L. P. MALYSHEV

Professor of the Tomsk Technological Institute

ANALYSIS AND SYNTHESIS

OF MECHANISMS

FROM THE POINT OF VIEW

OF THEIR STRUCTURE.

Proceedings of the Tomsk Technological Institute
Vol. 44. Issue 2.
1923.

Tomsk.
1923.

Introduction

Although since the time of F. Reuleaux (1875) there has existed, in the Kinematics of Mechanisms, a doctrine of their component parts, it must nevertheless be admitted that it is inadequate. Reuleaux’s doctrine of links, kinematic pairs, kinematic chains, and mechanisms is descriptive in character. Set forth in excellent language, it gives the reader much toward explaining the structure of a mechanism; but there is in this doctrine no method for investigating mechanisms from the standpoint of their structure, and still less for the synthetic construction of diagrams of new mechanisms; and the entire subsequent history of the science is proof of this thought.

It seems to me that the chief reason for the delay in the development of the theory of mechanisms in recent years lies in the fact that the role of kinematic pairs in the science has not been sufficiently clarified. This retards the structural investigation of mechanisms, as a result of which life outstrips science. In almost every issue of the major technical journals we see descriptions of new and novel automatic machines with interesting mechanisms, not only planar but also spatial; yet as a result of what synthetic activity these mechanisms appeared, we cannot even say. One is left merely to express amazement at the genius of the modern inventor, and then modestly to assimilate the idea embodied in the machine.

Science cannot limit itself to the role it has occupied up to now.

Five years ago I tried to set forth my method at the Mechanical Faculty of the Tomsk Technological Institute, and since then it has been included in the program of the course on the theory of mechanisms. Five years’ experience has led me to the conviction that I am on the right path, as a result of which I have resolved to publish my work.

The task I have set myself is a responsible one; perhaps it is even beyond the strength of me alone. Therefore I would feel great gratitude toward those readers who would help bring the method presented here into life. Indications of my mistakes and omissions would likewise bring nothing but benefit to the cause.

Tomsk,
May 4, 1923.

The Author.

Chapter I. General Information.

1. Historical Note.

In the study of mechanisms from the point of view of their structure there are at present large gaps. Whereas the theory of plane mechanisms has been more or less developed and, in any case, in science there is already a foundation on which the investigator may build his further inquiries, we know almost nothing about spatial mechanisms. The fate of this group of mechanisms is very instructive. There have been authors who tried to give schemes of such mechanisms—not a theory, but only schemes of certain mechanisms—as, for example, the outstanding representative of our science, F. Reuleaux ^3). However, their attempts ended in complete failure. A sufficiently cursory glance at these schemes is enough to feel in them the absence of theory and system. Criticism at once noted the errors of Reuleaux ^4), but what is remarkable is that in the criticism itself the absence of a rigorous method is striking, as a result of which one does not at all know how to proceed in some other case. Reuleaux, who first published his doctrine of mechanisms in 1875, evidently undertook a thorough revision of it and in 1900, i.e. after 25 years, issued a revised volume, which in many respects is more perfect than the first volume; however, in the theory of spatial mechanisms it still gives us nothing ^18). Another major scholar, L. Burmester, publishing in 1888 his excellent work on kinematics ^12), calls it “Volume I. Plane Motion,” while in the introduction he promises to issue “Volume 2. Spatial Motion.” However, he did not issue this volume. Criticism sees in this circumstance the exceptional difficulties of the doctrine of spatial mechanisms ^21).

The Russian scholar Kh. Gokhman found himself in exactly the same position. Issuing the first volume of his Kinematics of Machines ^14), he promises another three volumes, with a detailed investigation of the structure of mechanisms and the method of synthetic construction of schemes of new mechanisms. However, he too did not succeed in carrying out this intention. He issued only a small brochure on profiling the teeth of gear wheels ^15). Thus the ideas of Gokhman, expressed by him in the first volume in a general analytical form, remained undeveloped to the end, and it is now unclear whether Gokhman’s ideas could have been applied to practice or not.

Then, a whole series of scholars—Beck, Grashof, Grübler, Hartig, and others—while generally following the ideas of Reuleaux in the doctrine of the structure of mechanisms, confined themselves to the formal-descriptive side of the matter; the method of structural analysis of mechanisms did not yield, and thus the question of spatial mechanisms did not advance in the slightest. A different place is occupied by the work of Prof. P. Somov ¹¹), who points out a new procedure in the study of mechanisms. Namely, he counts the number of degrees of freedom in a kinematic chain and, in this way, determines the suitability or unsuitability of this chain.

This procedure, very interesting in essence, as it seems to me, remained without great practical success only because it was not developed to completion and therefore did not become a method for the study of mechanisms.

A particular case of a spatial mechanism, formed from rigid links coupled with one another by means of cylindrical hinges meeting at one point, had already been considered by Willis ²). An attempt to give this case a scientific explanation was made by Rittershaus ⁴).

At the same time, in the theory of plane mechanisms as well, the role of structure does not remain fully clarified. Many works may be noted in this direction as very valuable material. The works of Grübler, Burmester, Taubeles, Assur, and others led, in the end, to the fact that the majority of modern plane linkage mechanisms can be investigated from the point of view of structure quite satisfactorily. However, even here much room remains for further work. Thus, for example, the role of the link and of the kinematic pair still remains a riddle. It is clear to everyone that each link and each kinematic pair enter into the composition of a mechanism not merely by name. They substantially influence the motion of the mechanism, and nevertheless their significance remains concealed. Indeed, what will happen to a mechanism if in it, let us suppose, a cylindrical hinge is replaced by a rolling pair or by a Hooke joint? What significance do elastic bodies have for a mechanism?

These and many other questions connected with the doctrine of the structure of a mechanism must be acknowledged as unresolved even for plane mechanisms.

Finally, it is necessary also to note that in the theory of mechanisms there ought to be a method by which it would be possible to create schemes of mechanisms synthetically. Quite independently of what such synthesis will give to practice and how soon its results will appear, it must exist in science just as it exists in other branches of knowledge, for example, chemistry.

Attempts to create a synthesis of a mechanism were undertaken by the same F. Reuleaux, but, unfortunately, were not crowned with success. More will be said about this below.

Thus, at the present time the following questions may be considered mature and requiring solution:

1) A foundational doctrine of the structure of a mechanism, with clarification of the role of links and kinematic pairs.

2) The study not only of plane, but also of spatial mechanisms from the standpoint of their structure.

3) The construction of methods for the synthetic formation of diagrams of new mechanisms.

These questions constitute the subject of the present work.

2. Motion of coupled bodies parallel to a plane and in space.

Theoretical mechanics teaches that, in order to determine the position of a rigid body in space, it is necessary and sufficient to know the position of any three points of this body. Then the position of all the other points is determined by itself, since every new point brings with it into consideration 3 new coordinates and, at the same time, 3 new equations for determining these coordinates. Indeed, in a rigid body the distances between points are constant and known. Analytically these distances may be expressed by equations of the following type:

\[ (x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2 = AB^2 \tag{1} \]

where \(x_A, y_A, z_A\) and \(x_B, y_B, z_B\) are the coordinates of two points of the given body, \(A\) and \(B\). If 3 points of a rigid body are known, then every fourth point will be separated from the three named points by given distances, and therefore for it 3 equations of type (1) may be written.

Thus, in the motion of a rigid body in space, counting 3 coordinates for each point, it is necessary to determine 9 coordinates in order that the position of the body be determined; and for this there must exist 9 equations, independent of one another and connecting these coordinates. If a body has exactly 9 equations, it is called statically determinate. If more equations are given, a system with redundant constraints is obtained; it is immovable and, in the general case, statically indeterminate. If, finally, the number of equations is less than 9, we obtain a movable system, whose mobility will be the greater the fewer equations are given. The degree of mobility is characterized by the number of degrees of freedom of the body. Thus, if only 8 equations are given for determining the coordinates of 3 points of a body, then it possesses one degree of freedom; with 7 equations the body has 2 degrees of freedom, and so on.

In the motion of a rigid body parallel to a plane, it is necessary and sufficient to know the position of 2 of its points in order that the position of the whole figure be determined. In this case, obviously, counting 2 coordinates for each point, we must have in all 4 equations in order that all the coordinates be determined. And if, indeed, there are 4 equations independent of one another and connecting the coordinates of 2 points, we have a statically determinate system. If the number of equations is greater than 4, we obtain a system with redundant constraints; and if the number of equations is less than 4, the system will move with the greater number of degrees of freedom, the fewer equations are given.

For a rigid body there are always 3 equations expressing the constancy of the distances between three points. Thus, for the static determinacy of this body in space, 6 equations are lacking.

When a rigid body moves parallel to a plane, of the three equations just mentioned two turn into identities and, thus, there remains one equation characterizing the rigid body in this case. To determine the position of the body in this case, therefore, three equations are lacking.

In the case where what is introduced into consideration is not a rigid body, but some variable one, it is necessary to know precisely in what the variability of the body consists and, in accordance with this, to take into account how many equations remain for this body that connect the points under consideration. If, in spatial motion, two of them remain instead of three, then the number of equations for determining the position of the body must correspondingly be increased by one, and so on.

Let us now pass to the case of rigid bodies coupled with one another.

Let (Fig. 1) two bodies \(M\) and \(N\) move in space, with body \(M\) at point \(O\) movably joined to body \(N\), so that relative motion is possible between the two bodies. Here it is necessary to take into account three points for each body, i.e., a total of 6 points. Counting three coordinates for each point, we obtain altogether 18 unknowns, for the determination of which we have only 6 equations of type (1)—three in body \(M\) and the same number in body \(N\), if we assume both bodies to be rigid. Hence it follows that, for the static determinacy of two mutually coupled rigid bodies in space, as it were, 12 more equations are lacking.

Investigating the same case in the plane, we would obtain for rigid bodies a total of 8 unknown coordinates, with only 2 equations of distances, and for static determinacy, therefore, 6 equations would be lacking.

Looking closely at the grouping of the equations by which the position of bodies in the plane or in space is determined, we easily notice 2 groups of these equations. Some of them determine the distances between the points of each body separately. Thus, for a rigid body in space we counted 3 equations, and so on. Of the other group of equations it has only just been said that they must necessarily be additionally given in order that the motion of the bodies be determined. At the same time, up to now we have not touched at all upon the manner of coupling two bodies with one another, and have considered two bodies as though they were not coupled to one another. Meanwhile the coupling exists, and all those equations by which it is determined form precisely the second group.

The totality of the equations of the 1st group corresponding to one body characterizes a link of a mechanism, while the totality of the equations corresponding to the manner of joining two bodies with one another characterizes a kinematic pair.

If, in a group of coupled bodies, we denote the number of links by \(n\), the number of equations characterizing the links by \(k\), and the number of equations characterizing the kinematic pairs by \(m\), then, in spatial motion, we obtain:

a) for a statically determinate system

\[ \left. \begin{aligned} k + m &= 9n,\\ k + m &= 9n - 1 \end{aligned} \right\} \qquad \ldots (2) \]

b) for a system with one degree of freedom

and so on.

On the other hand, in plane motion:

a) for a statically determinate system
b) for a system with one degree of freedom

\[ \left. \begin{aligned} k + m &= 4n,\\ k + m &= 4n - 1 \end{aligned} \right\} \qquad \ldots (3) \]

In the case when all the links are rigid bodies, the number of equations characterizing the links in a spatial mechanism will be \(3n\), and in a plane mechanism \(n\). Then relations (2) and (3) will take the following form:

From equations (2)

\[ \left\{ \begin{aligned} k &= 6n,\\ k &= 6n - 1 \end{aligned} \right. \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

From equations (3)

\[ \left\{ \begin{aligned} k &= 3n,\\ k &= 3n - 1 \end{aligned} \right. \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5) \]

The character of the motion of a system depends on the number and quality of the equations that relate the coordinates of the points of this system. Leaving aside for the moment the qualitative role of the equations, let us turn our attention to the significance of their number. Suppose we have a system of \(n\) bodies, the motion of which is determined by the number of coordinates \(9n\), and suppose that \(9n\) equations are also given, essentially different from one another and having the following form:

\[ \left. \begin{aligned} 1)\;& f(x_1 y_1 z_1\; x_2 y_2 z_2 \ldots x_{3n} y_{3n} z_{3n}) = 0\\ 2)\;& \varphi(x_1 y_1 z_1\; x_2 y_2 z_2 \ldots x_{3n} y_{3n} z_{3n}) = 0\\ &\ldots \ldots \ldots \ldots \ldots \ldots \ldots\\ &\ldots \ldots \ldots \ldots \ldots \ldots \ldots\\ 9n)\;& \psi(x_1 y_1 z_1\; x_2 y_2 z_2 \ldots x_{3n} y_{3n} z_{3n}) = 0. \end{aligned} \right\} \qquad \ldots (6) \]

By some algebraic method these equations can be solved (for example, by successive elimination of unknowns), and thus all points are determined, and the given system will be immobile.

But if the number of given equations is less than \(9n\), then the equations cannot be solved, and the prescribed system will move. The character of this motion, in certain cases, is more or less determined. Let us examine this case in somewhat greater detail.

In the successive elimination of unknowns, after each elimination operation we shall obtain, as a remainder, a group of equations whose number will be one less and in which there remains an unk—

of unknowns is also less by one. Thus, the process of successive elimination may be represented in the following form:

	Equations.	Unknowns.
Given . . . . . . . . . . . . . . . . . . .	$9n - 1$	$9n$
Remains after the 1st elimination . . . . . .	$9n - 2$	$9n - 1$
“ after the 2nd elimination . . . . . .	$9n - 3$	$9n - 2$
. . . . . . . . . . . . . . . . . . .	. . .	. . .
“ after $9n - 3$ eliminations . . . . . .	$2$	$3$
“ after $9n - 2$ eliminations . . . . . .	$1$	$2$

Further elimination is impossible, since there remains one equation with two unknowns.

The last two lines of the little table given here are of some interest. Indeed, the process of eliminating unknowns can be arranged so that, in the end, there remain the coordinates of some one point; namely, in the penultimate case all three coordinates, and in the last case only two. Since at the same time in the penultimate case two equations remain, and in the last only one, then, according to the laws of analytic geometry, we obtain for the chosen point either a curve of double curvature, or a plane curve. This will be the trajectory of the point. Let us also note that the process of eliminating unknowns can be arranged in any way whatever, and as a result we may obtain dependencies for any of the points of the given system. On the basis of what has been set forth we conclude that, if a system of $n$ bodies is given and $9n - 1$ equations relating the $9n$ coordinates, then all points of this system describe definite trajectories, the equations of which, generally speaking, may be obtained by successive elimination of the unknown coordinates.

Let us now consider the case when for $9n$ coordinates there are in all $9n - 2$ equations. Applying, as before, the method of successive elimination of unknowns, we shall obtain in the end one equation with three unknowns. The process of elimination here too can be arranged so that precisely the three coordinates of one and the same point remain. Thus, as a result of the elimination, we arrive at the equation of a surface on which the given point is compelled to remain. What is true with respect to one point will also be true with respect to every other, and thus we arrive at the general conclusion that, if a system of bodies is given which has $9n$ coordinates, and if $9n - 2$ equations relating these coordinates are also given, then all points of the given system move along definite surfaces, the equations of which, in general, may be obtained by successive elimination of the unknowns.

The case in which \(9n - 3\) equations are given, after elimination, will lead us to one equation with four unknowns. However we arrange the process of elimination here, as a result we shall obtain the displacement of the coordinates of two points, or even of several. The motion turns out to go beyond the limits of geometrical representation. Only special points in this case turn out to lie on surfaces. It will be still more complicated if we further reduce the number of equations connecting the coordinates.

Let us note one very important special case. When a rigid body has one fixed point, then one can compose 6 equations for determining the coordinates. Namely, 3 equations for the link and 3 equations for the pair (the fixed point in the role of a spherical hinge). And nevertheless all points of the given body are constrained to remain on spherical surfaces. We shall speak of this again below.

3. On constraint conditions.

P. Somov proposed, in the study of mechanisms, to use the counting of degrees of freedom, rightly seeing in this a reliable path to the solution of many questions concerning the structure of mechanisms. However, for certain reasons it seems to me more fruitful to count not degrees of freedom, but constraint conditions. This is better because a constraint condition has a definite analytical and physical meaning, whereas a degree of freedom is only a general abstract concept. In many cases the peculiarities of a motion can be understood only by studying each constraint separately. This is quite possible to do, since a constraint condition is expressed by an equation. That is why all the subsequent investigation is built upon clarifying the role of constraint conditions in mechanisms and their practical significance.

By their nature, constraint conditions may be various. Let us note here 4 groups of them, namely: geometrical, mechanical, physical, and psychical conditions.

If the constraint of some element of a mechanism consists in the fact that some geometrical condition is imposed on its motion, we shall call it geometrical. Thus, for example, a segment may, during motion, preserve a constant length, or always remain at a given angle to a given plane; two segments or two angles may always remain proportional to one another, and the like. Analytically, geometrical constraint conditions are expressed by equations connecting the coordinates of points of the mechanism with its various geometrical elements. Thus, for example,

\[ x^2 + y^2 = a^2, \qquad r = k\varphi, \]

(where \(x\) and \(y\) are rectangular coordinates, and \(r\) and \(\varphi\) are polar), are geometrical constraints. (Here \(a\) and \(k\) are constant quantities.)

A mechanical constraint will be present when it imposes on the motion of the system some mechanical condition, such as, for example, a known relation between the forces acting on the system, a relation betw—

velocities of points of a mechanism, and the like. Centrifugal governors operate correctly only because the balls are governed in their motion by a dynamic constraint condition.

Analytically, mechanical constraint conditions may be expressed by equations of statics, kinematics, and dynamics. Thus, for example,

\[ \omega_1 = k\omega_2,\quad P + mg = O \]

(where \(\omega_1\) and \(\omega_2\) are the angular velocities of motion, and \(P\) is force, \(g\) acceleration), are mechanical conditions. (Here \(k\) and \(m\) are constant quantities).

As an example of a physical constraint, one may cite the dependence between the temperature and pressure of a gas in a vessel, or between the pressure and the height of the mercury column in a barometer, and the like.

Finally, it should also be noted that there exists a psychic constraint. Constraint conditions of this kind occur, of course, only in living organisms, which very often act in the role of a mechanism. When, for example, an artist, looking at the original, reproduces it on paper with a pencil, then, obviously, during such work the fingers are constrained in their motion and are forced to undergo definite displacements not only by the types of constraint conditions noted above, but also by psychic ones. In any case, the connection between the original and the copy is only psychic. We do not yet know how to express these constraints analytically; therefore the study of living organisms from the point of view of kinematics is possible insofar as psychic constraints do not function. In practice, as is known, such a study finds application in the so-called prosthetic art, where one has to replace the living members of the organism by artificial ones. Of all the types of constraint conditions listed above, the geometric ones have the greatest significance for kinematic study. They play an exceptional role in our mechanisms, as a result of which constraints precisely of this kind will most often appear below. But, of course, we shall not refuse to include in our consideration both mechanical and physical constraint conditions.

4. On degrees of freedom.

The degree of freedom may be called the number of missing constraint conditions needed in order for the position of a given system to be determined. Let, for example, (Fig. 2), in the plane of the drawing, a rigid rod \(AB\) be given, oscillating about a fixed point \(A\). The number of coordinates characterizing the motion of the rod is 4. The number of constraint conditions is 3, namely, two equations expressing the constancy of point \(A\) and one equation expressing the constancy of the distance \(AB\). Thus, for the rod, in order to determine its position in the plane, one constraint condition is lacking, i.e. it possesses one degree of freedom. Now let us attach to the rod \(AB\) another rod \(BC\), while preserving the same motion in the plane of the drawing (Fig. 3), and let the rod \(BC\) have motion with respect to the rod \(AB\). Then we obtain a different count. Namely, the rod \(BC\) with respect to \(AB\) has one degree of freedom in relative motion and, in addition, co-

together with the rod \(AB\), one more degree of freedom in translational motion. Thus, in the system we see 2 degrees of freedom. In exactly the same way we can obtain 3 degrees of freedom (Fig. 4), 4, 5, and so on.

If we were now to imagine that our rod swings not in a plane but in space, for which it would be necessary to introduce, for example, ball-and-socket hinges, then we would obtain a significantly greater number of degrees of freedom. Indeed, the rod shown in Fig. 2 has in space 9 coordinates characterizing its motion and 6 constraint conditions, i.e., it already possesses three degrees of freedom. The rod \(PC\) (Fig. 3) has 6 degrees of freedom in space, and so on.

As can be seen, we arrive at the conclusion that it is possible to obtain, in the motion of a system, 2, 3, and more degrees of freedom. This, however, is not so in reality and cannot be so. In nature, motion actually occurs with only one degree of freedom. Indeed, if a point moving in a plane possesses two degrees of freedom, then this means that it can occupy any position whatever in that plane. But in fact a point cannot be in several places at once, and at each moment of time it will be in some one place; that is, in motion it will necessarily trace a trajectory. And this is precisely what proves that a point actually has only one degree of freedom. Obviously, we must distinguish our conception of the motion of a point from its actual motion. The former is imperfect. Not knowing all the circumstances of the motion, we speak of the possibility of motion with two or more degrees of freedom, evidently being unable to take into account one or more constraint conditions.

What has been said about motion in a plane is likewise true for motion in space. Here too, however many freedoms we may give a body, it cannot at once proceed along several paths; under the influence, perhaps, of circumstances that are barely perceptible, it will follow some single path, i.e., all its points will describe trajectories. This applies without exception to all bodies of our world, and therefore it must be concluded in general that in our space there is motion with only one degree of freedom.

Suppose, for example, that a rigid body is freely thrown in space. So long as we say nothing about the circumstances of its flight, we consider that this body possesses six degrees of freedom. If we took into account the magnitude and direction of the initial velocity, air resistance, elastic changes in the body, the body’s mass, its form and moments of inertia, the acceleration of gravity, and other circumstances, we would find another 5 constraint conditions (no more and no fewer than the number of factors we took).

5. On the Mechanism.

On the basis of what has been set forth in the preceding paragraphs, we may proceed to the study of mechanisms. But what is a mechanism? Applied—

mechanics knows several definitions of a mechanism, yet none of them can satisfy the modern reader. Without dwelling on this question in too much detail, since considerable space has already been devoted to it in my course of Applied Mechanics (see issue I, Structure and Synthesis of Mechanisms), let us merely note that the most popular definition belongs to Reuleaux, who says that a mechanism is a closed kinematic chain, one link of which has been made immovable. This definition is already obsolete, and many mechanisms cannot be brought under it. Thus, for example, a sensitive balance is an open chain; the planimeter, the pantograph, and other mechanisms have no immovable link, but only an immovable point (bearing in mind, of course, Reuleaux’s conception). A deeper and more modern definition is that of I. Somov, according to which a mechanism is such a kinematic chain in which all points describe definite trajectories, i.e. each link of which has no more than one degree of freedom.

Two explanations should be added to this definition. First, it was shown above that motion in general has no more than one degree of freedom. Therefore, in Somov’s definition one must understand it in the sense that the one degree of freedom obtained in the mechanism is ensured only by those conditions of constraint which are known to us and which we control. If this is not so, then there will be no mechanism, even though the chain had only one degree of freedom under the influence of some conditions of constraint imposed by nature.

And secondly, it should be noted that even with the first correction present, nevertheless not all systems possessing one degree of freedom are mechanisms. A falling log, a flying bullet, a boy on sledges, even though they describe definite trajectories, cannot be called mechanisms. Into the definition of a mechanism it is necessary to include one of the most important features, namely the mechanical character of the motion, which consists in the fact that a given motion can be repeated any number of times; and this property leads to the so-called cycle in the motion. A mechanism, having at a given moment a definite configuration, periodically returns in its motion to the same configuration. If a falling log were somehow raised again to its former position and could then repeat its fall, it would become a mechanism. A bullet in a child’s gun is tied to a string—this is an approximation to a mechanism. A boy on sledges is governed by such conditions of constraint as cannot be fully known (psychical ones), and therefore this case is the most difficult of all to bring under the concept of a mechanism.

Taking into account the considerations expressed here, it seems rational in our time to separate the two concepts and to create two definitions instead of one.

Namely, a mechanism is such a construction in which, under the action of given conditions of constraint, one degree of freedom remains and motion with a cycle becomes evident.

A mechanical appliance is a structure in which, under the action of prescribed constraint conditions, one degree of freedom remains, but the motion proceeds without a cycle.

Thus, the hand rest of a lathe, the weight of a wall clock, water falling from a dam are mechanical appliances, whereas gear wheels, belt transmission, and the hinged quadrilateral are mechanisms.

In practical engineering one can very often encounter combinations of the one and the other.

Thus we arrive at the conclusion that, first, every mechanism in motion will always have only one degree of freedom and that, second, this sole degree of freedom may be determined by those constraints which depend entirely on the designer and are provided by his design. Or the designer may provide not all of the constraint conditions, and then all the missing conditions will be supplied by nature, so that in the motion of the mechanism there will nevertheless prove to be one degree of freedom. In the first case the designer knows all the constraint conditions, and therefore the motion of the mechanism will be quite intelligible and determinate. In the second case some constraint conditions are unknown, and therefore the motion will have an indeterminate character. This indeterminacy may be dangerous or quite harmless. Thus, for example, if we include in the chain of a steam-engine mechanism one more rod, then, from the intrinsic constraint conditions known to us, we obtain two degrees of freedom. If such a machine is set in motion, then the restriction of one degree of freedom will be left to the discretion of fate. Here there is a serious danger of obtaining unexpected motions, dangerous both for the machines and for the people servicing them. Now suppose another case. In the same mechanism of a steam engine the connecting rod is coupled with the crank and with the crosshead not by cylindrical, but by spherical hinges, as occurs in the steering mechanisms of automobiles. Owing to such an arrangement, the connecting rod, in addition to its principal motion, also obtains the possibility of rotating about its geometrical axis. This rotation is likewise a second degree of freedom, but it is quite harmless.

6. Elasticity of motion.

In our machines complete determinacy of motion is usually required. To a definite position of one link there must correspond quite definite positions of all the remaining links. This property entails rigidity of motion. The matter is not changed in the least even in the case when an elastic link, such as, for example, a spring, is included in the chain. There remains all the same the same determinacy and, consequently, rigidity. Meanwhile, quite often what is required of a mechanism is elasticity of motion, by which we shall understand a certain indeterminacy in the oscillations of the elastic links of the mechanism, i.e. the presence of several degrees of freedom. In order that the motion be elastic,

it is necessary that the mechanism contain elastic links and, moreover, that several degrees of freedom be left to the discretion of chance.

Let us clarify what has been said by examples. A simple carriage without rubber tires and without springs is a mechanism with one degree of freedom. In motion it possesses rigidity. A carriage without springs, but on rubber tires, possesses a certain elasticity, because it has elastic tires, whose limitation is not provided for by the design of the mechanism, but is left to the road itself to limit the form. Accidental jolts cause elastic oscillations of the carriage within the limits of relatively small deformations of the tire. A carriage on springs possesses greater elasticity, since its second degree of freedom, left to the will of chance, may also have a large amplitude. A carriage on tires and springs possesses double elasticity, since in it two degrees of freedom are left to the discretion of chance. In the movement of the human organism, the higher degrees of freedom act pleasantly upon the nervous system, and therefore their presence is desirable. However, the limits of the oscillations left to the will of chance must not only be known, but also limited in a definite manner. Rocking in a boat or on a swing, in a carriage or on soft furniture, is pleasant only up to a certain limit. Therefore the designer may admit the higher degrees of freedom only on condition that the limits of oscillation in these freedoms are appointed and guaranteed by him.

Elasticity is favorable not only for the human organism; it is definitely desirable also for machines themselves, provided only that the latter are subjected to jolts and impacts. In these cases elasticity, by softening impacts, protects the machine from rapid wear. As is known, in our time even machine tools are installed on rubber pads. Let us also note the springs of a locomotive and of freight cars, door springs, swinging frames in zincographs, and so forth.

Chapter II. Structure of the Mechanism.

7. Historical note.

The doctrine of the elements of a mechanism was first created by F. Reuleaux and set forth in his book Theoretische Kinematik (Vol. I, 1875; Vol. II, 1900). Unfortunately, Reuleaux, who deeply investigated many questions of applied kinematics, in this matter confined himself to the formal side of the case, and only in part touched upon the essence and role of the parts of a mechanism. He discerns in the composition of a mechanism links and kinematic pairs, gives names and classifications, and immediately passes on to examples. Having divided pairs into higher and lower, he distinguishes among the latter the cylindrical pair, the prismatic pair, and the screw pair, notes the property of reversibility of the lower pairs, investigates the motion of a body supported at 1, 2, 3, 4, and 5 points, and notes the possibility of so-called “force closure.”

The enormous importance that Reuleaux’s book had is fully deserved; however, in fairness, it cannot be attributed to the doctrine of the structure of the mechanism. It would take great effort to prove that this doctrine, in the present question, brought much benefit to pure science and to practical engineering. Reuleaux noted the simplest types of kinematic pairs, but even their role in the mechanism remained unclear. His attempt to combine mechanisms from these pairs ended in failure.

Only 25 years later did Reuleaux publish a revised volume of his kinematics, and it must be admitted that criticism regards this work as a model of logical construction[^21]. However, in the question of the structure of the mechanism, little changed even here. Here 3 kinds of rigid links are noted:

1. rigid in all directions,
2. rigid only in tension, and
3. rigid only in compression.

From the various pairwise combinations of these kinds, kinematic pairs are obtained:

1 and 1	1 and 2	1 and 3
2 and 2	2 and 3
3 and 3

In all, there are 6 classes of pairs, from which all machines are composed. Still, nothing is said about the essence of each link and each pair, and therefore Reuleaux was unable to eliminate the objections raised as early as 1875 by Rittershaus.

Grashof[^8] went comparatively further in the understanding of the essence of kinematic pairs. He distinguishes the following 11 cases:

1. One body, relative to another, rotates about a common straight line.

2. The same body, relative to another, translates along a common straight line.

3. The same body, relative to another, rotates about a common straight line and translates along it with a constant relation between rotation and translation.

4. The same, but with a variable relation.

5. The same body, relative to another, rotates about all axes perpendicular to a common plane.

6. The same body, relative to another, rotates about all axes passing through a common point.

7. The same body, relative to another, does the same as in item 4, but the relation changes according to a known law.

1. One body, with respect to another, without rotation, advances in such a way that one of its points follows a plane curve belonging to the other body.

2. ” ” ” ” the same, but the point follows a spatial curve.

3. ” ” ” ” without rotation advances in such a way that one of its points remains in a given plane of the other body.

4. ” ” ” ” the same, but the point remains on a given surface.

Grashof’s classification of pairs must be recognized as very valuable, although it is not compiled quite rigorously. Naturally, one could continue his enumeration further, indicating, for example,

1. ” ” ” ” advances with rotation in such a way that one of its points follows a prescribed plane curve, where:

a) the rotation takes place about one axis, b) about 2 axes, c) about 3 axes. Here there are still further varieties: d) the axis of rotation coincides with the direction of motion, e) is perpendicular to it, etc.

Grashof’s classification brings the reader to the essence of the matter, but it is made up of several accidental representatives of kinematic pairs and by no means exhausts all possible combinations.

Hohmann approached the question that interests us from another side, in his book Kinematics of Machines (1890). Hohmann distinguishes, in the relative motion of bodies, 6 elements—3 rotations about coordinate axes and 3 translational motions along these axes. In addition, according to the manner in which the bodies are coupled to one another, he distinguishes point, linear, and surface pairs, also singling out, as a special type, the trajectory pair. Then Hohmann has a division of pairs into complete and incomplete, simple and compound. Hohmann’s investigation, on the whole fairly thorough, nevertheless does not provide a solution to the very important practical question, namely, to what extent each pair constrains the motion of a mechanism, so that in the analysis of mechanisms the old gap remained. Moreover, Hohmann’s investigation is not finished even from the theoretical side. Indeed, Hohmann confines himself to 6 elements in the relative motion of two rigid bodies—three rotations and three translational motions. In reality the matter is more complex. For if we say that a body rotates about the axis \(X\), this means that all points of this body simultaneously participate in this rotation. Meanwhile a more complex picture is quite possible. Suppose, for example, that a certain body has 3 points \(A\), \(B\), and \(C\), and suppose that, indeed, point \(A\) rotates about the axis \(X\). At the very same time points \(B\) and \(C\) may perform quite different motions; point \(B\), for example, may move translationally along the axis \(Y\), while point \(C\) may remain in the plane \(XOZ\). But could this point \(C\) follow a pre-

of this trajectory instead of sliding along a given plane? Such a question is not answered by Goχman, and yet it is precisely on the solution of such questions that a correct conception of the essence of kinematic pairs and of the role they perform depends. It is true that in the motion of a rigid body there are only 6 elements, but how these elements are actually realized in kinematic pairs, and what results follow from this, is the question.

If we also note that in a kinematic pair rolling without sliding and rolling with sliding are possible, then we arrive at the conclusion that Goχman’s classification does not exhaust the whole complexity of the question of kinematic pairs.

8. Links.

Every body whose mutual arrangement of points is completely determined by the given constraint conditions, when it enters into the composition of a mechanism, is called its link. Links may be unchangeable and changeable. In unchangeable links the distances between all points are constant. To characterize the mutual position of the points of such a link it is necessary and sufficient to have three geometrical conditions in the form of constancy of the distances between three points. In plane motion it is sufficient to have only one condition of the same type. In changeable links the distances between points may change; however, in the mechanism there must be such a constraint condition by which the change of such a link is determined. If, for example, we insert into a mechanism a leaf spring or some spring of another type, then one of two things must be assumed—either no forces act upon the mechanism, and then the spring does not stretch at all and may be taken as a rigid body, or a load is given, under the influence of which the spring receives a quite definite deformation. In both cases the mutual arrangement of the points of the spring must be known to us, or determined. A free liquid cannot be a link of a mechanism. But if the liquid is poured into a vessel and retains its shape, or if it changes shape under the action of known constraint conditions (for example, mercury in a thermometer under the action of temperature), changes in the vessel while obeying the law \(PV = const\), or, falling freely from a plane, describes definite trajectories, then it may be a link of a mechanism, because in all these cases the mutual arrangement of the points is determined by the existing constraint conditions.

So-called flexible links—belts, ropes, chains—can be considered links only in the case when, by the very arrangement of the mechanism, the mutual arrangement of the points of these bodies is completely determined. The free end of a Galle chain in a hoisting machine, running off a sprocket and dangling in the air, cannot be a link of a mechanism. But if it is compelled to fold into festoons or into storage boxes under the action of given constraint conditions, it becomes a link. If in a link there are more constraint conditions than coordinates,

then so-called redundant constraints are obtained, which in the present case either turn into identities and therefore may be entirely disregarded, or else contradict the remaining conditions. We shall not consider the latter case at all, as incorrect; the former case indicates that redundant constraints in links introduce nothing new and therefore should not be taken into account.

Finally, it should also be noted that a link may have a different number of kinematic pairs by which it is connected with other links. Thus, for example, the rod \(AB\) (Fig. 5) has two kinematic pairs at points \(A\) and \(B\); the link \(BCD\) has 3 pairs, and so on. In studying, in the general case, the composition of a mechanism and the relation between the characteristic coordinates and the constraint conditions, we may proceed in two ways. First, one may take precisely the coordinates of the kinematic pairs as the characteristic coordinates—then in the link \(BC\), shown in Fig. 4, there would be 6 coordinates in spatial motion, while in the link \(BCD\) of Fig. 5 there would be 9 coordinates. Secondly, one may proceed differently, namely, assign to each link, independently of the number of pairs by which it is connected with other links, a definite number of unknowns; thus, for example, to a rigid body, 9 coordinates in spatial motion and 4 in plane motion.

It seems to me that the latter method is more reliable, and I shall use it.

9. Kinematic Pairs.

A kinematic pair, from the standpoint of construction, is a device by means of which two links are coupled with one another; and from the analytical standpoint both are the totality of constraint conditions by which the relative motion of the two links forming this pair is determined. It is very important to distinguish these constraint conditions from those by which the mutual arrangement of the points of one body is determined, since in this latter case we obtain, as indicated above, a link.

Whatever constraint conditions we impose on the relative motion of two bodies, it can ultimately be reduced to geometrical representations; therefore the study of kinematic pairs is most expediently conducted from the geometrical point of view.

As the basis of the investigation we shall adopt the following unshakable principles: 1) the geometrical elements are taken to be the point, line, and surface; and 2) in the spatial motion of a rigid body we consider it necessary and sufficient to know the position of three of its points, while in plane motion only two.

These simple foundations immediately lead logically to the conclusion that in the study of kinematic pairs it is necessary to take a sufficient number of points on each link (no author has done this up to now) and, in addition, for each such point to admit possible couplings with a point, line, or surface of another link (this work was begun, but not completed, by Grashof).

Here, obviously, a whole series of combinations arises. Namely, in the relative motion the points may be fixed; then they may slide along given curves and, finally, remain all the time on given surfaces. In particular cases the curves become straight lines, and the surfaces planes. In addition, surfaces and curves may roll over one another without sliding and with slipping.

Examining all these cases, in what follows we shall characterize them by the number of constraints corresponding to each individual case.

Thus we obtain the following types of mutual constraints of two links.

1. Fixed points.

a) One point of one link is fixed with respect to another link. According to the general rules of analytic geometry, this must correspond to 3 equations, i.e., 3 constraints in spatial motion and 2 in planar motion.

b) Two points of one link are fixed with respect to another link. Here one would have to count 6 constraints in spatial motion. However, in the case when both links are rigid bodies, one of these six constraints becomes an identity because the distance between the two points taken is constant and enters into the number of constraints of the link. Thus, for deformable links this case does indeed correspond to 6 constraints, while for rigid bodies only 5. In the case of planar motion, for two points there are 3 constraints, which together with one constraint of the rigid link give in all 4 constraints. It is obvious that a rigid body fastened in a plane to another by two fixed points loses relative mobility and the kinematic pair, as such, ceases to exist. We have obtained a rigid connection, and this case must be excluded. In spatial motion, however, this case remains valid.

c) Three points of one link are fixed with respect to another link. In this case, therefore, the whole body has no relative motion; a rigid connection is obtained, and the kinematic pair disappears.

1. Points of one link sliding along lines belonging to another link.

d) One point of one link slides along a line belonging to another link. Since a line in space is determined by two equations, and in a plane by one, for a spatial pair this case corresponds to 2 constraints, and for a planar pair to only one.

e) Two points of one link slide along lines belonging to another link. On the basis of the preceding, we obtain here 4 constraints for a spatial pair, and 2 for a planar one.

f) Three points of one link slide along lines belonging to another link. In this case we obtain, respectively, for a spa—

of spatial motion 6 constraints, and for plane motion—3. Together with the constraints of a rigid body we obtain, both for a spatial and for a plane pair, a fixed connection, owing to which the kinematic pair in fact disappears.

3. Points of one link sliding along surfaces belonging to another link.

ж) One point of one link slides along a surface belonging to another link. Since a surface is expressed by one equation, this case corresponds in spatial motion to one constraint, and in plane motion to none. Thus, in plane motion the kinematic pair disappears.

з) Two points of one body slide along surfaces belonging to another link. Correspondingly, in space 2 constraints are obtained, and in the plane none.

и) Three points on three surfaces give 3 constraints in space and none in the plane.

к), л) Two special cases. Case и) shows that, in order to determine the position of a link by three of its points, the constraints are insufficient if only one surface is assigned to each point. Indeed, in case и) we obtained only 3 constraints; adding to them three more constraints inherent in the link itself, we may have only 6 constraints in all, whereas for a rigid body it would have been necessary to have 8 of them. Owing to this, the possibility arises of further restricting the relative motion by surfaces or by other means. Since at present we are investigating the influence of surfaces on the restriction of the relative motion of links, the following further cases arise:

к) Four points of one link remain on surfaces belonging to another link. Here, in spatial motion, 4 constraints are obtained.

л) Five points of one link slide along five surfaces of another link. This case indicates the existence of 5 constraints in spatial motion.

м) 6 points of one link cannot simultaneously slide along 6 surfaces of another, since in this case we would obtain as many constraints as there are coordinates of a rigid body; the motion would not take place and the pair would in fact disappear. For deformable links this case is possible.

4. Rolling of links on one another.

In the complex phenomenon of the rolling of surfaces on one another, Thomson and Tait, in their book Treatise on Natural Philosophy, discern 3 characteristic kinds of motion, namely: spinning, rolling, and sliding.

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

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

When one point belonging to one surface runs through a whole series of points lying on another surface, sliding takes place.

What has been said with regard to surfaces also applies to curves. Thus, two lines touching one another at one point may, in relative motion, produce spinning, rolling, and sliding. Finally, it is interesting to note that lines drawn on surfaces can also roll one upon another, and certain peculiarities arise. It must be borne in mind that curves made, for example, of wire, and curves that are “traces” on surfaces (as the result of intersections with certain, still new, surfaces), as it were do not differ from one another in the number of constraint conditions; but traces on surfaces nevertheless have one superfluous constraint in the form of a common normal, which wire curves do not have. On the same basis, two surfaces touching one another at one point must be counted as 3 constraint conditions.

As for spinning, in its pure form it is a rotation about the given line and, consequently, may be likened to case b) of item 1.

We have already noted the cases of sliding above.

Let us now examine pure rolling and its combinations with spinning and sliding.

n) The surface of one link rolls over the surface of another link with spinning that is functionally independent of the rolling. 3 constraint conditions.

o) The same, but the spinning is functionally connected with the rolling. 4 constraint conditions.

p) One curve belonging to one link rolls over the surface of another link with independent spinning. 3 constraint conditions.

r) The same, but the spinning is connected with the rolling. 4 constraint conditions.

s) Two curves connected with one link roll over the surfaces of another link with spinning. The pair disappears in all cases.

t) One curve of one link rolls over a curve of another link. 4 constraint conditions.

u) A trace on the surface of one link rolls over a trace on the surface of another link. 5 constraint conditions.

f) In plane motion, a curve belonging to one link rolls over a curve belonging to another link. 2 constraint conditions.

x) In combinations of rolling with sliding it must be borne in mind that every element of sliding gives one degree of freedom, that is, lowers the number of constraint conditions by one, provided only that this sliding is not functionally connected with the rolling. Thus, when a surface rolls over a surface with spinning and sliding along one prescribed...

in this direction, without a functional dependence among all three kinds of motion, 2 constraint conditions are obtained. Other combinations can be made up in the same way.

In considering the various combinations of mutual coupling of links with one another, we must note that in kinematic pairs not only these combinations taken singly are possible, but also various groupings of them. It is only necessary that, in the sum between two coupled rigid links, there should not be more than 5 constraint conditions in the relative motion, since otherwise the pair disappears and the connection becomes rigid.

Thus, for one and the same pair we recognize the possibility of the simultaneous existence both of a fixed point, and of points following given trajectories, and of points remaining on given surfaces, and, finally, of known rolling conditions. Obviously, we must now compose the possible combinations from the elements just discussed above; for this purpose we shall proceed as follows:

1. we shall mark all the enumerated characteristic kinds of mutual coupling of points of two links,

2. we shall write down all possible cases in each such kind, as well as the number of coupled points and the corresponding number of constraint conditions,

3. we shall assign a letter, or sign, to each case.

In this way the following table is obtained:

Table 1.

Kinds of constraint conditions	No. of points	Number of constraint conditions	Sign	No. of points	Number of constraint conditions	Sign	No. of points	Number of constraint conditions	Sign	No. of points	Number of constraint conditions	Sign	No. of points	Number of constraint conditions	Sign	No. of points	Number of constraint conditions	Sign
Fixed points . . .	2	5	Ф	1	3	К	0	0	С
Points on given lines . .	2	4	Т	1	2	Л	0	0	Н
Points on given surfaces . .	5	5	Ю	4	4	У	3	3	И	2	2	Е	1	1	О	0	0	А
Rolling conditions . . . . .	—	5	Д	—	4	М	—	3	П	—	2	Б	—	1	Г	—	0	Р

Using the data of this table, we shall now compose kinematic pairs from its various elements, combining them with one another. Each combination will receive its own name; thus, for example, the kinematic pair KNER has one fixed point and 2 points on the given surfaces, etc.

In order to investigate all possible cases from the given elements, the following must be taken into account.

The mobility of one link relative to another is possible only when the number of constraint conditions for this link, together with the number of constraint conditions for the pair, is in sum less than the number of coordinates characterizing the link. Thus, for example, for a rigid body in spatial motion the total number of constraint conditions must not be greater than eight. But since there are 3 constraint conditions in the link itself, no more than five conditions remain for the kinematic pair. Thus, all combinations that will have more than 5 constraint conditions must be discarded.

The order of formation of kinematic pairs will be as follows. We shall take one condition from the first row and attach to it all possible conditions from the remaining rows. From such a construction, formally, a total of 324 pairs should result. However, the majority of them must be excluded, since they have too many constraint conditions.

As a result we obtain the following tables.

Table II. Kinematic pairs for rigid links.

No.	Link 1 of the table row	Total combinations	Possible combinations	Total	Excluded	Note
1	F	108	FNAR	1	107	*) See explanation below.
2	K	108	KTAR, ) KLOR, ) KLAG,* ) KLAR. KNER, KNOG, KNOR, KNAB, KNAG, KNAR.	10	98
3	S	108	STOR, STAG, STAR. SLIR, SLEG, SLER, SLOB, SLOG. SLOR, SLAP, SLAB, SLAG, SLAR. SNOR, SNUG, SNUR, SNIB, SNIG. SNIR, SNEP, SNEB, SNEG, SNER. SNOM, SNOP, SNOB, SNOG, SNOR. SNAD, SNAM, SNAP, SNAB, SNAR.	33	75
Total		324		44	280

Kinematic pairs listed in this table are suitable for coupling rigid links with one another, since each of them has no more than five constraints. They have been composed in the following way.

One must take some condition from the first row of Table I, for example, Ph. This condition means that the pair has two fixed points with 5 constraints. Thus, together with the three constraints possessed by a rigid link, we already obtain 8 constraints and, obviously, no further constraints can be added to this pair. Hence we naturally arrive at the conclusion that, with the sign Ph, only the signs N, A, and R from all the rows below can be combined, since they contain no constraints at all, and we obtain the pair PhNAR. All the remaining 107 combinations involving the sign Ph have to be excluded. This is what is recorded in the first row of Table II. Then we pass to the sign K of Table I and successively do the same with it, and so on.

Let us note a few further particulars.

The pairs KTAR, KLOR, and KLAG at first sight are impossible, since they have too many constraints. However, one should take into account what was said at the end of § 2. If a rigid body has one fixed point, then only 6 constraints are formed, and nevertheless all points of the body remain on spherical surfaces, as though 7 constraints were present. In this case the extra degree of freedom does not remove the point from the surface, but merely complicates the motion over this surface. Thus, the introduction of only one constraint (for example, the equation of a surface) already determines the trajectory of a point. This is why, in the presence of one fixed point, the trajectory of another point of the same body can be specified by only one equation, and not by two. Thanks to this it turns out that the pair KTAR has only 5 constraints,

KLOR likewise has 5 constraints,
KLAG likewise has 5 constraints,
KLAR has 4 constraints.

Finally, it should also be noted that the pair CNAR is excluded altogether, since in it there is no coupling whatever between the links.

Table II is very helpful in the classification and study of kinematic pairs. Thus, for example, it is immediately clear that all pairs which include the sign R have no rolling element. There are 18 of them in all. Conversely, 26 pairs are formed with rolling elements.

It is also not difficult to distribute the pairs according to the number of constraints. Incidentally, it is very convenient to assign pairs having one constraint to the 1st class, with two constraints to the 2nd class, and so on. Then the following classification is obtained:

Table III. Classification of kinematic pairs suitable for coupling rigid links.

Classes.	NAMES OF PAIRS.	Total.
1st class.	SNOR, SNAG.	2
2nd class.	SLAR, SNER, SNOG, SNAB.	4
3rd class.	KNAR, SLOR, SLAG, SNIR, SNEG, SNOB, SNAP.	7
4th class.	KLAR, KNOR, KNAG, STAR, SLER, SLOG, SLAB.	12
4th class.	SNUR, SNIG, SNEB, SNOP, SNAM.	12
5th class.	FNAR, KTAR, KLOR, KLAG, KNER, KNOG, KNAB.	19
5th class.	STOR, STAG, SLIR, SLEG, SLOB, SLAP, SNYUR.	19
5th class.	SNUG, SNIB, SNEP, SNOM, SNAD.	19
TOTAL		44

Although in the relative motion of two rigid bodies only pairs of the 5th class provide motion with one degree of freedom, because together with the constraint conditions of a link they give only 8 equations, nevertheless pairs of all the other classes may also occur in mechanisms. It is necessary for us only that in the mechanism the total numbers of coordinates and of constraint conditions differ from one another by one. Meanwhile it is entirely possible that some link has several kinematic pairs, as, for example, link \(BCD\) in Fig. 5.

It should finally be noted that the combinations presented above can be realized structurally in different ways. Suppose, for example, that the pair FNAR is given. Its essence consists in the fact that the coupled links have 2 common points, or an axis. This pair can be realized in various ways, and some of its forms are shown in Figs. 6–8. In exactly the same way, the constructions of the pair KLAR may be different, and some of them are shown in Figs. 9–11.

Applying the types of kinematic pairs described above to motion in a plane, we obtain certain special features. First of all, the case F in the plane is equivalent to a rigid connection and therefore must be excluded. Then all cases corresponding to points on surfaces must be excluded (Yu, U, I, E, and O). From the third line of Table I, only the combination A remains possible. Finally, from the rolling conditions only B, G, and R are possible.

Taking these remarks into account, we obtain for plane pairs the following types:

KNAR, STAR, SLAG, and SNAB according to 2 constraint conditions, and
SLAR, and SNAG according to one constraint condition.

By their designation, plane pairs do not coincide with spatial ones, although they may correspond to one and the same type. Thus, a simple cylindrical joint, according to the nomenclature of spatial pairs, should be called FNAR, whereas in plane motion it will be called KNAR. Meanwhile, in spatial motion KNAR is a spherical joint. To avoid any confusion, plane pairs are printed in italics.

It remains to clarify one more question: the kinds of kinematic pairs for variable links.

If the distances between points in a link vary, then the number of its constraint conditions decreases. Above, only rigid bodies were taken in the investigation, for each of which 3 constraint conditions were counted. We shall now consider links with two and with one constraint condition. Links that contain not a single constraint condition lose their significance, and we shall not consider them.

When constructing schemes of kinematic pairs that couple variable links, we must bear in mind that all pairs from Table II are suitable for them and remain valid. But, in addition, it becomes possible to construct several new combinations with the number of constraint conditions 6 and 7. The results are given in the following table:

Table IV. Additionally possible pairs for variable links.

Class	Pairs	Total
3rd class.	FLAR, FNOR, FNAG, KTOR, KTAG, KLOG, KNIR, KNEG, KNOB, KNAP, STER, STOG, STAB, SLUR, SLIG, SLEB, SLOP, SLAM, SNOG, SNUB, SNIR, SNEM, SNOD.	Total 23
7th class.	FTAR, FLOR, FLAG, FNER, FNOG, FNAB, KTER, KTOG, KTAB, KLIR, KLEG, KLOB, KLAP, KNUR, KNIG, KNEB, KNOP, KNAM, STIR, STER, STOB, STAP, SLYUR, SLUG, SLIB, SLEP, SLOM, SLAD, SNYUB, SNUP, SNIM, SLED.	Total 32

10. Constructions of kinematic pairs

In view of the fact that many kinematic pairs prove to be rather complex, on the one hand, and, on the other hand, taking into account that links may have not only two pairs but more, a certain practical difficulty arises, along with the possibility of confusion when constructing them. To put an end to this confusion once and for all, it must be accepted that a pair is a device by means of which two links are coupled with one another, however complex this device may be. Thus, for example, a transmission shaft is mounted in 10 bearings, but this does not mean that 10 pairs arise, since in these bearings only two bodies are nevertheless connected—the shaft and the fixed support. Therefore, it must be considered that, however many bearings there may be, in

with a support there exists only one kinematic pair. If the rod \(AB\) slides along the sides of a right angle (Fig. 12), then between it and the sides of the angle there again exists only one kinematic pair. Since two carriage wheels are rigidly mounted on an axle, they constitute one link, as a result of which there is one kinematic pair between the two wheels and the rails, and so on.

In the case where not two, but several links meet at one point, the question arises of how to determine the connection conditions for the kinematic pair. Then it is necessary to examine carefully which of the links, and with which ones, are coupled in this compound pair and, furthermore, of course, which pairs figure in this coupling. Thus, for example, when three links \(A\), \(B\), and \(C\) are joined at one point, all the pairs \(AB\), \(AC\), and \(BC\) are possible, and it is also possible that only some two of them exist. Suppose that in Fig. 13 a hinge is given in which these three bodies are coupled. Let link \(A\) be fixed; link \(B\) is coupled by a cylindrical hinge with link \(A\); on the other side, link \(C\) is also coupled by a cylindrical hinge with link \(A\). Thus, here we have 2 hinges of the 5th class, by means of which three links are coupled.

In Fig. 14 a connection is shown in which 3 links are grasped by three hinges. And although such a connection would be shown on the diagram by only one point, it nevertheless must be counted as 3 independent hinges.

By their construction, pairs may have a certain general form and then various special forms. It was already noted above how the pairs PhNAR and KLAR vary. Let us take also the pair SNOR. In general form it is the coupling of five points of one link with the same number of surfaces of another. In a special form this may be an ordinary slider with a guide, since all the surfaces of the pair SNOR may turn into planes and, moreover, all these planes may be parallel to one another. But under the same conditions the pair SNOR will also be satisfied by a screw with a nut and by a pin in a guide slot, as for example in the head of a sewing machine, and the like.

In view of the fact that practically every kinematic pair may be carried out in different ways, we shall not hereafter investigate each of them separately, still less in all modifications. This is predominantly a matter of practice. In general form, some of the spatial pairs are shown in Figures 15–30, and in Figures 31–36, diagrams of plane pairs.

In Fig. 15 an ordinary cylindrical hinge with the axis of rotation at point \(A\) is presented.

In Fig. 16 a spherical hinge with its center at point \(A\) is shown.

In Fig. 17 there is the same spherical hinge with its center at point \(A\), but a pin \(B\), sliding in a slot, is attached to the sphere.

Fig. 18. The same as in Fig. 17, but two pins \(B\) and \(C\) are attached to the sphere, each sliding in its own slot.

In Fig. 19 a spherical joint with its center at point \(A\) is shown. In addition, pin \(B\) slides in a slot, and point \(C\) slides along the given surface \(P\).

Fig. 20 depicts the following pair. Point \(A\) is made in the form of a spherical joint, while points \(B\) and \(C\) each slide along its own surface.

Fig. 21. A tripod. On link \(M\), three points \(A\), \(B\), and \(C\) slide along given surfaces.

Fig. 22. Link \(M\) has, on one side, a spherical joint with its center at point \(A\) and a pin \(B\) sliding in a slot, and on the other side, a surface \(CD\) rolling with slip along the given surface \(P\).

Fig. 23. Link \(M\) has point \(B\), sliding along the given surface \(P\), and, in addition, a spherical joint with its center at point \(A\).

Fig. 24. Point \(A\) is a spherical joint, while curve \(B\) rolls along the given surface \(P\).

Fig. 25. A sphere, as an example of a surface, is constrained to remain on the given surface \(P\).

Fig. 26. A body with a fixed spherical joint rolls along the given surface with slip.

Fig. 27. Link \(M\) has at point \(A\) a prescribed trajectory in the form of a guide; in addition, point \(B\) slides along the given surface \(P_1\), and finally, surface \(C\) rolls along the given surface \(P_2\) with slip.

Fig. 28. Three points of the given link slide along three given surfaces, and in addition one point (a fourth) slides along a given guide.

Fig. 29. Two points of the moving link slide along given guides.

Fig. 29-a. The same, but both guides coincide, or one is an extension of the other.

Fig. 30. Two points of the moving link \(A\) and \(B\) slide along given guides, while a third point \(C\) is constrained to remain on the given surface \(P\).

Fig. 30-a. The same, but both guides coincide, and, in addition, the surface is specified as a helical one.

Figs. 31--36 for plane motion, owing to their simplicity, do not require explanation.

For illustration, let us give some examples of kinematic pairs.

1. Sphere on a plane . . . . . . . . . . . . . . . . Pair SNAP.
2. Spherical joint . . . . . . . . . . . . . . . . . “ KNAR.
3. Tracing pin of a planimeter . . . . . . . . . . “ SLAR.
4. Tripod of a physical instrument . . . . . . . . “ SNIR.
5. Roller of a planimeter . . . . . . . . . . . . . “ SNAB.
6. Compass . . . . . . . . . . . . . . . . . . . . “ KNOR.
7. Between the teeth of cylindrical gear wheels . . “ SNAT.

1. Hooke’s joint consists of three bodies with two pairs of the type FNAR.

2. A railway axle on rails . . . . . . . . . . . . . . . . . SNAD.

Note: if the railway wheel did not have a flange on the rim, the pair obtained would be SNAM with an excess degree of freedom. On the other hand, the flange necessarily slips during rolling; otherwise motion would be impossible. The result would be a pair with six rolling conditions, which would stop the wheels.

1. A spring with a slide (Fig. 36-a). Pair FLAR (for variable links).

In exactly the same way, all other kinematic pairs may also be defined. Moreover, on the basis of their general forms, it is possible to construct new particular practical combinations.

11. Groups of joined links.

In the composition of complex mechanisms, some authors have long since noted such groups of mutually coupled links which can be separated from the mechanism, or added to it, and as a result a mechanism is again obtained. In Assur’s book, Investigation of Plane Bar Mechanisms, there is a historical survey of this question and a method for composing such groups for plane mechanisms. It must be noted that the whole history of this question relates exclusively to plane pin-jointed mechanisms with lower pairs (in Reuleaux’s terminology). Let us recall here that the so-called Sylvester dyad (Fig. 37), the three-lead link (Fig. 38), the four-lead group (Fig. 39), and then complex groups formed by the method of developing the lead, are known in the literature.²¹) As for spatial groups, unfortunately we find no references to them in the literature. Moreover, even with respect to plane groups, only pin-jointed ones have been investigated, and even in them there remains the question whether they are all exhausted by the method of developing the lead (Assur’s method).

Counting the constraints in groups gives an entirely different result. For us it is immaterial whether a plane or spatial mechanism is being investigated, and whether a plane or spatial group is being separated from it. Only one thing is important and quite definite. If we wish to separate some group from a mechanism in such a way that what remains is still a mechanism, then, obviously, the removed group must contain as many coordinates as constraints. Then, indeed, we obtain such a process.

In the mechanism there were \(n\) coordinates and \(n - 1\) constraints.
Removed: \(k\) coordinates and \(k\) constraints.
Remaining: \(n - k\) coordinates and \(n - k - 1\) constraints.

Whence we see that in the remainder there is obtained a system with one degree of freedom.

Let us now apply the method of constraints to various groups.

a) In a Sylvester dyad, for plane motion, there are in all 8 coordinates (2 links, with 4 coordinates for each link). There are also 8 constraint conditions, namely, 2 distance equations and 3 kinematic pairs of the type KNAP, with 2 conditions in each.

b) In a three-link element there are 4 links, i.e., in all 16 coordinates. Constraint conditions: 4 links give 4 conditions, and 6 pairs give 12 conditions. Consequently, altogether we have 16 constraint conditions.

c) A four-link group with 6 links has 24 coordinates. Constraint conditions: 6 links and 9 pairs give a total of 24.

d) A Sylvester dyad in a spatial mechanism would give an entirely different result. Namely, for its 18 coordinates there would be 6 constraint conditions in the links and 15 conditions in the kinematic pairs. Thus neither the attachment nor the separation of this dyad is possible in spatial mechanisms.

e) In spatial mechanisms there can be separated only those dyads in which the 3 kinematic pairs have, in sum, only 12 constraint conditions.

These may be, therefore, 3 pairs of the 4th class, two pairs of the 5th class and one of the 2nd, and so forth. A dyad having in its composition 2 pairs FNAP and one pair SLAP can be separated from a spatial mechanism.

f) In the same way we can also compose new groups satisfying the stated conditions. Let us name, for example, a triad in space (Fig. 40).

It has 27 coordinates in its three links. The constraint conditions—in the links, 9, and in the kinematic pairs there must be 18. Such pairs can be selected in various ways. For example, one may take 3 pairs of the 4th class and 2 pairs of the 3rd class, etc.

g) The four-link group shown in Fig. 41 is separated from plane mechanisms. It has 8 links and, consequently, 32 coordinates; here, 12 pairs account for 24 constraint conditions and 8 rods for another 8, for a total of 32 conditions.

The spatial group shown in Fig. 42 is separated from a mechanism, since it has 45 coordinates and the same number of constraint conditions, if we imagine all the pairs to be of the 5th class, for example, cylindrical hinges.

Then in the links there will be 15, and in the pairs 30 constraint conditions, which is what is required. In the study of groups, just as of mechanisms, a whole series of special cases is possible, when the number of constraint conditions decreases for various reasons. Thus, for example, the coincidence of two hinges into one, the parallelism of cylindrical hinges, their convergence at one point, the transition of a mechanism during motion from space into a plane, etc., are accompanied by a change in the quantitative relation between the number of coordinates and the number of constraint conditions, as a result of which a change in the motion also occurs. A moving system may come to rest; conversely, a fixed one may begin to move. More will be said about this below. (See p. 18).

In general form, the question of separating groups may be presented as follows. One rigid link in a plane mechanism has 4 coordinates and one constraint condition within itself. In order for it to be separable, in the kinematic pairs by which this link is attached to the mechanism there must be a total of 3 constraint conditions. This is quite possible if one pair has two conditions and the other one. This is why a rigid rod with a cylindrical hinge at one end and a slider at the other can freely be removed from the mechanism, or attached to it. As a result, one still obtains a certain mechanism.

One rigid link in a spatial mechanism has 9 coordinates and 3 constraint conditions within itself. Thus, for it to be possible to separate it from the mechanism, it is necessary that in the kinematic pairs of this link there be a total of 6 constraint conditions. If this is fulfilled, then such a link can freely be removed from the mechanism or attached to it.

\(n\) rigid links coupled to one another in plane motion have \(4n\) coordinates and \(n\) constraint conditions within themselves. By however many kinematic pairs these links may be attached to one another and to the mechanism, it is necessary that in these pairs there be, in all, \(3n\) constraint conditions. Then this group can be removed from the mechanism.

\(n\) rigid links coupled to one another in spatial motion have in all \(9n\) coordinates and \(3n\) constraint conditions within themselves. By however many kinematic pairs these links may be attached to one another and to the mechanism, it is necessary that in these pairs there be, in all, \(6n\) constraint conditions. Then this group can be removed from the mechanism or attached to it.

Example. Can the group represented in Fig. 43 be removed from the mechanism so that the remainder also proves to be a mechanism, if the hinges \(A, B, C\), and \(D\) are cylindrical, while \(K, L, M\), and \(N\) are spherical?

Since there are 5 bodies in the given group, there must be in all 30 constraint conditions in the kinematic pairs. Meanwhile, we have 20 conditions in the 4 cylindrical hinges and 12 in the 4 spherical ones, making a total of 32 constraint conditions. Thus this group cannot be removed. If in the given group there were 5 spherical and 3 cylindrical hinges, then exactly 2 constraint conditions would be subtracted and the group could be removed from the mechanism.

Chapter III. Analysis of the Mechanism.

12. The essence of mechanism analysis from the standpoint of its structure.

In the study of mechanisms up to the present time there has been a very serious gap. Whereas kinematic investigation has been developing successfully, already has its extensive literature, and has many methods for constructing and studying the trajectories, velocities, and accelerations of points

mechanisms—the study of the structure of mechanisms has scarcely moved forward. True, recently, especially thanks to L. Assur’s capital work, A Study of Plane Bar Mechanisms from the Point of View of Their Structure and Classification, much has been clarified, and with respect to plane mechanisms we have obtained a method of investigation. This method does not cover all combinations in plane mechanisms, but nevertheless it sheds much light on them, and therefore Assur’s merit in the history of our science must be acknowledged. As for spatial mechanisms, their analysis from the point of view of structure has in fact not even begun. The entire rather modest literature on this question reduces to two particular cases that have been examined more or less thoroughly. The first case is found already in R. Willis’s book Principles of Mechanism (2nd ed., 1870), where on pages 249–255 a short theory is set forth for constructing simple closed spatial chains formed of rigid bodies joined to one another by cylindrical joints converging at one point. References to this same case may also be found in the literature in other authors.

The other case likewise concerns simple closed spatial chains with cylindrical joints, but only without their convergence at a single point. In P. Somov we find a proof that such a chain in space must have 7 links, and in the plane 4.

Meanwhile, there remains a whole series of very important questions, not even touched upon, concerning precisely the structure of a mechanism.

First of all, when a new mechanism is being created, or the scheme of a proposed mechanism is being investigated, one must be certain that this scheme actually corresponds to a mechanism, since it may also correspond to an immovable system or to a loose system with several degrees of freedom. As is known, Reuleaux attempted, without a method, to create certain schemes of spatial mechanisms and, as it turned out, unsuccessfully. In science there must be a method by means of which the question posed could be solved reliably. This question, important in itself, acquires special practical interest because it leads organically to another question, with which it is connected—namely, under what conditions can a given mechanism work correctly? An answer to such a question is unthinkable without a detailed study of the structure of the mechanism, its links, and its kinematic pairs. Here there arises an entire series of modifications of the question posed, as for example: can the slider of a slider mechanism not rotate; is gear transmission possible without sliding between the teeth; is it not possible to introduce an elastic link into the mechanism, and so forth.

Then, as was already indicated above, in a mechanism individual links may also have more than one degree of freedom. These links must be found and examined, to determine to what extent the superfluous freedoms are harmless for the motion of the entire mechanism.

Another very important question is how to investigate the motion of such compound mechanisms, in which certain parts now begin to move, now come to rest. One must be able to decompose compound mechanisms in order to investigate the motion of each part.

13. Two-link mechanisms.

Since only two bodies figure in two-link mechanisms, in spatial motion we have only 18 coordinates; moreover, if one of the links remains fixed, it need not be considered at all, and we are left with only 9 coordinates. To determine these coordinates in a normal mechanism, 8 constraints must be formulated. In view of the fact that these conditions pertain partly to the link and partly to the single kinematic pair by means of which both links are coupled to one another, it is necessary to distinguish the conditions belonging to the link and those belonging to the pair. Obviously, one must first decide whether we are dealing with rigid or changeable links. Let us suppose, for the first time, that the links are rigid bodies. Then in the moving link there will be its own 3 constraints, and for the kinematic pair there remain, consequently, 5 conditions. This clearly shows that in the given case all pairs of the 5th class may be suitable. But from Table III it is seen that there are altogether 19 types of such pairs; whence we conclude that there may likewise be that many types of two-link spatial mechanisms formed from rigid bodies. Examples of such mechanisms may be the kinematic pairs of the 5th class themselves, since they are models of two rigid bodies coupled to one another (see Figs. 15, 18, 19, etc.).

In plane two-link mechanisms formed from rigid bodies, there will be only 8 coordinates in all, and only 2 intrinsic constraints in the links. Thus, discarding the fixed link with its 4 coordinates, we obtain, for the 4 coordinates of the remaining link, a single constraint. This shows that in the sole kinematic pair there may be 2 constraints.

Since there are only 4 such pairs, this means that it is possible to construct only 4 types of two-link plane mechanisms formed from rigid bodies.

Examples of such mechanisms may be the plane pairs themselves, such as: the cylindrical hinge, the slider, a wheel rolling in a plane along a given curve, and so forth. If a changeable link enters into the composition of a two-link mechanism, then, depending on how many constraints are lost in the rigid link when it is transformed into a changeable one, it will be necessary to require the same number of additional constraints from the kinematic pairs. Thus, if in a spatial two-link mechanism one link is fixed and the other is changeable, then two types of mechanisms are possible here. Namely, either in the changeable link there still remain 2 constraints out of three, or only one constraint remains in it. In the first case, for the 9 coordinates of the moving link, we obtain in it sa-

two conditions and, consequently, 6 conditions of constraint remain for the kinematic pair. In the second case the link has only one condition of its own and, consequently, 7 conditions of constraint remain for the kinematic pair. Above, in Table IV, there were given types of kinematic pairs with 6 and 7 conditions of constraint, which can now also be applied.

As an example, in Fig. 44 there is shown a mechanism consisting of a fixed frame and a cylindrical helical spring. A slot is cut in the frame, along which one end of the spring slides, while its other end is rigidly embedded in the frame.

In the spring there remains only one condition of constraint, namely the law of proportionality between the deformations of the points. Thanks to this, the kinematic pair must already have 7 conditions of constraint, which is what we see in the given mechanism.

In the case of a plane mechanism with a variable link, the following picture is obtained. The single condition of constraint in the link disappears and must be replaced by an additional condition in the kinematic pair. Thus, in plane mechanisms one has to add one condition to the pair. This can be achieved either by forcing one of the points of the link to move along a given trajectory, or by rigidly embedding one end of the link in the frame. Examples of such mechanisms may be a thermometer, an electric bell, spring scales, and also the one shown in 36-a, where point A is connected to the frame by a cylindrical hinge, and point B slides in the given slot.

14. Three-Link Mechanisms.

In studying three-link mechanisms it is necessary to take into account already three bodies, which in general spatial motion will have 27 coordinates. Taking one of these bodies as fixed, we obtain 18 coordinates subject to investigation. If the remaining two bodies are invariable, then in them themselves there are only 6 conditions of constraint and, thus, in the kinematic pairs there must be 11 conditions of constraint. Since in a three-link mechanism there can be only 3 pairs, the possibility arises here of the following combinations:

a) 2 pairs of the 5th class and one of the 1st.

b) 1 pair of the 5th class, 1 pair of the 4th, and 1 pair of the 2nd classes.

c) 1 pair of the 5th class and 2 pairs of the 3rd class.

d) 2 pairs of the 4th class and one of the 3rd.

Taking now into account that, according to Table III, there may be 19 different pairs of the 5th class, 12 pairs of the 4th class, and so on, we can obviously construct many mechanisms from three links. Namely, according to group a) there are altogether 380 original mechanisms, according to group b)—912, according to group c)—532 and, finally, according to group d)—546 mechanisms, altogether 2370 mechanisms, with permutations of pairs in this count being excluded.

Very many of these mechanisms are difficult to realize in practice because the actual mobility in the kinematic pair turns out to be very short-lived. However, on the other hand, very many mechanisms can easily be constructed. As examples we shall cite: a pair of toothed wheels in space (for example, a worm gear) (case a); a spirograph (see Fig. 45) (also case a); the mechanism shown in Fig. 46 (case b); the mechanism shown in Fig. 47 (case г).

In planar three-link mechanisms formed from rigid links, we have in all 12 coordinates; and, after subtracting the fixed link, only 8 coordinates. In the two movable links there will then be two intrinsic constraint conditions, and thus 5 constraint conditions remain for the 3 kinematic pairs. They can be distributed in only one way, namely: 2, 2, and 1. Since there are only 4 planar pairs of the 2nd class, and only 2 of the 1st class, 20 possible combinations of planar three-link mechanisms formed from rigid links can be compiled.

As an illustration of such mechanisms, let us cite a pair of planar cylindrical toothed wheels, a cam with a rod (Fig. 48), and a typewriter (Fig. 49).

Remark. Very often a formally correct mechanism gives a motion that is quite uninteresting, since in it all the links are in fact inactive except for some one link. Thus, for example, the mechanism shown in Fig. 50 has spherical hinges at points $A$ and $C$, and a cylindrical hinge at point $B$. In terms of the number of coordinates and constraint conditions it fully fits case c) of the classification given above. However, the motion will consist only in rotation about the geometric axis $AC$, and the hinge at point $B$ remains entirely unnecessary. If we were to change the order of placement of the hinges in the same mechanism and take the case in which the spherical hinges are at points $A$ and $B$, while at point $C$ we place a cylindrical hinge, then the only motion that remains in the mechanism will be that the rod $AB$ rotates about its own geometric axis.

Applying to the study of mechanisms the method based on counting constraint conditions, I obtained very satisfactory results and found answers to the questions posed above. In general outline this method consists in the following: for the given mechanism one must count the coordinates and the constraint conditions; then, if required, examine each constraint separately and determine its significance for the mechanism. In practical applications of this method it is necessary to keep the following in mind.

1. In counting the total number of coordinates in a mechanism, it is best to do this formally, without associating coordinates with particular points of the links. In my opinion, such a normal count should protect the investigator from many accidental errors. The customary division of a mechanism into links does not present any special—

labor, but it is already much more difficult to mark the points subject to investigation. Classroom practice has shown that the student becomes confused on this question even with simple problems. Therefore it is much more expedient to assign to each link the total number of coordinates characterizing it, independently of the kind of coupling of this link with the others. In this case a rigid body must always be counted as having 9 coordinates in spatial motion and 4 coordinates in plane motion. Thus, if we are given a mechanism of 4 rigid bodies coupled with one another in space, we must count in them a total of 36 coordinates. A plane mechanism consisting of 12 rigid links has 48 coordinates. The participation of a non-rigid link in a mechanism reduces the number of coordinates correspondingly by as many as the number of distances in the link that are taken as variable.

A fixed rigid link may be omitted from the count altogether, since it has 9 coordinates and just as many constraint conditions.

1. In calculating the total number of constraint conditions in a mechanism, it is best to indicate the number of conditions belonging to the links and the number of conditions belonging to the kinematic pairs. Thus, for each rigid link in spatial motion one must assign 3 constraint conditions, and in plane motion one. For each kinematic pair one must assign as many constraint conditions as there are units in the designation of the class of this pair; namely, for a pair of the 1st class one must assign one constraint condition, for a pair of the 2nd class, 2, and so on.

If variable links are included in the mechanism, then the number of constraint conditions is correspondingly reduced in that part which concerns the links. Depending on the degree of variability, either only 2 conditions are obtained in the link in spatial motion, or only one. In plane motion, obviously, we arrive at the complete elimination of the link from consideration if it proves to be variable. In fact, a rigid link in a plane has only one constraint condition, so that lowering this number is impossible without the link dropping out of consideration.

1. After the total count has been made, it is necessary to examine the particulars: whether any constraint conditions turn into identities, whether an excess of degrees of freedom is obtained under the given constraint conditions, and, if such an excess is obtained, where exactly it will appear in the mechanism and to what extent it may be considered undesirable.

2. Finally, it should be noted that if it were necessary to investigate more broadly the properties of a mechanism and to go beyond the limits of structural analysis alone, the method of constraint conditions could be very useful, since these conditions are, in essence, the equations by which the motion of the mechanism, the trajectories, velocities, and accelerations of its points are determined. With respect to the present work, these questions lie outside the tasks set here.

These examples, however, are interesting in another respect as well. From them it clearly follows that when the order of the kinematic pairs is rearranged, one obtains—

new mechanisms are obtained. In the counting of mechanisms given above, this was not taken into account, and thus one must conclude that the actual number of possible combinations of mechanisms must be still greater.

Let us take another example. Suppose a simple hoist is given (Fig. 51), consisting of a drum \(A\), a flexible connection \(BB\), and a fixed stand \(C\). This mechanism, being three-link with two moving links, has 18 coordinates to be considered. In its two links it is necessary to count 6 intrinsic conditions of their own, and consequently 11 conditions remain for the kinematic pairs. Five of them are present in the cylindrical pins of the pair \(AC\), 4 fall to the pair \(AB\) (the winding of the thread on the drum), and it remains necessary to obtain two more conditions of connection. If the load \(P\) dangles freely in the air, we shall not obtain the correct motion and shall be forced to refuse to call our construction a mechanism. If, however, the load \(P\), while tensioning the thread, always keeps it in a vertical position, then the point of its application moves, as it were, along a given trajectory, and we find precisely the two necessary conditions of connection in this trajectory. Here it is quite appropriate to recall the so-called trajectory pairs of Gohman.

In the kinematic pair \(AB\) we have assumed 4 conditions of connection, supposing that in the phenomenon of winding the thread on the drum (provided that the winding proceeds perfectly correctly and the thread is laid along a cylindrical helical line) we obtain the case of rolling of two curves over one another with four conditions of connection. Finally, in the flexible thread itself we have assumed 3 conditions of connection. This is because we suppose it to work only in tension. Otherwise the correctness of the motion would immediately be destroyed, and we would lose the mechanism.

If a variable link enters into the composition of a three-link mechanism, the number of conditions of connection in the kinematic pairs must be increased, as was stated in the preceding paragraph.

Suppose a spring hammer is given (Fig. 52), which we shall consider as a plane mechanism. The fixed frame of the hammer is denoted by the letter \(C\), the lever with the striker at its end by the letter \(B\), and the spring by the letter \(A\). In the two moving links of this hammer there are altogether 8 coordinates, and only one intrinsic condition of connection in the links, since the length of the spring changes. Therefore only 6 conditions of connection fall to the kinematic pairs, and it becomes possible to place cylindrical hinges in all three pairs.

15. Four-link mechanisms.

In four-link mechanisms of the spatial type, with one fixed link, only 27 coordinates are subject to investigation. If all the bodies are rigid, then there will be 9 intrinsic conditions of connection in the links and, thus, 17 conditions of connection remain for the kinematic pairs, which must be distributed among the four pairs.

This can be done in one of the following ways:

a) 3 kinematic pairs of the 5th class and one of the 2nd.

b) 2 pairs of the 5th class, one of the 4th, and one of the 3rd; and, finally,

c) one pair of the 5th class and 3 pairs of the 4th class.

Applying again the formal count of possible combinations, we find that for group a) a total of 3,876 mechanisms can be composed; for group b)—14,364; and, finally, for group c)—9,405, making 27,645 mechanisms in all.

If, in the kinematic pairs of a four-link mechanism formed from rigid links, when it moves in space there must be altogether 17 constraint conditions, then it is immediately clear that these pairs cannot be cylindrical joints, for they would have 20 constraint conditions. As examples of mechanisms of this group we shall give three. First, the sound-producing gramophone mechanism (Fig. 53), which is a representative of group a), since it has 3 pairs of the 5th class in the form of cylindrical joints and one pair of the 2nd class in the form of a needle sliding with its end along a prescribed trajectory. The count for this case gives, for 27 coordinates, 9 constraint conditions in the rigid links and 17 constraint conditions in the kinematic pairs. For the second example let us take the mechanism shown in Fig. 54. It consists of rigid rods coupled with one another at point \(A\) by a pair of type FNAR, at point \(B\)—KLAR, at point \(C\)—NNAR, and, finally, at point \(D\) again by a pair FNAR.

In the same way, for the 3rd example, the mechanism shown in Fig. 55 is taken. In it there are 3 pairs of type KLAR and one pair of type FNAR, by virtue of which this example can illustrate group c).

In plane four-link mechanisms formed from rigid links, we have in all, in the three moving links, 12 coordinates, while in these links themselves there are 3 intrinsic constraint conditions. Thus, for the 4 kinematic pairs there remain 8 constraint conditions. This problem has a single solution, namely that all the pairs must be of the 2nd class. But since there are only 4 types of such pairs, it is consequently possible to compose 256 types of mechanisms. The classical example of mechanisms of this group is the plane hinged quadrilateral, formed from rigid rods coupled with one another by means of cylindrical hinges. As another example one may take a belt drive. Here we have 2 pairs in the form of cylindrical hinges, and the two others in the form of pairs involving the winding of a thread, which, as was indicated above, also have, in plane motion, 2 constraint conditions. If, within a four-link mechanism, we have variable links, then, according to the preceding, we must correspondingly obtain other numbers of constraint conditions for the kinematic pairs. Thus, for example, a spatial four-link mechanism formed from rigid bodies coupled with one another by pairs of the 5th class cannot exist. In it, for 27 coordinates, there are 29 constraint conditions. In the mechanism shown in Fig. 56, with a plane arrangement, there will be 12 coordinates in three links and, at the same time, in the three links only one constraint condition, and in 5 pairs of the 2nd class 10, for a total of 11 conditions, which is what is required. With a spatial arrangement of this mechanism,

we would obtain 27 coordinates and 7 constraints in the links, owing to which, in five kinematic pairs, we would have to have 19 constraints. Only in this case will the mechanism operate correctly. It is true that, in the arrangement of the kinematic pairs \(A, B, C, D\), and \(E\), there may be such an order that some of the constraints may be reduced. This will be discussed in greater detail somewhat below (see p. 18).

16. Complex chains in four-link mechanisms.

Up to now we have considered only the so-called simple chains, i.e., those which are formed from simple kinematic pairs and simple links. Simple pairs are those in which only two links are joined, and simple links are those in which there are only two kinematic pairs. If, however, a link has more than two pairs, it is called complex and, correspondingly, if more than two links meet at one point, they form a complex pair. In schematic representations of mechanisms, especially plane ones, complex pairs may often be encountered. Thus, for example, in Fig. 40 a coupling is shown in which three links meet at point \(A\). On the other hand, in links it is also possible to encounter a number of pairs greater than only two. Already in the beam of ordinary scales with two pans there are 3 prisms, each of which is nothing other than a kinematic pair. In investigating two-link and three-link mechanisms we did not encounter complex pairs and complex links simply because they were impossible there. Indeed, there each link was already coupled with every other one, and therefore no further additional pairs could exist. But in four-link mechanisms such a possibility appears in view of the fact that four links, in pairwise couplings with one another, can give altogether 6 combinations, whereas in a simple four-link mechanism there are only four of them present. There remains, consequently, the possibility of an additional attachment of two more pairs. To clarify what has been said, let us take a simple hinged quadrilateral (Fig. 57), in which we shall denote the links by \(a, b, c\), and \(d\). The kinematic pairs in this quadrilateral will be \(ab, bc, cd\), and \(ad\). Meanwhile, there remains the possibility of two more pairs, namely \(ac\) and \(bd\). They are absent in the given mechanism.

In the investigation of complex pairs and links from the analytical point of view, everything remains as before. If a new pair appears as part of a link, it must bring with it the number of constraints belonging to it. If, at one point, the axes of rotation of two cylindrical joints meet, they do not turn into one, but still remain two, with the full 10 constraints in them. As an example of a four-link mechanism with a complex link, let us cite Amsler’s planimeter (Fig. 58). In the planimeter we note the lever \(AB\), the rod \(BE\), the roller \(D\), and the fixed stand with the drawing on it. We count the coordinates in the mechanism, as before, as 27; the constraints in the moving links are 9, so that 17 constraints remain for the kinematic pairs. Kinematic—

of the same pairs we note in mechanism 5, namely, 3 cylindrical hinge pairs at points $A$, $B$, and $C$, then one trajectory pair at point $E$, and, finally, a rolling pair at point $D$. Since the drawing is planar, the trajectory in the plane can be expressed by a single equation. Moreover, roller $K$ rolls over the paper with slipping, owing to which in this pair there is only one constraint condition. Thus, in the five pairs there are in fact 17 constraint conditions. In view of the unevenness of the surface of the drawing, both the trajectory and the rolling pair must be allowed the possibility of spatial displacements, and for this purpose the numbers of constraint conditions in the mechanism must be changed. In some constructions of planimeters, in order to provide precisely such spatial motions, hinge $B$ is made spherical. Then the two additional constraint conditions that appear, one in the trajectory of point $E$ and one in the rolling pair $D$, are compensated by a decrease of 2 constraint conditions in hinge $B$, and as a result a mechanism is still obtained.

17. Multi-link mechanisms.

In the history of our science, the question of simple multi-link chains formed of rigid bodies joined to one another by cylindrical hinges may be regarded as solved, and solved quite correctly. Namely, in space one must have 7 links connected successively with one another by cylindrical hinges, provided that these hinges are arranged arbitrarily. This question, from the standpoint of our method, can be clarified in the following way. Let us imagine a group of rigid bodies forming a simple chain. The number of links and the number of kinematic pairs are equal to one another and let them be denoted by $n$. Then, excluding one fixed link, we find that the total number of coordinates to be investigated will be $9(n-1)$. The number of constraint conditions in the links is $3(n-1)$ and, finally, the number of constraint conditions in the kinematic pairs, considering all of them as belonging to the 5th class, will be $5n$. Since the total number of constraint conditions in a mechanism must be one less than the total number of coordinates, we obtain the following formula:

$$ 3(n-1)+5n+1=9(n-1), $$

whence we obtain that indeed:

$$ n=7. $$

Investigating in a similar manner combinations composed of kinematic pairs of the 4th class, we would obtain:

$$ 3(n-1)+4n+1=9(n-1), $$

whence directly:

$$ 2n=7. $$

The result obtained in this case shows the impossibility of forming a mechanism from rigid bodies joined to one another by pairs of the 4th class, since the number of links turns out to be theoretically fractional.

It likewise turns out to be completely impossible to compose a mechanism of rigid bodies by means of pairs of the 3rd class, since the relation obtained is:

\[ 3n = 7. \]

In exactly the same way, a mechanism cannot be composed of other pairs either.
In plane mechanisms, under the assumption that all bodies are rigid and the pairs are of the 2nd class, we obtain the relation:

\[ n - 1 + 2n + 1 = 4(n - 1), \]

whence

\[ n = 4. \]

And if the mechanism is composed only of pairs of the 1st class, then we obtain:

\[ n - 1 + n + 1 = 4(n - 1), \]

whence:

\[ n = 2. \]

Both the one and the other cases are known in practice, since an example of the first is the ordinary hinged quadrilateral, while an example of the second may be the double-slider mechanism.

Clearly, many-link mechanisms cannot be obtained by this method.

The relations given here differ from the indications known in the literature in that they are a theoretical generalization. On the basis of these relations we speak not only of coupling by means of cylindrical hinges, as has been done up to now (see Willis, Reuleaux, Rittershaus, Somov, and others), but in general of coupling by means of pairs of the 5th class, then of the 4th, and so on.

In the general case, however, pairs of different classes may exist simultaneously in a mechanism, and therefore the relations given are insufficient. Moreover, we must admit the existence of both compound kinematic pairs and compound links. However, in counting the constraint conditions for compound pairs no peculiarity will arise, in view of the fact that we are obliged in any case to decompose such a pair into its component parts. As for compound links, here the question reduces only to ensuring that the additional kinematic pairs are all taken into account. Thus we arrive at a very simple general method of investigation. Let there be \(n\) rigid bodies in our mechanism. Discarding one fixed body, we obtain the total number of coordinates in the mechanism as \(9(n - 1)\). Taking up to 3 intrinsic conditions in each link, we find that altogether in the links there will be \(3(n - 1)\) constraint conditions. If we now denote by \(m_5\) the number of kinematic pairs of the 5th class, by \(m_4\) the number of pairs of the 4th class, and so on, then it is easy to write the following relation:

\[ 9(n - 1) = 3(n - 1) + 5m_5 + 4m_4 + 3m_3 + 2m_2 + m_1 + 1 \]

or:

\[ 6(n - 1) = 5m_5 + 4m_4 + 3m_3 + 2m_2 + m_1 + 1 \ldots\ldots (7) \]

To show how to use the formula given, let us take an example. Let there be given Amsler’s planimeter with supporting roller \(G\) (Fig. 59). In this mechanism there are in all 5 rigid links, of which one (the drawing table) we take as fixed. The remaining 4 links have altogether 36 coordinates. We see in the mechanism a total of 7 kinematic pairs, namely, at the pins \(F\) and \(C\)—pairs of the 5th class (cylindrical

hinges), at point \(A\)—a pair of the 4th class, at point \(B\) a ball-and-socket joint (of the 3rd class), at point \(E\) a trajectory pair of the 1st class, and at points \(D\) and \(G\) rolling pairs with slipping, both of the 1st class. Thus, by the formula we obtain:

\[ 6(5-1)=5\cdot 2+4\cdot 1+3\cdot 1+3\cdot 2+1;\ \text{or}\ 24=24. \]

The identity obtained proves the correctness of the structural composition of the mechanism.

In a completely similar way, for plane compound mechanisms one may obtain the following relation:

\[ \begin{gathered} 4(n-1)=n-1+2m_2+m_1+1\ \text{or}\\ 3(n-1)=2m_2+m_1+1 \ldots \tag{8} \end{gathered} \]

where \(n\) is the number of links in the mechanism,

\(m_2\) is the number of kinematic pairs of the 2nd class, and

\(m_1\) is the number of pairs of the 1st class.

Fig. 59-a represents Geisinger’s slider valve-distribution mechanism. It has in all 12 links and 16 kinematic pairs of the 2nd class

(at points \(A, B, C, F, G, H, L, M, D, N, P, Q\), and \(R\)—cylindrical hinges, and at points \(E, K\), and \(S\)—sliders).

Thus, by the formula we get:

\[ 3(12-1)=2\cdot 16+1, \]

which is quite correct.

Let us now turn to the solution of certain problems of a practical character and with formulations differing from those which have hitherto been given in the theoretical investigation.

Example 1. Is the slider \(E\) needed in the steering control of the Ollis system (Fig. 60)? To solve this problem we shall assume conditionally that the slider \(E\) is not needed and that point \(E\) is fixed. Then, temporarily discarding gear \(K\) from consideration, we find a total of 5 rigid links in plane motion, of which one is fixed, which gives a total of 16 coordinates. Since the mechanism has 6 pairs of the 2nd class (all hinges are cylindrical), there will be 12 constraint conditions in them. In addition, in the 4 movable links there are their own 4 constraint conditions, making a total of 16 conditions; whence we conclude that, with point \(E\) fixed, the whole system becomes immovable.

Let us now suppose that the slider is in the form of a special link. Then one link is added to the mechanism, the number of coordinates will already be 20, and the number of pairs of the 2nd class will also increase by one (the slider in the guide), so that in 7 pairs of the 2nd class there will already be 14 constraint conditions. Finally, in the 5 movable links there will be 5 constraint conditions, making a total of 19 conditions; whence we conclude that, with the presence of the slider, the system can move as a mechanism. If this system is considered together with gear \(G\), then we obtain one more link, whereby two kinematic pairs are added—one in the form of a cylindrical hinge (the axis of rotation of the gear), and the other a rolling pair between the teeth, belonging to the 1st class. The calculation will be as follows. In the 6 movable links there will be 24 coordinates; in them there are themselves...

of its own 6 constraint conditions; then in 8 pairs of the 2nd class there are 16 constraint conditions, and in one pair of the 1st class one condition, making a total of 23 conditions, which proves the possibility of mechanical motion. In view of the fact that, with the motion of the slider, the distance between the centers of the gear wheel and the pinion changes, it is necessary either to make the wheel spiral, or to connect the pinion with the slider $E$, forcing it also to undergo translational motions.

Example 2. Can the stone $K$ in Stephenson’s slotted valve gear mechanism fail to have play during motion? (Fig. 61). This problem may be understood in the following way: if the stone does not play during motion, then it is a simple cylindrical joint, i.e., it forms with the slot a pair of the 2nd class; but if it has play, then, besides rotation, it permits sliding along the slot and, therefore, must form a pair of the 1st class. To solve the problem, let us suppose that the stone does not slide and is taken as a pair of the 2nd class. Then in the mechanism we discern altogether 6 moving bodies, giving 24 coordinates and 6 intrinsic constraint conditions. In addition, in 9 pairs of the 2nd class there are 18 constraint conditions (in the pair $E$—a double joint), and thus an immovable system is obtained. Allowing play of the stone lowers the number of constraint conditions by one, thanks to which mechanical motion becomes possible.

Example 3. Is it correct that, in an automobile steering mechanism (Fig. 62), ball pins are placed at the points $B$ and $C$?

Since the mechanism is four-link and spatial, in its three moving links there are altogether 27 coordinates. Meanwhile, in the same links there are 9 intrinsic constraint conditions; furthermore, in two cylindrical joints there are 10 conditions, and in two spherical ones—6; thus a total of 25 constraint conditions is obtained, i.e., the system has one superfluous degree of freedom. Indeed, this freedom is easily seen in the fact that the rod $BC$ can rotate about its own geometrical axis. From the practical point of view, this superfluous degree of freedom threatens nothing and does not harm the principal motion of the mechanism. From the theoretical point of view, however, one of the ball pins at $B$ or $C$ should be replaced by a pair of the 4th class.

Example 4. Why, in the machine of the Wiggell system, are the links $M_1N_1$ and $M_2N_2$ placed at the two outer cylinders, while at the middle cylinder there is no such link? (Fig. 63). This machine operates as follows. Three cylinders are arranged parallel to one another, and the pistons are rigidly connected with the rods, so that the points $M_1$, $M_2$, and $M_3$ describe rectilinear trajectories. The role of the connecting rod is performed by the link $K$, with which all the rods are coupled, the outer ones by means of intermediate links, and the middle one directly. Finally, $AB$ is the crank. Thus, the peculiarity of this machine consists in the fact that all three cylinders act upon one crank.

In this machine there are 7 moving links and, consequently, there will be 28 coordinates; there are 7 intrinsic constraint conditions in the links, and in 10 pairs

of the 2nd class, 20, and in all this gives 27 constraint conditions. Such a ratio ensures correct mechanical motion. If a driver were also introduced into the middle mechanism, then we would obtain one additional link, on the one hand, and one pair of the 2nd class, on the other. In the system there would be a total of 32 coordinates, and constraint conditions—in the links 8 and in the kinematic pairs 22, in all 30. Thus, with the presence of a third driver in the machine, there would be two degrees of freedom, and we would not obtain mechanical motion.

Example 5. In simple hand beaters for beating lattices, a spatial four-link mechanism is used (Fig. 64), in which the link \(AB\) rotates about a vertical axis, the link \(BC\) is a connecting rod, and the link \(CD\) rotates about a horizontal axis. Disregarding the driver \(MN\), which couples the mechanism under consideration with the lattices, we find that, theoretically, in the mechanism, for three moving links and their 27 coordinates, there should be (with 9 constraint conditions in the links themselves) a total of 17 constraint conditions in the kinematic pairs. Hence it is clear that all 4 pairs cannot under any circumstances be cylindrical hinges. If two of them are left in the form of such hinges, then the third must be a pair of the 4th class, and the fourth a pair of the 3rd class. In practice, the execution of kinematic pairs of the 4th and 3rd classes is difficult, and they are replaced by simple cylindrical pins placed in loose holes, loose enough that the pin can rotate about all three coordinate axes. Dead travel and impacts result. In a noiseless beater it would be necessary to install one spherical hinge (3rd class), and the other, for example, of the KLAR type (4th class).

Example 6. Are springs needed on the valve spindles of the steam-distribution mechanism of Radovanovich (Fig. 65)?

Investigating the mechanism from the point of view of its structure, we find that it has 9 moving links and 12 kinematic pairs of the 2nd class, apart from rolling pairs along the surfaces \(S_1\) and \(S_2\). Ignoring these surfaces, we would obtain the following count:

Number of coordinates in the moving links: 36.

Number of constraint conditions in the links: 9, and in the kinematic pairs: 24; in all: 33.

Thus it is seen that, for the correctness of the motion of the mechanism, it is necessary to introduce two more constraint conditions, which are precisely the rolling pairs along the surfaces \(S_1\) and \(S_2\); moreover, on the one hand it is seen that here rolling will occur with slipping, since the pair must belong to the 1st class, and on the other hand it is clear that the presence of this pair is obligatory and must be guaranteed constructively. Hence we conclude that springs on the spindles are necessary; otherwise the rods \(GH\) and \(KI\), having moved away from their surfaces, will lose the kinematic correctness of their motion.

18. Special Cases of the Arrangement of Kinematic Pairs and Links.

In the motion of mechanisms, various special cases are possible, when the quantitative relation between the numbers of coordinates and constraint conditions changes, as a result of which the mechanism itself may turn either into an immovable system or into a system with several degrees of freedom. All these special cases may be divided into two principal classes: to the first will belong those cases in which the change in the quantitative relation between the numbers of coordinates and constraint conditions is constant for the given structure, and to the second those in which such a change occurs during the motion and only for certain configurations of the mechanism. From the analytical point of view, the occurrence of special cases can easily be explained by the fact that, during the motion of a mechanism, coincidences of all sorts are entirely possible among the quantities by which the motion is determined. Thus, for example, two different surfaces are expressed by two different equations, but when these surfaces coincide only one equation is obtained. In the same way, lines and points may coincide. Since, when coincidences occur, both coordinates and constraint conditions may vanish, there arises the possibility of a large number of special cases, of which it is sufficient here to examine only the most important.

a) First case. Coincidence of cylindrical joints in direction.

Let there be given two bodies \(A\) and \(B\) (Fig. 66), coupled with one another by two cylindrical joints \(M_1N_1\) and \(M_2N_2\). Applying the general method of investigation, we write for this case the number of coordinates in the two bodies as 18. Taking one of them as fixed, we shall find in the other link 9 coordinates.

The number of constraint conditions in the links is 3,
    “      “      “      in the pairs 10, in all 13.
Thus in the given system there are 4 redundant constraint conditions.

If we now suppose that both joints form continuations of one another, then in fact of the two joints one remains, while the other disappears. Then the enumeration assumes the following form:

    number of coordinates      9,
    “      constraints in the link 3,
    “      constraints in the pair 5, in all 8, i.e. as a result the system is transformed into a mechanism.

The same result is obtained if we take a larger number of joints. Let, for example (Fig. 67), body \(A\) be fastened to body \(B\) by three joints. With an arbitrary arrangement of these joints we obtain the following calculation:

    coordinates 9,
    constraints in the link 3 and in the pairs 15, and in all 18.

If, however, all three joints are extended into one straight line, then two joints disappear and again only 8 constraint conditions are obtained in all.

In the case where the coupling of several links is given, the calculation will be somewhat different. Indeed, let us imagine a plane mechanism (Fig. 68) in which points \(A\) and \(D\) are fixed. The count for this mechanism, under the condition that all pairs are cylindrical hinges (for plane motion—of the 2nd class), will be as follows:

coordinates of the moving links: 12,
constraint conditions in the links: 3, and in the pairs: 8, for a total of 11.

Now let us suppose that both hinges \(A\) and \(D\) coincide. In that case the following occurs. First, of the two hinges one remains, whereby two constraint conditions disappear; and, second, the two coordinates of point \(D\) disappear, and as a result one still obtains a certain new mechanism (Fig. 69).

In the four-link mechanism, however, in its spatial arrangement we would obtain the following picture. By the general method, as is known, for 27 coordinates there are altogether 29 constraint conditions. But if the hinges \(M_1N_1\) and \(M_2N_2\) are continuations of one another, the count must be changed. Indeed, we shall then obtain: coordinates still 27, and constraint conditions—in the links 9, then in three pairs as before 15, and in the fourth pair two conditions of a straight line in space. Thus the total turns out to be 26 conditions.

b) Case 2. Parallelism or coincidence of trajectories.

Let us take two links (Fig. 70), with the connection between them being effected by means of two pairs of translational motion, directed at some angle to one another. In this case the general count gives:

Number of coordinates for one moving link: 9.
Number of constraint conditions: in the link 3 and in the two pairs 10, for a total of 13.
Thus the connection is immovable, with four redundant constraints.

If, now, the guides of both sliders are arranged parallel to one another (Fig. 71), then one of the pairs may be disregarded altogether, since without it all points of the slider will move parallel to one another, and with it they will move in the same way. Consequently, it turns into an identity. As a result, the count will be as follows: coordinates again 9, and constraint conditions 3 and 5, for a total of 8, whereby the system will acquire motion with one degree of freedom.

Let us examine another case of coupling three links by means of translational pairs. Let a chain (Fig. 72) of three bodies \(A\), \(B\), and \(C\) be given, forming three pairs: \(AB\), \(BC\), and \(AC\). Counting by the general method would lead us to a certain error, since there are coincidences here. To study this case, as in general, it may be recommended to open the chain and see what results. On this basis let us assume that link \(A\) remains fixed; link \(B\) is coupled with it by means of pair \(AB\), and link \(C\) is also coupled with link \(B\), but there is no connection \(AC\) at all. Then the count will indicate to us that link \(C\) has two degrees of freedom, which is quite correct for the open chain. Now, obviously, if we close \(C\) and \(A\) by a pair of the 1st class, then we shall obtain a me-

mechanism. Thus, by letting the point \(P\) move in plane motion along a certain rectilinear trajectory (Fig. 73), we do indeed ensure the correct displacement of the entire system. Moreover, however, it is not difficult to prove that with such an arrangement all points of link \(C\) will move along lines parallel to the trajectory of point \(P\), exactly as though there existed a translational pair \(AC\). It should now be clear that the actual installation of such a pair is not compelled by the circumstances; and if precisely such a pair is installed, it does not bring with it the full number of constraint conditions proper to it, since in plane motion it has not two but only one constraint condition.

c) Third case. Convergence of the axes of rotation at one point.

This case is of special interest because several outstanding scholars have dealt with it. Thus, for example, Willis, Rittershaus, Reuleaux, and others, by theoretical considerations and by the construction of models, attempted to decide what significance the convergence of hinges at one point has, and to a certain extent achieved results. Nevertheless, this question still lacks complete generality and clarity. From the point of view of the method of constraint conditions, the case noted here can be illuminated. First of all, for convenience of exposition, let us consider open chains. Let (Fig. 74) an open chain be given, composed of rigid links coupled with one another by means of cylindrical hinges, these hinges being arranged arbitrarily in space. If the total number of links is \(n\), and one of them is fixed, then we obtain

coordinates \(9(n-1)\),

constraint conditions in the links \(3(n-1)\), and in the pairs \(5(n-1)\),

i.e., in all \(8(n-1)\).

Thus, in the system of the given open chain there prove to be in all \(n-1\) degrees of freedom.

Here it is very important to note that however the cylindrical hinges may be arranged, the result will be one and the same. It is also obtained in the special case when all the hinges converge at one point.

But as soon as we wish to pass from an open chain to a closed one and attach the last link to the fixed base, a difference between the two cases is immediately revealed. Namely, if the hinges are arranged arbitrarily in space, then the new hinge will bring with it the usual 5 constraint conditions, and we shall obtain in the end not \(n-1\) degrees of freedom, as was just found above, but only \(n-6\). Hence, among other things, it follows directly that if \(n=7\), then a system with one degree of freedom is obtained, which can be a mechanism.

But if all the hinges converge at one point, as shown in Fig. 75, then the newly attached hinge introduces fewer constraint conditions. Indeed, in this hinge one point has already been attached to the fixed base, and this point is precisely the common point. Thus, in order to determine the position of the hinge it is enough to attach one more point to the same fixed stand, and this introduces

only 2 new constraints if the link is a rigid body. Hence it follows that the last joint, in the case under consideration, introduces only 2 constraints instead of five, i.e. 3 constraints are lost because the joints intersect at one point.

In view of the special interest of the case analyzed here, let us trace, by means of several examples, the conclusion just drawn.

Let two bodies be given, coupled to one another by cylindrical joints, and let these joints intersect at one point.

A calculation by the general rule would give the following result. The coordinates of one link (with the exception of the other, taken as fixed) are in all 9; the constraints for the movable link are 3 and for two pairs of the 5th class are 10, making 13 in all. Thus we would have to recognize that the system has, as it were, 4 redundant constraints. Since, however, the cylindrical joints intersect in their extensions, 3 constraints must be discarded, and in the end there remain only 10 constraints instead of 13; consequently, in the given system there is only one redundant constraint.

Of course, from the point of view of analysis, it is entirely immaterial whether the joints converge at some actually existing point or only at some geometrical intersection in the prolongation of the directions of the two joints.

If we were now to take 3 links, then for them, according to what has been set out above, we would obtain:

coordinates in two links: 18,
constraints in the links: 6, and in three pairs: 15, making 21 in all.
Discarding 3 constraints, we again find 18 constraints.

That is, in the case where the cylindrical joints of three coupled bodies meet at one point, we obtain a statically determinate immovable system.

Further, with four links, all coupled likewise by cylindrical joints meeting at one point, we obtain:

coordinates in all: 27,
constraints: 26, i.e. a system that may be a mechanism.
With five links we would find:
coordinates: 36,
constraints: \(37 - 3 = 34\), i.e. in the given system there would be two degrees of freedom.

If an open chain is attached to a fixed base by a cylindrical joint of arbitrary direction, then no reduction of constraints occurs. Fig. 76 shows a chain in which there are, on average, three cylindrical joints meeting at one point; however, since this chain is attached to the fixed base by such joints which have different nonintersecting directions, no reduction of constraints occurs.

c) Fourth case. Cylindrical joints are parallel to one another.

Since the parallelism of cylindrical joints may be regarded as the limiting case of their intersection at a single point, when this point recedes to infinity, all the arguments of the preceding item are fully valid here as well. Just as with intersecting joints, in the case of their parallelism three constraint conditions are lost, provided that the joint by which the open chain is fastened to the fixed support is likewise parallel to the others. Otherwise no reduction in the constraint conditions will follow.

Let a four-link mechanism be given, consisting of rigid links coupled with one another by means of cylindrical joints. The calculation for this case has been made more than once. There are 27 coordinates, and 29 constraint conditions. If, however, in this mechanism all the joints are parallel to one another, then three constraint conditions drop out, and a properly moving mechanism is obtained.

To the question how many degrees of freedom a closed simple chain formed of 10 rigid bodies, coupled with one another by means of parallel cylindrical joints, has, the answer will be as follows:
there are 81 coordinates in the chain,
27 constraint conditions in the links, and 50 in the pairs, for a total of 77. Discarding 3 more conditions, we find 7 degrees of freedom.

d) Fifth case. The elements of motion in the kinematic pairs of one and the same link coincide.

If a mechanism includes a link that is fastened by several pairs to other links, and in all these pairs there is one and the same element of motion, then as many constraint conditions fall away as there are pairs in the link under consideration minus one. This circumstance is explained by the fact that the constraint conditions limiting the motion turn into identities.

e) Sixth case. Superposition of symmetrical parts of a mechanism upon one another.

During the motion of a mechanism it is entirely possible that its symmetrically arranged parts, at some instant, will be superposed upon one another in such a way as to coincide completely. As a result of such superposition, an entirely different relation may arise between the numbers of coordinates and constraint conditions, since under symmetrical folding a part of the mechanism is, as it were, excluded from action, and what remains will already have a reduced number of coordinates and constraint conditions. For the investigation of such cases it is best to carry out a new count after coincidence. Suppose, for example, that the Lanchester–Peaucellier mechanism is given (Fig. 77). It has 7 moving links, and since the mechanism itself is planar, 28 coordinates must enter into the investigation. In this mechanism all pairs are formed from cylindrical joints, owing to which the number of constraint conditions

there will be: in links 7, and in 10 pairs 20, hence 27 in all. From this we conclude that the given chain will move quite correctly in the plane.

(Note. In counting the number of kinematic pairs in this mechanism, it must be borne in mind that at the points \(A\), \(B\), \(C\), and \(E\) three rods each meet, and therefore form, at each of these points, not one pair but two pairs.)

Let us now note that, in the mechanism under consideration, some rods are equal to one another in length and are arranged symmetrically. Namely:

\[ AB = AC,\qquad BE = EC,\qquad BF = FC. \]

Thus, if during the motion the point \(B\) coincides with the point \(C\), then all the rods just indicated will coincide, and we shall obtain the chain represented in Fig. 78. But in this chain there are already only 4 moving links, giving 16 coordinates; the conditions of constraint in it will be: in links 5 and in five pairs 10, hence 14 in all. It follows from this that in the new chain there are two degrees of freedom, which is indeed the case in the mechanism under consideration, since the coincidence of the points \(B\) and \(C\) in it is possible.

The same picture is obtained also in ordinary hinged plane quadrilaterals when the dimensions of the rods have been chosen symmetrically. Let, for example, Fig. 79 give such a mechanism, in which

\[ AB = AD \quad \text{and} \quad BC = CD. \]

Having stretched out into one straight line, this mechanism may turn into a new one; namely, the link \(AB\) will remain fixed, while \(BC\), having folded together with the link \(CD\), will move as a simple crank.

19. Corrected Form of the Mechanism Formulae

On the basis of what was set forth in the preceding paragraph, it is evident that corrections must be introduced into the formulae of the mechanism given in § 17 for those special cases which may occur in a mechanism. At the same time, let us also draw attention to two further circumstances not provided for by those formulae. First of all, in mechanisms there may be not only rigid links, but also variable links, which must somehow be reflected in the formula. Next, in the formulae named it was assumed that the given conditions of constraint must necessarily provide one degree of freedom in the mechanism; but this is by no means obligatory, since the designer may create schemes with a larger number of degrees. Taking the circumstances noted into account, let us introduce 3 corrections into our formulae; then they will take the following form.

a) For spatial motion:

\[ 6(n-1)=5m_5+4m_4+3m_3+2m_2+m_1-k-p+q. \tag{9} \]

b) For plane motion:

\[ 3(n-1)=2m_2+m_1-k-p+q, \tag{10} \]

where: \(k\) denotes the number of conditions of constraint that drop out as a result of special cases, according to the indications of the preceding paragraph;

\(p\) shows how many constraint conditions must be introduced into the variable links in order for them to become solid; finally, \(q\) shows the number of degrees of freedom of the mechanism. The meaning of the remaining quantities is the same as in § 17.

20. Verification of the schemes of mechanisms proposed by F. Reuleaux in his book: Theoretische Kinematik. (Vol. I. 1875, pp. 555 and 494).

Scheme 1. Fig. 388. (fig. 80). Considering one link in the given chain as fixed, we obtain for the four remaining links a total of 36 coordinates. The constraint conditions will be: in the links, 12; then in four pairs of the 5th class, 20; and, finally, in one pair of the 4th class, 4; altogether 36. Thus we conclude that the chain will be immobile, unless there are special conditions. To verify this, let us open the chain at joint 3 and take link \(b\) as fixed. Then link \(c\) will have 5 degrees of freedom. With an arbitrary position of joint 3 in space, attaching the chain by it to the fixed base will introduce fully 5 constraint conditions, and thus all degrees of freedom will be absorbed.

Scheme 2. Fig. 389 (fig. 81). A six-link chain, for which, as before, one may count 45 coordinates. Constraint conditions: in the links, 15; in 4 pairs of the 5th class, 20; and in two pairs of the 4th class, 8; altogether 43 conditions. On this basis we conclude that in the given scheme there are clearly 2 degrees of freedom. Opening the chain at joint 1 and considering link \(a\) fixed, we find that link \(f\) has seven degrees of freedom. However we may position joint 1 when attaching the chain to the fixed base, we cannot introduce more than five constraint conditions, and therefore the chain cannot have fewer than two degrees of freedom. This is true for both schemes of figure 389.

Scheme 3. Fig. 390. (fig. 82). In three moving bodies there are 27 coordinates. Constraint conditions: in the links, 9; in two pairs of the 5th class, 10; and in two pairs of the 4th class, 8; altogether 27. Opening the chain at joint 6, we obtain 5 degrees of freedom for link \(e\). Now it depends on the manner of attachment how many constraint conditions joint 6 introduces. Indeed, if it is attached as shown in fig. 83, we obtain five constraint conditions in the joint, since not a single element of motion coincides. In this case the specified system proves to be immobile. If, however, we now suppose that pair 6 (fig. 82) permits not only rotation but also translation along the axis, then it will already belong to the 4th class; on the one hand, and on the other, link \(e\), in both pairs by which it is coupled with its neighbors, has parallel displacements, owing to which this link will have two degrees of freedom. If, instead of joint 6, we put a pair of the 4th class, as shown in fig. 84, then we obtain a mechanism with one degree of freedom.

A similar result would be obtained if we constructed a combination such as that shown in fig. 85, where pair 6 likewise has 4 constraint conditions and, in addition to rotation, also permits a longitudinal displacement of link \(e\).

Here the group of links $c$ and $e$ jointly proves, with respect to its neighbors, to be seized by pairs in which there is one and the same element of motion. It is precisely this whole group that can move horizontally with respect to the remaining part of the chain, owing to which one constraint condition drops out. The chain will again have two degrees of freedom. Transforming pair 6 into a simple cylindrical joint will enable the chain to move as a mechanism.

On the basis of what has been set forth, the scheme given by Reuleaux in Fig. 390 must be recognized as correct, and, at the same time, the modifications of this scheme shown in Robertson’s mechanisms in Figs. 391 and 392 are likewise correct. In the first of these mechanisms only 4 constraint conditions are left in pair 6, but one condition in the cross has been eliminated; in the second, pair 4 must have 5 constraint conditions, i.e., be a prismatic slide; otherwise the mechanism will not operate.

Scheme 4. Fig. 393. (fig. 86). Much to our regret, Reuleaux’s reference to Engineering proved incorrect, since on p. 158 for the year 1867 Brown’s mechanism was not to be found. Meanwhile it would be very important to see the original drawing of the mechanism, since, evidently, Reuleaux represented it incorrectly in his book. This may also be judged from the fact that at the end of his book, in note 53, Reuleaux indicates that this mechanism was built by him and appeared at the exhibition. The count gives the following. The coordinates of the four moving links are 36. The constraint conditions of the links are 12. In addition, the author indicates that pair $de$ is a spherical joint, i.e., has 3 constraint conditions. It is now clear that, for the correctness of the motion of the mechanism, in the remaining pairs there must be only $35 - 12 - 3 = 20$ constraint conditions. And since four of these pairs remain, they must all be representatives of the 5th class. Yet this is not visible in the scheme. It is as though the pairs $af$ and $ab$ permit not only rotation about the axis of the cylinder, but also translation along this axis. The scheme has been composed unclearly. It must be assumed that both doubtful pairs are simple cylindrical joints, and then the count will give a satisfactory result. The scheme should operate.

Scheme 5. Fig. 394. (fig. 87). This scheme is a reworking of the preceding one. In it, too, the arrangement is unclear. From the drawing no difference is visible between the pairs $af$, $ab$, and $cd$. But, if we count them as identical, we inevitably fall into error. In fact, let all these pairs belong to the 5th class. Then for 45 coordinates we have conditions: in the links, 15, and in six pairs of the 5th class, 30, altogether 45, which indicates the immobility of the system. Evidently, one of these pairs must belong to the 4th class. Most probably such a pair is the pair $cd$, since this mechanism is obtained from the preceding one, where the pairs $af$ and $ab$, by our admission, had to belong to the 5th class.

With respect to both schemes given in Figs. 393 and 394, one further supposition is possible. Namely, under the action of the forces of gravity, the links,

although they would have an excessive degree of freedom, will always descend to the lowest position, thanks to which it is also possible to introduce pairs of the 4th class, if the mechanisms are mounted vertically. Then, if the pairs \(af\) and \(ab\) are of the 4th class, the parts \(abce\) in the first mechanism and \(abc\) in the second can simply be taken out owing to the parallelism of the elements of motion. In this one sees one excessive degree of freedom. However, under the action of gravity a new constraint is introduced, compelling the system to assume the lowest position and to remain in it. Owing to this, the excessive degree of freedom cannot manifest itself. The same may be said of the pair \(de\). In one way or another, both schemes require fundamental clarification, because from the drawings one cannot judge the actual properties of the mechanisms.

Reuleaux’s further reasoning concerning simple polygonal chains formed by means of cylindrical pairs must be recognized as correct from the point of view of the constraint conditions. However, the scheme given in Fig. 400 (fig. 88) again raises doubts. Here a four-link mechanism is shown, in which all the pairs are translational, in the form of sliders. Since all the pairs are projected onto the plane of the drawing in their natural size, the mechanism must be recognized as plane. The general count for this case gives: 12 coordinates, 3 constraint conditions in the links and 8 in four pairs of the 2nd class, making in all 11 conditions, which is quite satisfactory. However, a more attentive investigation of this scheme leads to a different result, since here one constraint condition falls out. Indeed, let us open the chain at the pair \(cd\) and regard link \(c\) as fixed. Then we shall determine without difficulty that link \(b\) has one degree of freedom, link \(a\)—two, and link \(d\)—three degrees of freedom. The peculiarity of the motion consists in the fact that all the links move only translationally and there is not a single rotational motion; whence follows the parallelism of the trajectories of all points of each link. Now, fastening link \(d\) to link \(c\), one can prescribe only one trajectory for some point, and all the other points of link \(d\) will describe trajectories parallel to the prescribed one. Thus it follows that the introduction of the fourth slider is not necessitated by anything—here it is sufficient to have a pair of the 1st class, and not of the 2nd; and on this basis one may conclude that in the fourth slider there is an identical constraint, which in the count must be discarded. In such a case, for 12 coordinates in the mechanism there remain 10 constraint conditions, i.e. two degrees of freedom are indicated in the chain. Reuleaux makes a remark on this scheme that its application in practice is unknown to him, but that its correctness is obvious, owing to the fact that this scheme follows logically from the scheme of a hinged quadrilateral.

Without objecting to the application of such an interpretation of the translational pair, it must nevertheless be noted that the passage to the limit requires some discussion. What, for example, will result from the correct scheme given by Reuleaux immediately here on p. 560 under No. 398, if the same method of passage to the limit is applied to it? The mechanism will obviously cease

act, since it is transformed into a three-link articulated chain.

In fact, the opening of the chain shown in Fig. 400 has shown that link \(d\) has only translational displacements in the plane of the drawing, owing to which, with one new condition in the form of the trajectory of some point of this link, its correct motion is already obtained. Hence it follows that, for the fourth center (hinge) at infinity, likewise not two but only one constraint condition is sufficient.

Chapter IV. Synthesis of the Mechanism.

21. Historical Note.

The origin of the synthesis of mechanisms is lost in the most remote antiquity. It would be no exaggeration to say that the synthesis of mechanisms is as old as mankind. Indeed, it is even difficult to indicate when the first devices appeared for obtaining fire, catching animals, fish, and the like. More definite indications, based on literary materials, allow the following to be noted:

a) In ancient Egypt there were known various kinds of mechanical tools, water-lifting devices, vehicles, weaving and spinning appliances, and the like. (See E. Meyer, Geschichte des alten Aegyptens.)

b) In the time of Christ, Marcus Vitruvius Pollio, in his major work De architectura, left a whole series of sketches of pulleys, wheels, drums, a lifting waterwheel, and so forth, and, in addition, various practical combinations of these elements in the form of lifting devices, etc.

c) In the Middle Ages, Leonardo da Vinci, Georgius Agricola, Hieronymus Cardanus, Agostino Ramelli, and many others left in their notes a mass of material on the synthesis of mechanisms and a whole series of combined machines. Thus, for example, in Leonardo da Vinci we find beautifully executed sketches of ratchet, gear, and screw wheels, chains, cranks, and the like, as well as combinations of these elements in the form of machines: a lathe, a drawing machine, a machine for cutting files, a drive saw, and others.

(See Th. Beck. Beiträge zur Geschichte des Maschinenbaues).

d) The examples cited characterize what is called practical synthesis, when the scheme of a mechanism is created without any theoretical or mathematical argumentation. In the history of science another kind of synthesis is also known, a brilliant example of which may be the works of Prof. P. L. Chebyshev in constructing approximate straight-line mechanisms. Prof. Chebyshev found the dimensions of a mechanism by mathematically studying the deviation of the actual trajectory of a point from a straight line and determining the conditions under which these deviations would be the smallest.

e) All the examples given above, however interesting they may be, cannot serve as a guide to the synthetic construction of diagrams of new mechanisms. These examples merely show how one or another designer solved one or another problem, but from these examples it is not clear how a designer should act in general. F. Reuleaux came closer to the matter. In his work Theoretische Kinematik (1875 and 1900) he performs a twofold task for building a synthesis of mechanisms. First, instead of random sketches of individual elements of mechanisms, he gives a thorough systematic doctrine of the component parts of a mechanism, which, of course, considerably facilitates the synthetic activity of any designer. Second, he gives an entire special chapter on kinematic synthesis, where he describes its various kinds. Unfortunately, in this part Reuleaux’s work is limited to describing the possible kinds of synthesis and gives no indications whatever for the actual construction of diagrams of mechanisms. Moreover, as soon as Reuleaux himself tried to draw up diagrams of new mechanisms, he immediately became entangled and gave a whole series of incorrect combinations, as was already discussed above.

f) The synthesis of Gochman is much more detailed (Kinematics of Machines, 1890); he proposes special methods for forming kinematic pairs and methods for forming mechanisms. In the formation of kinematic pairs, Gochman sees four methods: 1) the method of axoids, 2) the method of envelope surfaces, 3) the method of trajectories, and 4) the method of the generating line. In addition, four derivative methods: 1) the method of machines, 2) the method of tangent planes, 3) the method of equidistants, and 4) the method of circular arcs. In the formation of diagrams of mechanisms he outlines:

1) the principle of closing pairs, 2) the principle of undetermined coefficients, 3) the principle of successive transformation of motion, and 4) the principle of decomposition of motion. Gochman’s methods bring the designer fairly close to the essence of the matter, especially in constructing kinematic pairs. But even here it remains unclear whether the author considered his methods exhaustive or only incidental. For the formation of simple kinematic pairs, the methods he enumerates may seemingly be recognized as exhaustive, but in complex pairs difficulty immediately arises. Let us take the pair SNEP. In it two points are constrained to remain on two given surfaces, and in addition there are three rolling conditions. To form such a pair one will probably have somehow to combine the method of envelope surfaces with the method of axoids; but how is this to be done? Moreover, Gochman’s exposition lacks precisely that practicality which is needed in this question. As for his principles of constructing new diagrams of mechanisms, they are still predominantly descriptive in character. True, Gochman does give the designer something with his principles, but the very fact that he proposes many methods and principles shows that he has no single, direct method, but only indications that can help in one or another methodological activity.

h) In an entirely different direction, and moreover very successfully, another line of work proceeded. A whole series of authors (Ampère, Sylvester, Burmester, Grübler, Taubeles, Assur and others) dealt with the question of how, from a given mechanism, by attaching additional parts to it, one may obtain other, more complex mechanisms. Without repeating here what is stated in Assur’s work, we shall only note that at the present time the following may be regarded as established:

1) A plane bar, pin-jointed, statically determinate system, deprived of one bar, is transformed into a mechanism; and, conversely, a plane bar, pin-jointed mechanism, strengthened by one bar, is transformed into a statically determinate immobile system.

2) In plane mechanisms, Sylvester’s dyad (Fig. 37), the three-lead link (Fig. 38), the four-lead group (Fig. 39), and groups formed by the method of developing a lead can be attached to a given mechanism; as a result, certain new, more complex mechanisms are obtained; and, conversely, the groups named here can be removed from complex mechanisms, as a result of which certain simpler mechanisms are obtained.

3) Systems with excess degrees of freedom can, by the introduction of additional constraints, be transformed into mechanisms; and likewise immobile systems with excess constraints can, by the removal of constraints, be transformed into mechanisms. (Here it is appropriate to recall the principle of possible displacements, as well as Prof. Zhukovsky’s method of the auxiliary lever.)

Unfortunately, it must be noted that the methods listed here concern exclusively plane mechanisms; spatial mechanisms are not discussed in them. Moreover, even among plane mechanisms, it is primarily bar, pin-jointed mechanisms that have been studied. True, at the present time in practical engineering there is a whole series of steam-distribution mechanisms of precisely this type, owing to which their investigation must be considered very important. However, the problem of investigating mechanisms as posed by the group of authors noted here is nevertheless too narrow. In fact, if some designer wishes to include a pair of toothed wheels in the kinematic chain of a steam-distribution mechanism, he will have to construct a new method of investigation. In addition, it must be noted that even in plane bar, pin-jointed mechanisms there remain ambiguities. Assur’s most complete investigation gives the impression of a well-developed particular case, developed methodically and possessing a certain practical significance, but not encompassing all combinations in mechanisms of this kind.

(See Ampère. Journ. de l’Ecole Polytechnique. Cah. XIII. T. VI. 1806.
Liguine. Sur les système de tiges articulées. Nouv. annal. de mathèm. 1875.
Burmester. Lehrbuch der Kinematik.)

Grübler. Allgemeine Eigenschaften der zwangläufig. ebenen kinematischen Ketten. Civiling. B. 29.

Taubeles. Ueber die Bildung eben. kinemat. Ketten. Techn. Blätter. Prag. 1887.

Assur. Study of plane bar mechanisms with lower pairs. Publ. of the St. Petersburg Polytechnic Institute. 1914.)

Indeed, almost exclusively mechanisms with lower pairs have been subjected to study. Moreover, many groups that can be joined to a mechanism or removed from it have not been investigated at all.

In any case it must be acknowledged that, by means of the methods noted here, one can obtain as many new schemes as desired of plane pin-jointed bar mechanisms formed from rigid links by coupling them by means of so-called lower pairs. In this lies the enormous merit of the authors listed here.

g) As for the recipe of S. Dunkerley—in his book: Mechanism (1910. See p. 14 and the preceding pages)—it is completely undeveloped and raises many doubts. This recipe is built on the use of the well-known formula for plane bar pin-jointed mechanisms and consists in counting the number of bars and joints. But the very question of the order in which the counting is to be carried out introduces confusion. On p. 14 the author gives a sample table illustrating the method he recommends. But if, in this very example, we merely interchange the order of the bars \(C\) and \(D\), we at once obtain an erroneous result, not coinciding with the truth. Namely:

\[ \begin{array}{cccc} D & d & h & 1 \\ C & f & g & \boxed{h} & 1 \end{array} \]

leads to an error.

i) An entirely exceptional fortune fell to the lot of toothed wheels. As early as 1694 De La Hire, then Camus in 1733–1752, Euler in 1754, gave a fundamentally substantiated method for constructing the teeth of cycloidal and involute gear wheels. Later Willis, Reuleaux, and many other scholars brought the problem of profiling teeth to such a state that now, not only for plane but also for spatial systems of simple cylindrical, conical, screw, rack, worm, and cam engagements, tooth profiles are constructed quite methodically, and the profile of one of the teeth may be taken arbitrarily. The method gives not only the conjugate profile, but also data for a critical assessment of the resulting engagement.

22. On methods of the synthetic construction of schemes of new mechanisms.

The field of application of mechanisms at the present time is very large. We see them in the sciences (laboratory instruments, measuring devices, etc.), in the arts (the piano keyboard, various mechanisms inside the modern organ, etc.), in industry (machine tools and machines), in everyday life (alarm clock, door lock,

meat grinder, etc.), in sports and amusements (swings of every possible type, wind-up toys, etc.), in military affairs (cannon, submarine mine, etc.), in office work (typewriter, arithmometer, etc.), in means of transport (automobile, airplane, etc.). Some of the categories noted here are extremely rich in content. Thus, for example, in industry we shall find a multitude of mechanisms for the most diverse operations of the mechanical, chemical, and mining specialties. We encounter a great variety of mechanisms in scientific settings, in military affairs, and so on. If we further subdivide the named fields, then one must note groups of hoisting machines, hammers, lathes, planers, etc.; moreover, in each of these groups there will be found a whole series of varieties: for example, lathes may have foot or mechanical drive, self-acting feed, may be turret lathes, automatic lathes, and so on.

The picture just drawn is sufficient to pose the question: is a unified synthesis for constructing the diagrams of new mechanisms possible, given such a diversity of mechanisms?

In order to answer this question, let us first of all distinguish, in every mechanism, its form and its content. We clearly see that, in content, or in its purpose, mechanisms are so varied that in all probability they cannot all be brought under a unified synthesis. And all the illustrations cited above concerned precisely the content of mechanisms, and not their form. Meanwhile, precisely in the question of the form of mechanisms the matter is considerably simpler. What a mechanism is intended for is one question; how it is combined from links and kinematic pairs is another. We know that, irrespective of its purpose, every mechanism must satisfy definite relations between the number of constraints and the number of coordinates. The designer, when composing the diagram of a mechanism, is concerned that one degree of freedom be ensured in it, namely by those constraints that depend on the designer himself. There may also be superfluous degrees of freedom in a mechanism, if the designer recognizes that they have no serious significance for the motion of the mechanism, and therefore the right of restricting these freedoms may be left to chance.

On the basis of what has been set forth in the preceding paragraphs, it may be said that all mechanisms without exception, from the point of view of their structure, must satisfy one and the same requirements. Since the number of different types of links and kinematic pairs is limited, a unified structural synthesis of mechanisms is possible, as is a method by means of which it would be possible to construct diagrams of any mechanisms whatsoever that are correct from the point of view of their structure.

On the other hand, the question arises: is a synthetic method for constructing the diagrams of new mechanisms needed in Applied Mechanics?

Such a method must exist, first, because in practical everyday technology there is a need for regular, methodical synthetic activity, in which, by compulsion and not by free-

one must solve particular problems by selection. Secondly, even with the inventor’s free creativity, the method can be of great help.

Thirdly, when a method is available, it becomes possible to construct such schemes whose significance is revealed only after they have been constructed. Finally, fourthly, it is known that in other fields of knowledge, such as, for example, chemistry, the synthetic method has long existed and enjoys the recognition of specialists. One may hope that in applied mechanics too it will not meet with a negative attitude.

In view of the fact that to every definite construction of a mechanism there correspond definite conditions of connection, the synthesis of a mechanism may have two basic kinds—practical and mathematical. Practical synthesis consists in the following: from elements known to him, the designer, by correctly coupling them, forms a chain that satisfies the requirements of the mechanism and the specified conditions. In this case the designer does not think about the conditions of connection; he need know only their number. In mathematical synthesis, on the contrary, it is necessary to formulate the very conditions of connection and, by investigating them, to find the form of mechanism that satisfies these conditions.

In view of the fact that in practice one has to combine not only simple but also complex mechanisms, it seems advisable to me to single out the so-called elementary synthesis, in which a scheme of a mechanism is constructed for only one operation; moreover, the designer, while composing a chain from elements known to him, need not even reflect on how his mechanism will be used. In elementary synthesis it is important only that the scheme being created be correct from the kinematic point of view.

The study of the method should rationally begin precisely with elementary synthesis and then proceed to the more complex practical synthesis.

A distinguishing feature of practical synthesis is a definite technical assignment that the mechanism must satisfy. Here, in the simplest case, a mechanism for one given operation is possible; then, in more complex cases, mechanisms for several operations, which may be arranged in parallel, i.e. simultaneously, and in sequence, when one operation replaces another after its completion.

Thus, we shall distinguish:

a) elementary synthesis;
b) practical synthesis for one operation;
c) practical synthesis for several parallel operations;
d) practical synthesis for several sequential operations;
e) mathematical synthesis.

23. Elementary Synthesis.

In constructing schemes of new mechanisms, the designer must feel confident that his methods are correct. However complex

whatever the mechanism may be, it always consists of parts, forming chains of links and kinematic pairs, and therefore it is very important to acquire skill in combining the elements of a mechanism so that a mechanism is obtained as a result. We shall not speak here about its purpose. We shall take various links—rods, cams, gear wheels, springs, etc.—and shall correctly couple them with one another by means of the pairs PHAR, KLOR, SNIG, STAR, etc., caring only that, as a result of our construction, the mechanism should possess the motions that interest us.

Here, naturally, the following questions arise:

a) To what shall we attach our mechanism? This is important in the sense that the mechanism must receive motion from somewhere, as from a source.

b) How can a new link be attached to a mechanism?

c) How can some links in a mechanism be replaced by others?

d) Finally, how are the constructed chains and mechanisms to be connected with one another?

Hence there arise four parts of elementary synthesis: the choice of the base to which the mechanism will be attached, then the attachment of various parts to this base, further modifications in the structure of the mechanism, and, finally, the connection of groups and entire mechanisms.

a) Base.

In setting about constructing the scheme of a mechanism in elementary synthesis, the designer is not constrained by anything. True, his problem is indefinite to the highest degree; he may begin his work with gear wheels, or a swinging pendulum, or the entire mechanism of a steam engine, or with something else. However, this indefiniteness plays no role here, since the problem of elementary synthesis consists only in setting forth the correct methods of coupling parts with one another into mechanisms. Nevertheless, be that as it may, the designer must begin his work with something. That simplest possible mechanism with which the designer wishes to begin his constructions we shall call the base. As such a base one may take: a transmission shaft, the shaft of an electric motor, or several more complex systems, such as, for example, a pair of gear wheels, the mechanism of a steam engine, a worker’s hand, etc.

One thing is important to note here, namely that the designer must know the base well. He must know how many degrees of freedom the base has; otherwise his synthesis will be unsuccessful.

If desired, the designer may create a base for himself; in that case he must check it according to the rules of analysis of mechanisms and make sure that it actually has as many degrees of freedom as he supposed. As was set forth above, this check must consist in counting the number of coordinates and the number of constraint conditions.

c) Attachments.

The method of attachment is known in the literature. Assur’s detailed investigation gives a complete picture of what had been done in this direction up to the most recent time. Noting that, in plane bar mechanisms with hinged joints, Silvester’s dyad, a three-link member, and a four-link group are attached to the mechanism, Assur continues the same work and, in the form of a method of development of the driving link, indicates a general procedure for attaching bars to plane hinged mechanisms, though only under the condition that they be gripped in lower pairs. After Assur’s work there still remained a large field of untouched mechanisms, namely: plane mechanisms with higher pairs, plane non-bar mechanisms, and, finally, all spatial mechanisms. By his method Assur could not encompass all these types of mechanisms, because in his investigation he proceeded not from general kinematic foundations, but from a particular case, which he undoubtedly elaborated and used excellently.

Proceeding from general kinematic foundations—the counting of coordinates in the links of a mechanism and of constraints—it is possible not to restrict the problem so narrowly. All the indicated types of mechanisms must be subordinated to a single method, by which any parts could be attached to a given base, regardless of whether a plane or spatial mechanism is formed, and even with equal success one could pass from a plane mechanism to a spatial one, or conversely.

For us it must be clear that to a mechanism possessing one degree of freedom one can attach only such groups in which the number of coordinates is equal to the number of constraints. Then, as the result of the attachment, there will again be obtained a system with one degree of freedom. If, however, to the mechanism we attach a group in which the number of constraints is greater by one than the number of coordinates, then the result will be an immovable structure. Conversely, if we attach a group in which the number of constraints is less by one than the number of coordinates, then we obtain a system with two degrees of freedom. In the same way, as a result of attachment, there may be obtained systems with redundant constraints and systems with many degrees of freedom. Let, for example, it be required to attach to the principal mechanism one rigid body. Since, in spatial motion, a rigid link is counted as having 9 coordinates, and its own constraints in it, as a link, number 3, there remain only 6 constraints for the kinematic pairs, which must be completely introduced. Hence it follows that one rigid link in a spatial mechanism can be attached to the base only by such pairs as, in total, give 6 constraints. For this purpose, therefore, the following combinations are suitable:

We grip the link with two pairs, of which

one is of the 5th class, the other of the 1st class,

or the 4th class, the other the 2nd,

or the 3rd class and the 3rd.

We grasp the link by three pairs, of which

one is of the 4th class, another of the 1st class, and the 3rd is of the 1st class,

or “ 3rd ” “ 2nd ” “ 1st ”,

or “ 2nd ” “ 2nd ” “ 2nd ”.

In the same way one can compose combinations for grasping by four or more pairs.

In the case of a plane mechanism, the grasping of one link to the base must satisfy other relations. Namely, in this case the link will have only 4 coordinates and one proper condition of connection. Thus in the kinematic pairs there must be only 3 conditions of connection, by virtue of which we obtain the possibility of composing the following combinations:

We grasp the link by two pairs, of which one is of the 2nd class, and the other of the 1st class.

We grasp the link by three pairs, which must all belong to the 1st class.

There can be no other combinations here.

When an additional link is coupled by means of two spherical joints in a spatial mechanism, the result obtained does not coincide with the count. The point is that a link which has two spherical joints acquires an axis of rotation passing through the centers of these joints, owing to which, in fact, the joined link has, as it were, one pair of the 5th class. Thus, a rod with two spherical joints at its ends, introduced into the composition of a spatial mechanism, will in reality have a motion with one degree of freedom in the form of rotation about its geometrical axis, but because of this the base will stop and cease to be a mechanism. On this ground, the joining of a rigid rod by means of two spherical joints may be considered incorrect.

Joining to the given base two rigid links, we obtain 18 additional coordinates, while in the links themselves there are 6 additional conditions of connection and, consequently, 12 conditions remain for the kinematic pairs. Here it is very important to note that among the kinematic pairs there is one which connects precisely these two newly joined links with one another. This pair enters into the composition of the joined group and may most often be regarded as known, since the designer, in planning the joining, must preliminarily prepare those groups which he intends to join, in accordance with what was set forth in § 11.

The total number of possible combinations for joining two rigid bodies is sufficiently large; therefore we shall confine ourselves to investigating only some of them, being certain that all the other combinations can be investigated in the same way.

Let the kinematic pair by which the two joined bodies are coupled to one another belong to the 5th class. Then in it there are 5 conditions of connection and, consequently, for those kinematic pairs by which the group—

will be used for attachment to the mechanism, 7 constraints remain. Hence it is evident that the following combinations will be suitable for proper attachment:

Attaching a group by two pairs, of which

one is of the 5th class, and the other of the 2nd class,

or ,, of the 4th ,, ,, of the 3rd ,,

Attaching a group by three pairs, of which

one is of the 5th class, the second of the 1st class, and the third of the 1st class.

or ,, of the 4th ,, ,, of the 2nd ,, ,, of the 1st ,,

or ,, of the 3rd ,, ,, of the 2nd ,, ,, of the 2nd ,,

or ,, of the 3rd ,, ,, of the 3rd ,, ,, of the 1st ,,

If the kinematic pair by which both bodies of the group being joined are coupled were of the 4th class, then all the combinations would change accordingly. In the same way one could find the possible combinations for cases in which the named pair belonged to the 3rd, 2nd, and 1st classes.

In plane mechanisms, two rigid links, when additionally attached, would bring with them 8 new coordinates and 2 constraints within themselves. Thus, 6 constraints would remain for the kinematic pairs here. If both attached links are coupled with one another by a pair of the 2nd class, then only 4 constraints are needed for attaching such a group to the mechanism. In this case the only possible variant is that both pairs by which the group is attached to the mechanism belong to the 2nd class. But if there are 3 such pairs, then one of them must be a pair of the 2nd class, and the remaining two pairs must be pairs of the 1st class. For this group to be attached by four pairs, it is necessary that all of them belong to the 1st class.

On the basis of what has been set forth it should be clear that groups with a larger number of links can also be attached to a mechanism without difficulty. A group of three rigid links, having 27 coordinates and 9 inherent constraints, requires 18 constraints in kinematic pairs, and so on. What is important is only to distribute these constraints among the pairs, i.e., from the constructive point of view, to choose the pairs rationally.

It should finally be noted that the reverse process is also possible, when one or several links are removed from a mechanism. The considerations will be the same. Finally, not only rigid bodies may be involved in attachment, but also variable links, whose presence will be reflected only in the fact that the number of constraints in the links themselves will prove smaller and, consequently, there must be correspondingly more of them in their pairs.

For illustration, let us show how a gradually lengthening chain of rods, connected to one another by Hooke joints, can be attached to an ordinary shaft.

Let such a shaft be given (Fig. 89), to which we attach another rod by a Hooke joint. Since within this joint there is a cross, which may be regarded as a rigid link with two cy-

cylindrical hinges, then in the present case 2 movable bodies with 18 coordinates and 6 constraint conditions in the links are added. For the kinematic pairs there remain, consequently, 12 conditions. Arranging the mechanism as shown in the figure, and recalling that in the case of concurrence of all cylindrical hinges at one point, 3 constraint conditions are lost (see p. 19), we obtain, instead of 15, only 12 constraint conditions in three cylindrical hinges (two in the cross and one fixed). Thus the attached group will act correctly.

By attaching to it one more rod by means of a new Hooke joint, we could repeat all our reasoning and would obtain that the mechanism represented in Fig. 90 should act correctly. In it, however, one more peculiarity may be observed. In fact, since the points that are the centers of two crosses remain fixed, the geometrical axis of the middle rod actually remains unchanged, and hence it follows that its support also is not needed. This mechanism may be represented as shown in Fig. 91.

The further attachment of rods by means of Hooke joints would lead us to the scheme represented in Fig. 92, and so on.

By successive attachments one can form mechanisms of any desired complexity. Let us give an example.

In Fig. 96 there is represented the scheme of a group composed in the following way.

At the points \(D, E, G, H, K, L\), and \(M\) there are cylindrical hinges. The gear wheel \(R\) is fastened to the link \(EG\) and is in mesh with the wheel \(U\), fixed on the link \(GK\). The chain transmission is composed as follows: the sprocket \(V\) is rigidly connected with the wheel \(U\), and the sprocket \(W\) is fastened to the link \(KL\). In counting the constraint conditions in this group it is necessary to bear in mind that at the point \(G\) three rods meet and here two cylindrical hinges must be counted. Thus, considering the whole group as plane, we find:

The number of links, including the flexible connection \((NP)\), is 7; consequently, the total number of coordinates is 28. The number of constraint conditions in the links is 7, and in the kinematic pairs 21, since there are 8 hinges of the 2nd class, 2 rolling pairs of the 2nd class, and one of the 1st class. A group composed in this way may therefore be attached to a plane mechanism. In Fig. 97 a spatial scheme is given. Here at points 1 and 3 there are pairs of the 4th class, and at points 2 and 4—of the fifth. There are altogether 3 movable links, owing to which we obtain the number of coordinates 27.

The number of constraint conditions in the links is 9 and in the pairs 18, i.e., also 27. Obviously, the scheme named is attached to a spatial mechanism.

In Fig. 98 an ordinary leaf spring is shown. If it is to be included in a plane mechanism, then, as is known, pairs of the 2nd class, for example cylindrical hinges, must be placed at points \(S\) and \(T\).

In Fig. 99 a diagram is presented of a composition made from the mechanism of the steam engine \(ABC\) and groups, composed in Figs. 96, 97, and 98, attached successively to this mechanism.

By attaching in the same manner one may compose a diagram of any desired complexity, up to infinity.

c) Modifications.

From the point of view of the conditions of connection, one further interesting operation with mechanisms is possible. Namely, all possible rearrangements of the parts of a mechanism, provided that the number and kind of kinematic pairs are preserved, are entirely admissible in essence, since in them the number of coordinates and the number of conditions of connection remain unchanged. However, as the result of each such rearrangement a certain new mechanism is obtained. One may go even further. One pair of the 5th class may be replaced by another pair of the same class, or by several pairs in which the total will be 5 conditions of connection. All such operations, in which a rearrangement of the parts of a mechanism is performed or they are replaced under the indispensable condition that the total number of coordinates and conditions of connection in the mechanism remain unchanged, are naturally to be called modifications of the conditions of connection in mechanisms. The lawfulness of modifications should be evident on the grounds that the number of coordinates in the mechanism during such an operation remains unchanged, while the conditions of connection change only qualitatively, but not quantitatively.

As an example, let us point to the known modifications of the Stephenson link-motion valve gear, when what is called the upper suspension, the middle suspension, or the lower suspension is made. Here all the parts of the mechanism remain unchanged, and only a rearrangement of some of them is performed.

Let us carry out a modification of another kind. Let a pair of toothed wheels, Fig. 93, be given, in which we shall mark the pairs by the numerals 1, 2, and 3, and shall call the wheels themselves \(A\) and \(B\). This mechanism is plane; in it there are two pairs of the 2nd class and one of the 1st (rolling between teeth). Let the first modification consist in the fact that the 2nd and 3rd pairs are interchanged, one with the other. We obtain the diagram of Fig. 94. Replacing now the rolling pair with sliding by a simple trajectory of a point along a given guide, we obtain the diagram shown in Fig. 95.

A more complex modification will be obtained when we wish to replace a rigid link by a variable one. In this case a modification is possible only when the class of the kinematic pairs can be raised. In fact, the introduction of a variable link will cause a reduction in the number of conditions of connection, which must obviously be compensated by a corresponding increase in the number of conditions of connection in the kinematic pairs. If such a raising of the class of the pairs is possible, then the modification itself is possible. Thus, for example, the substitution of an elastic spring for a rigid rod in plane mechanisms is possible only when there are pairs of the 1st class, one of which must immediately be repla—

by a pair of the 2nd class. It is true that there are other possibilities as well, but here the discussion concerns only modifications.

It is also of interest to calculate how many possible schemes of mechanisms are formed by modifications. A detailed answer to this question would take up a great deal of space, since all particular cases would have to be investigated. Therefore we shall confine ourselves only to general considerations.

Let us take the case of a four-link mechanism formed from rigid bodies. Three of them are movable, and one is fixed. Moreover, we shall take only plane mechanisms. In chains formed from four bodies, pairs of the 1st and 2nd classes may occur in the plane, and the number of pairs coupling the four bodies to one another may vary from four to six. Thus the following combinations are possible:

a)	2	2	2	2
b)	2	2	2	1	1
c)	2	2	1	1	1	1

where the numerals 2 and 1 denote the class of the kinematic pair, and each line indicates a group of possible pairs in the mechanism.

But since, as is known, there are four types of plane pairs of the 2nd class, and two types of the 1st class, there arises the possibility, by means of various combinations, of forming formally up to 256 combinations for each line, and consequently 768 in all. From this number of formally obtained schemes many will fall away as repetitions, and some as special cases of which mention was made above (item 18). However, in passing from plane to spatial schemes, where there must be considerably more combinations, it will not be an exaggeration if we determine the total number of possible, and at the same time original, schemes of four-link mechanisms formed from rigid links to be several hundred.

d) Connections.

Finally, attention should be paid to the method by which two separate mechanisms are joined together and then act as one. From the theoretical side the matter is as follows. Each of the mechanisms has one degree of freedom, and two of them therefore have two degrees of freedom; and therefore, in order properly to join two mechanisms into one, it is necessary to introduce one additional constraint. From the point of view of a practical method, it is evidently necessary only to know which parts will in fact introduce one additional constraint into the system of the two mechanisms.

If we choose one rigid body as the connecting link, then we shall be obliged to attach it only by such pairs as provide a sufficient number of constraints. Thus, for example, in spatial motion the additional rigid body will have 9 new coordinates and 3 of its own constraints. Obviously, there must be altogether 7 constraints in the kinematic pairs for the connection to be correct. In plane mechanisms, for the 4 new coordinates of the additional link, it will be necessary to have 4 constraints in the pairs, since in the additional link itself there is one constraint.

For example, let us imagine two toothed wheels that are not connected with one another and that it is required to connect them. The mechanism is plane; in the connecting rigid link there are 4 coordinates and one intrinsic constraint condition, owing to which 4 constraint conditions must be composed in the pairs. This can be done in various ways. Thus, for instance, connection is possible by means of a rod which is attached to the given wheels by two hinges. Each of such hinges, as is known, has 2 constraint conditions, and therefore the connection satisfies the stated requirement. In exactly the same way the wheels are connected by means of an additional intermediate wheel having a certain fixed axis of rotation. The peculiarity of this combination consists in the fact that it introduces already 3 new kinematic pairs with one link, two of these pairs being rolling pairs of the 1st class and one pair an ordinary hinge of the 2nd class. In the end we have again obtained in the pairs a total of 4 constraint conditions and, consequently, this combination is also correct.

24. Practical Synthesis of a Mechanism

Practical synthesis of a mechanism is distinguished by the basic feature that it has as its aim the creation of a scheme of a mechanism satisfying certain kinematic conditions set in advance.

The essence of practical synthesis consists in the fact that the designer selects links and kinematic pairs in such a way that, as a result of their coupling, the prescribed motion is obtained. This synthesis may be simple or complex. In the first case it is necessary to create the scheme of a mechanism that performs one prescribed operation. In complex synthesis, on the contrary, it is required to give the scheme of a mechanism for several operations. In this latter case there may be two types of mechanisms, namely: in one, the operations proceed simultaneously, or, as we shall say, in parallel; in the other, consecutively.

The mechanism of a steam engine has one operation. A lathe has two simultaneous operations, namely, the rotation of the workpiece being machined and the feed of the cutter. A planing machine has two consecutive operations, namely, the motion of the table with the workpiece being machined and then the displacement of the cutter by the amount of the shaving. In a typewriter we already discern a whole series of operations: the striking of a letter, the movement of the paper, the movement of the inking ribbon, the warning bell, the escapement of the carriage, the return, the shifting of the register, the shifting of the paper to another line, the clamping of the paper, the change to another color of ink, and others. Many of these mechanisms are entirely unconnected with one another. Thus, for example, the mechanism for shifting the paper to another line must be regarded as completely independent, performing one operation. On the contrary, the striking of a letter and the movement of the carriage are connected consecutively. While the letter is being printed, the carriage with the paper must remain motionless. After printing, however, the paper must automatically move. Here we see an example of consecutive

couplings. Finally, an example of parallel coupling may be the motion of a carriage and the motion of a dyeing belt.

In analyzing complex mechanisms, we shall always find that they consist of several simple ones, coupled with one another in one way or another. Thus, the design of the scheme of a complex mechanism ultimately reduces to the construction of several separate schemes for relatively simple mechanisms, each for a single operation.

a) Construction of the scheme of a mechanism for a single operation.

The simplest motion of a given body, performed with the aid of a separate mechanism, we shall call an operation. This definition should not be considered very strict. The designer himself may assign the operations. If, for example, it is required to impart a screw motion to body A, then this can be accomplished by one operation of the motion of a screw in a nut, or by two operations—translational and rotational motion separately.

In general, the construction of the scheme of a mechanism for one specified operation should be regarded as a task that is not difficult, especially if inexperienced designers are advised consciously to begin with uncomplicated operations. The simplest and at the same time important operations are the rotation of a given body about a given axis and translational displacement along a given axis. In this case there may be additional conditions, such as, for example, that the body must rotate with a specified number of revolutions per minute. In such a case it will be necessary to select a transmission.

Furthermore, just as in elementary synthesis, a base must be specified from which the mechanism receives motion. The presence of the base somewhat constrains the designer. Thus, for example, an electric motor as a base, or a transmission shaft, will somewhat complicate the question of the use of toothed wheels, and so forth.

Next it is necessary to determine whether the specified operation is performed exactly or approximately. In the case of an exact operation, transmissions by flexible connection and friction transmissions are unsuitable.

As for dynamic questions, such as, for example, the forces of inertia arising in the mechanism, and finally the forces of friction that arise, we shall not deal with them here in kinematic synthesis.

b) Practical synthesis of a mechanism for several parallel operations.

In constructing the scheme of a mechanism for several operations, the work consists in the following: a scheme is drawn up for each operation separately; then all the mechanisms are coupled with one another and, finally, are attached to the specified base.

Thus, in practical synthesis it is necessary to distinguish: 1) the choice of the base, 2) the preparation of mechanisms for each operation separately, and 3) the coupling of the prepared mechanisms. The first two parts of this synthesis

have already been covered above; therefore it remains to cover the question of connecting mechanisms.

In § 23, on elementary synthesis, a general method for connecting mechanisms was indicated, the main condition being that, when mechanisms are connected, the connecting parts must introduce additional constraints. If, for example, both prepared mechanisms each have one degree of freedom and, as a result of the connection, a mechanism also with one degree of freedom is to be obtained, then, in coupling these mechanisms, one superfluous constraint must be introduced. Hence follows the requirement that the connecting group itself have a number of coordinates one less than the number of constraints. This, of course, can be achieved in the most varied ways.

Such groups were discussed in the chapter on the analysis of mechanisms. Let us note here several common cases.

The meshing by a toothed wheel of two mechanisms is possible if three pairs are again introduced into the chain: the axis of rotation of the connecting wheel and two new meshing pairs. In this case, for plane mechanisms, 4 coordinates and 5 constraints are introduced, since in the link itself there is one constraint, in the cylindrical pair of rotation of the wheel there are 2 constraints, and in the two rolling pairs one each. In the same case, for spatial mechanisms there would be 9 new coordinates and 10 constraints (3 in the link, 5 in the pair FhAR, and one each in the pairs ChAG).

When pulleys belonging to two different mechanisms are coupled by a flexible connection, only two new pairs are introduced: in plane motion of the 2nd class, and in spatial motion of the 4th class. In such a case the plane problem gives, for 4 new coordinates, actually 5 constraints, and therefore coupling is possible. As for the spatial case, there 11 constraints are obtained for 9 coordinates (3 in the link and 8 in the pairs). Obviously, when coupling spatial mechanisms, the flexible connection must be freed by one degree. Thus, for example, a belt must be provided with freedom of twisting in crossed transmissions; for the same reason a Gall chain is not suitable for crossed transmissions, and the like.

When using friction wheels to couple mechanisms, it is necessary to take into account the considerations set forth in Chapter 2 concerning rolling conditions. A disk can roll over a surface with spinning, having three degrees of freedom; without spinning, two degrees of freedom (see Fig. 24, pair NNAB); and, finally, having, relative to the surface, not two rotations but only one, it obtains one degree of freedom. All this applies to a disk not connected with any other pairs. A friction wheel mounted on an axis of rotation, by the very presence of the pair FhAR, loses the possibility of spinning and lateral inclinations with respect to the surface on which it rolls. Therefore pure rolling of friction wheels mounted on axes forms a rolling pair of the 1st class both in plane and in spatial motion. The remaining condi-

The constraint conditions turn out to be identical with the conditions of the pair PHAP in spatial motion, or KHAP in plane motion.

When coupled by a rod, we obtain two new pairs, which must give, in spatial motion, 7 constraint conditions (with 3 conditions in the rod), and in plane motion 4 conditions (with one in the rod). This was already discussed above.

c) Practical synthesis of a mechanism for several successive operations.

The procedure for constructing the diagram in this case remains the same as before. Here, too, it is necessary to have a base, to make a blank for each operation separately, then to couple all the prepared mechanisms and attach them to the base. The peculiarity lies only in those parts by means of which the coupling is carried out. All these parts must be arranged so that they alternately couple with the mechanism and then disengage from it. The sequence of operations must be understood in such a way that one operation must cease completely, and only after this can the second operation begin. In a punching machine with automatic feeding of the sheet, it is necessary to finish the punching operation and to withdraw the punch; only then can the sheet be advanced.

Thus it is necessary now to couple the sheet-feeding mechanism, now to disengage it from the drive.

Let us note some of the most widespread parts for performing the indicated function.

ca) Anchor.

The anchor (Fig. 100) is a fork, one prong of which represents an arc of a circle described from the center \(O\), while the other prong may even be rectilinear. The fork oscillates about the center \(O\), and the gear wheel engaging with the anchor is compelled to rotate either by a spring or by the force of gravity of a load. Obviously, when the gear wheel rests with a tooth against the right leg of the fork, the pressure will be directed toward the center of oscillation of the anchor and will be annihilated by the resistance of the support. Thus the wheel with the load has encountered an additional condition of a force static constraint and has stopped. With the pendulum, on the contrary, nothing has happened, and it continues to move. When the tooth of the wheel slips off the right prong, another tooth will rest against the left prong, and both mechanisms will move simultaneously.

cb) Ratchet wheel and pawl.

In metalworking machines one may often see that the feed of the cutter is effected by means of an oscillating pawl, which (Fig. 101), when moving to the right, engages with the ratchet wheel, and when moving to the left, disengages. The essence of the constraint, which here appears and disappears, consists in the fact that the distance \(MN\) is alternately variable and constant. Essentially the same constraint arises in all kinds of stops.

cc) Combined cam.

This part is very widespread in complex machines. It may be encountered in almost all steam-distribution mechanisms, in many woodworking and metalworking machines, etc. Its essence consists in the fact that part of an ordinary cam is traced along the arc of a circle, owing to which the rod ceases to move, and thus disengagement is obtained, for part of the mechanism is stopped. The resulting constraint consists in the roller being at a constant distance from the center of the cam. The forms of such cams may be very varied. In Fig. 102 a type is shown that is ordinarily used in valve steam-distribution mechanisms.

cd) Combined guide slot.

Almost the same as was said of the combined cam may also be said of the combined slot. Here (Fig. 103) likewise part of the slot \(MN\) may be traced along the arc of a circle, owing to which the distance of the points of the guide from the center \(O\) is made constant and the link is stopped.

ce) Incomplete friction drive.

In Fig. 104 a friction hammer is shown, in which the engagement and disengagement of the rod is produced by means of an incomplete friction drive. The essence of the action is clear without explanation.

czh) Incomplete gear wheel.

In essence an incomplete gear wheel does not differ from an incomplete friction drive. In it the constraint in the form of a rolling pair between the teeth of the wheels now appears, now disappears. The practical difference consists in the fact that when the teeth are disengaged there is no guarantee that the subsequent engagement will occur correctly, and that the teeth will not strike one another. Obviously, in the mechanism there must be a special device to ensure this guarantee. For an incomplete gearing this condition is necessary.

cz) Couplings.

The connection of shafts by means of couplings is based on the fact that between the parts of two shafts, as separate mechanisms, a stop is created, owing to which an additional constraint arises.

ci) Stops.

All sorts of catches, bolts, locks, retainers, latches, and similar parts, most often in plane mechanisms, are used in order to make a variable distance constant, owing to which one constraint condition is introduced; and therefore all of them are suitable for coupling two mechanisms with one another.

Example. On the basis of what has been set forth, one may proceed to the construction of a scheme.

Suppose it is required to construct a scheme of a mechanism for performing the following consecutive operations.

1) The workpiece is raised along the \(Z\) axis by a given amount.

2) After this it rocks forward and backward along the \(X\) axis three times.
3) Finally, it receives a push along the \(Y\) axis.

As the base we shall take a rotating shaft, from which all the motion will proceed. Moreover, since the operations are sequential, parts must be selected for engaging and disengaging the connections. In this matter we are free, and therefore we may impose some condition upon ourselves. Let, for example, the machine be of a restrained style, and let all the individual mechanisms be coupled and uncoupled by means of cams. In that case each cam must perform two functions: on one part of its surface it engages with the base and sets its mechanism in motion, while on another it disengages, so that the mechanism belonging to it comes to a stop.

Let us now prepare each operation separately.

1) The lifting of the object can be accomplished by means of the mechanism shown in Fig. 102, where cam \(A\) is outlined so that its part \(MN\) is a circle described from the center of the cam with one radius (let us call it \(R_2\)), and the part \(KL\) is also a circle, but only of another radius (\(R_1\)). \(KM\) and \(LN\) are transition parts. With such an outline of the cam, the motion of the rod will be as follows. For some time it remains without motion in the lower position, then rises by the amount \(R_2 - R_1\), and again remains motionless, but now in the upper position, and finally descends to its initial lower position, from which the motion will be repeated again. If the \(Z\) axis is vertical, then the lowering of the rod can be produced by the force of gravity of the table and the object; but if this axis is inclined, or horizontal, then a spring must be introduced which would press the rod against the cam.

2) The second operation of the mechanism consists in the object rocking back and forth along the \(X\) axis three times. This operation can be performed by the combined cam shown in Fig. 104-a. Since part of the cam is outlined along the arc of a circle, during the whole time while the roller rolls along this particular part, the entire mechanism of this operation will remain inactive; but as soon as the roller goes along the wave-shaped projections of the cam, slide \(R\) of the mechanism will enter into alternating rocking motion, and, since three waves have been prepared on the cam, the slide will make the same number of complete oscillations.

The spring ensures reliable contact between the roller and the cam. When checking this mechanism scheme for the relation between the number of coordinates and the number of constraints, the construction of the roller must be indicated. If it is made as shown in Fig. 102, in the form of a separate link, then an extra link is introduced and pure rolling is formed, i.e., between the roller and the cam there will be a pair of the 2nd class. If, however, the construction is different—for example, the rod can bear against the cam without a roller—then the rolling will be with slipping, a pair of the 1st class is formed, and at the same time the extra link disappears. The first construction, when counted, gives, in a plane mechanism, 20 coordinates and 19 constraints.

3) Finally, the third operation of our machine consists in the workpiece receiving a blow along the \(Y\)-axis. In Fig. 105 there is shown a diagram of a mechanism that can perform the operation just named. The cam \(K\) is here outlined in such a way that over the greater part of it it has the arc of a circle, so that during this period the roller remains without translational motion. Then the rod \(CD\), owing to the abrupt drop on the cam, will fall, whereby the striker \(EG\) will produce the corresponding impact. Verification of the mechanism gives: 24 coordinates, constraints in the links 6, in the eight pairs of the 2nd class 16, and at point \(E\) (a pair of the 1st class) one, making 23 in all.

d) Assembly.

In proceeding to assemble the prepared parts, it is first of all necessary to arrange the mechanisms properly, since they act in different planes. Namely, the mechanism for the first and second operations may be left in the plane of the drawing, while the mechanism for the third operation will have to be placed in a plane perpendicular to the drawing; in doing so, it may be arranged vertically or horizontally. We shall adopt the latter. Further, all the cams must be coupled to the base so that a single mechanism is obtained with parallel motions of all the cams. The mutual arrangement of the cams must be such that the end of one operation corresponds to the beginning of another. In view of the fact that the relative disposition of the phases of motion of the cams must be strictly definite, coupling between the cams cannot be effected either by means of friction transmissions or by belts, since slipping in either case would completely disturb the correctness of the mutual operations. Gear and chain transmissions will be suitable.

In Figs. 106 and 107 the result of the synthesis is shown schematically in two projections.

The main shaft \(O_2\) sets in motion the cam of the first mechanism \(S\) and, at the same time, by means of a chain transmission, rotates the second shaft \(O_2\), on which the cam of the second mechanism \(N\) is keyed. From this shaft, by means of a pair of bevel gears, the motion is transmitted to shaft \(C\) with the cam of the third mechanism \(K\). This entire system of cams and gears forms one mechanism, which moves continuously in one direction. The prescribed operations are performed in the following way. The rod \(T\) sets in motion the table, which moves vertically, stops, and descends again. The lever \(MP\) of the second mechanism sets in motion the slide \(R\), which runs in guides belonging to the table. Finally, from the cam \(K\) of the third mechanism, in a horizontal plane, the striker \(EG\) is actuated. Thus, in fact, the workpiece \(X\), placed on the slide \(R\), is raised from below upward, then swings from right to left three times and, finally, receives a blow in a direction perpendicular to the plane of the drawing.

The machine is shown at the moment when the lifting of the table has ended and the oscillations are beginning.

25. The Mathematical Form of the Method for Constructing Schemes of New Mechanisms

Since every part of a mechanism corresponds to a definite number of constraint conditions, alongside the practical form there must also exist an analytical form of the method. In those particular cases where synthesis has as its aim the determination of certain geometrical elements, such as, for example, the curve along which the teeth of gear wheels are to be outlined, or a straight line in guiding mechanisms, and so forth, the method in analytical form can give, and has already given, useful results. In other cases, however, its significance seems to me at present not altogether clear. Nevertheless, it would be unfair to pass over it in silence, and, if only in brief form, the foundations of the method must be set forth here.

Its essence consists in the following.

Whereas, in the analysis of mechanisms, we formulate constraint conditions from given elements—links—and kinematic pairs, here, in synthesis, the process must proceed in the reverse order: namely, from given constraint conditions the elements of the mechanism must be determined. The problem in this form is very broad, since any equation may be taken as a constraint condition of a mechanism, provided only that the variable quantities are assigned the meaning of parameters of the mechanism. Indeed, let the equation be given:

\[ f(q_1, q_2, q_3 \ldots \ldots q_n)=0 \ldots \ldots (1) \]

where \(q_1, q_2, q_3\), etc., are arbitrary variable quantities. Substituting for them:

\[ x_1, y_1, z_1, x_2, y_2, z_2, \ldots t, p, T, \]

where: \(x_1, y_1, z_1\) are the coordinates of some one point of the mechanism,

\(x_2, y_2, z_2\) are the coordinates of another point,

\(t\)—time, \(p\)—pressure, \(T\)—temperature, and so forth, we obtain the general form of a constraint condition for a mechanism:

\[ f(x_1, y_1, z_1, x_2, y_2, z_2, \ldots t, p, T \ldots)=0 \ldots \ldots (2) \]

Of course, the meaning and significance of the obtained constraint condition depend essentially on the order in which the variables are replaced; thanks to this, from one given equation (1) there may be formed a whole series of constraint conditions (2), in their essence entirely unlike one another.

The principal requirement, of course, remains in force: namely, the number of constraint conditions in a mechanism must be less than the number of variable quantities that enter into the equations. The order of construction may remain the same as in the practical method. First it is rational to choose a basis, as simple as possible, and then to carry out operations of attachment and modification.

The basis may be specified by one equation or by a group of them. The equations of a circle, of a straight line, of the distance between two points, and also of the constancy of coordinates, in the form of ready-made solutions, are the simplest on the one hand and very widespread in mechanisms on the other. Therefore it is best to choose the basis from these equations.

a) Modifications.

In exactly the same way as in the practical method, here too it is possible to replace some constraint conditions by others, and as a result new combinations will be obtained. It is easiest to show by simple examples how such modifications may be carried out.

Let there be given a basis in the form of five equations:

\[ \begin{aligned} \text{I)}\quad & q_1^2+q_2^2=a^2,\\ \text{II)}\quad & q_3=mq_4-b,\\ \text{III)}\quad & (q_3-q_1)^2+(q_4-q_2)^2=c^2,\\ \text{IV)}\quad & (q_1-q_6)^2+(q_2-q_5)^2=d^2,\\ \text{V)}\quad & q_6=0 \ldots \ldots \ldots (3), \end{aligned} \]

which connect six variable quantities: \(q_1,q_2,q_3,q_4,q_5\) and \(q_6\), under the condition that the quantities \(a,b,c,d,m\) are considered constant.

Let us begin with the simplest case. Suppose that we shall construct the scheme of a plane mechanism and that all the variables are the coordinates of three points of the mechanism. It is still not yet evident which coordinate is which. However, not embarrassed by this for the time being, let us make an entirely arbitrary substitution, as indicated in the following table:

	\(q_1\)	\(q_2\)	\(q_3\)	\(q_4\)	\(q_5\)	\(q_6\)
instead of:	\(q_1\)	\(q_2\)	\(q_3\)	\(q_4\)	\(q_5\)	\(q_6\)
substitute:	\(x_2\)	\(y_2\)	\(x_3\)	\(y_3\)	\(y_1\)	\(x_1\)

Then we obtain the following group of equations:

\[ \begin{aligned} 1)\quad & x_1=0,\\ 2)\quad & (x_2-x_1)^2+(y_2-y_1)^2=d^2,\\ 3)\quad & x_2^2+y_2^2=a^2,\\ 4)\quad & x_3=my_3+b,\\ 5)\quad & (x_3-x_2)^2+(y_3-y_2)^2=c^2 \ldots (4). \end{aligned} \]

This group of equations corresponds to a definite mechanism, which can be constructed. Indeed, first of all let us note that the equations contain the coordinates of three points. Let these points be: \(A\) (with coordinates \(x_1,y_1\)), \(B\) (with \(x_2\) and \(y_2\)) and \(C\) (with \(x_3,y_3\)). Equations 1 show that point \(A\) slides along the axis of ordinates. Equation 3 shows that point \(B\) describes a circle. From equation 4 we see that point \(C\) moves along an inclined straight line, and equation 5 requires that the distance between \(B\) and \(C\) be constant. The distance \(AB\) is also constant, as follows from equation 2.

In Fig. 110 a scheme is presented which fulfills the conditions obtained. Here point \(A\) slides along the axis \(OY\), point \(B\) moves along the guide of a circle described from the center \(O\), and point \(C\) slides along an inclined slot.

Let us now carry out some modifications with the obtained scheme. First of all, let us discard the 2nd equation from group (4) and insert in its place the following:

\[ y_1=d \ldots \ldots \ldots \ldots \ldots (5) \]

In this case point \(A\) becomes fixed, and the distance \(AB\) becomes variable. The scheme of the mechanism satisfying the conditions of the first modification is shown in Fig. 111. Here the rod \(AB\)

is preserved only because it existed in the preceding mechanism; from the conditions of the first modification the existence of this rod is not apparent. It is as if it had been necessary to make point \(A\) solitary, and to eliminate rod \(AB\) altogether. A certain ambiguity of the structure being constructed depends on the fact that, from the given conditions, nothing can be said about the construction of the kinematic pairs. This will be discussed further in the last paragraph.

The mechanism shown in Fig. 111 can be executed structurally in another way (see Fig. 112), replacing the circular guide by a rod \(OB\) of constant length with its center of rotation at point \(O\). In addition, point \(C\) is seen here as connected with a slider moving along an inclined guide. There is no difference whatever in the motion of the rods of the mechanisms shown in Figs. 111 and 112.

Let us now carry out another modification. Suppose, for example, that the 4th equation from group (4) is replaced by

\[ \omega_{\mathrm{в}} = k\,\omega_{\mathrm{с}} \ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (6), \]

where \(\omega_{\mathrm{в}}\) is the angular velocity of rotation of rod \(OB\) (see Fig. 112) about center \(O\), and \(\omega_{\mathrm{с}}\) is the angular velocity of rotation of rod \(BC\) in absolute motion. If \(k\) is a constant quantity, then we may apply any transmission whatever—frictional, toothed, or with flexible linkage. In applying a toothed transmission, it must nevertheless be borne in mind that, according to equation (5) from group (4), the distance \(BC\) must remain constant. A scheme satisfying, more or less, all the conditions of this modification is shown in Fig. 113, where wheel \(M\) is fixed, and wheel \(N\) rolls on it. The resulting planetary transmission, for the chosen radii, as is known, imposes a certain condition on the coefficient \(k\).

Let us now try, from the same conditions, to carry out a larger modification with replacement of the equations and transition to a spatial scheme. For this we shall have to return to the original group of equations (3) and make another substitution of variables. Let now six variables belong to only two points of the mechanism, counting 3 coordinates for each of them.

For example

instead:	\(q_1\)	\(q_2\)	\(q_3\)	\(q_4\)	\(q_5\)	\(q_6\)
substitute:	\(z_1\)	\(y_1\)	\(z_2\)	\(x_2\)	\(y_2\)	\(x_1\)

in addition, we modify the 3rd equation in the following way:

Let it be

\[ \text{III)}\quad (q_3-q_1)^2+(d-q_2)^2=c^2 \]

Then the following system of equations is obtained:

\[ \left. \begin{aligned} \text{I)}\quad & z_1^2+y_1^2=a^2 \\ \text{II)}\quad & z_2=mx_2+b \\ \text{III)}\quad & (z_2-z_1)^2+(d-y_1)^2+x_2^2=c^2 \\ \text{IV)}\quad & y_2=d \\ \text{V)}\quad & x_1=O \end{aligned} \right\} \ .\ .\ (7) \]

These equations show that one point of the mechanism describes a circle in the plane \(YOZ\), while another point, at a distance \(d\) from the plane \(XOZ\), moves parallel to this plane along an inclined straight line. The scheme of such a mechanism may be represented by Figure 114.

b) Attachments.

When constructing the scheme of a mechanism analytically, one may proceed as follows. Having specified a base in which there are to be \(n\) coordinates and \(n-1\) constraints, we prepare a special group of \(k\) equations, in which there must be exactly the same number of unknown quantities. The coordinates of points of the base may also enter into these additional equations—only one thing is obligatory, namely that in each such additional equation, of which the group is composed, one or another of the new coordinates should enter. Otherwise, if even one of these additional equations contains only the coordinates of points of the base, this latter will turn into an immovable system. Having thus prepared an additional group of \(k\) equations and combining it with the equations of the base, we obtain as a result a new combination, to which there also corresponds a certain new mechanism, while the base will move without change.

This process may be represented as follows.

It was specified in the base: \(n\) coordinates and \(n-1\) constraints.
Prepared in the additional group: \(k\) „ \(k\) „ „
As a result of attachment: \(n+k\) „ \(n+k-1\) „ „

Another case is also entirely possible, when \(n\) coordinates and only \(n-2\) constraints are specified in the base. Then in the additional group to be attached there must be \(k\) coordinates and \(k+1\) constraints, and as a result one again obtains a correct motion.

In this case the attached group, by its constraints, restricts one extra degree of freedom of the base.

Let, for example, the base be specified in the form of the equations: \(x_1=y_1=z_1=0\)

\[ (x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2=a^2 \]

Since in it, for 6 coordinates, there are four constraints, in the group to be attached there may be one extra constraint. Let us prepare such a group. Suppose, for example, four equations with three new unknowns are given:

\[ (x_3-x_2)^2+(y_3-y_2)^2+(z_3-z_2)^2=b^2 \]

\[ x_3-x_2=m(y_3-y_1) \]

\[ z_3=z_2 \]

\[ y_3=c \]

This group, considered together with the equations of the base, gives 5 constraints for 6 coordinates, thanks to which it may correspond to a mechanism.

c) Couplings.

In an entirely similar manner, couplings of two groups of equations are also possible, each of which corresponds to some mechanism,

having a number of conditions one less than the number of coordinates. To join such groups it is necessary to compose one more equation, in which the coordinates of both groups would occur. Then, by considering all the equations jointly, one obtains a system corresponding to some new complex mechanism.

26. General remarks on the mathematical synthesis of mechanisms.

Although structural mathematical synthesis is possible, nevertheless, in constructing schemes of mechanisms from prescribed equations we have encountered certain essential difficulties, which, it is true, we circumvented, but not entirely smoothly.

In the chapter on the analysis of mechanisms it was established that a two-link mechanism with one fixed link will have, in space, 9 coordinates, and in the plane 4 coordinates; moreover, in order to provide one degree of freedom in the motion there must be, respectively, 8 and 3 constraint conditions. Even excluding from consideration those conditions which belong to the link itself, we still obtain, in a spatial mechanism, 5 equations, and in a plane mechanism, 2. A three-link mechanism with two moving links requires, in space, 18 coordinates and 17 equations, and in the plane 8 and, respectively, 7. Meanwhile, in the preceding paragraph, having at our disposal only five equations in all, we completed our exercise by constructing the scheme of a spatial four-link mechanism (fig. 114), for which formally not five but 26 equations are required.

Obviously, our constructions require clarification. Indeed, if point \(C\) (fig. 110) moves along an inclined straight line, then the substitution of a slider for this point (fig. 112) is an arbitrariness. If in plane motion the line \(BC\) and the material connecting rod are identical, then in a spatial mechanism the material connecting rod can be defined only by three points, while we had only two at our disposal. Here again an arbitrariness was admitted. Further, if point \(A\) describes a circle, then on this basis we created a crank, even in space. Of course, a single condition on the trajectory of a point is insufficient for constructing a crank, and therefore all the schemes shown in (figs. 110–114) must be called into question. These schemes satisfy the prescribed conditions, but they are not their unique solutions. To construct a spatial mechanism with three moving links one must require 26 equations. This problem will prove too complicated, all the more so because one must be certain that among the given equations there are no mutually dependent ones. The application of the functional determinant in the present case is complicated by the fact that the number of equations is usually less than the number of unknowns, whereas for forming the determinant these numbers must be equal.

Equation (6) has too primitive a form. If we wished to disclose it, we would encounter new difficulties.

Here it was simply assumed that all points of the link \(AB\) have one and the same angular velocity of motion. The same may be said of the link \(BC\).

Taking into account the circumstances noted here, one must come to the conclusion that, in constructing schemes of mechanisms, it is possible to proceed from given conditions of connection, but that in doing so the problem becomes extremely complicated in the number of unknowns, and its exact solution is very difficult even with a small number of links. That which, in the practical method, causes no difficulties at all here at once becomes burdensome. A simple cylindrical joint requires 5 equations, because it belongs to the 5th class. A single rod swinging in space on a ball-and-socket pin is determined in its motion by 8 equations, and so on.

These equations are extremely important, since from them one can proceed to the determination of all kinematic circumstances of the motion, to the calculation of the velocities and accelerations of points. However, on the basis of the considerations set forth, one must reject structural mathematical synthesis, considering that, in the present state of our knowledge, it is wholly undeveloped, although some particular problems, especially of a geometrical character and moreover with simple mechanisms, can be posed. An example of such activity is provided by the works of P. L. Chebyshev, which have a distinctly synthetic character. True, Chebyshev set himself in advance the scheme of the mechanism and then, by analysis, determined the dimensions of all its parts; but here we are once more convinced of the close connection between synthesis and analysis, and in Chebyshev’s work we see a splendid example of harmony between them.

In the history of science there have been other examples of mathematical synthesis as well; however, it must be acknowledged that we have no method by which one could generally construct schemes according to given equations, and there are serious reasons to fear that such a method cannot even be constructed.

LITERATURE.

1st group. Sources in which I found materials directly concerning the questions posed in this work:

1. P. Chebyshev. Theory of mechanisms known under the name of parallelograms. Collection, Vol. I, p. 8. 1854.
2. R. Willis. Principles of Mechanism. 2d Ed. 1870.
3. F. Reuleaux. Theoretische Kinematik 1875.
4. T. Rittershaus. Zur heutigen Schule der Kinematik. Civiling. 1875.
5. T. Beck. Ueber den Begriff “Maschine”. Civiling. 1877.
6. T. Beck. Noch ein Wort über den Begriff „Maschine“. Ibid.
7. F. Reuleaux. Ueber die Definition der Maschine. Ibid.
8. F. Grashof. Theorie der Getriebe. (Theoretische Maschinenlehre, Vol. II.) 1883.
9. M. Grübler. Allgemeine Eigenschaften der zwangläufigen ebenen kinematischen Ketten. Civiling. 1883.
10. E. Hartig. Ueber einige Allgemeinbegriffe der mechanischen Technik. Civiling. 1884.
11. P. Somov. On the degrees of freedom of a kinematic chain. Journal of the Russian Physico-Chemical Society. 1887.
12. L. Burmester. Lehrbuch der Kinematik 1889.
13. M. Grübler. Wandlungen der Kinematik in der Gegenwart.
14. H. Golovin. Kinematics of Machines. Vol. I. 1890.
15. H. Golovin. Kinematics of Machines. Vol. II. 1890.
16. Thomson and Tait. Treatise on Natural Philosophy. 1894.
17. P. Somov. On certain applications of the kinematics of variable bodies to articulated mechanisms. Proceedings of the Warsaw Polytechnic Institute. 1900.
18. F. Reuleaux. Theoretische Kinematik, Vol. II. 1900.
19. S. Dunkerley. Mechanism, 1904.
20. P. Somov. Theoretical Mechanics. 1904.
21. L. Assur. Investigation of plane rod mechanisms from the standpoint of their structure and classification. Proceedings of the St. Petersburg Polytechnic Institute. 1914.
22. L. Assur. The same. Supplement to Chapter II. Ibid. 1915.
23. W. Schell. Theorie der Bewegung und der Kräfte. 1879.

2nd group. Sources in which I found materials indirectly related to the questions investigated here.

1. D. Bobylev. On the motion of a surface touching another immovable surface. Transactions of the Imperial Academy of Sciences. 1867.
2. T. Suslov. On the question of the rolling of a surface over a surface. University News, Kiev. 1892.
3. A. Kennedy. The mechanics of machinery 1886.
4. M. Bour. Cours de mécanique et machines. (Cinématique. 2d ed. 1887).
5. H. Weiss. Grundsätze der Kinematik 1900.
6. W. Hartmann. Die Maschinengetriebe 1913.
7. T. Beck. Geschichte des Maschinenbaues.

3rd group. Sources from which I took examples.

1. H. Brown. 507 Bewegungsmechanismen. 1886.
2. G. Hiscox. Mechanical Movements. 1800.
3. C. Leist. Steuerungen der Dampfmaschinen. 1900.
4. C. Matschoss. Die Entwickelung der Dampfmaschine 1908.
5. N. Kartashov. Steam valve-gear mechanisms. 1914.
6. F. Dymarev. Manual for operators on the “Linotype” machine. 1922.

CONTENTS

Introduction.

Chapter I. General Information.

No.	Section	Page
1.	Historical note	1
2.	Motion of connected bodies parallel to a plane and in space	3
3.	On constraints	7
4.	On degrees of freedom	8
5.	On the mechanism	9
6.	Elasticity of motion	11

Chapter II. Structure of the Mechanism.

No.	Section	Page
7.	Historical note	12
8.	Links	15
9.	Kinematic pairs	16
10.	Design of kinematic pairs	24
11.	Groups of connected links	27

Chapter III. Analysis of the Mechanism.

No.	Section	Page
12.	The essence of the analysis of a mechanism from the standpoint of its structure	29
13.	Two-link mechanisms	31
14.	Three-link mechanisms	32
15.	Four-link mechanisms	35
16.	Complex chains in four-link mechanisms	37
17.	Multi-link mechanisms	38
18.	Special cases of the arrangement of kinematic pairs and links	43
19.	Corrected form of the formula for a mechanism	48
20.	Verification of mechanism diagrams proposed by F. Reuleaux	49

Chapter IV. Synthesis of the Mechanism.

No.	Section	Page
21.	Historical note	52
22.	On methods of synthetically constructing diagrams of new mechanisms	55
23.	Elementary synthesis	57
24.	Practical synthesis of the mechanism	65
25.	Mathematical form of the method for constructing diagrams of new mechanisms	72
26.	General remarks on the mathematical synthesis of mechanisms	76

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

(The book contains 16 pages of drawings.)

Gublit, Tomsk, No. 182. Print run 500 copies.

Typo-lithography of the Tomsk Linden Sibopis.

DRAWINGS FOR A. P. MALYSHEV’S WORK: “ANALYSIS AND SYNTHESIS OF MECHANISMS.”

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Figure 32 — STAR

Figure 33 — SLAG

Figure 32-a — STAR

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Figure 35 — SLAR

Figure 36 — SNAG

Figure 36-a — KLAR
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