ANALYSIS OF THE THEORETICAL ACCURACY OF OPERATION OF A CERTAIN GROUP OF SPATIAL MECHANISMS FOR GENERATING AN ELLIPSE

E. I. TOROPYGIN

(Presented by the scientific seminar of the departments of descriptive geometry and graphics of the higher educational institutions of Tomsk)

In automatic machines and computing mechanisms, cams, cylinders, and gear wheels of elliptical form have found wide application. If until recently the use of elliptical gears was hindered by the insufficient development of the theory of their calculation and of production technology [1], then with the appearance of the works of F. L. Litvin [2, 3, and others] the solution of these questions has been substantially advanced. As a result, in the preparation of working drawings of such parts there has arisen a considerable need for the mechanical drawing of ellipses. No less a need for the mechanical drawing of ellipses also exists in mathematical cartography [10], etc.

The great variety of existing mechanisms for the mechanical drawing of ellipses does not, however, give grounds for judging that this question has been completely solved and that for each individual case there is a recommended group of mechanisms. In the present-day literature, in the overwhelming majority of cases, there are only descriptions and drawings of one or another mechanism, but no guaranteed accuracy is indicated for drawing an ellipse for a given ratio of its axes and dimensions. Naturally, under these conditions it is often difficult to select a suitable mechanism. In order to fill this gap to some extent, the author has divided all mechanisms for drawing ellipses into two types: plane and spatial, of which, for a number of reasons [4], only spatial mechanisms are considered.

Many of the spatial mechanisms have such kinematic schemes that they predetermine in advance a distortion of the curve of the ellipse [4]. Depending on the nature of the distortion of the ellipse and on the principle of operation, the spatial mechanisms, in turn, are divided into four groups. The first group is based on transforming the ellipse by the method of similarity; the second—on laying off constant segments on normals to the given ellipse curve; the third—on laying off constant segments on the radius-vectors of the given ellipse curve; the fourth group includes all the remaining mechanisms that have no theoretical errors in drawing the curve.

An analysis of the operation of the mechanisms of the first group is given in [4]. The aim of the present article is briefly to illuminate the results of the study—

[[unclear: beginning of line]] investigations of the theoretical accuracy of operation of mechanisms of the third group. These results may also be used to determine the accuracy of operation of copying devices whose principle of action is analogous to the mechanisms under consideration.

In accordance with the classification we have adopted, the third group of mechanisms includes: 1. I. Günter’s trammel for drawing ellipses and other curves [5]. 2. P. P. Nikolaenko’s ellipsograph [6]. 3. N. A. Kozhemyakin’s trammel for drawing ellipses [7]. 4. V. F. Gumilevsky’s device for drawing ellipses [8].

The theoretical error of the mechanisms was determined by comparing the curves produced by these mechanisms with geometrically exact ellipse curves constructed in the same polar axes. Suppose that curve 1 with semiaxes \(a\) and \(b\) (Fig. 1) is the prescribed ellipse curve, relative to which equal distances \(d\) are laid off along the radius vectors, and the conchoid curve 2 is obtained, which is generated by the given mechanisms. Following the operating principle of the mechanisms of this group, we shall determine the theoretical error \(f\) from the radius vectors, having constructed ellipse 3 with the same semiaxes \(a+d\) and \(b+d\) as curve 2. For ellipse curve 1 we have

\[ x = a \cos t; \]

\[ y = b \sin t. \tag{1} \]

From Fig. 1 it is seen that

\[ \frac{y}{x} = \operatorname{tg}\varphi . \]

Substituting here the values of \(x\) and \(y\) from (1), we obtain

\[ \operatorname{tg} t = \frac{a}{b}\operatorname{tg}\varphi \tag{2} \]

or

\[ t = \operatorname{arc\,tg}\frac{a}{b}\operatorname{tg}\varphi . \tag{3} \]

On the basis of (3), expression (1) may be written as

\[ \left. \begin{aligned} x &= a \cos \operatorname{arc\,tg}\frac{a}{b}\operatorname{tg}\varphi,\\ y &= b \sin \operatorname{arc\,tg}\frac{a}{b}\operatorname{tg}\varphi . \end{aligned} \right\} \tag{4} \]

For ellipse 3, respectively,

\[ \left. \begin{aligned} x_2 &= (a+d)\cos \operatorname{arc\,tg}\frac{a+d}{b+d}\operatorname{tg}\varphi,\\ y_2 &= (b+d)\sin \operatorname{arc\,tg}\frac{a+d}{b+d}\operatorname{tg}\varphi . \end{aligned} \right\} \tag{5} \]

Fig. 1. Comparison of curve 2, generated by the mechanisms, with curve 3 of the exact ellipse.

Visible annotations in the figure:

\[ a = 120\ \text{mm}, \qquad b = 40\ \text{mm}, \qquad d = 40\ \text{mm}. \]

Calculated value:

\[ f = 11.4\ \text{mm}. \]

Graphical value:

\[ f = 12.1\ \text{mm}. \]

Axes: \(x\), \(y\).

The required quantity \(f\), in order to simplify the solution of the problem, will be expressed as follows:

\[ f=m-d, \tag{6} \]

where \(m\) is the distance between ellipses 1 and 3, measured along the radius vectors. The quantity \(m\) can be expressed as

\[ m=\sqrt{(x_2-x)^2+(y_2-y)^2}. \]

Substituting here the values \(x, x_2, y, y_2\) from (4) and (5), we obtain

\[ m=\sqrt{ \left[ (a+d)\cos\arctan\frac{a+d}{b+d}\tan\varphi - a\cos\arctan\frac{a}{b}\tan\varphi \right]^2 + \left[ (b+d)\sin\arctan\frac{a+d}{b+d}\tan\varphi - b\sin\arctan\frac{a}{b}\tan\varphi \right]^2 }. \tag{7} \]

By assigning different values to the angle \(\varphi\), it is possible, for any parameters \(a\), \(b\), and \(d\), to determine the quantity \(m\). From the found value \(m\), with the aid of (6), one can find \(f\).

From Fig. 1 it is evident that, as the angle \(\varphi\) changes from \(0\) to \(90^\circ\), the value of the theoretical error \(f\) changes from zero to a maximum and vice versa. Of greatest interest to us is the maximum value of \(f\). To determine it, it is necessary to find the value of the angle \(\varphi\) at which it will occur, substitute this value of the angle into (7), and from (7) pass to (6). For this purpose we take the derivative of \(f\) with respect to \(\varphi\), first substituting the value of \(m\) from (7) into (6):

\[ \frac{df}{d\varphi} = \frac{1}{m} \Biggl\{ \left[ (a+d)\cos\arctan\frac{a+d}{b+d}\tan\varphi - a\cos\arctan\frac{a}{b}\tan\varphi \right] \]

\[ \times \left[ -(a+d)\sin\arctan\frac{a+d}{b+d}\tan\varphi \, \frac{1}{1+\dfrac{(a+d)^2}{(b+d)^2}\tan^2\varphi} \, \frac{a+d}{b+d} \, \frac{1}{\cos^2\varphi} \right. \]

\[ \left. + a\sin\arctan\frac{a}{b}\tan\varphi \, \frac{1}{1+\dfrac{a^2}{b^2}\tan^2\varphi} \, \frac{a}{b} \, \frac{1}{\cos^2\varphi} \right] \]

\[ + \left[ (b+d)\sin\arctan\frac{a+d}{b+d}\tan\varphi - b\sin\arctan\frac{a}{b}\tan\varphi \right] \]

\[ \times \left[ (b+d)\cos\arctan\frac{a+d}{b+d}\tan\varphi \, \frac{1}{1+\dfrac{(a+d)^2}{(b+d)^2}\tan^2\varphi} \, \frac{a+d}{b+d} \, \frac{1}{\cos^2\varphi} \right. \]

\[ \left. - b\cos\arctan\frac{a}{b}\tan\varphi \, \frac{1}{1+\dfrac{a^2}{b^2}\tan^2\varphi} \, \frac{a}{b} \, \frac{1}{\cos^2\varphi} \right] \Biggr\} =0. \]

The expression \(\frac{1}{m}\) is equal to zero. Equating to zero the expression enclosed in braces, we write it in the following form:

\[ \begin{aligned} &\left[(a+b)\cos\arctan\frac{a+d}{b+d}\tan\varphi -a\cos\arctan\frac{a}{b}\tan\varphi\right]\times \\ &\times\left\{ a^{2}b\sin\arctan\frac{a}{b}\tan\varphi \left[(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi\right]\right. \\ &\left.\qquad -(a+d)^{2}(b+d)\sin\arctan\frac{a+d}{b+d}\tan\varphi \left(b^{2}+a^{2}\tan^{2}\varphi\right) \right\}+ \\ &+\left[(b+d)\sin\arctan\frac{a+d}{b+d}\tan\varphi -b\sin\arctan\frac{a}{b}\tan\varphi\right]\times \\ &\times\left\{ (a+d)(b+d)^{2}\cos\arctan\frac{a+d}{b+d}\tan\varphi \left(b^{2}+a^{2}\tan^{2}\varphi\right)\right. \\ &\left.\qquad -ab^{2}\cos\arctan\frac{a}{b}\tan\varphi \left[(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi\right] \right\}=0. \end{aligned} \tag{8} \]

It is also evident from Fig. 1 that

\[ \tan t_{1}=\frac{a+d}{b+d}\tan\varphi . \tag{9} \]

Taking (2) and (9) into account, we make the following substitution of trigonometric functions

\[ \begin{aligned} \sin\arctan\frac{a+d}{b+d}\tan\varphi &=\sin t_{1} =\frac{(a+d)\tan\varphi} {\sqrt{(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi}},\\[6pt] \cos\arctan\frac{a+d}{b+d}\tan\varphi &=\cos t_{1} =\frac{b+d} {\sqrt{(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi}},\\[6pt] \sin\arctan\frac{a}{b}\tan\varphi &=\sin t =\frac{a\tan\varphi} {\sqrt{b^{2}+a^{2}\tan^{2}\varphi}},\\[6pt] \cos\arctan\frac{a}{b}\tan\varphi &=\cos t =\frac{b} {\sqrt{b^{2}+a^{2}\tan^{2}\varphi}} . \end{aligned} \tag{10} \]

On the basis of (10), expression (8) may be represented as

\[ \left[ \frac{(a+d)(b+d)} {\sqrt{(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi}} - \frac{ab} {\sqrt{b^{2}+a^{2}\tan^{2}\varphi}} \right]\times \]

\[ \times\left\{ \frac{ a^{3}b\tan\varphi\left[(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi\right] \sqrt{(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi} } { \sqrt{b^{2}+a^{2}\tan^{2}\varphi}\, \sqrt{(b+d)^{2}+(a+d)^{2}\tan^{2}\varphi} } -\right. \]

\[ -\frac{(a+d)^3(b+d)\operatorname{tg}z\,(b^2+a^2\operatorname{tg}^2 z)\sqrt{b^2+a^2\operatorname{tg}^2 z}} {\sqrt{b^2+a^2\operatorname{tg}^2 z}\sqrt{(b+d)^2+(a+d)^2\operatorname{tg}^2\varphi}} + \]

\[ +\operatorname{tg}z\, \frac{(a+d)(b+d)^3(b^2-a^2\operatorname{tg}^2 z)\sqrt{b^2+a^2\operatorname{tg}^2 z}} {\sqrt{b^2+a^2\operatorname{tg}^2 z}\sqrt{(b+d)^2+(a+d)^2\operatorname{tg}^2\varphi}} \]

\[ -\operatorname{tg}z\, \frac{ ab^3\left[(b+d)^2+(a+d)^2\operatorname{tg}^2 z\right]\sqrt{(b+a)^2+(a+d)^2\operatorname{tg}^2 z} } {\sqrt{b^2+a^2\operatorname{tg}^2 z}\sqrt{(b+d)^2+(a+d)^2\operatorname{tg}^2\varphi}} =0. \tag{11} \]

The part of equation (11) enclosed in square brackets, i.e.

\[ \left[ \frac{(a+d)(b+d)} {\sqrt{(b+d)^2+(a+d)^2\operatorname{tg}^2 z}} - \frac{ab} {\sqrt{b^2+a^2\operatorname{tg}^2\varphi}} \right]=0 \]

has no real roots. Let us examine its other part, enclosed in braces. To do this we shall perform the corresponding transformation and represent it as follows:

\[ \begin{aligned} &a^2(a+d)^2\left\{(a+d)^4(a^2-b^2)^2b^2 -(b+d)^2a^4\left[(a+d)^2-(b+d)^2\right]^2\right\}\operatorname{tg}^6 z \\ &+3(a+d)^2(b+d)^2a^2b^2\left\{(a+d)^2(a^2-b^2)^2 -a^2\left[(a+d)^2-(b+d)^2\right]^2\right\}\operatorname{tg}^4 z \\ &+3(a+d)^2(b+d)^2a^2b^2\left\{(b+d)^2(a^2-b^2)^2 -b^2\left[(a+d)^2-(b+d)^2\right]^2\right\}\operatorname{tg}^2 z \\ &+(b+d)^2b^2\left\{(b+d)^4a^2(a^2-b^2)^2 -(a+d)^2b^4\left[(a+d)^2-(b+d)^2\right]^2\right\}=0. \end{aligned} \tag{12} \]

Let us make the following substitution:

\[ \left. \begin{aligned} &a^2(a+d)^2\left\{(a+d)^4(a^2-b^2)^2b^2 -(b+d)^2a^4\left[(a+d)^2-(b+d)^2\right]^2\right\}=A,\\ &3(a+d)^2(b+d)^2a^2b^2\left\{(a+d)^2(a^2-b^2)^2 -a^2\left[(a+d)^2-(b+d)^2\right]^2\right\}=B,\\ &3(a+d)^2(b+d)^2a^2b^2\left\{(b+d)^2(a^2-b^2)^2 -b^2\left[(a+d)^2-(b+d)^2\right]^2\right\}=C,\\ &(b+d)^2b^2\left\{(b+d)^4a^2(a^2-b^2)^2 -(a+d)^2b^4\left[(a+d)^2-(b+d)^2\right]^2\right\}=D,\\ &\operatorname{tg}^2 z=z_0 . \end{aligned} \right\} \tag{13} \]

Then equation (12) is written in the following form:

\[ Az_0^3+Bz_0^2+Cz_0+D=0. \tag{14} \]

We shall solve the resulting equation of the third degree by Cardano’s formula [9]. To do this, divide equation (14) by \(A\) and obtain

\[ z_0^3+\frac{B}{A}z_0^2+\frac{C}{A}z_0+\frac{D}{A}=0. \tag{15} \]

Let us make the following substitution of coefficients:

\[ \frac{B}{A}=\gamma;\qquad \frac{C}{A}=\Theta;\qquad \frac{D}{A}=\varepsilon. \tag{16} \]

Then equation (15) is written as

\[ z_0^3+\eta z_0^2+\Theta z_0+\varepsilon=0 . \tag{17} \]

We replace the unknown \(z_0\) by a new unknown \(z\), which is related to the unknown \(z_0\) by the equality

\[ z_0=z-\frac{\eta}{3}. \tag{18} \]

After this, equation (17) takes the form of an incomplete cubic equation and is written as follows:

\[ z^3+\frac{3\Theta-\eta^2}{3}z+ \frac{\eta(2\eta^2-9\Theta)+27\varepsilon}{27}=0 . \tag{19} \]

We make the substitution

\[ \frac{3\Theta-\eta^2}{3}=p;\qquad \frac{\eta(2\eta^2-9\Theta)+27\varepsilon}{27}=q . \tag{20} \]

The incomplete cubic equation (19) takes the form

\[ z^3+pz+q=0 . \tag{21} \]

Equation (21) has three roots. Let one of these roots be \(z_1\). Cardano’s formula, expressing the roots of equation (21) through its coefficients by means of square and cube radicals, is written as follows:

\[ z_1= \sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}. \tag{22} \]

From the root \(z_1\) found, by means of (18) we pass to \(z_0\), and from \(z_0\) by means of (13) we pass to the angle \(\varphi\). In Fig. 2 is shown the variation of the angle \(\varphi\) as a function of the variable quantities \(a\) and \(d\) for \(b=\mathrm{const}\), where the indicated parameters \(a\) and \(d\) are reduced quantities, i.e., expressed in terms of \(b\). From this figure it is seen that, with increasing \(a\), the value of the angle \(\varphi\), for one and the same value of \(d\), decreases; moreover, this decrease of the angle \(\varphi\) lags somewhat behind the increase of \(a\). From the same figure it is seen that, with increasing \(d\), the value of the angle \(\varphi\), for one and the same value of \(a\), also increases; moreover, in passing from smaller values of \(a\) to larger ones this increase becomes more intensive.

Fig. 2. Dependence of the angle \(\varphi\), at which the error of mechanisms for generating ellipses assumes its maximum value, on \(a\) and \(d\).

For the found value of the angle \(\varphi\), in accordance with (7) \(m\) assumes its maximum value. From this value of \(m\), with the aid of

(6), we determine the sought value \(f\). This value \(f\) represents the maximum deviation, measured along radius vectors, between the curve obtained by the mechanisms and the point of the curve of the ellipse constructed on the same semiaxes. The problem has been solved in general form.

In order to accelerate the process of determining the absolute values \(f\) for particular values of \(a\) and \(d\) at \(b=\mathrm{const}\), the values of \(f\) were calculated from the formulas given above, and the data were expressed graphically. In Fig. 3 the dependence of the absolute value \(f\) on the variables \(a\) and \(d\), at \(b=\mathrm{const}\), is presented. Analysis of the character of the curves shows that, as \(a\) increases, the absolute error \(f\), for a constant value of \(d\), increases; moreover, in passing from smaller values of \(d\) to larger ones, this increase becomes more intensive. Similarly, as \(d\) increases, the absolute error \(f\), for a constant value of \(a\), increases, and this increase also becomes more intensive in passing from smaller values of \(a\) to larger ones.

Fig. 3. Dependence of the absolute error \(f\) on the variables \(a\) and \(d\).

In mechanical-engineering practice, it is more often not the absolute value of the deviation from the nominal dimension that is of interest, but the relative one. In our case, as applied to parts having an elliptical form, one may take, for example, as the measure of deviation from the nominal dimension the ratio of the absolute error \(f\) to the major semiaxis of ellipse 3, which is also the semiaxis of conchoid 2 (Fig. 1). In Fig. 4 the graphical dependence of the relative error of the mechanisms, \(f_{\mathrm{rel}}\), on the variables \(a\) and \(d\), at \(b=\mathrm{const}\), is presented, where the ratio of the absolute error \(f\) to the value \(a+d\) is taken as the measure of distortion of the exact curve of the ellipse. For convenience of reading, the relative error \(f_{\mathrm{rel}}\) is represented in the graph by a value one thousand times greater than its actual value.

Analysis of the curves shows that, with an increase in \(a\), \(f_{\mathrm{rel}}\) increases for a constant value of \(d\). In passing from smaller values of \(d\) to larger ones, for \(a=\mathrm{const}\), \(f_{\mathrm{rel}}\) at first increases rapidly and then, having reached a certain value—a maximum—begins gradually to decrease. Points I, II, III, IV represent the maximum values of \(f_{\mathrm{rel}}\) for the corresponding values of \(a\). Figure 5 presents the graphical dependence of \(f_{\mathrm{rel.\,max}}\) on the variable \(a\). The value \(d\) is not taken into account in this case. If necessary, it can be determined from Fig. 4.

In order to check the correctness of the analytical and graphical expressions given, let us consider a specific example (Fig. 1). Let \(a=120\) mm, \(b=40\) mm, and \(d=40\) mm. Since the error of the mechanisms was determined with the parameters of the curves reduced to one parameter, namely to \(b\), we shall do the following:

Fig. 4. Dependence of the relative error \(f_{\mathrm{rel}}\) on the variable quantities \(a\) and \(d\).

Fig. 5. Dependence of the maximum relative error \(f_{\mathrm{rel\,max}}\) on the variable \(a\).

\[ a=\frac{120\,\mathrm{mm}}{40\,\mathrm{mm}}=3;\qquad d=\frac{40\,\mathrm{mm}}{40\,\mathrm{mm}}=1. \]

In this case the conversion scale of the units will have the value

\[ M=\frac{40\,\mathrm{mm}}{1}=40\,\mathrm{mm}. \tag{23} \]

From Fig. 2 we determine the angle \(\varphi\) at which the value of the error \(f\) will be maximal: \(\varphi \cong 25^\circ\). From the graph given in Fig. 3, we find the absolute error \(f \cong 0.3\). On the basis of (23), for our specific example the error of the mechanisms is

\[ f \cong 0.3\cdot 40\,\mathrm{mm}\cong 12\,\mathrm{mm}. \]

If a more accurate solution of the problem is necessary, one should proceed as follows. In accordance with (13), determine the coefficients \(A, B,\)

$C$ and $D$. For our example, with $a=3$, $b=1$, and $d=1$, they are, respectively:
$A=-43.5917\cdot 10^5$; $B=-4.70016\cdot 10^5$; $C=1.93536\cdot 10^5$; $D=0.34560\cdot 10^5$. According to (16), find $\eta$, $\Theta$, and $\varepsilon$: $\eta=0.10782$; $\Theta=-0.04440$; $\varepsilon=-0.00793$. From (20), determine the values of $p$ and $q$: $p=-0.04827$; $q=-0.00631$. By formula (22), compute one of the roots $z_1=0.238$. Equate $z_1$ to $z$ and, from equation (18), determine $z_0=0.2021$. In accordance with (13), find $\operatorname{tg}\varphi=\sqrt{z_0}=\sqrt{0.2021}=0.45$, and $\varphi=24^\circ14'$. By formula (7), find $m=1.2975$. With the aid of (23), express $m$ in mm: $m=1.2975\cdot 40\ \text{mm}=51.9\ \text{mm}$. And finally, from equation (6), find the value of the absolute error $f=51.9-40=11.9\ \text{mm}$. The relative error for the present example is

\[ f_{\mathrm{rel}}=\frac{f}{a+d}=\frac{11.9}{120+40}=0.0744. \]

With accuracy sufficient for practical purposes, this same value $f_{\mathrm{rel}}$ can be determined from Fig. 4.

Thus, as a result of the analysis of the theoretical accuracy of operation of spatial mechanisms of the third group for generating an ellipse, the values of their absolute and relative errors have been found, which makes it possible to determine the range of practical application of these mechanisms.

REFERENCES

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