KINEMATIC CHAIN AS A TOOL FOR ANALYZING CERTAIN MATHEMATICAL FUNCTIONS¹

D. G. STANKO

In the general theory of machines and mechanisms, a kinematic chain is considered from the point of view of structure and classification; kinematic and dynamic analysis is carried out, and the synthesis of the chain is touched upon to some extent.

Basically, the problem is reduced to analyzing mechanisms with varying degrees of mathematical formalization. Of course, such work is necessary, but it should not be exhausted by this alone.

Sufficient attention is paid to the kinematic chain, but the question of what the kinematic chain itself can provide, as a movable geometric image, is usually not addressed. Let us approach the kinematic chain from a somewhat different point of view.

Fig. 1

Suppose that we have a certain kinematic chain (Figs. 1, 2), consisting of a crank of radius \(r\) and a connecting rod of length \(l\). At point \(A\) we have

Fig. 2

joint, and at point \(B\)—a sliding pair with a joint. Let us introduce the notation that we shall use below:

\(\alpha\) — the angle between the initial position and the crank radius. As the crank rotates, this angle changes in magnitude;

\(\rho\) — the variable radius vector—the straight line joining the center of rotation of the crank with the position of the slider—a quantity variable in fractions of the crank radius;

\(\alpha_1\) — the angle formed by the position of the crank and the radius vector \(\rho\)—a variable quantity;

\(\beta\) — the angle formed by the connecting rod \(l\) and the radius vector \(\rho\)—a variable quantity;

\(c\) — the minimum distance from the center of rotation of the crank to the trajectory of point \(B\), in other words, the perpendicular dropped from the center \(O\) to the trajectory of point \(B\) (the trajectory of point \(B\) is a straight line). This quantity is constant;

\(\rho_0\) — the quantity \(c\), expressed in fractions of the crank radius;

\(\varphi\) — the variable angle with vertex at point \(O\), formed by the radius vector \(\rho\) and \(\rho_0\);

\(p\) — the perpendicular dropped from point \(A\) to the radius vector \(\rho\)—a variable quantity.

Let us express the equation of the trajectory of the slider through the radius vector \(\rho\) (see Fig. 1):

\[ \rho=\frac{c}{\cos\varphi}. \tag{1} \]

On the other hand, the value \(\rho\) may be obtained as a function of \(r, l, \alpha\), and \(\beta\). Then

\[ \rho=r.\cos\alpha_1+l.\cos\beta. \tag{2} \]

Equating the right-hand sides of the functions, we have

\[ \frac{c}{\cos\varphi}=r.\cos\alpha_1+l.\cos\beta \]

or

\[ r.\cos\alpha_1.\cos\varphi+l.\cos\beta.\cos\varphi-c=0. \tag{3} \]

The angle \(\beta\) can be expressed through the angle \(\alpha_1\), since

\[ p=r.\sin\alpha_1=l.\sin\beta=l\sqrt{1-\cos^2\beta}, \]

whence

\[ \cos\beta=\frac{1}{l}\sqrt{\,l^2-r^2\sin^2\alpha_1\,}. \]

After substituting \(\cos\beta\), equations (2) and (3) take the form

\[ \rho=r.\cos\alpha_1+\sqrt{\,l^2-r^2\sin^2\alpha_1\,}, \tag{4} \]

\[ r.\cos\alpha_1.\cos\varphi+\cos\varphi\sqrt{\,l^2-r^2\sin^2\alpha_1\,}-C=0. \tag{5} \]

In order that the investigation have a general character, let us abstract our mechanism. To do this, we take the crank radius as unity, i.e. \(r=1\); we take the length of the connecting rod \(l\) in fractions of the crank radius, then \(l=r.k\), or, for \(r=1\), \(l=k\). We shall likewise consider the quantity \(\rho_0=c\) in fractions of the crank radius, or, in other words, in fractions of unity. Now functions (2), (3), (4), (5) will be written as

\[ \rho=\cos\alpha_1+k\cos\beta, \tag{I} \]

\[ \cos\alpha_1.\cos\varphi+k\cos\beta.\cos\varphi-\rho_0=0, \tag{II} \]

\[ \rho=\cos\alpha_1+\sqrt{k^2-\sin^2\alpha_1}, \tag{III} \]

\[ \cos\alpha_1\cdot\cos\varphi+\cos\varphi\sqrt{k^2-\sin^2\alpha_1}-\rho_0=0, \tag{IV} \]

\[ \rho=\sin\alpha_1=k\sin\beta. \tag{V} \]

Into the resulting group of functions one may introduce the angle of rotation \(\alpha\), replacing the angle \(\varphi\), since

\[ \varphi=\alpha-\alpha_1. \]

Then

\[ \cos\alpha_1\cdot\cos(\alpha-\alpha_1)+k\cos\beta\cdot\cos(\alpha-\alpha_1)-\rho_0=0, \tag{VI} \]

\[ \cos\alpha_1\cdot\cos(\alpha-\alpha_1)+\cos(\alpha-\alpha_1)\cdot\sqrt{k^2-\sin^2\alpha_1}-\rho_0=0. \tag{VII} \]

Thus we obtain a functional dependence among a number of quantities \(\alpha_1,\beta,\varphi,\alpha\) and certain constants \(\rho_0\) and \(k\). Moreover, we have dependences both in explicit and in implicit form. Let us proceed to the analysis of the functions obtained from I to VII. For this purpose we shall make use of a movable geometric image. Several variants are possible here.

Variant 1. \(k<\rho_0<2;\quad k>1.\)

We shall move our kinematic chain by rotating the crank or by moving the slider (see Figs. 2, 3). The minimum value of \(\rho\) will be equal to \(\rho_0\). The position of the mechanism corresponding to \(\rho_0\) is shown separately in Fig. 2.

Fig. 3

We have

\[ \rho_{\min}=\rho_0 \]

or

\[ \rho_{\min}=\cos\alpha_0+k\cos\beta_0 \]

\[ \rho_{\min}=\cos\alpha_0+\sqrt{k^2-\sin^2\alpha_0}. \]

The values of the angles \(\alpha_0\) and \(\beta_0\) are not difficult to find, since the sides of the triangle \(OA_0B_0\) are known and are equal to \(1,k_0\), and \(\rho_0\). At \(\rho=\rho_{\min}\) we have the minimum value of the angle \(\varphi\), which is equal to zero, that is

\[ \varphi_{\min}=0. \]

The maximum value of \(\rho\) will occur at \(\rho_{\min}\), which gives

\[ \rho_{\max}=\sin\alpha_0=k\sin\beta_0. \]

The dead positions of the mechanism (see Fig. 3) at the points \(B'\) and \(B'_1\) give the values of the maximum for \(\rho\), namely

\[ \rho_{\max}=1+k. \]

At \(\rho_{\max}\) we have the maximum value of the angle \(\varphi\). From the triangles \(OB'B_0\) or \(OB'_1B_0\) (see Fig. 3) we find \(\varphi_{\max}\) through \(\cos\varphi_{\max}\), since

\[ \cos\varphi_{\max}=\frac{\rho_0}{1+k}. \]

If \(OB_0\) is taken as the zero position, then the maximum value of the crank rotation angle \(\alpha\) will correspond to the position of the connecting rod that is parallel to \(OB_0\). Expressing this by means of the trigonometric relation, we obtain

\[ \cos\alpha_{\max}=\rho_0-k. \]

The value \(\alpha_{\max}\) does not coincide with the position of the mechanism for \(\varphi_{\max}\).

The path traversed by the slider is determined by the dead positions of the mechanism and is equal to \(B'B'_{11}\). The magnitude of half this path is

\[ B_0B'=B_0B'_1=(1+k)\sin\varphi_{\max}, \]

and in this case

\[ \alpha=\varphi_{\max}. \]

How the angle \(\alpha\) changes as the slider moves from the upper dead position to the lower one, and conversely, can be judged from the arc with arrows shown in Fig. 3. We note the following feature: when \(\rho\) changes from \(B'\) to \(B'_1\) and from \(B'_1\) to \(B'\), the change in \(\alpha\) can occur in two ways, namely, the downward motion is along arc I; and if, for arc II, the end is taken as the beginning and the crank is rotated, then we likewise obtain downward motion. An analogous picture is obtained for motion from below upward, which occurs when the crank is rotated along arc II, or else along arc I when the beginning and end of the arc are interchanged. If the angle \(\varphi\) and the radius vector \(\rho\), when changing from \(B'\) to \(B'_1\), occupy correspondingly different positions, then the angle \(\alpha\) at \(\varphi_{\max}\) will correspond to two variable radius vectors \(\rho_{\max}\) and to some \(\rho'\). The latter can be expressed in terms of known quantities by solving the problem purely geometrically.

Of some interest is the character of the change in the position of the point \(B\), if the crank is rotated from the position \(A\) to the position \(A''\). In this case the character of the change of \(\varphi\) and \(\rho\) as functions of the angle of rotation of the crank is represented in Fig. 3 (on the right) in the form of straight lines with arrows. When the crank is rotated clockwise, the point \(B\) will occupy the positions \(B\to B'\to B''\), and when rotated counterclockwise, \(B''\to B'_1\to B\). Keeping the direction of rotation of the crank unchanged, the displacement of the point \(B\) may take place along another path, namely, clockwise \(B\to B'_1\to B''\), counterclockwise \(B''\to B'\to B\).

Let us note that the arcs showing the direction of rotation of the point \(A\) when the point \(B\) moves from its extreme positions upward and downward, as it were, reflect the character of the motion of the point \(B\) when the crank is rotated from the position \(A\) to the position \(A''\), and, conversely, from the position \(A''\) to \(A\). Proceeding from the motion of the mechanism, one could obtain a number of relations between \(\alpha,\alpha_1,\beta,\varphi,\rho,p\), construct these relations in the coordinates “\(xy\),” and obtain a number of curves expressed both in the ordinary and in parametric form. The numerical value can be obtained purely graphically from the drawing. If greater accuracy is required, the values of the quantities can be computed analytically. The entire analysis is carried out under the control of the mechanism.

Let us proceed to the subsequent analysis of our functions.

Variant 2. \(\rho_0 = k = 1\) (Fig. 4).

Let us note the special features of the functions in this case.

We have

\[ \alpha_1=\beta, \]

\[ \alpha_0=\beta_0. \]

Substituting the values of the angles into functions I and II, after transformation we obtain

\[ \rho=\cos \alpha_1(1+k), \]

\[ \cos \alpha_1 \cos \varphi(1+k)-\rho_0=0. \]

The remaining functions (III, IV, V, VI, VII) will have their previous form.

Fig. 4

The value of \(\varphi_{\max}=60^\circ,\ \alpha_{1\max}=45^\circ,\ \alpha_{\max}=90^\circ\).

Everything else stated in the first variant can be extended to the second variant.

In what follows, in the investigation we shall note only what is characteristic of the given variant, assuming that everything else has already been discussed earlier.

Variant 3. \(\rho=k>1\) (Fig. 5).

In this variant the form of the function is preserved. The value of the maximum angle of rotation

\[ \alpha_{\max}=90^\circ, \]

that is, the same as in variant 2.

Variant 4. \(\rho=k<1\) (Fig. 6).

We find the minimum value of \(\rho\).

We have

\[ \rho_{\min}=\rho_0 \]

or

\[ \rho_{\min}=\cos \alpha_0-k\cos(180^\circ-\beta^\circ), \]

\[ \rho_{\min}=\cos \alpha_0+k\cos\beta. \]

Fig. 5

Fig. 6

Thus, although the triangle has become obtuse-angled with obtuse angle \(\beta_0\), the form of the function does not change.

Among the special features of this variant one may note that, in a certain position, \(B_{\mathrm{н}}\) and \(B'_{\mathrm{н}}\) of the connecting rod are located on the straight-line trajectory of point \(B\). Let us find the value of the radius vector \(\rho = OB_{\mathrm{н}}\) corresponding to this characteristic position. According to the adopted notation, we have

\[ \overline{OB'_{\mathrm{н}}}=r=1;\qquad \overline{B'_{\mathrm{н}}B}=l=k;\qquad OB_0=c=\rho_0. \]

We solve the problem geometrically. From triangle \(OB_{\mathrm{н}}B_0\), the value of \(\rho\) is equal to

\[ \rho=\sqrt{\rho_0^2+\overline{B_0B_{\mathrm{н}}}^{\,2}}. \tag{a} \]

In turn, we have the equalities

\[ B_0B_{\mathrm{н}}=B_0B'_{\mathrm{н}}-k, \]

\[ B_0B'_{\mathrm{н}}=\sin \alpha=\sqrt{1-\cos^2\alpha}, \]

\[ \cos\alpha=\rho_0. \]

After substitution and transformation in equation \((a)\), we obtain

\[ \rho=1-2k\sqrt{1-\rho^2}+k^2. \]

At this value of \(\rho\), the crank at point \(B'_{\mathrm{н}}\) intersects the trajectory of point \(B\).

Variant 5. \(k>\rho_0>1;\ \rho_0<k;\ \rho_0>1\) (Fig. 7).

Fig. 7

If, in the variants considered, the kinematic chain could move within an angle \(\alpha\) from the zero position of no more than \(90^\circ\), then in variant 5 the value \(\alpha_{\max}\) is greater than \(90^\circ\). As in variant 4, we have obtuse triangles formed by the crank radius, the connecting rod, and the radius vector \(\rho\), but now the obtuse angle is \(\alpha\).

We shall find an expression for the minimum value of \(\rho\); we have that

\[ \rho_{\min}=\rho_0, \]

\[ \rho_{\min}=k\cos\beta_0-\cos(180-\alpha_0), \]

\[ \rho_{\min}=\cos\alpha_0+k\cos\beta_0, \]

that is, the quantity \(\rho_{\min}\) has the same expression as in the preceding variants.

The value of \(\cos\alpha_{\max}\) will be negative, since

\[ \cos\alpha_{\max}=\rho_0-k, \]

where \(\rho_0\) is less than \(k\).

Variant 6. \(\rho_0=k<1/2\) (Fig. 8).

A characteristic feature of this variant is that the angle \(\alpha\) on the arc \(A_2A'_2\) and the radius vector \(\rho\) on the segment of the straight line \(B_nB'_n\) have a discontinuity,

Fig. 8

otherwise our functions, for the given coefficients, would turn out to be discontinuous.

From the triangle \(OB_nB_0\) we find the value \(\rho_{\min}\):

\[ \rho_{\min}=1-k. \]

\(\rho_{\min}\) is determined by the dead position of the mechanism. In contrast to the similar variants, \(\rho_{\min}\) is not equal to \(\rho_0\).

The value of the segment \(B_nB'_n\) is

\[ B_nB'_n=2\sqrt{\rho_{\min}^2-\rho_0^2} =2\sqrt{1-2k-k^2-\rho_0^2}. \]

The value \(\varphi_{\min}\) can be expressed through the trigonometric relations

\[ \cos\varphi_{\min}=\frac{\rho_0}{1-k}. \]

The minimum value \(\varphi_{\min}\) does not coincide with the crank angle \(\alpha_{\min}\). The latter is determined from the following relation:

\[ \cos \alpha_{\min}=2\rho_0. \]

As the arc arrows show, the nature of the change in the angle \(\alpha\) as the radius vector passes from the point \(B'\) to \(B_h\) differs somewhat from the uniform variants. We shall observe an analogous picture in the displacements of the point \(B\), if the crank is rotated from position \(A\) to position \(A_2\), since the arc arrows as it were represent the motion of the point \(B\).

Variant 7. \(\rho_0>k<1;\ \rho_0>\dfrac{1}{2};\ k<0.5\) (Fig. 9).

Unlike variant 6, here \(\alpha_{\max}\) is not \(90^\circ\), but smaller.

Fig. 9

The values of \(\alpha_{\min}\) and \(\alpha_{\max}\) in variant 7 can be expressed as follows:

\[ \cos \alpha_{\min}=k+\rho_0, \]

\[ \cos \alpha_{\max}=\rho_0-k. \]

All the remaining features are the same as in variant 6.

Variant 8. \(\rho=k=0.5\) (Fig. 10).

This variant is limiting. If \(\rho=k<0.5\), then the function will be discontinuous (see Fig. 8); but if \(\rho=k>0.5\), the functions will be continuous. A special feature is that in the zero position (\(A_0\) and \(B_0\)) the angles \(\alpha_0\) and \(\beta_0\) are equal to zero; the mechanism occupies a dead-center position.

Thus, using the kinematic chain, we have been able to analyze the functions for various values of certain constants \(k\) and \(\rho_0\).

If the arrangement of the quadrants is considered in accordance with Figure 11, then, in the analysis, the mechanism covered only quadrants I and IV and only partially, in variant 3, all four quadrants. All the investigations given above will be valid if the radius vector \(\rho\) is taken as the independent variable.

Fig. 10

In order for the mathematical investigation of the functions to be complete, it is necessary to analyze how the functions behave with independent variables \(\alpha\) and \(\varphi\). This problem is also solved with the aid of the mechanism by a simple mirror reflection of the drawing. Then quadrants I and IV will, respectively, reflect the change of the functions in quadrants III and II. In Figure 11, the shaded part of the quadrants shows that, with the independent variable \(\varphi\), we have a periodic discontinuous function. Within \(2\pi\) there is a discontinuity of the function at two points.

In variants 6 and 7, however, with independent variable \(\varphi\) (see Fig. 12), there is a discontinuity of the function at four points. If \(\alpha\) is taken as the independent variable, then for the variants having \(\rho_0 = k\), the function is continuous. The function is also continuous for variant 5, when \(k > \rho_0 > 1; \rho_0 > 1\). In variant 1 the function is discontinuous. The region of discontinuity is the unshaded part of the circumference (see Fig. 13) at two points. In variants 6 and 7, with independent variable \(\alpha\), the region of discontinuity is at four points (see Fig. 14). The regions of discontinuity for the independent variables \(\alpha\) and \(\varphi\) are not equal in magnitude.

Let us proceed to the analysis of the following variants:

Variants 9, 10, 11, shown in Figs. 15, 16, 17, respectively, have
\(\rho = 1,\ k = 2;\ k > \rho_0 > 1,\ \rho_0 = k - 1;\ k > 1 > \rho_0,\ \rho_0 = k - 1\).
A characteristic feature of these variants, in contrast to those analyzed, is that, with independent variable \(\rho\), when the latter changes within the limits from \(B'\) to \(B'_1\) and from \(B'_1\) to \(B'\), the change of the angle \(\alpha\) will occur over a full crank circle, and the whole process will extend over more than \(2\pi\) (see the arcs with arrows). In the initial position, when \(\rho = \rho_0 = k - 1\), the values of the angles \(\alpha_1\) and \(\beta\) are equal to zero.

With independent variables \(\alpha\) and \(\varphi\), the complete change of the functions will be given by a mirror reflection of the drawing, as was discussed earlier.

For a comparative evaluation, we present a summary of the results of the analysis of the variants examined with independent variable \(\rho\) (see Table No. 1). In the same way one could obtain a summary of the main parameters of the functions with independent variables \(\varphi\) and \(\alpha\), starting from the simple description of the motion of the kinematic chain. It is also possible to obtain the numerical value of all intermediate quantities.

Fig. 11

Fig. 12

Let us proceed to a further analysis of the variants.

Variant 12. \(\rho_0 = 0;\ k > 2\) (Fig. 18).

Fig. 13

Fig. 14

For the value \(\rho_0 = 0\), the radius vector \(\rho\) turns into a certain sliding vector; in this case the value of the angle \(\varphi\) is equal to zero, and the angle \(\alpha_1 = \alpha\). If the adopted notation for the angles is retained, then the value of the functions is written as follows:

\[ \rho = \cos \alpha + k \cos \beta, \]

\[ \rho = \cos \alpha + \sqrt{\sin^2 \alpha - k}, \]

\[ \rho = \sin \alpha = k \sin \beta. \]

Table No. 1

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Some changes have occurred because the angle \(\varphi\) is equal to zero and the angle \(\alpha=\alpha_1\). The dead positions of the mechanism will give the values:

\[ \rho_{\max}=1+k, \]

\[ \rho_{\min}=k-1. \]

Variant 13, with \(\rho=0,\ k=2\) (Fig. 19), introduces nothing new. The values of the maximum and minimum radius vector in this case will be different and equal to

\[ \rho_{\max}=k;\qquad \rho_{\min}=1. \]

Variant 14. \(\rho_0=0;\ k<1\) (Fig. 20).

In variant 14 we likewise have a sliding vector \(\rho\). When point \(B\) is moved from the upper position \(B'\) to the lower \(B'_1\), the angle \(\alpha\) changes in accordance with the arcs with arrows, which corresponds to the independent variable \(\rho\).

Fig. 19

Fig. 20

The function has the same character as in variant 12. The maximum value of \(\alpha\) and, correspondingly, the radius vector in this case are equal to

\[ \sin\alpha_{\max}=k, \]

\[ \rho'=\sqrt{1-k^2}, \]

where the point \(B'_0\) is the point of intersection of the straight-line trajectories of point \(B\). The angles \(\alpha_1\) and \(\beta\) remain as before, while the values of the angles \(\psi'\), \(\varphi_{\max}\), and \(\alpha'_{\max}\) will be different, since they are related to the radius vector \(\rho'_0\).

Let us trace how the functional dependence referred to \(\rho'_0\) is written.

Equation (I) remains unchanged, i.e.,

\[ \rho=\cos\alpha_1+k\cos\beta. \tag{I'} \]

In equation (II), the value of the angle \(\varphi\) is replaced by \(\varphi'\).

According to Figures 24 and 25, respectively, we have

\[ \varphi=\varphi' + \psi, \]

\[ \varphi=\varphi' - \psi, \]

where the angle \(\psi\) expresses the angle of inclination of the trajectory of point \(B\) and is a constant quantity. In other words,

\[ \psi=\mathrm{const}, \]

expressing \(\rho_0\) through \(\rho_0'\), we obtain

\[ \rho_0=\rho_0'\cdot \cos\psi=\mathrm{const}. \]

Substituting the values of \(\varphi\) and \(\rho_0\) into the second equation, we obtain

\[ \cos\alpha_1\cos(\varphi' \pm \psi)+k\cos\beta\cos(\varphi' \pm \psi)-\rho_0'\cos\psi=0. \tag{II'} \]

The form of equation (III) remains unchanged,

i.e.

\[ \rho=\cos\alpha_1+\sqrt{\sin^2\alpha_1-k}. \tag{III'} \]

Into equation IV we introduce the value of the angle \(\varphi\) and \(\rho\). Then we obtain,

\[ \cos\alpha_1\cdot\cos(\varphi' \pm \psi)+\cos(\varphi' \pm \psi)\sqrt{\sin^2\alpha_1-k^2}-\rho_0'\cos\psi=0. \tag{IV'} \]

Equation IV remains unchanged, i.e.

\[ p=\sin\alpha_1=k\sin\beta. \tag{V'} \]

Into the obtained functions we introduce the angle of rotation of the crank, \(\alpha'\). We have

\[ \alpha=\alpha' + \psi, \]

\[ \varphi=\alpha' \mp \psi-\alpha. \]

With independent variable \(\alpha\), the function will have a discontinuity in two places (the dashed arcs). In order to use the mechanism for analysis of the functions, it is necessary to rotate the mechanism through \(180^\circ\).

Variant 15. \(\rho_0=0,\quad k=1\) (Fig. \(20'\)).

The peculiarity of this variant consists in the following: the angles \(\alpha=\beta\), \(\rho_{\min}=0\), \(\rho_{\max}=2\). With independent variable \(\alpha\) on the arc \(\pi\), we have a discontinuity of the radius vector \(\rho=0\). The displacement of point \(B\) from its extreme positions is accomplished at a crank rotation angle of \(180^\circ\).

Variant 16. \(\rho_0=\infty,\quad k=\infty\) (Fig. 21).

In this variant the mechanism has a connecting rod equal to infinity. Obviously, the radius vector will also be equal to infinity.

Since the radius vector is connected with the crank and connecting rod, the end of the radius vector will perform some real displacement along a straight line removed to infinity from the center of rotation of the crank. The mechanism that will satisfy the given functional relationship is shown in Fig. 22. It is the so-called Wolf coulisse, or, in other words, a slider-crank mechanism with an infinitely long connecting rod.

Let us now trace what form the functions under investigation will take. We have:

\[ \rho=(1-\cos\alpha)+\infty, \]

\[ \rho=1+\infty, \]

\[ \rho_{\max}=2+\infty, \]

\[ \rho_{\min}=1+\infty=0. \]

Fig. 20′

Fig. 21

Fig. 22

As we see, infinities participate in the functional dependence. One might say that, in studying the function, we took and brought the end of the radius vector closer and moved the center of rotation of the crank off to infinity; in other words, we inverted our mechanism.

Variant 17. \(\rho_0=0,\ k=0\) (Fig. 23).

In this case the mechanism has no connecting rod; only one crank remains. The trajectory of point \(B\) would have to pass through the center of rotation of the crank.

The value of the radius will occur at two points, for \(\alpha=\dfrac{\pi}{2}\) and \(\pi\), i.e. \(\rho=\pm 1\).

Thus, by means of a kinematic chain an analysis of the functional dependence has been carried out.

Fig. 23

Let us make a few generalizations:

1. The dead positions of the mechanism give a minimum or maximum for \(\rho\) and \(\varphi\).
2. The position of the connecting rod parallel to \(\rho_0\) gives a maximum of \(\alpha\).
3. With the smallest radius vector we have a maximum of \(\rho\).
4. The way in which the function changes for different independent variables \(\rho,\varphi,\alpha\) gives the character of the motion of the mechanism.
5. The entire analysis of the function was carried out for different values of \(k\) and \(\rho_0\). But the analysis of the function may be regarded as a consequence of a parallel translation of the trajectory of point \(B\).

Let us somewhat extend the domain of analysis of functions by means of a kinematic chain. Let us imagine that the trajectory of point \(B\) is traversed along a certain broken straight line. Proceeding from the previous reasoning, we have the trajectory of point \(B\) along two intersecting straight lines having the minimal radius vector \(\rho_0\). With respect to \(\rho_0\) we retain all the previous notation.

Let us refer all the functions under consideration to a certain radius vector \(\rho_0'\), which is the straight line from the intersection of the center of rotation of the crank \(O\) with the point \(B'_0\) (see Figs. 24, 25). Substituting the values of \(\varphi\) and \(\rho_0\) into equations VI and VII, we have

\[ \cos \alpha_1 \cos(\alpha' \pm \psi-\alpha_1) + k \cos \beta \cos(\alpha' \pm \psi-\alpha_1) - \rho_0' \cos \psi, \tag{VI} \]

\[ \cos \alpha_1 \cos(\alpha' \pm \psi-\alpha_1) + \cos(\alpha' \pm \psi-\alpha_1)\sqrt{\sin^2\alpha_1-k^2} - \rho_0' \cos \psi. \tag{VII'} \]

In analogous fashion, these functions can be analyzed by means of a kinematic chain, since it is not difficult to construct a mechanism by specifying one parameter or another. Let us note several characteristic points provided by the chain (see Figs. 24 and 25). The maximum value of \(\rho\) is, as before, determined by the dead positions of the mechanism, i.e.

\[ \rho'_{\max}=1+k. \]

For \(\rho'_{\max}\) we have the maximum of the angle \(\varphi'\).
In contrast to the previously examined variants, the value of \(\rho_0\) will be replaced by its value.
Then we have

\[ \cos \varphi'_{\max}=\frac{\rho'_0\cos\psi}{1+k}. \]

Fig. 24

Taking the value \(\rho_0\) as the zero position for the angle of rotation of the crank, we shall find the maximum value \(\alpha'_{\max}\). This value will occur when the connecting rod is positioned perpendicular to the trajectory of point \(B\).
It is not difficult to find that

\[ \cos \alpha_{\max}=\rho_0\cos\psi-k. \]

The arcs with arrows show how the value of \(\alpha\) will change when the independent variable is \(\rho\), as point \(B\) moves upward and downward. If

Fig. 25

the independent variable is the angle of rotation \(\alpha'\), then the character of the motion of point \(B\), as the end of the vector \(\rho\), and, correspondingly, the change of the angle \(\varphi'\) in this case will proceed according to the straight arrows shown on the right in Figs. 24 and 25.

The functional dependence expressed by equations (I′)—(VII′) can be analyzed for various values. A characteristic feature is that, for \(\psi=90^\circ\), we shall have a crank-and-slider mechanism, as in Figs. 18–20, rotated through \(90^\circ\). But in this case \(\rho_0'\) will not be equal to zero.

The analysis of the function can be carried out not only by varying the angle \(\psi\), but also by varying the values of \(\rho_0\) and \(k\) within the widest limits. With a broken-line trajectory of point \(B\), the change of the angle \(\alpha', \varphi_1 \varphi'\) is continuous, although the end of the vector \(\rho\) will undergo a change at the point of transfer. In other words, the trajectory of point \(B\) runs not along one straight line, but along two straight lines.

Fig. 26

It is obvious that the analysis of the function can be carried out for different values of \(\psi\).

Figure 26 shows the kinematic chain for the analysis of functions for

\[ k>1,\quad \rho_0'>k,\quad \psi(\mathrm{I}),\quad \psi_{\mathrm{III}}=\psi_{\mathrm{IV}}=0. \]

Correspondingly, in Fig. 27,

\[ k>1;\quad \rho_0'>k,\quad \psi_{\mathrm{I}},\ \psi_{\mathrm{IV}}; \]

in Fig. 28,

\[ k<1,\quad \rho_0'>k>1,\quad \psi_{\mathrm{II}};\quad \psi_{\mathrm{III}}=\psi_{\mathrm{IV}}=0; \]

in Fig. 29,

\[ k=1;\quad \rho_0'>1,\quad \psi_{\mathrm{I}};\ \psi_{\mathrm{III}}; \]

in Figs. 30 and 31,

\[ k=\infty;\quad \rho_0'=\infty;\quad \psi_{\mathrm{IV}};\quad \psi_{\mathrm{I}}=\psi_{\mathrm{II}}=0; \]

in Figs. 32 and 33,

\[ k=\infty;\quad \rho_0'=\infty;\quad \psi_{\mathrm{II}};\ \psi_{\mathrm{III}}. \]

Fig. 27

Fig. 28

Fig. 29

Fig. 30

Fig. 31

Fig. 32

It is not difficult, proceeding from the movable geometric image, to describe the characteristic properties of the functions under investigation, taking as independent various quantities entering into the function. We pass to the conclusion.

The purpose of the present work has been to show that a kinematic chain, in itself, can be used for the analysis of certain mathematical functions—in the form of a movable geometric image.

Such operations as finding the maximum and minimum of a function are performed purely automatically. The character of the functional variation of many variables in their mutual interrelation is also easily represented by the mechanism, something that is perhaps difficult to achieve by any other means.

Fig. 33

When a kinematic chain is used, the problem can be solved purely graphically, but at the same time this very problem, using the mechanism, can also be solved purely analytically. The analytical solution is carried out under the control of the mechanism, thereby avoiding the possibility of errors.

As was indicated, a kinematic chain is a means for the analysis of certain mathematical functions. If, however, one approaches the question in principle, from the standpoint of formulating the problem, then the word “certain” could be replaced by “a very great many.” In fact, there exists an innumerable number of kinematic chains with one and with the \(n\)-th degree of freedom. All these chains in some way represent the analysis of functions. But even the case considered here of a kinematic chain of the “crank-and-curvilinear-slider mechanism” type can be considerably extended by causing the slider to move along curves of various kinds. As kinematic chains are studied, properties will be revealed that can be used for solving various kinds of mathematical problems.

The abstraction of kinematic chains is applicable not only in solving mathematical problems, but also, to a certain extent, in solving the problem of the synthesis of mechanisms.

It may be expected that the proposed method will find an appropriate place in the solution of both mathematical and technical problems.

1. Editor’s note. The article is published as a contribution to discussion. ↩