degree-of-freedom has been known to be 230 [6]. In this paper, the 8-bar Assur groups with multiple joints are dealt with, and the number of 872 is veri®ed, sketches of which are not shown here for the limitation of pages. The authors do believe that it will oer a strong instrument for systematically enumerating mechanisms with multiple joints. AcknowledgementsÐThis project is supported by the National Natural Science Foundation of China.
REFERENCES 1. 2. 3. 4. 5. 6.
Manolescu, N. I., in Proc. of 5th World Cong. on TMM, 1979, p. 514. Mruthyunjaya, T. S., Mechanism and Machine Theory, 1979, 14221. Manolescu, N. I., Mechanism Machine Theory, 1973, 183. Wei-qing, Tsao, in Proceeding of International Conference on Engineering Design, Vol. WDK 6, Rome, 1981, p. 311. Weiqing, Cao, Publishing House of Science (Beijing), 1989, 5. Woo, L. S., Trans. ASME, Series B, 1967, 159.
ZusammenfassungÐIn diesem Beitrag wird das Prinzip der Komposition fuÈr kinimatishen Ketten mit multipelen Gelenkn zunaÈchst diskutiert. Der Faktor fuÈr multipelen Gelenkn wird de®niert und dargestellt, so daû die strukturelle Analyse und Synthese koÈnnen vereinfachen werden. Der Type Synthese fuÈr 6-Stangen und 8-Stangen Assur Gruppen mit multipelen Gelenkn wir erfuÈllen, und allen Typen mit unabhaÈngigen Strukuren 10 fuÈr 6-Stangen und 872 fuÈr 8-Stangen werden bekommen. # 1998 Elsevier Science Ltd. All rights reserved
Mech. Mach. Theory Vol. 33, No. 8, pp. 1127±1133, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00117-1
SYSTEMICS OF ASSUR GROUPS WITH MULTIPLE JOINTS CHU JINKUI and CAO WEIQING Department of Mechanical Engineering, Xi'an University of Technology, Xi'an 710048, People's Republic of China AbstractÐIn this paper, the composition principle of kinematic chain (K.C.) with multiple joints (M.J.) is ®rstly discussed, and a factor of M.J. about kinematic chain with M.J. is de®ned, so that the structural analysis and synthesis of the K.C. with M.J. could be simpli®ed. The 6-bar and 8-bar Assur groups with M.J. are also investigated. All the independent types of the Assur groups with M.J. are obtained. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
In general, multiple joints (M.J.) may exist in dierent structures of mechanisms, especially in epicyclic gear trains and gear-link mechanisms, but up to now, reports on the synthesis of mechanisms with M.J. are rather few. In refs [1±3], which are circulated world-wide, this topic has somewhat been dealt with. Theoretically speaking, relevant solutions are far from satisfactory. Therefore, further research on the synthesis problem about mechanisms with M.J. are necessary. The second author of this article had got all structures for up to 8-bar Assur groups with simple joints only, as shown in ref. [4]. Based on various theories mentioned in ref. [4], the formation principle of M.J. and the synthesis of Assur groups with M.J. are developed here. Lemma 1 In a simple-joint kinematic chain (K.C.), if there is a link with joint elements more than 2, and all the distances among these joint elements are shrunk to zero, then one M.J. is formed and this link is eliminated. An M.J. formatted by a link with k + 1 joint elements is equivalent to k simple joints. The number of M.J. composed of k simple joints would be denoted by Pk(k e 2). As shown in Fig. 1(a), there are three joint elements on link 4, they would be shrunk into one M.J., so k = 2, P2=1. In Fig. 1(b), there are four joint elements on link 5, they would be shrunk into one M.J., so k = 3, P3=1. In the K.C. with Pk(k = 2, 3, 4, . . .) M.J., if one of these M.J. is replaced by one link with k + 1 joint elements, the modi®ed K.C. can be obtained, the number of links, the number of kinematic pairs and the number of mobility here in the modi®ed K.C. increase by one. The reverse procedure is also true. Lemma 2 On basis of Lemma 1, in a closed K.C. with M.J., one formula can be obtained as below: X ini 2P ÿ ; 1 i2
where i is the number of joint elements of a link, P is the total number of kinematic pairs of single and multiple joints in the K.C., ni is the number of links with i joint elements in the K.C., is called the M.J. factor of the K.C., and we can have the following relation: X k ÿ 1Pk : 2 k2
Proving that according to Lemma 1, replacing all the Pk(k = 2, 3, 4, . . .) M.J. in the K.C. by links with K + 1 joint elements, a modi®ed K.C., which only has simple joints, can be obtained here. The number of links and kinematic pairs increase by Dn, DP, respectively, and we have 1127
1128
C. Jinkui and C. Weiqing
Fig. 1. Composition of M.J.
Dn X
X
Dnk1
k2
Pk DP
k2
Dn DP:
3
Based on graph theory, in the modi®ed K.C., let nj be the number of the links with j joint-elements, q be the number of kinematic pairs, we can get X jnj 2q j2
X j2
jnj
X
ini
i2
X k 1Dnk1 k2
2q 2 P
X
!
Pk :
k2
Rearranging the above formulas using formula (3) we obtain X X X ini ÿ 2P 2 Pk ÿ k 1Pk i2
k2
k2
X ÿ k ÿ 1Pk ÿ: k2
Therefore,
X
ini 2P ÿ :
i2
For example, in Fig. 2, the K.C. in Fig. 2(a) is the modi®ed K.C. of the K.C. in Fig. 2(b±d).
SYNTHESIS OF INTERNAL CHAIN OF ASSUR GROUPS
Among the Assur groups, except those containing only the ``suspension link'' [4], all their internal chains are closed chains. Suppose the number of links in the internal closed chain is n', the number of the pairs is P', the number of mobility is w' and M.J. factor in the internal chain is I, then n 02 n 03 n 0i n 0
4
2n 02 3n 03 in 0i 2p 0 ÿ I :
5
On the basis of formulas (4) and (5), and 3n' ÿ 2P ' = w', we get
Systemics of Assur groups with multiple joints
1129
Fig. 2. M.J. factor and modi®ed K.C.
n 02 n 04 2n 05 3n 06
i ÿ 3n 0i w 0 I :
6
According to Equation (6), the maximum value of I can be obtained I max n 02 max ÿ w 0 n 0 ÿ w 0 :
7
On the basis of formulas (4) and (6), the distributions of the number of links in the internal chain may be obtained. As shwon in Fig. 2(a), the distributions of the number of the link, ni00, in the modi®ed K.C. can be obtained by using the following transformations, and this would be true for k e 3. n300 n 03 I
ni00 n 0i
i 6 3:
8
According to the distributions of the number of links ni00(i>2), the contract mappings of the modi®ed K.C. can be drawn, and distributions of n200 on the dierent edges of the correspondent contract mapping can be speci®ed also, so the graphs of modi®ed K.C. are obtained. In order to avoid appearance of the overconstraint subchain, the distributions of n200 should satisfy the following formula: X ni00 4 nc00 2 L ÿ 1 9 i3
where L is the number of loops of the subchain in the contract mapping; nc00 is the number of the links that should be shrinked for forming M.J. in the subchain. Lastly, converting I links with three joint elements into M.J. in the modi®ed K.C., the structural graphs of the internal chain with M.J. can be obtained. SYNTHESIS OF THE EXTERNAL CHAIN
Let the M.J. factor of the Assur group be , since M.J. may appear in both the internal and external chain, we have I E
10
where I is the M.J. factor in the internal chain, E is the M.J. factor in the external chain. According to the structural properties of the Assur group, there are two types of external chains. The ®rst one is called the k1 type, which is a ``suspension link'' [4], and consists of two pairs at both ends and one constraint. The second one is the k2 type, and consists of a simple external pair with two constraints. There are two pairs on a suspension link, one pair is called the external pair, another is called the internal pair. The external pair can be connected to the ground link, input link or the link of the adjacent Assur group. The internal pair is directly connected to the internal chain. Hence, we have: E E1 E2
11
where nE1 is the number of M.J. formed by connecting the internal pair of the suspension link
1130
C. Jinkui and C. Weiqing
to the internal chain, nE2 is the number of M.J. formed by connecting the external pair to each other. Let the number of the external pair be r, the number of external pairs with one and two constraints be k1 and k2, respectively, the number of links in the Assur group be n, the mobility of the internal chain be w', on the basis of the structural properties of the Assur group [4], we have r k1 k2
12
n 1r2 2
13
n n 0 k1
14
w 0 k1 2k2
15
E1 k1
16
r E2 : 2
17
and in an external chain, we get
In order to avoid the appearance of the decomposable Assur group and the overconstrainted subchain, the synthesis rules stated below about external chains [4] are also suitable for the synthesis of an Assur group with M.J. In the internal chain of an Assur group, the number of the external constrains is w 002s , any subchain with binary links that can be receptible should satisfy the following equations: n 02s ÿ 1 w 002s 2s n 02s 1
18
n 0s ÿ 2 n 02s 1
19
where n 02s is the number of links of the subchain, n 0s is the number of links in a loop to which the subchain belongs.
Example Synthesizing an 8-bar Assur group. Given: r = 3, k2=3, k1=0, I=2, E=0. Since k1=0, n = n' = 8, k2=3, we have w' = 6. From formulas (4) and (6), we get n 02 n 03 n 0i 8 n 02 n 04 2n 05 i ÿ 3n 0i 6 2: So one distribution is obtained below: n 02 8, n 03 n 04 0: On the basis of formula (8), the distributions about modi®ed K.C. would be: n 002 8, n 003 n 0 I 2,
n 004 n 005 0:
Based on the distribution, the contract mapping of the modi®ed K.C. [in Fig. 3(a)] can be drawn. Since n 002 8, according to formula (9), three distributions (0, 4, 4), (3, 1, 4), (3, 2, 3) of binary links are obtained. The distribution (0, 4, 4) is eliminated, since it can produce a decomposition subchain. So, two kinds of the modi®ed K.C. are obtained [Fig. 3(b)]. The internal chains are obtained by shrinking two links with three joint elements of the modi®ed K.C. [Fig. 3(c)]. Figure 3(d) shows the synthesis of the external chains, which are obtained by connecting three k2 type external pairs to the internal chain on the basis of formulas (18) and (19).
Systemics of Assur groups with multiple joints
1131
Fig. 3. I=2, k2=3.
Example Synthesizing an 8-bar Assur group. Given: = 6, r = 4, k1=4, k2=0. Since k1=4, n' = 8 ÿ 4 = 4, so there are only four bars in the internal chain [Fig. 4(a)], and I=0. Based on formula (16), we have E1=4 [Fig. 4(b)]. Owing to r = 4, on the basis of formula (17), we have E2=2. The structural graph of an Assur group is shown in Fig. 4(c). STRUCTURAL SYNTHESIS OF 6-BAR ASSUR GROUP
From formulas (4), (6), (14) and (15), some formulas about a 6-bar Assur group could be developed as below: n 02 n 03 n 0i 6 ÿ k1
n 02 n 04 2n 05 i ÿ 3n 0i I k1 2k2 : When nI=0, the valid value of one set of parameters k1 and k2 about the internal chain are listed below: 1. k1=1, k2=2, the internal chain is a pentagon. As a result of synthesis of the external chain, two independent graphs [Fig. 5(c) and (d)] are obtained. 2. k1=2, k2=1, the internal chain is a quadrilateral. As a result of synthesis of the external chain, six independent structural graphs [Fig. 5(e)±(j)] are obtained. 3. k1=4, k2=0, the internal chain is a single-open-chain. As a result of synthesis of the external chain, two independent structural graphs are obtained [Fig. 5(k), Fig. 5(l)]. When I=1, there is only one group of the valid value k1=0, k2=2. Firstly the modi®ed internal K.C. is synthesized, and then the internal chain can be obtained from the modi®ed K.C. As a result of the synthesis of the external chain, the two independent structural graphs are obtained [Fig. 5(a) and (b)]. Summing up the above results, 12 types of independent structural graphs are obtained (Fig. 5, Table 1). STRUCTURAL SYNTHESIS OF 8-BAR ASSUR GROUPS
From the synthesis of the 8-bar Assur group with M.J., since the minimum value of the mobility in the internal chain is 4, from formula (7), when n' = n = 8, we get I max 4: On the basis of formulas (4) and (6), seven kinds of contract mappings of the modi®ed internal
Fig. 4. I=0, k1=4.
1132
C. Jinkui and C. Weiqing Table 1.
Fig. 5. Structural graphs of 6-bar Assur groups with M.J.
Table 2.
Systemics of Assur groups with multiple joints
1133
chains are obtained. From these mappings, 25 types of the modi®ed internal chains could be developed. From these modi®ed chains, 83 types of the internal chains are deduced. Among them 50 types of the internal chains with I=1, 2, 3, 4 are shown in Table 2, other 33 types of the internal chains with I=0, had already been discussed in references [4, 5]. Based on the 83 internal chains, the synthesis of the external chains are developed according to formulas (18) and (19). Thus, all the independent structural graphs of 8-bar Assur groups, 872, could be obtained. CONCLUSION
The concept of Assur groups developed by the Soviet kinematician provides an idea of building blocks for the structures of mechanisms. The author had once solved the problem of enumeration and classi®cation for simple joints of 8-bar Assur groups resulting in a number of structures of 173 [4]. The number of structures for ten-bar linkages with simple joints and one degree-of-freedom has been known to be 230 [6]. In this paper, the 8-bar Assur groups with multiple joints are dealt with, and the number of 872 is veri®ed, sketches of which are not shown here for the limitation of pages. The authors do believe that it will oer a strong instrument for systematically enumerating mechanisms with multiple joints. AcknowledgementsÐThis project is supported by the National Natural Science Foundation of China.
REFERENCES 1. 2. 3. 4. 5. 6.
Manolescu, N. I., in Proc. of 5th World Cong. on TMM, 1979, p. 514. Mruthyunjaya, T. S., Mechanism and Machine Theory, 1979, 14221. Manolescu, N. I., Mechanism Machine Theory, 1973, 183. Wei-qing, Tsao, in Proceeding of International Conference on Engineering Design, Vol. WDK 6, Rome, 1981, p. 311. Weiqing, Cao, Publishing House of Science (Beijing), 1989, 5. Woo, L. S., Trans. ASME, Series B, 1967, 159.
ZusammenfassungÐIn diesem Beitrag wird das Prinzip der Komposition fuÈr kinimatishen Ketten mit multipelen Gelenkn zunaÈchst diskutiert. Der Faktor fuÈr multipelen Gelenkn wird de®niert und dargestellt, so daû die strukturelle Analyse und Synthese koÈnnen vereinfachen werden. Der Type Synthese fuÈr 6-Stangen und 8-Stangen Assur Gruppen mit multipelen Gelenkn wir erfuÈllen, und allen Typen mit unabhaÈngigen Strukuren 10 fuÈr 6-Stangen und 872 fuÈr 8-Stangen werden bekommen. # 1998 Elsevier Science Ltd. All rights reserved