Experimental Verification of Certain Parameters of a Spring Motor Obtained on the Basis of Its Geometry

V. I. Kalyagina, Yu. Ya. Kovylyn

(Presented by the scientific-methodological seminar of the Department of Applied Mechanics)

The object of the investigation was the motors of the “Baltika,” “Polet,” and “Slava” clocks with factory spiral springs made of K40TYu alloy at the Leningrad Steel-Rolling Plant. All parameters of the motors, except the lengths of the springs, conformed to the industry standard ON6—183—67 of NIICHASPROM. The spring lengths are given in Table 1.

The following notation is used in the work:

\(h, b\) — thickness and width of the spring strip;

\(r, R\) — theoretical radii of the arbor and the barrel, determined through the design dimensions \(r_0\) and \(R_0\) (Fig. 1) by the formulas:

\[ r = r_0 + 1.5h; \qquad R = R_0 - 0.5h. \]

The length of the working portion of the spiral spring is taken to be

\[ L = L_0 - L_{\mathrm{в}} - L_{\mathrm{н}}, \]

Fig. 1

Table 1

where

\(L_0\) — design length of the spring,

\(L_{\mathrm{в}}, L_{\mathrm{н}}\) — length of the inner and outer nonworking sections used for fastening the ends of the spring to the arbor and the barrel.

In the experiment the length \(L_{\mathrm{v}}\) was taken equal to the length of the inner coil of the spring, closely adjacent to the arbor both in the wound and in the unwound states (Fig. 1), i.e. \(L_{\mathrm{v}}\approx 2\pi(r_0+0.5h)\). The segment \(L_{\mathrm{n}}\) was taken equal to the length of the normal lenticular overlay.

A number of parameters of the spring motor could be obtained both theoretically and from experiment. These include (Fig. 2):

Fig. 2

\(n_{\mathrm{w}}\) — the number of turns of the winding arbor \([1, 2]\) required to transfer the spring from the unwound state to the wound state;

\(i_{\max}\) — the number of coils of the spring in the wound state;

\(i_{\mathrm{un}}\) — the number of coils of the unwound spring;

\(n_{\max}^{\mathrm{exp}}\) — the number of turns of the arbor when winding the spring from the free state to the fully wound state.

Thus, on the one hand, we have

\[ n_{\mathrm{w}}^{\mathrm{theor}}=i_{\max}^{\mathrm{theor}}-i_{\mathrm{un}}^{\mathrm{theor}}, \tag{1} \]

where \([1,2]\)

\[ i_{\max}^{\mathrm{theor}}=-\frac{r}{h}+\sqrt{\left(\frac{r}{h}\right)^2+\frac{L}{\pi h}}, \tag{2} \]

\[ i_{\max}^{\mathrm{theor}}=\frac{R}{h}-\sqrt{\left(\frac{R}{h}\right)^2-\frac{L}{\pi h}}, \tag{3} \]

and \([2]\)

\[ n_{\max}^{\mathrm{theor}}=\frac{m}{m-1}\cdot n_{\mathrm{work}}. \tag{4} \]

Here \(n_{\mathrm{work}}\) is the number of turns of the arbor during unwinding of the spring within the working (rectilinear) segment of its characteristic;

\[ m=\frac{M_{\max}}{M_{\min}}, \]

\(M_{\max}\) and \(M_{\min}\) are the maximum (at \(n=n_{\max}\)) and minimum (at \(n=n_{\max}-n_{\mathrm{work}}\)) torques of the ideal spring characteristic.

Formulas (1), (2), (3) are obtained from well-known simple geometric relations in the spring motor. Formula (4) is obtained if one follows L. E. Andreeva’s hypothesis [2] that the curvilinear section of the release diagram deviates from the ideal characteristic in proportion to the number of revolutions (the line \(A_1B_1\) in Fig. 2 passes approximately through the middle of the loop).

On the other hand, the numbers \(n_{\mathrm{w}}^{\mathrm{exp}}\), \(n_{\mathrm{op}}^{\mathrm{exp}}\) can be taken from diagrams recorded on a Technographe instrument; the number of turns of the spring in the released state \(i_{\mathrm{sp}}^{\mathrm{exp}}\) can be counted directly; the experimental number of turns of the spring in the wound state can be determined as

\[ i_{\max}^{\mathrm{exp}} = n_{\mathrm{w}}^{\mathrm{exp}} + i_{\mathrm{sp}}^{\mathrm{exp}}, \tag{5} \]

and the number of revolutions \(n_{\max}^{\mathrm{exp}}\) as

\[ n_{\max}^{\mathrm{exp}} = i_{\max}^{\mathrm{exp}} - i_{\mathrm{fr}}^{\mathrm{exp}} . \tag{6} \]

Fig. 3

The number of turns \(i_{\mathrm{fr}}^{\mathrm{exp}}\) of the spring in the free state is determined through the angle of mutual rotation of its end sections over the working portion [2].

Figure 3 shows a free spring with an overlay in a rectangular coordinate system. The ordinate axis is drawn parallel to the section of the spring at point \(O\). The working length of the spring corresponds to the portion from point \(O\) to point \(B\). At the outer end of the spring, point \(C\) was found (in the general case, one of the points \(C, C', C'', C'''\)), at which the angle \(\varphi_1 = K \cdot \frac{\pi}{2}\); \(K\) is an integer. With the aid of a polar-coordinate grid, the central angle \(\varphi_2\) of the arc \(CB\) was determined. Finally,

\[ i_{\mathrm{fr}}^{\mathrm{exp}} = \frac{\varphi_1 + \varphi_2}{2\pi}. \]

Table 2 gives the theoretical and experimental values of the parameters under discussion. Comparison of the results shows satisfactory agreement between the quantities \(n_{\mathrm{w}}^{\mathrm{theor}}\) and \(n_{\mathrm{w}}^{\mathrm{exp}}\), \(i_{\max}^{\mathrm{theor}}\) and \(i_{\max}^{\mathrm{exp}}\), \(i_{\mathrm{sp}}^{\mathrm{theor}}\) and \(i_{\mathrm{sp}}^{\mathrm{exp}}\). The maximum discrepancy does not exceed 5%. This provides a basis, in practical calculations of motors, for using formulas (1), (2), (3).

If, in the calculation by formula (4), we use the experimental values of \(m\) and \(n_{\mathrm{op}}\) (Table 2), then we find values of \(n_{\max}^{\mathrm{theor}}\) that differ substantially from the quantities \(n_{\max}^{\mathrm{exp}}\) determined in accordance with (6). (When calculating the coefficient \(m\), the values of the moments \(M_{\max}\), \(M_{\min}\) were determined from diagrams recorded on a Technographe instrument, and the scales along the moment axis were established with the aid of a Cary instrument.)

Nevertheless, the indicated difference between the quantities \(n_{\max}^{\mathrm{theor}}\) and \(n_{\max}^{\mathrm{exp}}\) is not noted in most of the works known to us [1, 2, etc.]

and is not taken into account. Only in the work of F. V. Drozdov [3, p. 44], in Fig. 24, is the indicated difference reflected. However, in the calculations this noted fact is not taken into account in that work either. This may lead to a significant deviation of the actual parameters of the motor from the calculated ones.

Table 2

Comparison of theoretical and experimental parameter values for the motor of a “Baltika” watch. The data are averaged over five springs

The difference can apparently be explained by the approximate nature of the hypothesis concerning the proportional inclination of the release curves.

References

1. T. A. Gevondyan. Spring Motors. Moscow, Oborongiz, 1956.
2. L. E. Andreeva. Elastic Elements of Instruments. Moscow, Mashgiz, 1962.
3. F. V. Drozdov. Instrument Parts. Moscow, Oborongiz, 1948.