A procedure to count the number of planar mechanisms subject to design constraints from kinematic chains

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Mechanism and Machine Theory 43 (2008) 676–694

Mechanism and Machine Theory www.elsevier.com/locate/mechmt

A procedure to count the number of planar mechanisms subject to design constraints from kinematic chains Chih-Ching Hung a, Hong-Sen Yan b, Gordon R. Pennock a

c,*

Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC Department of Mechanical Engineering, DaYeh University, Datsuen, Chamghua 515, Taiwan, ROC c School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA

b

Received 20 September 2006; received in revised form 6 June 2007; accepted 9 June 2007 Available online 10 August 2007

Abstract This paper presents a systematic procedure to count the number of planar mechanisms subject to design constraints from the candidate kinematic chains. The procedure is based on well-known principles that can be found in graph theory and combinatorial mathematics. A link-path is employed to include the design constraints that a number of specified joints must correspond to a number of specified links. Also, modified permutation groups, generating function, and Polya’s theory are used to count the number of non-isomorphic mechanisms with the required design constraints. Then the pattern inventory is used for the conceptual design of the identified mechanisms. The procedure can identify all of the non-isomorphic mechanisms in a specified kinematic chain. In addition, the procedure can be used to determine the isomorphic mechanisms in a straightforward manner. Three practical examples are included in the paper to illustrate the systematic nature of the proposed procedure; namely: the differential-type south pointing chariot, the Watt kinematic chain, and a variablestroke engine. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Conceptual design; Design constraints; Candidate kinematic chains; Non-isomorphic mechanisms; Graph theory; Combinatorial mathematics; Linkage path code; Modified permutation groups

1. Introduction The conceptual design of mechanisms is a very important stage in the kinematic synthesis of mechanisms. A mechanism is defined in this paper as a planar kinematic chain with one link fixed (referred to as the ground link). Several pioneer researchers in mechanism design have studied the number, type, and structural synthesis of kinematic chains and opened the door to the conceptual design of mechanisms [1–4]. Numerous design methodologies have been proposed, during the past 40 years, to develop techniques for the conceptual design *

Corresponding author. Tel.: +1 765 494 5728; fax: +1 765 494 0539. E-mail addresses: [email protected] (C.-C. Hung), [email protected] (H.-S. Yan), [email protected] (G.R. Pennock). 0094-114X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2007.06.009

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of mechanisms [5–10]. Many of these methodologies have been employed successfully in practical applications; for example, epicyclic gear trains [11–17], variable-stroke engines [18–20], window regulating mechanisms [21], and suspension systems [22]. The traditional approach is to identify the non-isomorphic, or feasible, mechanisms, from kinematic chains, based on inspection or observation [23]. The isomorphic, or non-feasible, mechanisms are deleted from the search, subject to the design constraints. Such an approach, however, is ineffective when the design problem becomes too complex. Therefore, techniques based on graph theory and combinatorial mathematics [24–28] have been developed to determine the number of non-isomorphic mechanisms. Dobrjanskyj and Freudenstein [3] applied Polya’s theory to analyze the structural classification of singleloop spatial mechanisms. Freudenstein, in his discussion of [4], showed that this theory could be used to enumerate distinct ways of adding binary link vertices to contract graphs. He also applied the theory to the structural classification of mechanisms [29]. Then Buchsbaum and Freudenstein [30] used network concepts and combinatorial analysis to synthesize geared kinematic chains, and applied Polya’s theory to prove the results. Freudenstein and Woo [31] employed the theory to the enumeration of graphs representing the kinematic structure of mechanisms. Yan and Hwang [32] proposed an algorithm, based on combinatorial theory and the concept of permutation groups, to count the number of non-isomorphic mechanisms with the required number and types of links and joints from the candidate kinematic chains. Then Yan and Hung [33] presented an approach, based on the generating function and Polya’s theory, to identify and count the number of non-isomorphic mechanisms from the kinematic chains. The chains were subject to the design constraints of adjacency relation and/or incidence relation between links, between joints, or between links and joints. Yan and Hung were able to specify additional design constraints by extending their work to include modified permutation groups and using variables to indicate the remaining links, or joints, in these groups [34]. In the research to date, the design constraints (that typically include the number and types of links and joints, and the adjacency and incidence relations of links and joints) can be solved by well-known techniques. These techniques, in general, are based on graph theory and permutation group handling processes. As an example, consider the wheel damping mechanism [22] shown in Fig. 1a. Initially, the following three design constraints are specified: (i) the ground link 1 (denoted as Lg) must be a ternary link; (ii) the piston 5 (denoted as LP) must be a binary link; and (iii) the cylinder 6 (denoted as LCy) must be a binary link. The kinematic chain corresponding to this mechanism is the Stephenson chain, see Fig. 1b, which is a (6, 7) kinematic chain; i.e., six links connected by seven revolute joints. From the algorithm presented in [32] there are seven mechanisms which correspond to this kinematic chain, see Fig. 1c–i. Now assume that the design constraints also specify that the piston must be adjacent to the ground link. Based on the approach presented in Refs. [33,34], there are five mechanisms which satisfy all four constraints; i.e., the mechanisms shown in Fig. 1c, d, f, h and i. If the design constraints also specify that the cylinder must be adjacent to the piston by a prismatic joint then there is only one mechanism which will satisfy all five design constraints; i.e., the mechanism shown in Fig. 1c. It is interesting to note, however, that the existing literature has not addressed the problem where the design constraints require that specified links must correspond to specified joints (for example, the cylinder must be adjacent to the piston by a prismatic joint). To the knowledge of the authors this paper makes an original contribution to this problem. In fact, a major contribution of the paper is a procedure to count the number of non-isomorphic mechanisms with the required design constraints from candidate kinematic chains. This systematic procedure is based on the mathematical techniques of the generating function and Polya’s theory. For a comprehensive background and a detailed description of the procedure, the reader is referred to the thesis by Hung [35]. The paper is arranged as follows. Section 2 presents a brief review of some of the important terminology and definitions from graph theory and combinatorial mathematics. Then Section 3 presents the generating function, Polya’s theory, and the pattern inventory to count the number of non-isomorphic mechanisms. Section 4 presents the procedure to count the number of identified mechanisms subject to design constraints from the candidate kinematic chains. Section 5 presents three practical examples to illustrate the systematic procedure. The examples are the differential-type south pointing chariot [36,37], the Watt kinematic chain, and a variable-stroke engine [18]. Finally, Section 6 presents some important conclusions and suggestions for future research in the conceptual design of mechanisms.

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Fig. 1. Wheel damping mechanism, Stephenson chain, and corresponding mechanisms.

2. Terminology and definitions 2.1. Adjacency relation In graph theory, links are denoted by vertices and joints are denoted by edges. The edge connection between vertices corresponds to the joint connection between links. If the vertices u and v are the endpoints of an edge, then they are said to be adjacent or neighbors [28]. For example, Fig. 1b shows that links 1 and 3 have an adjacency relation and joints a and b have an adjacency relation. 2.2. Incidence relation If the vertex w is an endpoint of edge e then w and e are said to be incident [28]. For example, Fig. 1b shows that link 1 and joint b have an incidence relation. 2.3. Permutation groups In the process of applying combinatorial theory to the conceptual design of a mechanism, the links and the joints of the mechanism are transformed into the link set and the joint set [25,26,32–35]. Then permutation

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groups can be used to demonstrate the symmetry of the topological structure of the mechanism. Three permutation groups for a labeled kinematic chain will be defined here, namely, a link-group, a joint-group, and a chain-group. A permutation p is a bijection (i.e., one-to-one and onto) of a finite set S onto itself. For example, the sequence (b, c, a, d) is a permutation of the set S = (a, b, c, d) in which a is transformed into b (written as a ! b), also b ! c, c ! a, and d ! d. In this permutation, a ! b ! c ! a forms a cycle, denoted as [a b c], with a length of three and d ! d forms another cycle [d] with a length of one. The cyclic representation of this permutation is denoted as p = [a b c][d]. Let L = (1, 2, 3, . . . , n) be the set of labels of the links in a kinematic chain. Applying a permutation p of L is equivalent to relabeling the links of the kinematic chain. In general, the original kinematic chain and the relabeled kinematic chain are isomorphic. For example, the permutation p = [1][2 6][3 5][4] transforms the Stephenson kinematic chain shown in Fig. 1b into the isomorphic chain shown in Fig. 2a. For some special permutations, the relabeled kinematic chain is the same as the original chain; i.e., not only the link adjacency but also the labels of the links are the same. In terms of graph theory, these two chains are referred as automorphic chains. For example, the permutation p = [1 4][2 3][5 6] transforms the Stephenson kinematic chain shown in Fig. 1b into the automorphic chain shown in Fig. 2b. The set of special permutations, which relabel the links of a kinematic chain and transform the chain into an automorphic chain, forms a group. This group is referred to as the link-group of the kinematic chain and is denoted as GL. For example, the link-group of the Stephenson kinematic chain shown in Fig. 1b is GL ¼ fP L1 ; P L2 ; P L3 ; P L4 g

ð1Þ

where P L1 ¼ ½1½2½3½4½5½6

ð2aÞ

P L2 ¼ ½1½2 3½4½5 6 P L3 ¼ ½1 4½2½3½5½6 and

ð2bÞ ð2cÞ

P L4 ¼ ½1 4½2 3½5 6

ð2dÞ

Similarly, let J = (a, b, c, . . .) be the set of labels of the joints in a kinematic chain, then there exists a set of permutations which transforms the kinematic chain into an automorphic chain. These permutations form the joint-group of the kinematic chain and is denoted as GJ. For example, the joint-group of the Stephenson kinematic chain shown in Fig. 1b is GJ ¼ fP J 1 ; P J 2 ; P J 3 ; P J 4 g

ð3Þ

where P J 1 ¼ ½a½b½c½d½e½f ½g

ð4aÞ

P J 2 ¼ ½a f ½b g½c½e d P J 3 ¼ ½a½bd½c½eg½f  and

ð4bÞ ð4cÞ

P J 4 ¼ ½a f ½b e½c½dg

ð4dÞ

(a) The Isomorphic Chain.

(b) The Automorphic Chain.

Fig. 2. The isomorphic and automorphic chains.

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If the links and the joints of a kinematic chain are labeled simultaneously then an analogous group, referred to as the chain-group, can be obtained and is denoted as GC. For example, the chain-group of the Stephenson kinematic chain shown in Fig. 1b is GC ¼ fP C1 ; P C2 ; P C3 ; P C4 g

ð5Þ

where P C1 ¼ ½1½2½3½4½5½6½a½b½c½d½e½f ½g

ð6aÞ

P C2 ¼ ½1½2 3½4½5 6½a f ½b g½c½e d

ð6bÞ

P C3 ¼ ½1 4½2½3½5½6½a½b d½c½e g½f  and

ð6cÞ

P C4 ¼ ½1 4½2 3½5 6½a f ½b e½c½d g

ð6dÞ

A procedure to determine the three permutation groups (i.e., the link-group, the joint-group, and the chaingroup) of a candidate kinematic chain was proposed by Yan and Hwang [32]. The procedure is based on an algorithm to identify the permutation groups. The first step in the algorithm is to define the set of labels of the links in the kinematic chain. The second step is to define all possible link permutations, the link adjacency matrix (LAM), and the labeled link adjacency matrix (LLAM) of the kinematic chain. The third step is to apply each of the possible link permutations on the LAM. If the resulting LAM is the same as the original LAM then this permutation is the number of the link-group of the kinematic chain. The fourth, and final, step is to apply each of the possible link permutation on the LLAM. The joint-group and the chain-group are obtained by observing the transformations of non-diagonal elements of the LLAM. This four step procedure will be used in Section 4 to count the number of mechanisms subject to a set of design constraints from the candidate kinematic chains. 2.4. Cycle index Let G denote the permutation group and jGj denote the number of permutations of the set S. Each permutation p in G can be written uniquely as a product of disjoint cycles where a disjoint cycle is the number of cycles in a permutation. For example, the permutation p = [1][2 3][4][5 6] indicates the product of the four disjoint cycles [1], [2 3], [4], and [5 6]. The cycle structure representation of a permutation is defined as b2 bk xb1 1 x2    xk in which xk is a dummy variable for a cycle with length k, and the superscript bk is the number of cycles with length k. For example, the permutation p = [1][2 3][4][5 6] has the cycle structure representation x21 x22 . The cycle index of the permutation group G, denoted as PG(x1, x2, . . . , xk), is defined as the summation of the cycle structure representation of all the permutations (which constitute the elements of the group) divided by the number of permutations [24–27,32–35]. Therefore, the cycle index of the permutation group G can be written as P G ðx1 ; x2 ; . . . ; xk Þ ¼

1 X b1 b2 x x    xbk k jGj p2G 1 2

ð7Þ

For example, the cycle index of the link-group of the Stephenson kinematic chain shown in Fig. 1b, obtained from Eqs. (1) and (2), is 1 P G ðx1 ; x2 Þ ¼ ðx61 þ x21 x22 þ x41 x2 þ x32 Þ 4

ð8Þ

Similarly, the cycle index of the joint-group of the Stephenson kinematic chain, obtained from Eqs. (3) and (4), is 1 P G ðy 1 ; y 2 Þ ¼ ðy 71 þ 2y 1 y 32 þ y 31 y 22 Þ 4

ð9Þ

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The representation of the cycle index of the chain-group is obtained by applying two dummy variables, say x and y, to indicate the links and the joints of the kinematic chain, respectively. Therefore, the cycle index of the chain-group of the Stephenson kinematic chain can be written from Eqs. (5) and (6) as 1 P G ðx1 ; x2 ; y 1 ; y 2 Þ ¼ ðx61 y 71 þ x21 x22 y 1 y 32 þ x41 x2 y 31 y 22 þ x32 y 1 y 32 Þ 4

ð10Þ

2.5. A walk, a loop, and a path A walk of a kinematic chain is defined as an alternating sequence of links and joints, beginning and ending with a link, in which each joint is incident with the two links immediately preceding and following that joint [24]. For example, the sequence 1-b-3-a-3 shown in Fig. 1b is a walk. A walk is defined as closed if the beginning and the ending of the kinematic chain have the same link. A closed walk of a kinematic chain with n links is defined as a loop, if the n links are distinct for n P 3. For example, the closed walk 1-b-3-d-4-e-2-g-1 shown in Fig. 1b is a loop. A walk of a kinematic chain is defined as a path if the links in the chain are distinct; i.e., each link of the path is unique. For example, the sequence 1-b-3-a-6 shown in Fig. 1b is a path. The length of a path is defined as the number of joints in the path. For example, the sequence 1-b-3-a-6 shown in Fig. 1b is a path with length 2. Two paths are said to be the same if they are a counter permutation of each other. A counter permutation indicates an inversion of a path that has the same links as the original path. For example, the two paths 1-b-3-a-6 and 6-a-3-b-1 shown in Fig. 1b can be regarded as the same path. A link-path is defined as a path that begins and ends with a link, and a joint-path is defined as a path that begins and ends with a joint. 2.6. Linkage path code The linkage path code of a kinematic chain with n links is defined as an ordered sequence of numbers {p0, p1, . . . , pi, . . . , pn1, pn2} in which pi is the total number of paths with length i in the chain [24]. The linkage path code of a given kinematic chain can be obtained directly, according to the definition, by inspecting the structure of the chain. As an example, consider the four-bar kinematic chain shown in Fig. 3. The paths with zero length are 1, 2, 3, and 4; i.e., p0 = 4; the paths with length one are 1-a-2, 1-b-4, 2-c-3, and 3-d-4; i.e., p1 = 4; the paths with length two are 1-a-2-c-3, 1-b-4-d-3, 2-a-1-b-4, and 2-c-3-d-4; i.e., p2 = 4; and the paths with length three are 1-a-2-c-3-d-4, 1-b-4-d-3-c-2, 2-a-1-b-4-d-3, and 3-c-2-a-1-b-4; i.e., p3 = 4. Therefore, the linkage path code of the four-bar kinematic chain is {4, 4, 4, 4}. To obtain the linkage path code of a kinematic chain directly, requires that all possible paths in the chain must be identified. As the number of links in the chain increases, however, the numeration of the paths becomes more difficult. A method which can simplify the inspection procedure for obtaining the linkage path code of a given kinematic chain will be presented here. Consider a kinematic chain with n links. The corresponding linkage path code can be expressed as {p0, p1, . . . , pi, . . . , pn1, pn2} where the first number of the code (p0) is the number of links in the chain and the second number of the code (p1) is the number of joints in the chain. The third number of the code is obtained from the relation: p2 ¼

n X J i ðJ i  1Þ 2 i¼1

ð11Þ

Fig. 3. The four-bar kinematic chain.

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where Ji is the number of joints in the ith link. Finally, the number of the code for i P 3 can be obtained from the relation: pi ¼

pi2 X

ðuk  ak Þðvk  bk Þ  iLi

ð12Þ

k¼1

where uk and vk are the number of joints of the end links of the path k with length i  2 (uk 6 vk), ak and bk are the corresponding number of joints which are incident to the end links and connecting two links of the path k with length i  2, and Li is the number of i-bar loops in the chain. For example, the linkage path code of the Stephenson kinematic chain shown in Fig. 1b is {6, 7, 10, 13, 14, 6}. The generating function, Polya’s theory and the pattern inventory to count the number of non-isomorphic identified mechanisms will be presented in the following section. 3. The number of non-isomorphic identified mechanisms 3.1. Generating function Let (a0, a1, a2, . . . , ar) be a sequence of real numbers [25–27,33–35]. The generating function of this sequence, which represents all possible ways of selection, can be written as F ðxÞ ¼ a0 x þ a1 x2 þ a2 x3 þ    þ ar xr

ð13Þ

where x, x2, x3, . . . , xr is a sequence of functions of x that are used as indicators. For example, from the three distinct objects a, b, and c there are three ways to choose one object, namely a or b or c. Similarly, there are three ways to choose two objects, namely either a and b, or b and c, or c and a, from these three objects. Note that there is only one way to choose three objects; i.e., abc. The generating function can be written from Eq. (13) as F ðxÞ ¼ ð1 þ axÞð1 þ bxÞð1 þ cxÞ ¼ 1 þ ða þ b þ cÞx þ ðab þ bc þ caÞx2 þ ðabcÞx3

ð14Þ

where the factor (1 + ax) means that for object a, the two ways of selection are to choose a or not to choose a. A similar interpretation can be given to the factors (1 + bx) and (1 + cx). In the generating function, the product means and, and the plus means or. As a result, the coefficients of the powers of x show all the possible ways of selection. 3.2. Polya’s theory Let S denote the set of elements in a candidate kinematic chain, T denote the set of types (t) of these elements, G denote the permutation group of the set S, and PG(x1, x2, . . . , xk) denote the cycle index of G (where k is the number of elements in the cycle). Then the pattern inventory [25–28,32–35] of the identified mechanisms can be written as X X X X  I ¼ PG t; t2 ; t3 ; . . . ; tk ð15Þ Eq. (15) can be modified to count the number of mechanisms when the links and the joints are assigned simultaneously. In order to do this, let S denote the set of links and joints in the kinematic chain, U denote the set of types of links, and V denote the set of types of joints. The chain-group of S is denoted as GC, and PG(x1, x2, . . . , xk; y1, y2, . . . , yk) is the cycle index of GC. Therefore, the pattern inventory of the identified mechanisms can now be written as X X X  X X X I ¼ PG u; u2 ; u3 ; . . . ; v; v2 ; v3 ; . . . ð16Þ where the coefficient of each term in this polynomial represents the number of non-isomorphic mechanisms.

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Two examples will be presented here to illustrate Polya’s theory. Eq. (15) will be applied to the first example to obtain the non-isomorphic mechanisms with specified links or specified joints. Then Eq. (16) will be applied to the second example to obtain the non-isomorphic mechanisms with specified links and specified joints. Example 3.1. For the Stephenson kinematic chain shown in Fig. 1b, the problem is to count the number of non-isomorphic mechanisms that satisfy the following two design constraints: (i) a ground link (denoted as Lg); and (ii) five moving links (denoted as LK). Let SL = (1, 2, 3, 4, 5, 6) be the set of labels of the links, T = {LK, Lg} be the set of types of the links, and GJ = {PL1, PL2, PL3, PL4} be the permutation groups of the set SL . Then the cycle index of the Stephenson kinematic chain, from Eq. (7), is 1 P G ðx1 ; x2 Þ ¼ ðx61 þ x21 x22 þ x41 x2 þ x32 Þ 4

ð17Þ

where the dummy variable x indicates the type of links in the chain; i.e., the ground link and the moving links. Substituting x1 = Lg + LK and x2 ¼ L2g þ L2K into Eq. (17), and using Eq. (15), the pattern inventory is 1 6 2 2 4 3 I ¼ ½ðLg þ LK Þ þ ðLg þ LK Þ ðL2g þ L2K Þ þ ðLg þ LK Þ ðL2g þ L2K Þ þ ðL2g þ L2K Þ  4

ð18aÞ

which can be written as I ¼ L6K þ 3L5K Lg þ 7L4K L2g þ 8L3K L3g þ 7L2K L4g þ 3LK L5g þ L6g

ð18bÞ

LiK Ljg

The coefficient of the term in Eq. (18b) represents the total number of non-isomorphic mechanisms (with i moving links and j ground links) which satisfy the two design constraints. For this example, i = 5 and j = 1, therefore, there are three non-isomorphic mechanisms. The three mechanisms are referred to as the Stephenson I, the Stephenson II, and the Stephenson III mechanisms, see Fig. 4a–c. Example 3.2. For the Stephenson chain shown in Fig. 1b, the problem is to count the number of nonisomorphic mechanisms that satisfy the following three design constraints: (i) two prismatic joints (denoted as JP), (ii) five revolute joints (denoted as JR), and (iii) a ground link (denoted as Lg). The cycle index of the chain-group for the Stephenson kinematic chain, from Eq. (7), is 1 P G ðx1 ; x2 ; y 1 ; y 2 Þ ¼ ðx61 y 71 þ x21 x22 y 1 y 32 þ x41 x2 y 31 y 22 þ x32 y 1 y 32 Þ 4

ð19Þ

where the dummy variable x indicates the type of links in the chain, and the dummy variable y indicates the type of joints in the chain. Substituting x1 = Lg + LK, x2 ¼ L2g þ L2K , y1 = JP + JR, and y 2 ¼ J 2P þ J 2R into Eq. (19), and using Eq. (16), the pattern inventory is 1h I ¼ ðLg þ LK Þ6 ðJ P þ J R Þ7 þ ðLg þ LK Þ2 ðL2g þ L2K Þ2 ðJ P þ J R ÞðJ 2P þ J 2R Þ3 4 i þ ðLg þ LK Þ4 ðL2g þ L2K ÞðJ P þ J R Þ3 ðJ 2P þ J 2R Þ2 þ ðL2g þ L2K Þ3 ðJ P þ J R ÞðJ 2P þ J 2R Þ3

(a) Stephenson I.

(b) Stephenson II.

(c) Stephenson III.

Fig. 4. The three Stephenson mechanisms.

ð20Þ

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The coefficient of the term J 2P J 5R L5K L1g in Eq. (20) is obtained from a formal Taylor series expansion and can be written as    5  5  2   1 d d d dI ð21aÞ 2!5!5! dLg dL5K dJ 5R dJ 2P or, using short-hand notation, as   1 d13 I 2!5!5! ðdJ 2P ÞðdJ 5R ÞðdL5K ÞðdLg Þ

ð21bÞ

where dJ 2P is the second derivative of the pattern inventory with respect to JP, dJ 5R is the fifth derivative of this result with respect to JR, dL5K is the fifth derivative of this result with respect to LK, and dLg is the first derivative of this result with respect to Lg. Performing this set of differentiations, the coefficient in Eq. (21b) can be shown to be 152. Then substituting this value into Eq. (20), the total number of Stephenson mechanisms which satisfy the three design constraints can be shown to be 38. Due to space considerations, the thirty-eight mechanisms will not be included in this paper. The following section presents the procedure to count the number of non-isomorphic mechanisms subject to design constraints from the candidate kinematic chains. The work shows that the mechanisms with specified types of links and joints can be obtained in a systematic manner. 4. Procedure to count the number of mechanisms A flowchart to illustrate the procedure to count the number of non-isomorphic mechanisms subject to design constraints from the candidate kinematic chains is shown in Fig. 5. The broken lines in the flowchart (for the generating function, linkage path code, and design constraints feedback to permutation groups) indicate that they are not required for all design problems. For example, they are not required when the design constraints only consider the types and numbers of links and joints. If the design constraints consider the adjacency relation, incident relation, or some joints corresponding to some links, then they must be included in the procedure. The systematic procedure to obtain the number of non-isomorphic identified mechanisms subject to the specified design constraints from the candidate kinematic chains is: Step 1. Identify the permutation groups (i.e., link-group, joint-group, and chain-group) of the candidate kinematic chains by applying the link adjacency matrix (LAM) and the labeled link adjacency matrix

Candidate Kinematic Chains Kinematic Matrices Modified Permutation Groups Permutation Groups Cycle Structure Representations Cycle Index Generating Function Design Constraints Linkage Path Code Polya’s Theory Coefficients of Equation Number of Mechanisms Fig. 5. The flowchart to count the number of identified mechanisms.

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(LLAM). In the examples presented in this paper, the permutation groups are obtained from the algorithm proposed by Yan and Hwang [32]. Step 2. Determine the cycle index of the permutation groups, see Eq. (7). Step 3. Assign the design constraints to the cycle index, see Refs. [32–35]. If the design constraints include the adjacency relation, the incident relation, or some specified joints corresponding to some specified links, then the permutation group is redefined. Apply Eqs. (11) and (12) to obtain the linkage path code of the kinematic chains. The modified permutation group can then be obtained by using the link-path (or the joint-path) to show some joints corresponding to some links. The generating function, see Eqs. (13) and (14), can now be applied to show the adjacency relation, the incident relation, or some joints corresponding to some links into the cycle index. In this way, the identified mechanisms with the required design constraints can be obtained. Step 4. Obtain the number of non-isomorphic identified mechanisms from the corresponding coefficients that are identified from the pattern inventory polynomial, see Eqs. (15) and (16). Three practical examples are provided in the following section to illustrate the step-by-step approach of this systematic procedure. The first example is the differential-type south pointing chariot, the second example is the Watt kinematic chain, and the final example is a variable-stroke engine. 5. Three practical examples Example 5.1. Consider the differential-type south pointing chariot shown in Fig. 6a. This mechanism has two degrees of freedom; i.e., two inputs are required for a unique output [37]. The candidate kinematic chain is the (5, 6) kinematic chain shown in Fig. 6b which has four lower pairs and two higher pairs. The problem is to count the number of non-isomorphic mechanisms from this chain subject to the following four design constraints: (i) the ground link (denoted as Lg) must be a multiple link (i.e., a fixed link with more than two joints); (ii) there must be three gears (denoted as LG1, LG2, and LG3); (iii) there must be two gear joints (denoted as JG1 and JG2); and (iv) three revolute joints (denoted as JR1, JR2, and JR3) must be incident with the ground link. The first step is to identify the permutation groups of the candidate kinematic chain. The chain-group of this kinematic chain, see Eq. (5), is GC ¼ fP C1 ; P C2 ; P C3 ; P C4 ; P C5 ; P C6 ; P C7 ; P C8 ; P C9 ; P C10 ; P C11 ; P C12 g

ð22Þ

where P C1 ¼ ½1½2½3½4½5½a½b½c½d½e½f  P C2 ¼ ½1½3½2 4½5½a b½e f ½c½d

(a) The Differential-Type South Pointing Chariot.

ð23aÞ ð23bÞ

(b) The (5, 6) Kinematic Chain.

Fig. 6. The south pointing chariot and the candidate kinematic chain.

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P C3 ¼ ½1½3½2 5½4½a c½e d½b½f  P C4 ¼ ½1½3½2½4 5½a½e½b c½d f 

ð23cÞ ð23dÞ

P C5 ¼ ½1½3½2 4 5½a b c½e f d P C6 ¼ ½1½3½2 5 4½a c b½e d f 

ð23eÞ ð23fÞ

P C7 ¼ ½1 3½2 4½5½a f ½b e½c d P C8 ¼ ½1 3½2 5½4½a d½c e½b f 

ð23gÞ ð23hÞ

P C9 ¼ ½1 3½2½4 5½a e½b d½c f 

ð23iÞ

P C10 ¼ ½1 3½2 4 5½a d b e c f  P C11 ¼ ½1 3½2 5 4½a f c e b d

ð23jÞ ð23kÞ

and

P C12 ¼ ½1 3½2½4½5½a e½c d½b f 

ð23lÞ

The second step is to determine the cycle index of the permutation groups. From Eq. (7), the cycle index of the kinematic chain can be written as PG ¼

1 2 3 6 ðx y z þ 3x21 y 1 y 2 z21 z22 þ 2x21 y 3 z23 þ 3x2 y 1 y 2 z32 þ 2x2 y 3 z6 þ x2 y 31 z32 Þ 12 1 1 1

ð24Þ

where the dummy variable x indicates the multiple links; y indicates the binary links; and z indicates the joints. The third step is to assign the design constraints to the cycle index. Since the second and third design constraints imply that the two gear joints are incident with the three gears then the link-path must be determined. Three gears and two gear joints form a link-path with a length of two. Substituting n = 5, J1 = J3 = 3, and J2 = J4 = J5 = 2 into Eq. (11), the third number of the code is     3ð3  1Þ 2ð2  1Þ p2 ¼ ð2Þ þ ð3Þ ¼9 ð25Þ 2 2 The nine link-paths are: 1-a-2-e-3, 1-b-4-f-3, 1-c-5-d-3, 2-a-1-b-4, 2-a-1-c-5, 2-e-3-f-4, 2-e-3-d-5, 4-b-1-c-5, and 4-f-3-d-5. Since the ground link must be a multiple link then the link-paths that contain two multiple links (1 and 3) must be deleted from consideration. Therefore, six link-paths are introduced into the modified permutation group. Based on the procedure to find the modified permutation group, presented in Ref. [33], the modified chain-group can be written as

 G0C ¼ P 0C1 ; P 0C2 ; P 0C3 ; P 0C4 ; P 0C5 ; P 0C6 ; P 0C7 ; P 0C8 ; P 0C9 ; P 0C10 ; P 0C11 ; P 0C12 ð26Þ where P 0C1 ¼ ½2-a-1-b-4½2-a-1-c-5½2-e-3-f -4½2-e-3-d-5½4-b-1-c-5 ½4-f -3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4  P 0C2 P 0C3 P 0C4

¼ ½2-a-1-b-4½2-a-1-c-5 4-b-1-c-5½2-e-3-f -4 ½2-e-3-d-5 4-f -3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4 

ð27bÞ

¼ ½2-a-1-b-4 4-b-1-c-5½2-a-1-c-5½2-e-3-d-5 ½2-e-3-f -4 4-f -3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4 

ð27cÞ

¼ ½2-a-1-b-4 2-a-1-c-5½2-e-3-f -4 2-e-3-d-5

½4-b-1-c-5½4-f -3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4  P 0C5 ¼ ½2-a-1-b-4 2-a-1-c-5 4-b-1-c-5 P 0C6

ð27aÞ

ð27dÞ

½2-e-3-f -4 2-e-3-d-5 4-f -3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4  ¼ ½2-a-1-b-4 4-b-1-c-5 2-a-1-c-5

ð27eÞ

½2-e-3-f -4 4-f -3-d-5 2-e-3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4 

ð27fÞ

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P 0C7 ¼ ½2-a-1-b-4 2-e-3-f -4½2-a-1-c-5 4-f -3-d-5 ½2-e-3-d-5 4-b-1-c-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4  P 0C8

¼ ½2-a-1-b-4 4-f -3-d-5½2-e-3-f -4 4-b-1-c-5 ½2-a-1-c-5 2-e-3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4 

687

ð27gÞ ð27hÞ

P 0C9

¼ ½2-a-1-b-4 2-e-3-d-5½2-a-1-c-5 2-e-3-f -4 ½4-b-1-c-5 4-f -3-d-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4    2-e-3-f -4 2-a-1-c-5 4-f -3-d-5 P 0C10 ¼ 2-a-1-b-4 2-e-3-d-5 4-b-1-c-5 ½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4    2-e-3-f -4 4-b-1-c-5 2-e-3-d-5 0 P C11 ¼ 2-a-1-b-4 4-f -3-d-5 2-a-1-c-5 ½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4 

ð27iÞ

ð27jÞ

ð27kÞ

and P 0C12 ¼ ½2-a-1-b-4 2-e-3-f -4½2-a-1-c-5 2-e-3-d-5 ½4-f -3-d-5 4-b-1-c-5½V L2 ½V L3 ½V J 1 ½V J 2 ½V J 3 ½V J 4 

ð27lÞ

According to the four design constraints, the two gear joints are incident with the three gears. Therefore, the two variables VL2 and VL3 indicate the remaining binary link and ternary link, respectively. The remaining joints are denoted by the variables V J 1 , V J 2 , V J 3 , and V J 4 . The cycle index of the kinematic chain, see Eq. (24), must be modified after obtaining the modified permutation groups. The cycle index of the modified chain-group of the kinematic chain, from Eq. (7), is P 0G ¼

1 6 ½x y z1 w4 þ 3x21 x22 y 1 z1 w41 þ 2x23 y 1 z1 w41 þ 4x32 y 1 z1 w41 þ 2x6 y 1 z1 w41  12 1 1 1

ð28Þ

The dummy variable x indicates the link-path with the three links and the two joints; y indicates the remaining binary link; z indicates the remaining ternary link; and w indicates the revolute or prismatic joints. The fourth, and final, step is to obtain the number of non-isomorphic identified mechanisms. The procedure is to apply the generating function to demonstrate the inventory of the dummy variables. The link-path is denoted as A which includes LG and JG. Also, let LK and JK denote a link and a joint, respectively, which are not assigned in the design constraints. Therefore, in this example there is one JK which is chosen to be a revolute joint from practical considerations. Substituting x1 = A + 1, x2 = A2 + 1, x3 = A3 + 1, x6 = A6 + 1, y1 = LK, z1 = Lg + LK, and w1 = JR + JK into Eq. (28), the pattern inventory can be written as 1 h 6 4 2 2 4 ðA þ 1Þ ðLK ÞðLg þ LK ÞðJ R þ J K Þ þ 3ðA þ 1Þ ðA2 þ 1Þ ðLK ÞðLg þ LK ÞðJ R þ J K Þ 12 2 4 3 4 þ 2ðA3 þ 1Þ ðLK ÞðLg þ LK ÞðJ R þ J K Þ þ 4ðA2 þ 1Þ ðLK ÞðLg þ LK ÞðJ R þ J K Þ i 4 þ 2ðA6 þ 1ÞðLK ÞðLg þ LK ÞðJ R þ J K Þ

I¼

ð29Þ

The coefficient of the term A1 J 4R LK L1g in Eq. (29) is obtained from a formal Taylor series expansion, and using the same notation as in Eq. (21a), can be written as   1 d7 I 4! ðdAÞðdJ 4R ÞðdLK ÞðdLg Þ

ð30Þ

where dA is the first derivative of the pattern inventory with respect to A, dJ 4R is the fourth derivative of this result with respect to JR, dLK is the first derivative of this result with respect to LK, and dLg is the first derivative of this result with respect to Lg. Performing this set of differentiations, the coefficient in Eq. (30) is 12.

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Substituting this value into Eq. (29), the total number of non-isomorphic mechanisms which satisfy the four design constraints is one. This mechanism is shown in Fig. 7. Example 5.2. Consider the Watt kinematic chain shown in Fig. 8 which is a (6, 7) kinematic chain. The problem is to count the number of non-isomorphic mechanisms from this chain subject to the following five design constraints: (i) the ground link must be a multiple link (denoted as Lg); (ii) the input link (denoted as LI) must be adjacent to the ground link by a revolute joint (denoted as JR); (iii) the output link (denoted as LO) must be adjacent to the ground link by a prismatic joint (denoted as JP); (iv) there must be at least one prismatic joint but no more than two prismatic joints in order to avoid a dead center configuration; and (v) no binary link can have more than one prismatic joint. The chain-group of the Watt kinematic chain, see Eq. (5), is GC ¼ fP C1 ; P C2 ; P C3 ; P C4 g

ð31Þ

where P C1 ¼ ½1½2½3½4½5½6½a½b½c½d½e½f ½g

ð32aÞ

P C2 ¼ ½1 2½3 4½5 6½a f ½b g½c½d½e P C3 ¼ ½1½2½3 5½4 6½a b½c e½d½f g and

ð32bÞ ð32cÞ

P C4 ¼ ½1 2½3 6½4 5½a g½b f ½c e½d

ð32dÞ

According to the second and third design constraints, the input link and the output link have to be adjacent to the ground link by revolute and prismatic joints, respectively. Substituting n = 6, J1 = J2 = 3, and J3 = J4 = J5 = J6 = 2 into Eq. (11), the third number of the code is     3ð3  1Þ 2ð2  1Þ p2 ¼ ð2Þ þ ð4Þ ¼ 10 ð33Þ 2 2 The ten link-paths are: 1-d-2-f-6, 1-d-2-g-4, 1-a-5-c-6, 1-b-3-e-4, 2-d-1-a-5, 2-d-1-b-3, 2-f-6-c-5, 2-g-4-e-3, 3-b-1a-5, and 4-g-2-f-6. Since the ground link must be a multiple link and the input link can not be adjacent to the output link then the link-paths that contain two multiple links (i.e., links 1 and 2) must be deleted from consideration. Therefore, two link-paths are introduced into the modified permutation group and the modified chain-group can be written as G0C ¼ fP 0C1 ; P 0C2 ; P 0C3 ; P 0C4 g

ð34Þ

where P 0C1 ¼ ½3-b-1-a-5½4-g-2-f -6½V L3 ½V L21 ½V L22 ½c½d½e½V J 1 ½V J 2 

ð35aÞ

P 0C2 ¼ ½3-b-1-a-5 4-g-2-f -6½V L3 ½V L21 V L22 ½c½d½e½V J 1 ½V J 2 

ð35bÞ

P 0C3

¼ ½3-b-1-a-5½4-g-2-f -6½V L3 ½V L21 V L22 ½ce½d½V J 1 ½V J 2 

and

P 0C4 ¼ ½3-b-1-a-5 4-g-2-f -6½V L3 ½V L21 V L22 ½c e½d½V J 1 ½V J 2 

Fig. 7. The mechanism which satisfies the specified design constraints.

ð35cÞ ð35dÞ

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689

Fig. 8. The Watt kinematic chain.

The cycle index of the modified chain-group of the Watt kinematic chain, from Eq. (7), is 1 P 0G ¼ ðx21 y 1 z21 w51 þ x2 y 1 z2 w51 þ x21 y 1 z2 w31 w2 þ x2 y 1 z2 w31 w2 Þ 4

ð36Þ

The dummy variable x indicates the link-path with three links (i.e., the ground link, the input link, and the output link) and two joints (i.e., a revolute joint and a prismatic joint); y indicates the remaining ternary link; z indicates the remaining binary links; and w indicates the revolute or prismatic joints. Finally, apply the generating function to demonstrate the inventory of the dummy variables. The link-path is denoted as B and includes Lg, LI, LO, JR, and JP. Substituting x1 = B + 1, x2 = B2 + 1, y1 = LKt, z1 = LKb, z2 ¼ L2Kb , w1 = JR + JP, and w2 ¼ J 2R þ J 2P into Eq. (36), the pattern inventory can be written as 1h 2 2 5 5 I ¼ ðB þ 1Þ ðLKt ÞðLKb Þ ðJ R þ J P Þ þ ðB2 þ 1ÞðLKt ÞðL2Kb ÞðJ R þ J P Þ 4 i ð37Þ þðB þ 1Þ2 ðLKt ÞðL2Kb ÞðJ R þ J P Þ3 ðJ 2R þ J 2P Þ þ ðB2 þ 1ÞðLKt ÞðLKb Þ2 ðJ R þ J P Þ3 ðJ 2R þ J 2P Þ

Case (i). If the mechanism is to have only one prismatic joint then the coefficient of the term B1 J 5R LKt L2Kb in Eq. (37) is obtained from a formal Taylor series expansion. Using the same notation as in Eq. (21a), the result can be written as   1 d9 I 5!2! ðdBÞðdJ 5R ÞðdLKt ÞðdL2Kb Þ

ð38Þ

where dB is the first derivative of the pattern inventory with respect to B, dJ 5R is the fifth derivative of this result with respect to JR, dLKt is the first derivative of this result with respect to LKt, and dL2Kb is the second derivative of this result with respect to LKb. Performing this set of differentiations, the coefficient in Eq. (38) is 4. Substituting this value into Eq. (37), the total number of non-isomorphic mechanisms which satisfy the design constraints is one. This mechanism; i.e., the Watt mechanism with one prismatic joint, is shown in Fig. 9. Case (ii). If the mechanism is to have two prismatic joints then the coefficient of the term B1 J 4R J 1P LKt L2Kb in Eq. (37) is obtained from a formal Taylor series expansion. Using the same notation as in Eq. (21a), the result can be written as " # 1 d9 I ð39Þ 4!2! ðdBÞðd4 J R ÞðdJ P ÞðdLKt Þðd2 LKb Þ where dB is the first derivative of the pattern inventory with respect to B, dJ 4R is the fourth derivative of this result with respect to JR, dJP is the first derivative of this result with respect to JP, dLKt is the first derivative of this result with respect to LKt, and dL2Kb is the second derivative of this result with respect to LKb. Performing this set of differentiations, the coefficient in Eq. (39) is 16. Substituting this value into Eq. (37), the

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Fig. 9. The Watt mechanism with one prismatic joint.

total number of non-isomorphic mechanisms which satisfy the design constraints is four. The four mechanisms; i.e., the Watt mechanisms with two prismatic joints, are shown in Fig. 10a–d. Example 5.3. Consider the variable-stroke engine shown in Fig. 11a. The candidate kinematic chain is the (8, 10) kinematic chain shown in Fig. 11b. The problem is to count the number of non-isomorphic mechanisms from this chain subject to the following six design constraints: (i) the ground link 1 (denoted as Lg) must be a quaternary link; (ii) the crank 2 (denoted as LCr) must be a binary link adjacent to the ground link by a revolute joint; (iii) the piston 8 (denoted as LP) must be a binary link adjacent to the ground link by a prismatic joint; (iv) the connecting rod 7 (denoted as LC) must be a binary link adjacent to the piston by a revolute joint; (v) the maximum number of prismatic joints cannot exceed two; and (vi) no link can have more than one prismatic joint. The chain-group of the kinematic chain, see Eq. (5), is GC ¼ fP C1 ; P C2 g

ð40Þ

JP

JR

JR

JR LI

JR Lg

JR

LO JP

(c) The Third Identified Mechanism.

(d) The Fourth Identified Mechanism.

Fig. 10. The four Watt mechanisms with two prismatic joints.

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691

where P C1 ¼ ½1½2½3½4½5½6½7½8½a½b½c½d½e½f ½g½h½i½j P C2 ¼ ½1½2 8½3 7½4 6½5½a b½c f ½d e½g j½h i

and

ð41aÞ ð41bÞ

According to the design constraints, the crank and the piston have to be adjacent to the ground link by revolute and prismatic joints, respectively. Also, the connecting rod has to be adjacent to the piston by a revolute joint. From Eq. (12), the fourth number of the code is p1 X p3 ¼ ðuk  1Þðvk  1Þ  3L3

ð42aÞ

k¼1

which can be written as p3 ¼ 2ð4  1Þð2  1Þ þ 2ð4  1Þð3  1Þ þ 2ð2  1Þð2  1Þ þ 4ð2  1Þð3  1Þ ¼ 28

ð42bÞ

Since the ground link must be a quaternary link, and the crank, the piston, and the connecting rod must be a binary link then the link-path that contains more than one multiple link must be deleted from consideration. Therefore, only two of the 28 link-paths satisfy the specified design constraints. These two link-paths are introduced into the modified permutation group and the modified chain-group can be written as G0C ¼ fP 0C1 ; P 0C2 g

ð43Þ

where P 0C1 ¼ ½8-b-1-a-2-c-3½2-a-1-b-8-f -7½4½5½6½d½e½g½h½i½j½V J  and

ð44aÞ

P 0C2

ð44bÞ

¼ ½8-b-1-a-2-c-3 2-a-1-b-8-f -7½4 6½5½d e½g j½h i½V J 

The cycle index of the modified chain-group of the kinematic chain, from Eq. (7), is 1 P 0G ¼ ðx21 y 21 z1 w1 u41 v21 t1 þ x2 y 2 z1 w1 u22 v2 t1 Þ 2

ð45Þ

The dummy variable x indicates the link-path with four links (the ground link, crank, piston and connecting rod) and three joints (two revolute joints and a prismatic joint); y indicates the ternary links; z indicates the binary links; w indicates the remaining binary link; u indicates that one prismatic joint is not incident to the quaternary link; v indicates that one prismatic joint is incident to the quaternary link; and t indicates the remaining joint. Finally, apply the generating function to demonstrate the inventory of the dummy variables. The link-path is denoted as C and includes Lg, LCr, LP, LC, JR, and JP. In this example there are six JK joints which are chosen to be revolute joints for practical considerations. Substituting x1 = C + 1, x2 = C2 + 1, y1 = LKt,

(a) A Variable-Stroke Engine.

(b) The (8, 10) Kinematic Chain.

Fig. 11. The variable-stroke engine and the candidate kinematic chain.

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Fig. 12. The eight-bar mechanisms which satisfy the specified design constraints.

y 2 ¼ L2Kt , z1 = LKb, w1 = LK, u1 = JP + JK, u2 ¼ J 2P þ J 2K , v1 = JKv, v2 ¼ J 2Kv , and t1 = JKt into Eq. (46), the pattern inventory can be written as 1 2 2 4 2 2 I ¼ ½ðC þ 1Þ ðLKt Þ ðLKb ÞðLK ÞðJ P þ J K Þ ðJ Kv Þ ðJ Kt Þ þ ðC 2 þ 1ÞðL2Kt ÞðLKb ÞðLK ÞðJ 2P þ J 2K Þ ðJ 2Kv ÞðJ Kt Þ 2

ð46Þ

The coefficient of the term C 1 J 3K J 1P J 2Kv J 1Kt L2Kt L1Kb L1K in Eq. (46) is obtained from a formal Taylor series expansion. Using the same notation as in Eq. (21a), the result can be written as 1 d12 I 3 2 3!2!2! ðdCÞðdJ K ÞðdJ P ÞðdJ Kv ÞðdJ Kt ÞðdL2Kt ÞðdLKb ÞðdLK Þ

ð47Þ

where dC is the first derivative of the pattern inventory with respect to C, dJ 3K is the third of derivative of this result with respect to JK, dJP is the first derivative of this result with respect to JP, dJ 2Kv is the second derivative of this result with respect to JKv, dJKt is the first derivative of this result with respect to JKt, dL2Kt is the second derivative of this result with respect to LKt, dLKb is the first derivative of this result with respect to LKb, and dLK is the first derivative of this result with respect to LK. Performing this set of differentiations, the coefficient in Eq. (47) is eight. Substituting this value into Eq. (46), the total number of non-isomorphic mechanisms which satisfy the six design constraints is four. The four mechanisms are shown in Fig. 12a–d. Now consider the two evaluation criteria presented in Ref. [18]; namely: (i) prismatic joints cannot exist in the drive loop; i.e., the loop that contains the crank; and (ii) the size of the control link cannot be too large, for a given stroke, in order to avoid high piston side thrust. From the first criterion, the mechanism shown in Fig. 12a is isomorphic and must be eliminated from consideration. From the second criterion; the mechanisms shown in Fig. 12b and c are also isomorphic and must be eliminated from consideration. Therefore, the only non-isomorphic mechanism which satisfies both evaluation criteria is the mechanism shown in Fig. 12d. The three examples in this section clearly illustrate that the approach presented in this paper can identify the number of non-isomorphic mechanisms from the candidate kinematic chains in a systematic manner.

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693

6. Conclusions This paper presents a procedure to determine the number of non-isomorphic mechanisms which satisfy a set of required design constraints from the candidate kinematic chains. This systematic procedure is based on modified permutation groups, generating function, and Polya’s theory and is believed to be an original contribution to the kinematics literature on the conceptual design of planar mechanisms. The design constraints include: (i) the types of links and joints; (ii) the numbers of links and joints; (iii) the adjacency relation of links and joints; (iv) the incidence relation of links and joints; and (v) some specified joints correspond to some specified links. Traditionally, most approaches use an observation method to determine the number of nonisomorphic mechanisms. This paper applies the concept of a link-path to present the design constraints with a number of specified joints corresponding to a number of specified links. The procedure not only counts the number of non-isomorphic mechanisms accurately but also simplifies the conceptual design of mechanisms. Designers can use this procedure, in the conceptual stage, to obtain a complete atlas of planar mechanisms with specified number of links and types of joints. A future research activity is to computerize this systematic approach in order to solve complex design problems in an efficient manner. The authors plan on developing a flowchart, writing a computer program, and illustrating the generality of a completely computerized code by several practical examples. This will greatly advance the state-of-the art for the conceptual design of both planar and spatial mechanisms. References [1] F. Freudenstein, L. Dobrjanskyj, On a theory for the type synthesis of mechanisms, in: Proceedings of the 11th International Congress of Applied Mechanics, Springer, Berlin, 1964, pp. 420–428. [2] F.R.E. Crossley, The permutations of kinematic chains of eight member or less from the graph-theoretic viewpoint, in: W.A. Shaw (Ed.), Developments in Theoretical and Applied Mechanisms, vol. 2, Pergamon Press, Oxford, 1965, pp. 467–489. [3] L. Dobrjanskyj, F. Freudenstein, Some applications of graph theory to the structural analysis of mechanisms, ASME Journal of Engineering for Industry, Series B 89 (1967) 153–158. [4] L.S. Woo, Type synthesis of plane linkages, ASME Journal of Engineering for Industry, Series B 89 (1967) 159–172. [5] R.C. Johnson, Application of number synthesis to practical problems in creative design, ASME Paper No. 65-WA-MD9, 1965. [6] M. Huang, A.H. Soni, Application of linear and nonlinear graphs in structural synthesis of kinematic chains, ASME Journal of Engineering for Industry, Series B 95 (1973) 525–532. [7] F. Freudenstein, E.R. Maki, The creation of mechanisms according to kinematic structure and function, Journal of Environment and Planning B 6 (4) (1979) 375–391. [8] H.S. Yan, C.H. Hsu, A method for the type synthesis of new mechanisms, Journal of the Chinese Society of Mechanical Engineers, Taipei 4 (1983) 11–23. [9] H.S. Yan, A methodology for creative mechanism design, Mechanism and Machine Theory 27 (1992) 235–242. [10] H.S. Yan, Creative Design of Mechanical Devices, Springer, Singapore, 1998. [11] R.C. Johnson, K. Towfigh, Creative design of epicyclic gear trans using number synthesis, ASME Journal of Engineering for Industry, Series B 89 (1967) 309–314. [12] N.I. Manolescu, P. Antonescu, Structural synthesis of planetary mechanisms used in automatic transmissions, ASME Paper No. 68MECH-44, 1968. [13] F. Freudenstein, An application of Boolean algebra to the motion of epicyclic drives, ASME Journal of Engineering for Industry, Series B 93 (1971) 176–182. [14] L.W. Tsai, An application of linkage characteristic polynomials to the topological synthesis of epicyclic gear trains, ASME Journal of Engineering for Industry, Series B 109 (1987) 329–336. [15] H.S. Yan, L.C. Hsieh, Concept design of planetary gear trains for infinitely variable transmissions, in: Proceedings of the 1989 International Conference on Engineering Design, Harrogate, UK, pp. 757–766, 1989. [16] J.U. Kim, B.M. Kwak, Application of edge permutation group to structural synthesis of epicyclic gear trains, Mechanism and Machine Theory 25 (1990) 563–574. [17] C.H. Hsu, K.T. Lam, A new graph representation for automatic kinematic analysis of spur-planetary gear trains, ASME Journal of Mechanical Design 114 (1992) 196–200. [18] F. Freudenstein, E.R. Maki, Development of an optimum variable-stroke internal-combustion engine from the viewpoint of kinematic structure, ASME Journal of Mechanisms, Transmissions, and Automation in Design 105 (1983) 259–266. [19] F. Freudenstein, E.R. Maki, Kinematic structure of mechanisms for fixed and variable-stroke axial-piston reciprocating machines, ASME Journal of Mechanisms, Transmissions, and Automation in Design 106 (1984) 355–364. [20] Z. Yu, T.W. Lee, Kinematic structural and functional analysis of wobble-plate engines, ASME Journal of Mechanisms, Transmissions, and Automation in Design 107 (1986) 226–236.

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