Eng.-Mech. A. V. Verkhovsky

A four-link spatial mechanism with cylindrical joints whose axes are not parallel and do not intersect at one point, and its investigation.

(With the article, 1 page of drawings).

The general form of a spatial mechanism with cylindrical joints, as is known, is the seven-link mechanism. Such a seven-link mechanism may possess any arrangement of the axes of the joints relative to one another, as well as any ratio of the lengths of the links.

A distinctive proof of this may be found in the article by T. Rittershaus, “Zur heutigen schule der Kinematik,” Civiling, 1873, and also in the article by Prof. Malyshev, “Analysis and synthesis of mechanisms from the point of view of their structure,” Izvestia T. T. I., 1923, p. 45.

Passing to particular cases of spatial mechanisms with cylindrical joints, one may point, in the same article by Rittershaus, to a six-link mechanism which will indeed be a mechanism, i.e. a system with one degree of freedom, only when certain particular conditions are observed.

Further, one may point to the long-known four-link mechanism, all the axes of which are parallel to one another, and also to the four-link mechanism all the axes of which intersect at one point. If one makes a count of the constraints for these two mechanisms according to the system of Prof. Malyshev (described in the article cited above), then in these mechanisms there will prove to be three superfluous constraints each. It is obvious that in these mechanisms, by virtue of certain particular conditions, these superfluous constraints turn into identities, since otherwise these mechanisms could not possess mobility. In the article cited, this question is examined in detail.

When I was an assistant in the Cabinet of Applied Mechanics of the Tomsk Technological Institute, Prof. Malyshev proposed that I consider the question of whether the two four-link mechanisms cited above are the only possible ones, or whether there may also exist other four-link spatial mechanisms with cylindrical joints. One such mechanism was constructed by me, and the axes of its joints turned out to be non-parallel and not intersecting at one point.

A model of this mechanism may be seen in the Museum of Applied Mechanics of the T. T. I.

Judging by the external appearance of the mechanism, the particular condition for its mobility was the equality of opposite links and of the angles of the skew axes of the joints in these links. It was not possible to determine directly in what manner the three superfluous constraints in this mechanism turn into identities, and therefore, in order to explain the mobility of this mechanism, it was necessary to resort to a special method, in the form of the mathematical analysis given below.

For convenience in representing the mechanism, let us note that each separate link of a mechanism with cylindrical joints may be schematically represented in the form of a certain segment, which must be taken equal to the shortest distance between the axes of the cylindrical joints adjacent to the link, and of the angle between these axes. Thus, for example, the link shown in the...

To the article by A. V. Verkhovsky, “A Four-link Spatial Mechanism
with Cylindrical Joints.”

Drawing 1.

Drawing 2.

Drawing 3.

Drawing 4.

Drawing 5.

Principal Misprints

in drawing 1 is characterized by the length of the rod “\(r\),” and by the angle between the axes “\(A\)” and “\(B\),” which we shall denote by the letter “\(a\),” retaining in each link the same indices for “\(r\)” and “\(a\).”

Taking four such links, one can construct, by coinciding the axes of neighboring links, the mechanism shown in drawing 2.

The latter will not yet be the general form of a four-link mechanism, since in its construction a certain particular condition has already been observed—namely, the coincidence on the axes of the mechanism of the ends of adjacent links.

If this condition is not observed, the general form of the four-link mechanism shown in drawing 3 is obtained.

The object of the present work is to investigate the mechanism shown in drawing 2.

Attempts to investigate the mechanism shown in drawing 3 were unsuccessful, since in its analysis insoluble equations of high degrees are obtained.

For convenience of investigation we shall divide the mechanism shown in drawing 2 into two parts, so that in each part there would be two adjacent links. We shall transfer each of these parts into rectangular coordinate systems, so that the axis remaining between the rods would coincide with the \(Y\)-axis, and the rods themselves would lie in the plane \(XOZ\), with one of the rods coinciding with the \(Z\)-axis.

Such a coincidence is presented in drawings 4 and 5. In order that these two parts of the mechanism could again be combined into a single whole, the simultaneous fulfillment of the following conditions is necessary:

\[ t = t' \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (I) \]

\[ \angle CA = \angle A'C' \ldots \ldots \ldots \ldots \ldots \ldots \ldots (II) \]

\[ \angle tC = \angle t'C' \ldots \ldots \ldots \ldots \ldots \ldots \ldots (III) \]

\[ \angle tA = \angle t'A' \ldots \ldots \ldots \ldots \ldots \ldots \ldots (IV) \]

(In these formulas “\(t\)” and “\(t'\)” are the distances between the ends of the links.) In addition, in order that the system thus obtained should be a mechanism, i.e., should possess mobility, it is necessary that at least one of the written equations contain quantities which could be variable.

The problem now reduces to finding the dependencies between the unknowns of these equations and the constant elements of the links, i.e., the lengths of the rods and the angles between adjacent axes of the links, and then, by eliminating certain unknowns, to find the conditions which the elements of the links must satisfy.

Let us transform equation I.

From drawings 4 and 5 it follows:

\[ t^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \operatorname{Cos}\omega b;\quad t'^2 = r_3^2 + r_4^2 - 2 r_3 r_4 \operatorname{Cos}\omega d \]

equating “\(t\)” to “\(t'\)” we obtain

\[ r_1^2 + r_2^2 - 2 r_1 r_2 \operatorname{Cos}\omega b = r_3^2 + r_4^2 - 2 r_3 r_4 \operatorname{Cos}\omega d \ldots \ldots \ldots (1-a) \]

in this equation, instead of the variables “\(t\)” and “\(t'\),” new variables “\(\omega b\)” and “\(\omega d\)” have been introduced (not necessarily equal to one another).

For the transformation of equation II we shall use three trigonometric relations.

$$ \pm \operatorname{Cos} CA=\operatorname{Cos} AX\operatorname{Cos} CX+\operatorname{Cos} AY\operatorname{Cos} CY+\operatorname{Cos} AZ\operatorname{Cos} CZ \ldots \ldots (1) $$

$$ \operatorname{Cos}^{2} AX+\operatorname{Cos}^{2} AY+\operatorname{Cos}^{2} AZ=1 \ldots \ldots \ldots (2) $$

and from the condition of perpendicularity of the axis \(A\) to the rod \(r_1\), we obtain the third relation.

$$ \pm \operatorname{Cos} r_1X \operatorname{Cos} AX+\operatorname{Cos}\omega b\operatorname{Cos} AZ=0 \ldots \ldots (3) $$

In view of the coincidence of some elements of the links with the coordinate axes, we shall have:

$$ \left. \begin{aligned} \operatorname{Cos} CX&=\operatorname{Sin}\alpha_2\\ \operatorname{Cos} AY&=\operatorname{Cos}\alpha_1\\ \operatorname{Cos} CY&=\operatorname{Cos}\alpha_2\\ \operatorname{Cos} CZ&=0\\ \operatorname{Cos} r_1X&=\operatorname{Sin}\omega b \end{aligned} \right\}\ldots \ldots \ldots (4) $$

Substituting these values into equations (1), (2), and (3), we rewrite them as follows:

$$ \operatorname{Cos} CA=\pm AX\operatorname{Sin}\alpha_2+\operatorname{Cos}\alpha_1\operatorname{Cos}\alpha_2 \ldots \ldots (1\text{-}a) $$

$$ \operatorname{Cos}^{2} AX+\operatorname{Cos}^{2}\alpha_1+\operatorname{Cos}^{2} AZ=1 \ldots \ldots (2\text{-}a) $$

$$ \pm \operatorname{Sin}\omega b\operatorname{Cos} AX+\operatorname{Cos}\omega b\operatorname{Cos} AZ=0 \ldots \ldots (3\text{-}a) $$

From equation (3-a) we have:

$$ \operatorname{Cos} AX=\pm\frac{\operatorname{Cos}\omega b\operatorname{Cos} AZ}{\operatorname{Sin}\omega b} $$

or

$$ \operatorname{Cos} AX=\pm\frac{\operatorname{Cos} AX}{\operatorname{tg}\omega b}\ldots \ldots (5) $$

From equation (2-a) we determine \(\operatorname{Cos} AZ\):

$$ \operatorname{Cos} AZ=\pm\sqrt{1-\operatorname{Cos}^{2}\alpha_1-\operatorname{Cos}^{2} AX}\ldots \ldots (6) $$

Eliminating \(\operatorname{Cos} AZ\) from equations (4) and (5), we obtain:

$$ \operatorname{Cos} AX=\pm\operatorname{Sin}\alpha_1\operatorname{Cos}\omega b \ldots \ldots (7) $$

Substituting the obtained value \(\operatorname{Cos} AX\) into equation (1-a), we find:

$$ \operatorname{Cos} CA=\pm \operatorname{Sin}\alpha_1\operatorname{Cos}\omega b\operatorname{Sin}\alpha_2+\operatorname{Cos}\alpha_1\operatorname{Cos}\alpha_2 \ldots \ldots (8) $$

In an analogous manner, for the second part of the mechanism, for \(\operatorname{Cos} A'C'\) we find

$$ \operatorname{Cos} A'C'=\pm \operatorname{Sin}\alpha_4\operatorname{Sin}\alpha_3\operatorname{Cos}\omega d+\operatorname{Cos}\alpha_3\operatorname{Cos}\alpha_4 \ldots \ldots (9) $$

To satisfy condition II, we equate equations (8) and (9):

$$ \pm \operatorname{Sin}\alpha_2\operatorname{Sin}\alpha_1\operatorname{Cos}\omega b+\operatorname{Cos}\alpha_1\operatorname{Cos}\alpha_2 = \pm \operatorname{Sin}\alpha_4\operatorname{Sin}\alpha_3\operatorname{Cos}\omega d+\operatorname{Cos}\alpha_3\operatorname{Cos}\alpha_4 \quad (II\text{-}a) $$

Further, for transforming conditions III and IV we have:

$$ \operatorname{Cos} tC=\pm \operatorname{Cos} tX\operatorname{Cos} CX \ldots \ldots (10) $$

the other two terms vanish because

$$ \operatorname{Cos} tY=0 \quad \text{and} \quad \operatorname{Cos} CZ=0 $$

Using equalities (4), we transform equation (10)

$$ \operatorname{Cos} tC=\pm \operatorname{Cos} tX\operatorname{Sin}\alpha_2 \ldots \ldots (10\text{-}a) $$

Then in an analogous way we may write

\[ \cos t' A'=\pm \cos t' X' \cos A'X' \dots \tag{11} \]

or

\[ \cos t' A'=\pm \cos t' X' \sin a_4 \dots \tag{11-a} \]

In exactly the same way we shall have

\[ \cos tA=\pm \cos tX\cos AX \mp \cos tZ\cos AZ \dots \tag{12} \]

using equation (7), and also determining from equations (7) and (6):

\[ \cos AZ=-\sin a_1 \sin \omega b, \]

we obtain from equation (12)

\[ \cos tA=\mp \cos tX \sin a_1 \cos \omega b+\cos tZ \sin a_1 \sin \omega b \dots \tag{12-a} \]

and analogously:

\[ \cos t'C'=\mp \cos t'X' \sin a_3 \cos \omega d+\cos t'Z' \sin a_3 \sin \omega d \dots \tag{13-a} \]

To satisfy conditions III and IV, we equate equations (10-a) with (13-a), and equations (11-a) with (12-a):

\[ \cos tX\sin a_2=\mp \cos t'X'\sin a_3\cos \omega d+\cos t'Z'\sin a_3\sin \omega d \tag{III-a} \]

and

\[ \cos t'X'\sin a_4=\mp \cos tX\sin a_1\cos \omega b+\cos tZ\sin a_1\sin \omega b \tag{IV-a} \]

From (III-a) we have

\[ \pm \frac{\sin a_2}{\sin a_3} = \frac{\cos t'X'\cos \omega d+\cos t'Z'\sin \omega d}{\cos tX}. \]

But since \(\cos t'Z'=\sin t'X'\), we obtain

\[ \pm \frac{\sin a_2}{\sin a_3} = \frac{\cos t'X'\cos \omega d+\sin t'X'\sin \omega d}{\cos tX} \]

or

\[ \pm \frac{\sin a_2}{\sin a_3} = \frac{\cos (t'X'-\omega d)}{\cos tX} \dots \tag{14} \]

From (IV-a), analogously, we obtain, observing that \(\cos tZ=\sin tX\),

\[ \pm \frac{\sin a_4}{\sin a_1} = \frac{\cos (tX-\omega b)}{\cos t'X'} \dots \tag{15} \]

multiplying (14) by (15), we shall have

\[ \pm \frac{\sin a_2 \sin a_4}{\sin a_3 \sin a_1} = \frac{\cos (t'X'-\omega d)}{\cos tX} \cdot \frac{\cos (tX-\omega b)}{\cos t'X'} \dots \tag{16} \]

Further, from the triangles whose sides form the segments \(r_1, r_2, t\) and \(r_3, r_4, t\), we have

\[ \frac{r_3}{r_1} = \frac{\sin tr_1}{\sin tr_3} = \frac{\sin(90^\circ-\omega b+tX)}{\sin tZ} = \frac{\cos(tX-\omega b)}{\cos tX} \dots \tag{17} \]

\[ \frac{r_4}{r_3} = \frac{\cos(t'X'-\omega d)}{\cos t'X'} \dots \tag{18} \]

multiplying (18) by (17), we obtain

\[ \frac{r_2 r_4}{r_1 r_3} = \frac{\cos(tX-\omega b)}{\cos tX} \cdot \frac{\cos(t'X'-\omega d)}{\cos t'X'} \dots \tag{19} \]

Comparing equation (16) with (19), and in equation (16 taking the values of adjacent angles of the difference, we obtain:

\[ \frac{\sin \alpha_2 \sin \alpha_4}{\sin \alpha_3 \sin \alpha_1} = \frac{r_2 r_1}{r_3 r_1}. \tag{20} \]

We have obtained the first equation which connects the constant elements of the mechanism with one another.

Now, eliminating from equations (I-a) and (II-a) \(\cos \omega b\), and solving them with respect to \(\cos \omega d\), we obtain:

\[ \cos \omega d = \frac{ -\cos \alpha_1 \cos \alpha_2 -\cos \alpha_3 \cos \alpha_4 -\sin \alpha_1 \sin \alpha_2 \frac{r_3^2+r_4^2-r_1^2-r_2^2}{2r_1r_2} }{ \sin \alpha_3 \sin \alpha_4 -\frac{r_3 r_4}{r_1 r_2}\sin \alpha_1 \sin \alpha_2 }. \tag{21} \]

Since the right-hand side of the equation obtained consists exclusively of quantities characterizing the invariable elements of the mechanism, while the left-hand side must be a variable quantity, since it contains the variable angle \(\omega d\), in order to satisfy this condition it is necessary that the right-hand side represent an indeterminacy of the form \(\frac{\infty}{\infty}\) or \(\frac{0}{0}\).

Restricting ourselves to the latter indeterminacy, since the former gives no interesting results, we equate the numerator and denominator of equation (21) to zero:

\[ -\cos \alpha_1 \cos \alpha_2 -\cos \alpha_3 \cos \alpha_4 -\sin \alpha_1 \sin \alpha_2 \frac{r_3^2r_4^2+r_1^2r_2^2}{2r_1r_2} =0 \tag{22} \]

and

\[ \frac{\sin \alpha_3 \sin \alpha_4}{\sin \alpha_1 \sin \alpha_2} = \frac{r_1 r_4}{r_3 r_2}. \tag{23} \]

Thus, for the time being, we have for the solution of the proposed problem three equations: (20), (22), and (23). But these are not yet all the equations, since they are insufficient for the observance of one more condition—the non-divergence of the ends of adjacent links on the axes \(A\) and \(C\).

Such divergence on the axes \(B\) and \(D\) cannot occur, since the ends of the links on these axes are joined with the origin of coordinates. In order to bring the axes \(A\) and \(C\) into the same position, let us divide the mechanism along the other axes. If we divide the mechanism not along the axes \(A\) and \(C\), but along the axes \(B\) and \(D\), and carry out the same operation, we obtain three more completely analogous equations:

\[ \frac{\sin \alpha_1 \sin \alpha_3}{\sin \alpha_2 \sin \alpha_4} = \frac{r_1 r_3}{r_2 r_4}. \tag{24} \]

\[ -\cos \alpha_4 \cos \alpha_1 -\cos \alpha_2 \cos \alpha_3 -\sin \alpha_4 \sin \alpha_1 \frac{r_3^2+r_2^2-r_4^2-r_1^2}{2r_4r_1} =0. \tag{25} \]

\[ \frac{\sin \alpha_1 \sin \alpha_3}{\sin \alpha_4 \sin \alpha_1} = \frac{r_2 r_3}{r_4 r_1}. \tag{26} \]

Thus six equations have been obtained, connecting the eight elements of the mechanism; hence, in the general case, only two elements can be assigned arbitrarily, while the values of the remaining ones are obtained from the equations. The solution of these equations in general form leads to equations of high degrees which do not admit solution and, moreover, does not enter into the problem posed.

The problem, as was said, consists in proving mathematically the mobility of the proposed mechanism.

For this purpose let us somewhat modify the equations obtained.

Multiplying and dividing equations (20) and (23), we obtain:

\[ \frac{\operatorname{Sin}\alpha_4}{\operatorname{Sin}\alpha_1}=\frac{r_4}{r_1} \tag{27} \]

\[ \frac{\operatorname{Sin}\alpha_2}{\operatorname{Sin}\alpha_3}=\frac{r_2}{r_3} \tag{28} \]

Multiplying and dividing equations (24) and (26), we obtain:

\[ \frac{\operatorname{Sin}\alpha_3}{\operatorname{Sin}\alpha_4}=\frac{r_3}{r_4} \tag{29} \]

\[ \frac{\operatorname{Sin}\alpha_1}{\operatorname{Sin}\alpha_2}=\frac{r_1}{r_2} \tag{30} \]

Now, instead of the former six equations, there are six: (22) (25) (27) (28) (29) (30).

It was indicated above that the proposed mechanism possesses equality of opposite rods, i.e. \(r_1=r_3,\ r_2=r_4\), and also equality of the angles between the axes in opposite links, i.e. \(\alpha_1=\alpha_3\) and \(\alpha_2=\alpha_4\).

It is not difficult to see that, with these relations between the elements of the mechanism, all six equations are satisfied; by this the mobility of the mechanism under consideration is mathematically proved.

Equations (27) (28) (29) (30) indicate that the lengths of the rods are related as the \(\operatorname{Sin}\)’s of the angles between the axes in the same links.

Using this, it is not difficult, from the mechanism considered, to obtain, as a particular case, a mechanism with parallel axes, or, in other words, a plane four-link mechanism.

For this, the angles of the links, in order that they become parallel, should be set equal to zero, i.e.

\[ \alpha_1=\alpha_2=\alpha_3=\alpha_4=0. \]

Then, as before, equations (22) and (25) are satisfied, and from equations (27) (28) (29) (30) we conclude that the ratio of the rods turns into an indeterminacy of the form

\[ \frac{0}{0}, \]

therefore they all may be chosen of arbitrary length. All this, as is known, actually takes place in plane four-link mechanisms.

To obtain a mechanism with axes intersecting at one point, it is necessary, as is not difficult to notice, to set the lengths of the rods equal to zero, i.e.

\[ r_1=r_2=r_3=r_4=0. \]

Then equations (22) and (25) are also satisfied, and from the four other equations we conclude that the ratios of the \(\operatorname{Sin}\)’s of the angles turn into an indeterminacy; consequently such a mechanism is possible for any angles, which also, as is known, is quite correct.

The mechanism shown in (drawing 2) has two forms: the first—open, and the second—“crossed.” One or the other form is determined by the choice of signs in the six equations given.

Mechanisms of these two forms can transmit rotational motion between two skew axes.

Just as in the Hooke joint, here too uniformity of transmission will be absent, i.e., with uniform rotation of the link serving as the driving crank, nonuniform rotation of the driven crank is obtained. The derivation of the equations relating the velocities presents no difficulty, but, in view of its great length, is omitted here.

It is also interesting to note that both of the described types of mechanism have no dead points.

Turning now to the mechanism shown in (Fig. 3), an analysis of which could not be carried out, it should be noted that, as the constructed models have shown, for certain values of the elements of the links it is possible. For it, all the requirements imposed on the mechanism considered earlier must be observed, and, in addition, the equality of the segments formed by the ends of the rods on the opposite axes must be fulfilled.

This article, without giving a complete analysis of the mechanisms described, proposes an entirely new four-link mechanism and indicates its internal connection with previously known four-link mechanisms.