the creation of mechanisms according to kinematic ... · with link 2 y gear integral with link 6...

Environment and Planning B, 1979, volume 6, pages 375-391

The creation of mechanisms according to kinematic structure and function

F Freudenstein Department of Mechanical Engineering, Columbia University, New York, NY 10027

E R Maki Mechanical Research Department, General Motors Research Laboratories, General Motors Corporation, Warren, Michigan 48090, USA Received 28 September 1979

Abstract. The classification of mechanisms is reviewed and their kinematic structure and graph representation discussed. Mechanism creation based on the separation of structure from function is illustrated for shaft couplings and other devices. The use and significance of this procedure in mechanical design and interpretation of patents is shown.

1 Introduction Perhaps the most difficult phase of mechanical design is the conceptual phase—for example the creation of a mechanism which has to achieve a given function. Two approaches have been developed for this problem in the course of a long historyM. The first is the creation of atlases of mechanisms grouped according to function (Jones et al, 1968), and these remain the primary source of ideas for mechanisms. The second involves an abstract representation of the structure of mechanisms somewhat in the spirit of the symbolic representation of chemical compounds. A few attempts of limited success have also been made to represent functional aspects of mechanisms symbolically (Reuleaux, 1876; Franke, 1951).

Beginning in the mid-nineteen-sixties, the abstract representation of kinematic structure has been investigated with the aid of graph theory(2). This is a powerful tool for the creation of mechanisms in a relatively simple, semisystematic manner. An approach which appears to be promising in this connection is the creation of mechanisms by the separation of structure from function.

In this paper the method is described in the simplest possible terms and its application to several kinds of mechanisms is illustrated. The technique, which is partly analytical and partly empirical, requires a willingness to approach an old subject in an initially unfamiliar manner. For those willing to try this approach, the rewards should be well worth the effort.

2 Separation of structure and function The basic idea underlying the creation of mechanisms according to this method is the separation of structure from function. The kinematic structure of a mechanism is essentially the answer to the question: "Which link connects which other link by what type of joint?" Kinematic structure can be enumerated in an essentially systematic, unbiased fashion as a function of the degree of freedom of the mechanism, the nature of the desired motion (plane or spatial, number of moving parts) and a parameter representing an indication of the complexity of the mechanism. Each

(1) (Artobolevskii, 1956; Assur, 1913-1916; Babbage, 1826; Jones et al, 1968; Reuleaux, 1876) W (Buchsbaum and Freudenstein, 1970; Crossley, 1965; Davies, 1968; Dobrjanskyj and Freudenstein, 1967; Freudenstein, 1971; Freudenstein and Dobrjanskyj, 1965; Freudenstein and Woo, 1974; Woo, 1967)

376 F Freudenstein, E R Maki

structure obtained in this way can then be sketched and evaluated with respect to the functional requirements of the mechanism. Potentially acceptable mechanisms can then be evaluated in depth until a final design is achieved. A more formal outline of the method is given in table 1. The abstract representation of kinematic structure is now considered.

Table 1. Creation of mechanisms.

(a) Structure 1. Determine freedom (F), number of moving members, complexity (L-m&) and nature of motion (plane, X = 3; spatial, X = 6). 2. Find structures of kinematic chains from tables. 3. Label structures according to types of joints and choice of fixed link in as many different ways as possible. 4. For each structure, sketch corresponding mechanism.

(b) Function Determine the functional requirements for the desired mechanism and the relationship to kinematic structure.

(c) Structure and function Use functional requirements to screen out unacceptable mechanisms. Remaining mechanisms represent potential solutions and can then be evaluated in greater detail.

3 The representation of kinematic structure 3.1 Joints The joints in a mechanism can be classified according to the degree of freedom permitted by the joint. Figure 1 shows the joints in which the degrees of freedom in rotation and translation are independent (uncoupled). The most familiar of these are , the turning pair or pin joint (R), the sliding pair (P), and the spherical pair or ball joint (S). Joints in which the degrees of freedom in translation and rotation are not independent (coupled) are shown in figure 2. The most familiar is the helical pair or screw-and-nut (H). It is useful to consider the connection between gear teeth and the connection between cam and follower as joints, and these are included in figure 2.

3.2 The degree of freedom of mechanisms This is a significant property because of the constraint on the structure of the mechanism imposed by the required degree of freedom. Let us define the following variables: F is the degree of freedom of a mechanism; / is the number of links of a mechanism including the fixed link (all links are

considered as rigid bodies having at least two joints—if several machine parts are assembled as a rigid part the assembly is considered as a single link);

/ is the number of joints of a mechanism with each joint assumed as binary, that is, connecting two links (joints connecting more than two links will be discussed separately);

fi is the degree of freedom of /th joint, as shown in figures 1 and 2; this is the freedom of the relative motion between the connected links;

X is the degree of freedom of the space within which the mechanism operates. For plane motion and motion on a surface, X = 3, and for spatial motions, X = 6; exceptional cases are discussed later;

Lm& is the number of independent circuits or closed loops in the mechanism.

Degree Rotational Translational Schematic Comments of freedom (/) of element pair

t { LUt 11

Q / /rn / j

sphere between parallel planes or concentric spheres, or coaxial cylinders

ball in cylinder

cylinder between parallel planes

ball joint (S)

sphere in slotted cylinder

plane joint (E)

slotted sphere

torus

cylindrical joint (C)

roller in slot

S 3 S 3

turning pair (R) or pin joint

sliding pair (P)

Figure 1. Kinematic elements with uncoupled rotational and translational freedoms.

Degree Kinematic Comments of freedom (/) element

m '/,

Od

helical pair (H)

circular slider in circular slot

gear pair (G)

cam pan-

Degree Kinematic Comments of freedom (/) element

noncircular gear pair

constant-breadth cam

constant-breadth cam

pvzz^ZZ

/// tT/7/. &

Figure 2. Kinematic elements with coupled rotational and translational freedoms.


The following degree-of-freedom equations then apply to a large class of mechanisms:

F = X ( / - / - l ) + £ ft, i ~ 1

^ind = /-/+! , which combine and give

I / , = F + X L i n d . 'I = 1

(1)

(2)

(3) For gear trains it can be shown (Buchsbaum and Freudenstein, 1970) that these

equations specialize to the following:

/ = F + 1 + L i n d , (4)

j = F+2Lmd, (5)

/G = And , (6)

/R = F+Zind, (7)

where /G , /R denote the number of gear pairs and pin joints, respectively, of the gear train.

The determination of the degree of freedom for the plane slider-crank mechanism, shown in figure 3, is as follows:

/ 1 = 1 = 4, £ ft= 1 + 1 + 1 + 1 = 4 , and X = 3 ,

i = I

hence F = 3 ( 4 - 4 - 1 ) + 4 = 1. In the application of these equations certain rules [(a)-(c) below] need to be kept

in mind. (a) Joints connecting n links, where n > 2, are called multiple joints and are counted as (n — \) binary joints. (b) Certain linkages commonly thought of as having one degree of freedom have an F-value greater than one. This can occur, for example, in spatial linkages having links with two joints of the types spherical - spherical (S-S), spherical-cylindrical (S-C), and spherical-planar (S-E). Such links can have a freedom of rotation about the joint axis, which is independent of the motion of the mechanism as a whole, for example, the S-E joints on the coupler link of the swash-plate drive shown in figure 4. (c) Mechanisms exist—some highly significant—which do not obey the general degree-of-freedom equations given above. These are mechanisms which depend on special dimensions or proportions for their mobility. The most familiar example is probably the lazy-tong linkage with its parallelogram proportions. The rhombic drive (Flynn et al, 1968) used in Stirling engines represents another case (figure 5). The movability of this drive derives from the symmetry of the mechanism about the Y-Y axis.

^ ^ » \K//LU

//////y// i * -

Figure 3. Slider-crank mechanism and graph.

Creation of mechanisms by use of kinematic structure and function 379

Mechanisms with mixed plane/spatial motions (variable X) are usually exceptions as well. In spatial linkages, special cases are often associated with parallel, intersecting, or perpendicular joint axes. There are no simple rules which will predict whether a mechanism obeys equation (1) or not. Here experience is very helpful. The procedures outlined in this article are limited to mechanisms which obey the general degree-of-freedom equations, and it is important to be aware of this restriction.

swash plate, RESP spatial linkage (single loop for each piston)

Figure 4. Swash-plate drive and graph.

gear integral with link 2 y

gear integral with link 6

Figure 5. Rhombic drive and graph.

3.3 Graph representation of the kinematic structure of mechanisms The graph of a mechanism with binary joints is defined as a graph in which the links correspond to vertices, the joints to edges and the joint-connection of links to the edge-connection of vertices. All edges are labeled according to joint type and the fixed link is labeled as well.

The correspondence between graphs and mechanisms is shown in table 2. The graph representation of a slider-crank mechanism is illustrated in figure 3 and a swash-plate drive graph is shown in figure 4. The sketching of the graph of a mechanism is straightforward. The inverse—the sketching of a mechanism given its graph—is not difficult, but requires some practice to obtain good proportions.

Different mechanisms have different graphs (the technical term is nonisomorphic graphs). The definition of difference used here means that there is no way in which the graphs of different mechanisms can be drawn to look identical. It follows that an


atlas of graphs can be used to create a very large variety of mechanisms. Advantages of the graph representation of kinematic structure are summarized in table 3.

Table 2. Correspondence between graphs and mechanisms.

Mechanism Graph Symbol

Links (/) vertices (v) Joints (/) edges (e) Joined links edge-connected vertices Different joints labeled edges

R, a pin joint P, a sliding joint S, a spherical joint E, a plane joint C, a cylindrical joint

Grounded element fixed link Same mechanisms isomorphic graphs Different mechanisms nonisomorphic graphs Independent loops independent loops

Table 3. Advantages of graph representation of mechanisms.

1. Network properties of graphs are directly applicable to mechanisms, for example, And = e-v + l = / - / + 1.

2. Unique identification of kinematic structure can be made (joint sequence for single loops or characteristic polynomial for multiple loops).

3. A single atlas of graph structures can be used to enumerate mechanisms. 4. It leads to creation of mechanisms by the separation of structure and function. 5. It leads to an automatic kinematic and dynamic analysis of mechanisms.

3.4 Kinematic structure of mechanisms with up to six links The kinematic structure of mechanisms with up to six links is shown in table 4, which is adapted from Buchsbaum and Freudenstein (1970). In the table, v denotes the number of vertices, e the number of edges, and LDS (the local degree structure) the number of edges incident at each vertex. Graphs, which can represent mechanisms, must obey restrictions associated with degree-of-freedom constraints. In table 4, the collection of graphs given in Buchsbaum and Freudenstein (1970) has been reduced according to the following guidelines. (a) Since each link possesses two or more joints, all vertices must be incident at least at two edges. (b) Mechanisms consisting of submechanisms connected by a single joint or link (usually a common base) are omitted. (c) Mechanisms corresponding to nonplanar graphs are omitted. This is somewhat arbitrary, but, in view of the complexity of such mechanisms, we have thought it reasonable to exclude them.

There are other more specialized restrictions depending on the mechanisms involved. For plane mechanisms, for example, it can be shown that the number of vertices, v, cannot be less than £(3 + e). We shall not concern ourselves further with such restrictions here.

It is noteworthy that fifty-seven graphs represent the structure of nearly all mechanisms having six links or less, which obey the general degree-of-freedom equations.


Table 4.

Number

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

Kinematic structure of mechanisms with up to

V

3

4

4

4

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6 a Local degree

e

3

4

5

6

5

6

6

7

7

7

8

8

9

6

7

7

7

8

8

8

8

8

8

8

8

8

9

9

9

LDSa

222

2222

3322

3333

22222

33222-1

33222-2

33332

43322

44222

43333

44332

44433

222222

332222-1

332222-2

332222-3

333322-1

333322-2

333322-3

333322-4

433222-1

433222-2

433222-3

442222-1

442222-2

333333

433332-1

433332-2

structure.

Graph

A

n s M

O O Q &

#

6 ® 6 w

0 0 o o $

0 *

$

£ 0 a $

® $

$

o

six links.

Number

30

31

32

• 33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

V

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

e

9

9

9

9

9

9

9

9

9

9

10

10

10

10

10

10

10

10

10

10

10

11

11

11

11

11

12

12

LDSa

433332-3

443322-1

443322-2

443322-3

443322-4

443322-5

444222

533322 ,

543222

552222

443333-1

443333-2

444332-1

444332-2

444332-3

444422

533333

543332-1

543332-2!

544322

553322

444433

544333

544432 i i

553333

554332

444444

554433

Graph

0 #

0 $

0 10: O i ^

^

$

$

£ $

$

$

#

$

^

$

1$!

® 0 3$I

#

$ 3 ^

0 |^[


4 The creation of mechanisms We illustrate the creation of mechanisms with several examples, in increasing order of complexity.

4.1 Four-link structures Suppose one wishes to create all plane four-link, single-degree-of-freedom mechanisms with pin joints (R) and up to two sliding joints (P). If our method is valid, it should be possible to reinvent many of the most basic linkage configurations.

Number Structure

2 R 3

R<l-JR

1 ® R 4

Mechanism

M- I %

Comments

plane four-bar linkage

2 'R 3

•n-1 R 4

(a) turning-block linkage (b) swinging-block linkage

2 R

R

1

3

P

R 4

R _

a plane slider-crank mechanism

2 . R . 3

o 1 P 4

Scotch yoke

2 R 3 t—~~1

R P

1 4> 1 P 4

cardanic motion

2 ^ R R 3

P

P 4

2 P 3

R R

7 4 * 1 P 4

4 1

inverse cardanic

Oldham coupling

Rap son slide (conchoidal motion)

Figure 6. Four-link single-degree-of-freedom mechanisms with turning pairs (R) and up to two sliding pairs (P).


In equation (1), we have F = 1, X = 3, £/} = / (/ = 1,..., /) (since pin joints and sliding joints both have one degree of freedom). This gives F — 1 = 3 ( 4 - / - 1)+/ , or / = 4 and Llnd = 1. From table 4, with v = e = 4, only one structure is found— the square. The edges (R or P) and the vertex denoting the fixed link now need to be labeled in as many different ways as possible.

The result of this effort is shown in figure 6. There are seven mechanisms, all of which are basic plane linkages. Notice that in sketching a sliding pair it is arbitrary which link has the slider and which link has the slotted link. Structure 2, for example, includes both the turning-block linkage and the swinging-block linkage, depending on this choice of pair representation.

The skill required in sketching the mechanism corresponding to a given graph is illustrated in structure 6. A straightforward sketch yields the inverse cardanic motion shown. If we did not know it to begin with, however, we might not have thought of the construction of the Oldham coupling. In this respect creativity and ingenuity are most helpful.

4.2 Six-link structures If we were to limit ourselves to plane single-degree-of-freedom linkages with turning pairs and sliding pairs, then structures with six vertices and seven edges would be obtained, corresponding to mechanisms with six links and seven joints. There are three structures corresponding to this specification in table 4, that is structures 15, 16, and 17. Of these, structure 15 is rejected, because the three-link loop would be rigid. The remaining two graphs together with some randomly selected six-link mechanisms, are shown in figure 7(a)-(g). The number of distinct ways of labeling the two graphs (with R, P joints) is quite large and the creation of all possible mechanisms is beyond the scope of this introduction.

One significant aspect of figure 7 is the similarity of structure of mechanisms with entirely different functions. This fact can be useful both in the creation of mechanisms

Figure 7. Some six-link structures and mechanisms: (a) Watt; (b) Stephenson; (c) opposed-piston engine; (d) window-awning guide; (e) internal-combustion engine (Valentine, 1933); (f) power-bar press (Spachner, 1970); (g) invisible hinge (Selby Furniture Hardware Co. Inc.).

384 F Freudenstein, E.R Maki

and in concept evaluation, including patents. If each mechanical patent were cross-indexed with a simple structural formula, the search for prior art would be greatly simplified. Single-loop devices are completely defined by the joint sequence, beginning and ending with the fixed joints for example. Two-loop devices, such as the six-bar linkages above, can be defined by the joint sequences in the two loops together with some simple notational conventions. In this way any similarity which may exist between mechanisms in two entirely different fields will be recognized automatically.

4.3 Gear drives and transmissions The creation of planetary gear drives and gear transmissions according to the above procedures has been carried out by Buchsbaum and Freudenstein (1970) for gear drives with six members or less. This procedure is simplified with the aid of equations (4)-(7) and the added labeling of the edges representing turning pairs according to the axis location of the pair. As expected, the mechanisms created in this way are generally known, but include some rare configurations and one (structure 2200-1-3—the bicycloidal crank) discovered independently in recent times (Buchsbaum and Freudenstein, 1970). In the case of gear drives (as well as generally) the structural representation leads automatically to the formulation of the displacement equations and provides insight regarding the structural similarity of or difference between complex mechanisms, such as coupled epicyclic gear trains and transmissions.

4.4 Constant-velocity shaft couplings Constant-velocity (C-V) shaft couplings illustrate the creation of spatial mechanisms and the relation of functional considerations to kinematic structure (Hunt, 1973).

C-V shaft couplings are used in automobiles, trucks, and general machinery to transmit a 1:1 angular-velocity ratio between nonparallel, intersecting shafts. There are two basic groups of types: the ball types (Rzeppa, Bendix-Weiss, etc) and the linkage types. The former are characterized by localized contact at the joints and the latter by surface contact. We shall limit the creation to the latter types, and for the sake of simplicity and brevity, we shall restrict ourselves to single-loop, spatial mechanisms (X = 6) with pin joints (R), sliding pairs (P), cylindrical pairs (C), ball joints (S), and plane joints (E).

From the construction of C-V couplings it is known that they are single-degree-of-freedom mechanisms and that the 1 :1 angular-velocity transmission is associated with the symmetry of the coupling about the plane bisecting the shaft axes. Although it is conceivable that a single-loop coupling may exist which violates this principle, this

Lind = j-l+\ = 1

hence j = I , I = 5 or 7

lfi= 1

R fi\ R

O 6 Z 7

2/x + 2 / Y + / z = 5

••• fx = h = /z = 1

R, P: / = 1

( b ) / = / = 5 E, S: / = 3 (c) / = / = 7

Figure 8. Constant-velocity shaft couplings; single-loop spatial mechanisms with R, P, C, E, S joints.


paper is not concerned with this possibility. (A multiple Hooke joint can function in this fashion but it does not obey the general freedom equation (1).

Since there is one independent loop the number of links and joints are equal. Figure 8(a) shows the two connected shafts coupled to ground by turning pairs [that is, figure 8(a) shows two joints and three links]. The rest of the mechanism remains to be created. For single-loop spatial mechanisms equation (3) predicts £/J- = 7 (/ = 1,..., /). Hence / = / < 7. Thus, owing to the symmetry, / = / = 3, 5, or 7. The case / = 3 is rejected because the third joint would require a degree of freedom of 5, which is greater than that of any of the admissible joint types mentioned above. Thus I = j = 5 or 7. The corresponding graph structures are shown in figures 8(b) and 8(c). In these, vertex 1 represents ground and vertices 2, 3 the connected shafts.

Let X, Y, Z denote the unknown joints required to complete the graph and let their freedoms be denoted by / x , / Y , fz respectively. Since X/j = 7 (/ = 1, ...,/), the degree-of-freedom constraints give 2 / X + / Y

= 5 when / = 5, and 2 / x + 2 /y+ /z = 5 when / = 7. The solutions to these equations are (/x

= 1, /Y = 3) or (/x = .2, fY — 1) for / = 5, and fx = fY = fz = 1 for / = 7.

The X, Y, Z now need to be associated with the joints R, P, C, S, E in as many different ways as possible. There are twelve distinct combinations. The corresponding structures are shown in table 5 and some of the associated shaft couplings in figures 9(a)-(g). Of the twelve structures, six [figures 9(a), (b), (c), (d), (f), (g)] are

Table 5. Constant-velocity structures.

Joint combinations

1. RRERR 2. RRSRR 3. RPEPR 4. RPSPR 5. RCRCR 6. RCPCR

Coupling type

Tracta Clemens — Altmann Myard —

Joint combinations

7. RRRRRRR 8. RRRPRRR 9. RRPRPRR

10. RRPPPRR ll.RPRRRPR 12. RPRPRPR

Coupling type

Myard, Voss, Wachter and Rieger — Derby, S. W. Industries — — —

/ = 1 . / = 3 / = 1

(a) RRERR (Tracta)

/ = 1 / = 1

(b) RRSRR (Clemens)

f=\ / = 1

/ = 1 / = 1 (c) RPSPR (Altmann)

/ = 1

/ = 1

/ = 2 ^

/ = 1

M=1

/ = 1 (d) RCRCR (Myard)

/ = 1.

/ =

£=2^4

f=l (e) RCPCR

1

^ > / = l

/ = 1

= 1

X / = 2

/ = 1

(f) RRRRRRR (Myard, Voss, Wachter and Rieger) (g) RRPRPRR (Derby, S. W. Industries)

Figure 9. Constant-velocity shaft couplings.


known, whereas the remainder appear to us to be new. We have made no attempt to determine the practicality of the potentially new structures, but it is interesting to observe that all the existing structures of which we are aware are included in the enumeration.

4.5 Dwell-type intermittent-motion mechanisms As a final illustration, we consider certain intermittent-motion mechanisms with rigid elements capable of providing a momentary dwell. Momentary dwells are used in production operations, which include splicing, labeling, and sealing, for example. We consider some of the many types which have been invented in the course of many years and attempt to create others. Intermittent mechanisms have been invented for finite dwells and for momentary dwells. Some consist only of rigid links, others include belts or chains. In this case we shall limit ourselves to mechanisms with rigid links. The basic types include those described in paragraphs (a)-(e) below.

planet 1 gear 1 (integral with crank)

crank

gear 2

(a)! perspective

housing

schematic (b)

sector gear 1

gear 3 (output)

• planet gear

cam follower -

cam groove

planet cage (input)

(c)

output

rotation produces

gP'S) oscillation

input

Figure 10. Dwell-type intermittent-motion mechanisms: (a) two-gear drive, courtesy of Machine Design (Talbourdet, 1948); (b) three-gear drive (Spotts, 1959); (c) cam and planetary, courtesy of Product Engineering (Anonymous, 1965); (d) screw and gear, courtesy of Machine Design (Talbourdet, 1950); (e) rotary step mechanism, courtesy of Industrial Press (Bickford, 1972).


(a) The two-gear drive (Talbourdet, 1948) shown in figure 10(a). This consists of a four-bar drag linkage with planet 1 centered at C. It is rigidly mounted on the coupler. Output gear A is pivoted at D. (b) The three-gear drive (Spotts, 1959) shown in figure 10(b). This consists of a four-bar crank-and-rocker linkage with three gears: gear 1, integral with the crank and centered at B; gears 2 and 3 pivoted at C and D. Gear 3 is the output. (c) A cam and planetary gear drive (Anonymous, 1965) shown in figure 10(c). This consists of an input arm, CD planetary sector-gear 1, pivoted at C and guided by means of a roller riding on a fixed cam. Gear 2, the output, is pivoted at D. (d) The screw and gear (Talbourdet, 1950) shown in figure 10(d). This consists of a worm and worm gear, the worm carried on a shaft, which rotates and translates. The translation and the rotation are controlled by a cam-and-slot arrangement. (e) A rotary step mechanism (Steinke, 1969; Bickford, 1972) shown in figure 10(e). This consists essentially of a spherical crank-and-rocker linkage of wobble-plate proportions, the input being the rotation of the 'fixed' link of the spherical crank-and-rocker. The output is the rotation of the rocker link. The driving crank of the crank-and-rocker is guided by a planetary bevel gear coaxial with the driving crank and meshing with a fixed ring gear.

There are other dwell motions (Talbourdet, 1948; 1950), but the above five types are representative. We may now ask whether we can create such mechanisms in a more or less systematic fashion, which would require translating the functional requirements into structural requirements.

On studying these mechanisms for a while, it becomes apparent that one way of characterizing the functional aspects of their operation is as follows. Each drive consists of two mechanisms: (1) A two-degree-of-freedom linkage in which there are two adjacent members, at least one of which is capable of complete rotation [links CD and BC in figures 10(a)-(c); link BC in figures 10(a) and (b) and sector-gear 1 in figure 10(c)]. In the case of the screw and gear, the worm rotation is unlimited and its reciprocation corresponds to oscillation. (2) A single-degree-of-freedom linkage, which also has two adjacent members capable of the same rotation and/or oscillation. By coupling the corresponding members in

Table 6. Dwell-type intermittent-motion mechanisms.

Mechanism

Two-gear drive

Three-gear drive

Cam and planetary

Screw and gear

Rotary step mechanism

Two-degree-of-freedom linkage

Simple differential gear train (gears 1, 2 and arm CD)



Worm and shaft

Spherical crank-and-rocker with rotating

One-degree-of-freedom linkage

Drag linkage ABCD

Crank-and-rocker ABCD

Cam-and-follower linkage

Cam-and-slot mechanism

Planetary-gear pair with fixed ring gear

Rotating link

CD

CD

CD

Worm (rotating)

'Fixed' link of spherical

Adjacent link with rotating or oscillating component

BC

BC

Sector-gear 1

Worm (reciprocating)

Output crank of spherical linkage

'fixed' link and crank linkage


each linkage, the degree of freedom of the combination is reduced to one, and the output motion becomes the sum or difference of the rotation and oscillation. Appropriate proportions can then produce a dwell. Table 6 shows the results of applying this interpretation to the preceding five mechanisms, (a)-(e).

With this in mind, the creation of dwell motions involves the creation of two-degree-of-freedom mechanisms with one or two members capable of rotation/oscillation (reciprocation), and the creation of single-degree-of-freedom mechanisms with members having similar rotation/oscillation capability. Although we have made no attempt to create all potential dwell motions, we have created several [figures l l (a)-(e)] , mostly well known (Volmer, 1957).

(d) (e)

Figure 11. Dwell-type intermittent-motion mechanisms obtained by structural analysis: (a) two-gear crank and slider linkage; (b) two-gear crank and slider linkage; (c) two-gear crank and rocker linkage; (d) two-gear turning-block linkage; (e) two-gear swinging-block linkage.

5 Concept evaluation The evaluation of new concepts and patents involving mechanisms can be greatly facilitated by an examination of their kinematic structure. Since the kinematic structure defines the 'core' of the mechanism, it helps eliminate the nonessentials, including prose, from the essential mechanical description. It brings out similarities and differences with respect to other structures and this bears on novelty and originality.

Creation of mechanisms by use of kinematic structure and funct ion 389

And finally it seems to us that if one can identify the kinematic structure of a patent, one generally understands the patent. If a consistent kinematic structure cannot be identified, it is possible that the patent has not yet been fully understood, or that the inventor has not fully understood his creation.

We illustrate the examination of kinematic structure with respect to one patent selected essentially at random: "Mechanism for converting rotary motion into reciprocating motion" (Boxall, 1939). Figure 12(a) (Boxall's figure 4) illustrates the basic concept. The patent describes this figure as follows:

"A plunger 10 of a high-speed reciprocating pump is slidable in a cylinder ... supported by a frame member 12. It is required to reciprocate the plunger 10 by rotating a shaft 13 which is borne in a bearing .... carried by a frame member 15. The frame members .... are supported in a housing 17 of any convenient construction.

The end of the shaft 13 is formed with an eccentric recess containing a cup 18, which receives the ball-shaped end of 19 of a cylindrical rod 20, the other end of which is forked at 21 to engage a tubular gudgeon-pin 22 upon which the tongue-end 23 of a bent coupling rod 24 is pivoted. The other end of the rod is formed with a ball-end 25 which engages a cup 26 in the end of the plunger 10.

The cylindrical rod 20 is slidable in a diametral bore in a ball 27, which is supported for tilting movement in part-spherical recesses in plates 28 secured to a frame-member 16."

We now ask what is the basic structure of this device? Figure 12(b) illustrates schematically the basic mechanism consisting of ground 15, drive shaft 13, connected by ball joint 18-19 to the cylindrical rod 20, which is supported by ball 27 and pin-jointed at 22 to the bent coupling rod 24, which is ball jointed at 25, 26 to plunger 10 the axis of which is parallel but offset from that of drive shaft 13.

(a) *l (c) Figure 12. The Boxall patent (Boxall, 1939): (a) figure 4 of the patent; (b) schematic of the mechanism; (c) structural diagram.

(a) (b) Figure 13. (a) basic two-loop shaft coupling; (b) a symmetrical zero-stroke structure.


The structural diagram of the mechanism [figure 12(c)] shows that it is a two-loop, spatial six-link mechanism with the joint freedoms adding up to 14, rather than the expected 13 [X = 6, / = 6, / = 7, E/j- = 14 (/ = 1,...,/), F — 2]. This can readily be accounted for. The extra freedom derives from the freedom of rotation of the ball 27 about the axis of the cylindrical rod 20. Insofar as the kinematics are concerned, therefore, the cylindrical rod could have been replaced by a prismatic rod. The mechanism, therefore, is one which does not depend on special proportions for its mobility and would be included in our creation procedure.

Is this a novel form of mechanism? Or rather, was it a novel mechanism at the time of the invention (1939)? Two-loop spatial mechanisms are relatively rare, at least in comparison to single-loop, spatial mechanisms. They do, however, have significant applications, for example, vehicle suspensions, linkage-type differentials, and shaft couplings with a grounded intermediate member. The latter have been used, for example, in hermetically sealed shaft couplings.

A basic two-loop shaft-coupling configuration is the symmetric two-loop structure shown in figure 13(a). If the output R-R pair were replaced by a ball-ended connecting-rod and slider (S-S-P), as shown in figure 13(b), the piston would have zero stroke. Both figure 13(a) and figure 13(b) represent mechanisms in which there are dimensional restrictions (for example in figure 13(b) the R-R pair axes intersect). In order to obtain a finite stroke/an asymmetrical construction is required, which in turn requires removal of the dimensional restrictions, thus giving X/} = 13 (/ = 1,...,/). One example of such a construction is the Boxall patent.

Inasmuch as the structure of the patent is derived as an asymmetrical structure derived from a symmetrical, zero-stroke structure, the size-to-stroke ratio is likely to be quite large in terms of current performance requirements. Based on the state of the art in 1939, however, the award of a patent for the mechanism described by Boxall seems justified.

6 Summary A procedure has been outlined for the creation of mechanisms according to kinematic structure and function. This procedure is based on the separation of the structure of a mechanism from its function. The structures of mechanisms can be enumerated in an essentially systematic, unbiased fashion. Usefulness of the method in establishing the total number of mechanisms in a given class with a specified number and type of joints is illustrated with four-link planar mechanisms and spatial mechanisms with five and seven links. Dwell-type intermittent-motion mechanisms obtained by structural analysis are shown. The application of this procedure to the examination of an existing mechanism, such as found in the patent literature, is illustrated and shown to be a practical method for identifying that mechanism's true identity.

References Anonymous, 1965 "Varies speed cyclically in each revolution" Product Engineering 36 (6 December)

92 Artobolevskii 11, 1956 "Evolution de la theorie de la structure et de la classification de mecanismes

en Russie" Actes de 8ieme Congres International d'Histoire des Sciences, Florence, Italy, , 3-9 September. Volume 1 pp 980-983 ;

Assur L V, 1913-1916 "Investigation of plane hinged mechanisms with lower pairs from the point of view of their structure and classification" Izvestiya S-Peterburgskogo Politekhnicheskogo Instituta Imperatora Petra Velikago 1913 20 part I, 329-385; 20 part II, 581-635; 1914 21 187-283, 475-573; 22 177-257; 1915 23 1-169

Babbage C, 1826 "On a method of expressing by signs the action of machinery" Philosophical Transactions of the Royal Society 2 250

Bickford J H, 1972 Mechanisms for Intermittent Motion (Industrial Press, New York); Steinke patent is illustrated in figures 10-24 and 10-25, p 147


Boxall F C, 1939 "Mechanism for converting rotary motion into reciprocating motion" US patent 2,173,247(31 October)

Buchsbaum F, Freudenstein F, 1970 "Synthesis of kinematic structure of geared kinematic chains and other mechanisms" Journal of Mechanisms 5 357-392

Crossley F R E, 1965 "The permutations of kinematic chains of eight members or less from the graph-theoretic viewpoint" in Developments in Theoretical and Applied Mechanisms, Volume 2 Ed. W A Shaw (Pergamon Press, Oxford) pp 467-486

Davies T H, 1968 "An extension of Manolescu's classification of planar kinematic chains and mechanisms of mobility M > 1, using graph theory" Journal of Mechanisms 2 87-100

Dobrjanskyj L, Freudenstein F, 1967 "Some applications of graph theory to the structural analysis of mechanisms" Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 89B 153-158

Flynn G Jr, Percival W, Heffner F E, 1968 "GMR Stirling thermal engine—part of the Stirling engine story—1960 chapter" Society of Automotive Engineers, Transactions 68 665-685

Franke R, 1951 Vom Aufbau der Getriebe, Band I (VDI, Diisseldorf) Freudenstein F, 1971 "An application of Boolean algebra to the motion of epicyclic drives"

Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 93B 176-182

Freudenstein F, Dobrjanskyj L, 1965 "On a theory for the type synthesis of mechanisms" Proceedings of the 11th International Congress of Applied Mechanics (Springer, Berlin) pp 420-428

Freudenstein F, Woo L S, 1974 "Kinematic structure of mechanisms" in Basic Questions of Design Theory (North-Holland, Amsterdam) pp 241-164

Hunt K H, 1973 "Constant-velocity shaft couplings: a general theory" Transactions, Journal of the American Society of Mechanical Engineers, Journal of Engineering for Industry 95B 455-464

Jones F D, Horton H L, Newell J A (Eds), 1968 Ingenious Mechanisms for Designers and Inventors (Industrial Press, New York)

Reuleaux F, 1876 Kinematics of Machinery: Outline of a Theory of Machines translated and edited by A B W Kennedy (Macmillan, London)

Selby Furniture Hardware Co. Inc. 1977 Catalog, 15/21 East 22nd Street, New York, NY 10010 Spachner S A, 1970 "Use of a four-bar linkage as a slide drive for mechanical presses" SME paper

MF-70-216, Society of Manufacturing Engineers, Dearborn, Mich. 48128 Spotts M F, 1959 "Kinematic properties of the three-gear drive" Journal of the Franklin Institute

268(6)464-473 Steinke J, 1969 "Rotary step mechanism" US patent 3,475,977 (4 November) Talbourdet G J, 1948 "Intermittent Mechanisms II" Machine Design 20 (October) 135-138;

two-gear drive is illustrated in figures 12 and 13, p 137 Talbourdet G J, 1950 "Intermittent mechanisms" Machine Design 22 (October) 121-125; screw-

and-gear drive is illustrated in figure 9, p 121 Valentine WP, 1933 "Internal combustion engine" US patent 1,912,604 (June) Volmer J, 1957 "Raderkurbelgetriebe" VDI Forschungsheft 461B 52-55 Woo L S, 1967 "Type synthesis of plane linkages" Transactions of the American Society of .

Mechanical Engineers, Journal of Engineering for Industry 89B 159 -172 j

the creation of mechanisms according to kinematic ... · with link 2 y gear integral with link 6...

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