A. V. VERKHOVSKY.

THEORY OF SPATIAL AND PLANAR KINEMATIC PAIRS.

Introduction.

Every mechanism, from the point of view of its structure, represents a certain combination of links connected with one another by kinematic pairs.

Since a kinematic pair, depending on its design, determines the motion of the links of a mechanism, it is quite natural that, in questions concerning the study of mechanisms from the point of view of their structure, the study of the properties of kinematic pairs comes to the fore.

A coherent theory of kinematic pairs, deeply illuminating their kinematic essence, was created comparatively recently by Prof. Malyshev and set forth in his work “Analysis and Synthesis of Mechanisms from the Point of View of Their Structure” (printed in the Proceedings of the Tomsk Technological Institute for 1923).

This work, in its essence, is a brilliant synthesis of all the investigations by various authors that had existed on this question up to that time.

The chief merit of Prof. Malyshev in this work is that he established with complete clarity that the role of a kinematic pair in a mechanism, in its structural investigation, is determined by the number of constraint conditions present in it.

Hence Prof. Malyshev’s classification of kinematic pairs is also constructed according to the number of constraint conditions.

As the basis of his investigation Prof. Malyshev took geometrical schemes of kinematic pairs, independently of their constructive embodiment, thereby attaining great clarity in revealing the kinematic essence of each individual pair.

Among the shortcomings of this classification, in my opinion, should be included the circumstance that rolling pairs are not sufficiently differentiated in it.

Wishing to fill this gap, I attempted to develop the theory of rolling kinematic pairs separately, but it turned out that the principle which I had laid at the basis of the investigation proved equally applicable also to the analysis of other kinds of kinematic pairs.

Thus I arrived at a new classification of kinematic pairs.

This classification is based on the relative motions that may occur in the contacting surfaces of a pair under the influence of external forces acting on the pair.

The proposed classification, of course, does not possess the geometrical clarity inherent in Prof. Malyshev’s classification, but it does illuminate the kinematic pair from a new angle.

Consequently, the classification set forth below does not exclude or replace Prof. Malyshev’s classification; rather, I would say, it supplements it, deepening the conception of the kinematic pair and expanding the interesting category of rolling pairs.

Theory of kinematic pairs based on the types of relative motion of the links

Let us note, before anything else, one very important circumstance necessary for the development of the subsequent reasoning. It is not difficult to see that the links of mechanisms, being rigid bodies, can in practice come into contact with one another only by means of their so-called working surfaces.

Between two working surfaces of connected links, the following types of relative motion may in general appear, namely:

1. Sliding.
2. Rolling.
3. Spinning.

These types of relative motion require, for further constructions, a more detailed analysis.

When one surface rotates relative to another about the common normal drawn through the point of contact of the surfaces, spinning is obtained.

When one surface rotates relative to another about some line lying in the common tangent plane, and the said line passes through the point of contact, rolling is obtained.

When one point belonging to one surface traverses an entire series of points lying on the other surface, sliding takes place.

Let us first consider sliding.

Depending on the form of the working surfaces, and also on the operating conditions of the kinematic pair, two cases of sliding should be distinguished:

a) when the presence of sliding in the pair entails one degree of freedom;

b) when the presence of sliding entails two degrees of freedom.

To clarify these cases, let us consider the motion of a point arbitrarily taken on the working surface of one link and in contact with the working surface of the other link of the pair. If, owing to the particular form of the working surfaces, the point under consideration is able at each given instant to slide only along one line lying on the surface of the other link, then such a case of sliding will characterize “linear mobility” and will possess only one degree of freedom. The presence in a pair of sliding in one direction will be denoted by the symbol \(C_1\). Here the letter “\(C\)” will indicate that there is “sliding” in the pair, while the subscript attached to the letter “\(1\)” will indicate that this sliding can take place in one direction and corresponds to one degree of freedom. Examples are the sliding that occurs in a cylindrical hinge or in a prismatic slide, and the sliding between the threads of a screw and a nut.

Further, a case is possible in which the point under consideration, given the corresponding form of the working surfaces, will be able to slide in different directions.

In this case the presence of sliding in the pair will bring with it two degrees of freedom, and we shall denote this circumstance by the symbol \(C_2\). Here the letter “\(C\)” will denote the circumstance that sliding is present in the pair, while the subscript attached to the letter “\(2\)” will mean that this sliding has two degrees of freedom.

Similarly to what has just been said, for rolling we shall also have two cases.

a) If rolling, owing to the corresponding form of the surfaces, is able to take place only in one direction (for example, a cylinder rolling on a plane), then the presence of the possibility of such rolling in the pair will introduce one degree of freedom. Its presence in the pair shall be denoted by the symbol \(K_1\), where the letter “\(K\)” will denote the presence of rolling in the pair, and the subscript attached to the letter “\(1\)” will denote one degree of freedom;

b) in the case where the form of the working surfaces permits rolling to occur in different directions (for example, a sphere on a plane), such rolling will possess two degrees of freedom. The corresponding symbol for it will be \(K_2\), where the letter “\(K\)” will denote the presence of rolling in the pair, and the subscript attached to the letter “\(2\)” will indicate that this rolling has two degrees of freedom.

It remains to consider spinning.

Spinning is rotation about a common normal drawn to the surfaces. Since through the point under consideration, chosen on one of the surfaces and in contact with the other surface, only one common normal can be drawn at each instant of time, it follows that, for spinning to take place at the point under consideration, at any given instant we shall have only one axis about which spinning can occur, provided that this is not obstructed by other parts of the working surfaces.

Thus, if by virtue of the corresponding form of the working surfaces spinning in the pair under consideration is possible, then in this case spinning will always introduce only one degree of freedom.

Its symbol will be \(B_1\), where the letter \(B\) denotes spinning, and the subscript 1 denotes one degree of freedom.

Thus the list of types of relative motion, or of the factors determining a kinematic pair, is very short, namely:

\[ C_1 \quad C_2 \quad K_1 \quad K_2 \quad B_1 \]

From these types of relative motion, or “factors,” any kinematic pair can be composed. Thus, for example, the formula

\[ K_2 \quad C_2 \quad B_1 \]

represents a certain elementary kinematic pair. The letters and subscripts in this formula show that in this pair rolling can occur in any direction, as can sliding, and, in addition, there is the possibility of spinning. The sum of all the subscripts shows that this elementary kinematic pair has five degrees of freedom. Such a pair is obtained in the case of contact between two bodies (links) by surfaces of different curvature (for example, a sphere on a plane). To determine the number of constraints present in the pair, it is necessary to subtract the number of its degrees of freedom (the sum of the subscripts) from the number six; this follows from the consideration that the greatest possible number of degrees of freedom is six, and that the presence in the pair of fewer than six degrees of freedom is due to the presence in it of the corresponding number of constraints.

For the example given, the pair \(K_2 \ C_2 \ B_1\) will have \(6 - 5 = 1\) constraint, i.e., one constraint.

Let us note that finding the number of constraints in pairs is extremely important, since in the analysis of a mechanism from the point of view of its structure it is not the number of degrees of freedom of the pair that figures, but the number of constraints present in it.

When composing formulas for elementary kinematic pairs, it must be borne in mind that the presence in the formula of \(K_2\) does not allow the presence in the formula of the symbol \(K_1\), since the concept \(K_2\) already includes ...

itself the concept \(K_1\). Likewise, the presence of \(C_2\) in the formula does not, by analogous considerations, permit the presence in it of the sign \(C_1\).

Let us also note here one special case. If, in an elementary kinematic pair, one type of motion is functionally connected with another type of motion, then the functionally connected type of motion does not introduce a new degree of freedom into the pair. Thus, for example, in a heel made in the form of a surface of revolution (see Fig. 1), sliding may occur at all points taken on the working surface, except for the point lying on the axis of rotation. At this last point only spinning can occur. Sliding at the remaining points will here be functionally connected with this spinning; therefore this pair cannot be denoted by \((C_1 B_1)\), but must be denoted only by \(B_1\) or \(C_1\).

Fig. 1.

Now let us compile a table of all possible elementary kinematic pairs constructed according to the proposed principle, arranging them by the number of degrees of freedom.

Table I.

Table of elementary kinematic pairs

Let us give examples of some elementary kinematic pairs.

\(C_2 B_1\)—a ball-and-socket joint (three degrees of freedom).

$C_2$—a circular cylinder sliding inside another similar cylinder (two degrees of freedom), or a plane sliding along another plane (two degrees of freedom).

$C_1$—a prismatic body sliding inside another body having the same prismatic cavity (one degree of freedom).

$K_1$—a cylinder rolling on a plane or on another cylinder without slipping.

Or a cone rolling on another cone or on a plane without slipping.

$C_2 K_1$—a cylinder rolling on another cylinder, with slipping possible both in the direction of rolling and along the generators.

$C_1 K_1$—a cylinder rolling on another cylinder, with slipping possible only in the direction of rolling and impossible along the generators, or conversely.

$B_1$ or $C_1$—a cylindrical joint (one degree of freedom). In the case of a trunnion rotating in a bearing, it may also be denoted by the sign “$B_1$.”

$K_2 C_2 B_1$—this pair represents the most general case of rolling of one surface on another, with the presence of slipping and pivoting.

As an example—a ball on a plane.

Figures of elementary kinematic pairs are given below.

Up to now we have considered only elementary kinematic pairs. From these elementary pairs, compound pairs may be formed by the simple connection, in one joint of two links, of several elementary pairs.

In making such a connection, only one condition must be observed: that the number of constraint conditions in the resulting pair not exceed five. Thus, for example, in Fig. 2 there is shown a kinematic pair which is composed of three elementary kinematic pairs of the type $K_2 C_2 B_1$.

The formula for such a kinematic pair may be written as follows: $3\ K_2 C_2 B_1$, where the numeral 3 standing in front indicates that the pair consists of three elementary pairs of the type $K_2 C_2 B_1$.

Fig. 2.

The number of constraints of a compound pair is equal to the sum of the numbers of constraints of the component pairs. Thus, in order to determine the number of constraint conditions of a compound pair, it is first necessary to determine the number of these conditions in the component pairs.

In the example given, we shall have: the number of constraint conditions for the pair $K_2 C_2 B_1$ will be

$$ 6 - 5 = 1. $$

Since three such elementary pairs are taken in the compound pair of Fig. 2, the number of constraints in it will be: $1 \times 3 = 3$.

Another example: the pair shown in Fig. 3 is a compound pair, in which: \(A\) is a spherical joint, \(B\) and \(C\) are balls sliding along certain surfaces. This pair can be expressed by the formula:

\[ C_2 B_1 + 2 K_2 C_2 B_1 \]

For determining the number of constraints, in this case one may write the formula:

\[ (6 - n_1) + q(6 - n_2) = 6 - 3 + 2(6 - 5) = 5 \text{ constraints} \]

in this formula \(n_1\) is the sum of the indices attached to the letters in the elementary pair \(C_2 B_1\).

Fig. 3.

\(n_2\) is the sum of the indices attached to the letters in the elementary pair \(K_2 C_2 B_1\).

\(q\) is the number showing how many times the elementary pair \(K_2 C_2 B_1\) is taken as an addend.

When determining the number of constraints, it is convenient to use a table of elementary pairs. In this table, opposite the elementary pairs, in the corresponding column, the numbers of constraints are indicated.

When determining the number of constraints of a compound pair, one must take from the table the corresponding numbers that stand in the column of the number of constraints opposite the elementary pairs taken for constructing the compound pair, and add these numbers.

The total number of kinematic pairs, both compound and elementary (corresponding to the system of their formation just considered), will be determined if, using the table of elementary pairs, one forms from them the possible combinations; moreover, in forming these combinations it should be borne in mind that the resulting pairs must not have more than five constraint conditions. Then the number of possible kinematic pairs will be equal to 61.

The number of possible kinematic pairs obtained here is greater than in the system of Prof. Malyshev, where it was equal to 41. This occurred owing to the greater detailing of rolling pairs.

Thus, for example, the pair S N A P in Prof. Malyshev’s system represents all rolling pairs having three constraint conditions, whereas under the new subdivision several such pairs are considered (see the table of elementary pairs of Malyshev: “Proceedings of the Tomsk Technological Institute” for 1923).

As for the elementary pairs of the first three lines of Prof. Malyshev’s table of pairs, they have not disappeared without a trace, but, having taken constructive form, have also entered the new table.

Thus, for example, let us consider the kinematic pair of class II, called S L A R according to Prof. Malyshev’s notation. This pair

is characterized by a single point situated on a line belonging to the other body.

Structurally (physically), this kinematic pair may be made as shown in Fig. 4, i.e., in the form of a ball belonging to one link and sliding in a cylindrical groove belonging to the other link.

According to the notation proposed here, this kinematic pair is expressed by the formula:

\[ C_2\ K_1\ B_1 \]

since the points of the surface of the ball, being in the cylindrical groove, can slide in any direction; in the pair there is \(\left(C_2\right)\); further, the ball can roll only in the direction of the axis of the cylinder, which is denoted by the sign \(K_1\); and, finally, spinning can also take place here. It is self-evident that, according to this notation, we must also obtain a pair of the second class.

Fig. 4.

The actual number of degrees of freedom, determined by the sum of the indices attached to the letters, will be four.

The number of constraints will therefore be:

\[ 6 - 4 = 2 \]

which means that the pair belongs to the second class.

Thus, the question of how the kinematic pairs containing the elements of the second and third rows of Prof. Malyshev’s table were reconstructed has, in what has been said, received some clarification. Let us also touch upon the first row of the same table, “fixed points.”

Thus, for example, the ball-and-socket joint (the pair K N A R) belonging to the first row (which, by its construction, cannot be replaced by a rolling pair) was replaced, in the new system, by the symbol \(C_2 B_1\). If the pair K N A R is regarded as a suspension on a point (for example, the suspension of a magnetic needle on a point), then it will be necessary here to examine the point in greater detail. The connection of the point with the socket may, upon closer examination, reveal several designs. Two such variants are shown in Figs. 5 and 6.

Since (as is seen in Fig. 5) tangency (contact) over a spherical surface participates in the mating, the kinematic pair may quite legitimately be called, according to Prof. Malyshev’s notation, K N A R, and according to the notation proposed here, \(C_2 B_1\).

The kinematic pair shown in Fig. 6, respectively, may be called \(CHAG\) and \(C_2 K_2 B_1\). Moreover, if one considers that in this pair, owing to the presence of friction, no sliding occurs, but only rolling and spinning take place, then, accordingly, this pair will receive the name \(CHAP\) and \(K_1 B_1\).

Fig. 5.

Fig. 6.

Let us merely sum up what has been said.

The new method of determining the number of constraints of kinematic pairs consists in the following: from consideration of the arrangement of the mating working surfaces of the pair and from consideration of the form of these surfaces, one finds what kinds of motion \((C_1, C_2, K_1, K_2, B_1)\) can occur. According to the formula compiled in accordance with these kinds of motion, the sum of the indices for each elementary pair is calculated, and this sum is subtracted from the number six. The result is the number of constraints present in the elementary pair.

In the case of a compound pair, the same operation is performed separately for each elementary pair, and the obtained results are added.

When designing a new kinematic pair for a prescribed number of constraints, it is necessary to compile such a formula for the pair which, when the number of constraints is calculated, would satisfy the prescribed number. In doing so, one may use the table of elementary kinematic pairs, in which the number of constraints is entered for each elementary pair in the corresponding column.

It is also possible, to facilitate the task, to compile a table of all 61 kinematic pairs, indicating in it, for each pair, the corresponding number of constraints. Then the problem is reduced to selecting from this table a kinematic pair possessing the prescribed number of constraints. Determining, for the selected pair, the corresponding arrangement of the mating working surfaces and the form of the working surfaces themselves presents no difficulty. For one and the same number of constraints, there will be several kinematic pairs to choose from, depending on the class of the pair. Here the choice must be made exclusively by the designer himself. At the same time, no great difficulties are to be expected, because there are only 17 elementary pairs from which compound pairs are formed, and becoming acquainted with all of them, with even the briefest exercises, presents no difficulty.

It must also be added that compound pairs are used extremely rarely in mechanical engineering.

Thus, one will usually have to operate only with the 17 elementary pairs; moreover, the choice of the kinematic pair, in

case, if the number of constraints with which it must be endowed is known, it will be necessary to construct it from no more than five kinematic pairs, as is evident from the table of elementary kinematic pairs.

On the basis of the method set forth here, the following definition of a kinematic pair may be given:

A kinematic pair is a conjugation of two rigid bodies (links), characterized by the number of degrees of freedom, which is equal to the number of manifested kinds of relative motion \((C_1\ C_2\ K_1\ K_2\ B_1)\). The number of constraints of the pair is determined by subtracting the number of degrees of freedom of the pair from the maximum possible number of degrees of freedom, which for spatial motion is equal to six.

Changes in the number of degrees of freedom of a pair under the action of external forces on it

From observations of the operation of kinematic pairs in machines, numerous examples may be cited in which the number of degrees of freedom of a pair changes with a change in the forces acting on the links of the pair. Thus, for example, the wheel of a locomotive rolling along a rail under normal operating conditions constitutes a rolling pair \((K_1)\); but if the driver admits an excessive amount of steam into the engine cylinders, then, simultaneously with the rolling of the wheel along the rail, sliding of the wheel may also occur, i.e., instead of the pair \((K_1)\) we shall have the pair \((K_1\ C_1)\). In exactly the same way, a bicycle wheel rolling along a road under normal operating conditions constitutes the pair \((K_2\ B_1)\), but if, during braking or on a sharp turn, the tire of the wheel slides along the road, then the former pair \((K_2\ B_1)\) will pass into the pair \((K_2\ C_1\ B_1)\), or even into the pair \((K_2\ C_2\ B_1)\).

The so-called closure of a pair by the force of friction has long been known (Reuleaux already pointed it out); here we shall try to elucidate the phenomena occurring in this case from the point of view of the number of degrees of freedom and the number of constraint conditions of the pair.

The force of friction between the working surfaces of a pair, under the condition that the force applied to the links cannot overcome it, is the same kind of constraint and the same “kinematic” factor as the circumstance that the contacting surfaces of which the kinematic pair consists resist their mutual penetration and thereby limit the relative mobility of the links, which is what determines the kinematic pair.

If the surfaces are destroyed by their action upon one another, then the kinematic pair itself will also be destroyed, or, at least, some of its constraints. In exactly the same way, if an external force overcomes that kind of friction which, in the normal operation of the pair, could not be overcome, then one or several constraints (frictional constraints) will be destroyed, and a new kinematic pair will result,

with a smaller number of constraints. This kinematic pair in some cases may again be easily restored, while in others the indicated rupture of “frictional constraints” may entail further destruction of the pair and even, in general, of the entire mechanism. Along with this, the rupture of a “frictional constraint” is sometimes used to protect a mechanism from breakage in cases of a sudden increase in the forces acting on the links of the mechanism.

Thus, the number of degrees of freedom of a pair is determined by the number of constraints that can be broken by forces known in advance and applied to the links of the pair, or, to formulate it somewhat differently: by the number of constraints whose rupture we provide for under the given operating conditions of the pair. Such constraints, whose rupture or destruction we allow, or anticipate, are in most cases frictional constraints; but sometimes a certain crushing of the working surface is also allowed: for example, when a wheel moves along an earth road. The study of a kinematic pair apart from the actual conditions of its operation can hardly give us a complete idea of it. The purpose of every kinematic pair (in practice) consists not only in transmitting motion from link to link, but also in transmitting force. Thus it is incorrect to consider a kinematic pair in isolation from the forces acting on it and arising in it.

Between the working surfaces of a pair, under the conditions of its actual operation, a friction force will inevitably arise.

From all that has been said it is clear that, until the driving active forces applied to the links of the pair have broken in it some number of constraints, the kinematic pair will represent, as it were, a rigid connection.

To clarify what has been said, let us give a very illustrative example (although one taken not from the field of machine building).

Suppose that we have a large round stone lying on another similar stone. Suppose further that a bird has alighted on the edge of the upper stone. The bird’s weight will be too small to overcome the resistance of rolling friction between the contacting surfaces of the stones. For the force of the bird’s weight, the connection of the stones under consideration will be rigid. But if a physically strong person applies his force to the stone, then perhaps he will be able to rock the upper stone somewhat, i.e., to break the constraint of rolling friction. Rolling will appear as one of the kinds of relative motion, and for the force of the person this connection of the stones will be a pair \((K_2)\). If not one person but ten apply their force to the stone, then perhaps they will be able to produce some rotation between the surfaces of the stones; then the connection of the stones will already be a kinematic pair \((K_2 B_1)\).

Continuing further—one hundred people may perhaps even be able to move the upper stone over the lower one, i.e., they will already break the constraint of sliding friction, and sliding will appear as a kind of relative—

relative motion. Then this pair will be \((K_2\, C_2\, B_1)\). Only one sixth constraint remains unbroken, namely the resistance of the surfaces of the stones to penetration into one another (resistance to crushing). In exactly the same way, the front wheel of an automobile rolling along a road represents the pair \(K_2\, B_1\), and we base our calculation on this when studying the motion of an automobile. On a sharp turn, owing to the centrifugal force arising in the whole automobile, the front wheel may begin to slip sideways; the sliding-friction constraint will be broken—the kinematic pair will turn into \(K_2\, B_1\, C_1\). This breaking of the constraint \(C_1\) may entail the wrecking of the automobile, whereas the breaking of the constraints \(K_2\) and \(B_1\) in the case we have considered was allowed deliberately and was determined by the conditions of proper operation of the automobile.

Thus, we ourselves allow the breaking of certain constraints in the pair; moreover, we allow such a state in which these constraints are in the process of continual breaking.

This phenomenon, however, does not proceed without leaving a trace with respect to the integrity of the entire pair as such—as a certain aggregate of broken and unbroken constraints. It is known that every kinematic pair gradually wears out and, in the end, becomes completely deranged (fails). Thus, for example, a railway rail and the tire of a wheel wear out because the rolling-friction constraint is continuously being broken between them; the bearings of all kinds of machines wear out because in them there is a continuous breaking of the sliding-friction constraint, and so on.

Thus, from all that has been said it follows that the number of degrees of freedom of a pair is determined not only by the form of its working surfaces but, generally speaking, by the number of constraints broken in it. Moreover, each broken constraint manifests itself in the pair as the corresponding kind of relative motion, and is indicated in the formula of the pair by the introduction into this formula of the corresponding symbol. Unbroken constraints may be proposed to be called the active constraints of the pair.

Generally speaking, in a kinematic pair the rolling-friction constraint can most easily be broken (since this friction is usually less than other kinds of friction), but this “breaking” of rolling friction can occur only in so-called rolling pairs, i.e., in such pairs whose working surfaces permit the manifestation of rolling. In other pairs, in which rolling cannot occur by virtue of the special arrangement of their working surfaces, spinning or sliding will take place (for example, in a ball-and-socket joint, in a cylindrical pin, in a prismatic slide, etc.).

For the sake of generality of presentation, let us note that in these pairs rolling cannot occur only so long as there remains un-

the broken “sixth constraint” present in all pairs and which we formerly called the constraint of resistance to crushing of the surfaces.

It was mentioned above (in passing) that in certain cases a considerable deformation (crushing) of its working surfaces occurs in a pair. This deformation may be plastic (a dirt road) or elastic (an automobile tire).

The consideration of this deformation, however, should not be included in the theory of kinematic pairs, in order to avoid complicating that theory. These deformations are better referred not to the whole pair, but to the individual link of the pair in which they take place, and should be taken into account, in analyzing the whole mechanism as a whole, as a deformation of a link of the mechanism; moreover, this deformation will almost always be functionally connected with the forces acting on the links of the mechanism.

Proceeding from the point of view set forth in this paragraph, the following definition of a kinematic pair may be given: a kinematic pair is the mating of two rigid bodies (links), characterized by the number of unbroken constraints.

Explanation of the drawings of examples of elementary kinematic pairs.

Most of the drawings placed in Table II and depicting elementary kinematic pairs are self-explanatory. Only some of them require additional explanation.

Thus, for example: 1) the drawing relating to the pair \((K_2 C_2)\) depicts the coupling of two links of a chain with one another. At first glance it seems that rotation can easily be excited in this joint. However, as soon as we try to produce rotation in this pair, this elementary pair immediately disappears (is destroyed), and instead of it a complex pair \((2 K_2 C_2 B_1)\) is formed.

This is explained by the fact that, as soon as rotation begins between the links, the links will cease to touch one another at one point (in one region) and will begin to touch at two points (more precisely, at two regions separated from one another), and this circumstance determines the occurrence of a complex pair in place of an elementary one.

All this will hold under the condition that the curvature of the surfaces at the initial point of their contact is the same.

2) The drawings relating to the pair \(K_2 B_1\) depict this pair under the condition that the pair is closed by sliding friction (the drawings show the mating of a magnetic needle with a “point,” a bicycle ball with a ring, an automobile wheel with a road).

3) The drawing relating to the pair \(K_1 B_1\), and depicting the front cylinder of a road roller, may also be assigned to the pairs \(K_1 B_1 C_1\) and \(K_1 C_2 B_1\), provided that to the roller will be appli-

forces are applied, capable of first breaking the constraint \(C_1\), and then also \(C_2\). In addition, for the pair \(K_1 C_2 B_1\) a separate drawing is given, showing a ball sliding in a groove having a circular cross-section.

4) The drawing relating to the pair \(C_1 B_1\) depicts a tractor track. Here spinning and sliding along the track lugs can quite easily be excited. Sliding in a direction perpendicular to the lugs can appear only in the presence of a large force. Although this sliding does occur to some extent while the tractor is operating, nevertheless under normal operating conditions no complete rupture of the constraint is observed here, and the sliding may be expressed as a percentage of the distance traveled; that is, it is functionally connected until complete slipping of the track occurs.

Let us note that in this example it was necessary to resort to a ribbed surface. The traces produced on the soil surface by the lugs should not be confused with the phenomenon of the soil being crushed by the supporting surface of the track (which may or may not occur). The traces obtained here do not differ in principle from the traces produced by the friction of metal surfaces against one another and visible only under a microscope.

5) The drawing relating to the pair \(K_2 C_1\) depicts a wheel of barrel-shaped form; ribs (lugs) are present on the rim of the wheel. Here rolling can quite easily be excited in either direction, as can sliding in the direction of the ribs. Spinning and sliding in the other direction are impeded. This pair can turn into the pair \(K_2 C_1 B_1\) if the friction of spinning is overcome.

6) For the pair \(K_1 C_1\) two examples are given:

a) A cylinder rolling with sliding along a cylindrical surface.

b) A double cone rolling along a wedge-shaped groove. In this example spinning also takes place, but it is functionally connected with the other kinds of motion present in this pair, and therefore it is not included in the formula for the pair. Generally speaking, a functionally connected kind of motion could also be introduced into the formula for the pair, but then no index should be placed after the letter denoting this kind of motion. Thus, for example, for the case under consideration one could write \((K_1 C_1 B)\). The absence of an index at “\(B\)” would mean that spinning, although it occurs in the pair, does not bring with it an additional degree of freedom.

Planar Kinematic Pairs

Although planar kinematic pairs are particular cases of spatial pairs, it is nevertheless more convenient to consider them separately.

In planar pairs only the following kinds of relative motions are possible: \(K_1\) and \(C_1\).

A free plane body moving in a plane has three degrees of freedom; hence the number of degrees of freedom of a planar pair cannot be more than two. The number of constraints likewise cannot be more than two, since already with three constraints relative motion will be impossible, because then none of the three possible degrees of freedom will remain. Bearing all this in mind, from the above-listed kinds of relative motions one may compose the following planar pairs, taking into account that the sum of the indices in elementary pairs (corresponding to the number of degrees of freedom) must not be more than two. These pairs will be:

Elementary kinematic pairs—\(K_1\); \(C_1\); \(K_1 C_1\).

In determining the number of constraints in these planar elementary pairs, the sum of the indices must be subtracted from the number three.

In composing a complex planar kinematic pair it must be borne in mind that the number of constraints is determined as the sum of the numbers of constraints of the elementary pairs that make it up. Taking this into account, only one complex planar kinematic pair can be composed, namely: \(2K_1 C_1\).

Thus, in planar motion only four pairs in all are possible (three elementary and one complex). Examples of these pairs are given in Fig. 7.

Fig. 7.

In this figure the pair \(C_1\) is represented in three variants: 1) in the form of a cylindrical hinge, 2) a slot stone, and 3) a slider. The fact that these three variants have one and the same nature and are varieties of one and the same kinematic pair was shown by Reuleaux, who gave a method by which, from one form of this pair, its other forms can be obtained. (See F. Reuleaux: Theoretische Kinematik § 73, Die geschränkte Schubkurbelkette).

Therefore there is nothing surprising in the fact that in the system proposed here as well, being constructed on the constructive features of the working surfaces, all these forms of one and the same kinematic pair have been denoted by one sign.

To avoid ambiguities that may arise when considering the various constructions of planar pairs, let us dwell on some details of these constructions.

Both the annular and the rectilinear sliders may slide not only along a one-sided guide, as was the case in Fig. 7; in certain cases the pair is arranged so that these sliders slide between two parallel guides, as shown in Fig. 8.

It is not difficult to see that in this case the influence of one guide on the motion of the slider will be identical to the influence of the other guide. The motion of the slider will be determined entirely by only one guide—practically, by the one against which the slider is pressed at any given moment.

This pair, from the standpoint of both kinematics and dynamics, will not differ in any way from the pairs of the same name shown in Fig. 7. The only difference here will be that the slider will be pressed all the time against one guide, whereas in the pairs shown in Fig. 8 the direction of the normal force pressing the slider may change, and the slider may alternately be pressed now against one guide, now against the other.

Fig. 8.

Similarly, in the pair \(2' K_1 C_1\) shown in Fig. 9, the motion of link \(A\) will be determined by only two points of contact, and not by three—namely, by those two points at which body \(A\) is pressed against the guides. The influence of the third point of contact will be identical to that of the other two, provided that the guides are made as envelopes of body \(A\) during its motion.

Fig. 9.

Applying Prof. Malyshev’s method of geometric schemes to the formation of planar kinematic pairs, we shall ultimately obtain the same kinematic pairs as those shown in Fig. 7. But the process of their formation will be somewhat different.

Table II.

Continuation of Table II.

Designation	Examples
\(K_1, B_1\)
\(C_1 K_1 B_1\) \(C_2 K_1 B_1\)
\(C_2 K_1 B_1\)
\(K_2 C_1\) \(K_2 C_1 B_1\)
\(C_1 B_1\)
\(C_1\)
\(K_1\)
\(B_1\)