ON THE DEFINITION, CLASSIFICATION, AND SYMBOLISM OF KINEMATIC PAIRS

Yu. Ya. Kovylyin

(Presented by the scientific-methodological seminar of the departments of theoretical and applied mechanics)

1. In the literature on the theory of mechanisms and machines, unified definitions, classifications, and conventional designations of links, kinematic pairs, kinematic chains, mechanisms, etc., have still not been adopted. The abnormality of such a situation is quite obvious: not only is the educational process seriously and, it must be said plainly, unjustifiably complicated (especially in the system of correspondence education), but at times communication among direct specialists becomes impossible without preliminary agreement on terminology, classifications, and the like. This has been pointed out more than once, in particular in speeches at the Second All-Union Conference on the fundamental problems of the theory of mechanisms and machines and in periodical publications.

At the present time, in our opinion, there is every possibility of developing and adopting unified systems of terms, classifications, and designations in the theory of mechanisms and machines. It can be properly realized if not only the corresponding organizations of the Academy of Sciences of the USSR but also broad circles of the scientific and technical public take part in the work.

In the present article the intention is to consider certain questions connected with the concept of a “kinematic pair.”

1. Let us first of all dwell on the definition of a kinematic pair. In the literature it is reduced to one of the following:

a) the connection of two links¹;

b) the movable connection of two links;

c) the movable connection of two bodies, restricting their relative motions;

d) two bodies connected with one another movably;

e) two links mutually in contact by their elements, restricting the relative motions of these links;

f) the aggregate of elements which, as a result of mutual contact, restrict the relative motions of those links to which they belong;

g) a set of two elements, each belonging to one link, which, as a result of mutual contact, constrain the relative motions of the links.

Different authors also invest the concept of an “element” with different content:

a point, a line, a surface (or a set of these) of one link that comes into contact with another link; a part of a link that is in direct contact with the corresponding part of another link.

According to definitions (a), (b), and (c), a kinematic pair is interpreted as a certain device (joint, coupling, node), i.e., in the sense in which this is generally accepted in mechanical-engineering practice and in the literature on the calculation and design of machines.

It follows from definition (a) that the links entering into a kinematic pair undoubtedly move relative to one another, since otherwise we would have one link and not two; at the same time, it is clear that they cannot have as many relative motions as would be possible if the links were free bodies, otherwise there could be no question of any connection between them.

Definition (a) does not necessarily require direct contact of the links, which makes it possible, for example, to regard assemblies with rolling friction, providing relative rotary, translational, and screw motions, as the corresponding kinematic pairs (conditionally). Otherwise, even a simple lever mounted on rolling bearings would, taking into account the intermediate rolling bodies, always have to be considered a multi-link mechanism, whereas it is usually considered a two-link one.

The erroneous conclusion does not follow from definition (a) that a kinematic pair imposes constraints on the relative motions only of the links that enter into it, for in a mechanism, as is known, a kinematic pair may partially constrain the motions of other links as well.

Finally, definition (a) is advantageously distinguished from all the others by its extreme brevity.

In definition (b) the word “movable” has been added. This addition, on the one hand, in no way makes definition (a) more precise (“butter oil”), for, as shown above, even without this addition it is impossible to suppose that the links entering into a kinematic pair do not move relative to one another.

On the other hand, the addition of the word “movable” entails the unconditional necessity of an additional explanation that it characterizes only relative mobility. Otherwise, excluding the possibility of tautology, one may think that a kinematic pair is only such a connection of links as moves in space (for example, a crank–connecting rod, but not a crank–frame).

Definition (c) is very imprecise: first, it follows from it that a link is a single body, whereas in the general case, as is known, a link may include several material bodies (including, for example, lubricant filling cavities); second, with such a formulation the concept of a link is inevitably introduced after the definition of a kinematic pair, and this violates the logical sequence of exposition; finally, unless the meaning of the word “movable” and the degree of constraint of relative motions are specially clarified, devices that have never been regarded as kinematic pairs may also satisfy it; such as, for example, the connection of the rod and the crosshead in the mechanism of a steam ma-

1. Proceedings of TPI, vol. 107.

—of the tire moves (relative to the frame) and restricts the relative motions of the bodies being connected (completely).

In definitions (г), (д), (е), and (ж), a kinematic pair is interpreted as a certain aggregate of two bodies, links, or elements. Each of these definitions has one or another of the shortcomings noted above in the analysis of definitions (б) and (в), and, in addition, others are added. Thus, for example, if one proceeds from definitions (г) and (д), one might think that four bodies (links) can form only two kinematic pairs, whereas in fact they may form two, three, or four. Definitions (е) and (ж) in one case (element—a point, a line...) are too abstract, while in the other (element—a part of a link...) they require additional explanation regarding the dimensions (volume) of the element.

Thus we arrive at the conclusion that, of all the definitions considered, the first is the best.

A kinematic pair is the connection of two links.

In this form it is given, for example, in the book by Prof. M. V. Semenov, Structure of Mechanisms (Fizmatgiz, 1959).

1. Let us now turn to consideration of classifications of kinematic pairs.

According to the principal modern classifications, all kinematic pairs are divided into five classes either by the number of degrees of freedom in the relative motion (i.e., by the degree of mobility), or by the number of constraint conditions. Formally, these two classifications are equivalent, but in practice classification by degree of mobility is more convenient and natural, in particular for the following reasons: the very name “kinematic pair” already indicates a connection with motion; greater visual clarity (the possible motions can be indicated on the drawing by arrows); the class number of a kinematic pair does not change depending on whether it is formed by links of a plane or a spatial mechanism; each class has its own name—“single-mobility,” “two-mobility,” ..., “five-mobility”—which clearly indicates the feature underlying the classification; the mobility formulas (structural formulas) of kinematic chains and mechanisms have a more symmetrical form.

Thus, in our view, there are sufficiently weighty grounds for giving preference to the division of kinematic pairs into classes, proposed by Corresponding Member of the USSR Academy of Sciences V. V. Dobrovolsky,¹ according to degree of mobility.

Within each class, kinematic pairs may be divided into types according to the form, mutual orientation, and mutual conditioning of the instantaneous relative motions permitted by them for the links included in them.

1. In addition to the classification considered above, the division of kinematic pairs into lower and higher pairs has become widespread. The definitions of these concepts found in the literature may be reduced to the following:

lower kinematic pairs are those in which the links are in contact along surfaces;

higher kinematic pairs are those in which the links are in contact along individual lines and points.

Kinematic pairs possessing simultaneously the characteristics of both lower and higher pairs are assigned by some authors to a special category—mixed pairs.

¹ In the works of V. V. Dobrovolsky, instead of the word “class,” the word “order” is used.

Considering various kinematic pairs, it is easy to establish that in some of them (for example, in five-mobility pairs) the character of the contact of the links is entirely predetermined by the structure of the relative motions being realized; in others it depends on the design embodiment. Thus, for example, revolute and prismatic kinematic pairs, depending on their design, according to these definitions may be lower, higher, or mixed.

Since such a division reflects only the character of the contact of links apart from any definite connection with their possible relative motions, it is evident that, for the theory of mechanisms and machines, there is no need for it. It might be useful to some extent for the calculation and design of parts of mechanisms and machines, but even in this case, instead of the terms “lower,” “higher,” and “mixed,” it is more convenient to use the precise terms “surface,” “linear,” “point,” “surface-linear,” etc.

Noting these circumstances, some authors propose abandoning altogether the division of kinematic pairs into lower and higher and excluding these terms from use. Meanwhile, a large part of scholars, irrespective of the existing definitions, successfully use the division of kinematic pairs into lower and higher not only for a general characterization of their properties, but also for the purpose of a systematic exposition of the problems of the theory of the corresponding mechanisms. In doing so, they classify without any reservations as lower kinematic pairs the revolute, prismatic, screw, cylindrical, spherical, and planar pairs¹, and as higher—all the remaining ones.

It is easy to see that with such a division the terms “lower” and “higher” are directly connected with the character of the relative motions admitted by the corresponding kinematic pairs: lower pairs admit relatively simple motions of the links, as a result of which the realization of prescribed forms of motion by means of mechanisms with only lower kinematic pairs, as is known, is a problem of exceptional difficulty; higher pairs, on the contrary, possess considerably greater kinematic possibilities, in view of which, with mechanisms containing them, even with a minimal number of links, one can comparatively easily reproduce practically any forms of motion.

On the other hand, such a division of kinematic pairs is directly connected with the possibilities of their structural embodiment: lower pairs can be made in such a way that contact of the links will be only over surfaces; in higher pairs, contact along individual lines and points is inevitable.

From the standpoint of the theory of mechanisms, the principal feature is evidently the character of the relative motions. The character of the possible contact here is, as it were, a kind of measure by means of which it is convenient to judge the character of the relative motions of the links of the pair.

Summarizing all the considerations set forth above, one may propose the following definitions:

lower kinematic pairs are those in which the relative motions of the links can be reproduced with contact even only over surfaces;

higher kinematic pairs are those in which the relative motions of the links can be reproduced only in the presence of their contact along individual lines and points.

According to these definitions, the lower kinematic pairs, regardless of their structural design, include the revolute, prismatic, screw, cylindrical, spherical, and plane pairs, while all the others belong to the higher pairs. The category of mixed kinematic pairs is completely eliminated.

1. In conclusion, let us dwell on questions of symbolism in the designation of kinematic pairs.

Symbolic designations are of interest from the standpoint of the possibility of expressing, in generalized form, the most important properties of kinematic pairs and of shortening the text, especially when dealing with kinematic pairs that do not have definite names. In practice it is important that the symbol be as brief as possible and at the same time unambiguously reflect all the basic characteristics of the kinematic pair: the number and type of the permitted relative motions, their mutual orientation, and the presence of a functional relation between them.

Among the symbolic systems proposed at various times, the one that best satisfies these requirements is the system of symbols developed by Corresponding Member of the Academy of Sciences of the USSR V. V. Dobrovolsky¹. However, when composing

a symbol according to V. V. Dobrovolsky, it is necessary to consider finite displacements, which in a number of cases proves to be an extremely laborious matter, and for the student simply beyond his powers. This shortcoming is to a considerable degree eliminated if, in composing the symbol, one proceeds from taking account of instantaneous relative motions.

To simplify the symbolism, it is also advisable not to use the sign $\perp$ (“perpendicular”), assuming that the adjacent letters B and P denote, respectively, instantaneous rotational and prismatic relative motions of the links, oriented mutually perpen-

perpendicular axes. Otherwise the corresponding two letters are separated by the sign \(||\) (parallel).

Letters denoting motions between which there are some functional dependencies, as in the symbolism of V. V. Dobrovolsky, should be enclosed in parentheses. The number of these dependencies may be indicated by the corresponding index in the exponent (the index “1” is omitted). To indicate the links that make up a kinematic pair, its symbol may be supplied with a subscript composed of the corresponding numbers of the links.

In writing the symbol, the letters \(В\), in our opinion, should be written first, since rotational motion in engineering is fundamental.

The degree of mobility of a kinematic pair is equal to the number of letters \(В\) and \(П\) in its symbol minus the number of functional dependencies between the relative motions of the links.

To illustrate all the considerations set forth above, we provide a table of the most frequently encountered kinematic pairs, indicating their symbols and other characteristics.

The table gives the figure numbers according to the textbook by Academician I. I. Artobolevsky, Theory of Mechanisms and Machines (GTTI, 1953).

1. See, for example, his book Theory of Mechanisms (Mashgiz, 1953). ↩↩↩