PROCEEDINGS

of the Tomsk Polytechnic Institute named after S. M. Kirov, holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Volume 263　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1975

DETERMINATION OF THE NUMBER OF FREE TURNS OF SPIRAL MAINSPRINGS FOR WRISTWATCHES

V. I. Kalyagina, Yu. Ya. Kovylin

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

The number of turns of a mainspring \(i_{\mathrm{sv}}\) in the free state (outside the barrel) is a parameter that largely predetermines the mechanical characteristic of the spring motor, its dimensions and weight. T. A. Gevondyan showed [1] that the value \(i_{\mathrm{sv}}\) depends on the elastoplastic properties of the spring strip, on the ratio \(\frac{r}{h}\) of the arbor radius to the strip thickness, and on the number of turns \(i_{\max}\) of the wound spring. This dependence is expressed by a rather complicated formula, and its use in design calculations leads to cumbersome and laborious computations.

In design calculations it is more convenient to use the empirical [2] dependence of \(i_{\mathrm{sv}}\) on the number of arbor revolutions \(n_{\max}\) required to transfer the spring from the wound to the free state. For springs made of steels U8A and 70S2KhA, the formula [2] was proposed earlier

\[ i_{\mathrm{sv}} \approx i_{\mathrm{sv}\,\max}\left(1-e^{-\alpha n_{\max}}\right), \tag{1} \]

where \(i_{\mathrm{sv}\,\max}\) and \(\alpha\) depend on the steel grade and on the spring stabilization regime.

Formula (1) provides high accuracy: the maximum error is approximately 5% over the entire range of values of \(n_{\max}\) having practical significance.

At present, a high-strength anticorrosive and antimagnetic alloy of the K40TYu type has come into wide use. In this connection, it became necessary to carry out additional studies of springs made from the indicated alloy.

The experimental studies were carried out on the equipment of NIICHASPROM (Moscow). The objects of investigation were spiral springs manufactured by the Leningrad Steel-Rolling Plant according to the existing technology. The parameters of interest of the springs were recorded after running-in in standard barrels for 1000 wind—unwind cycles on a test stand. As a result of the experiments, a set of experimental data \(i_{\mathrm{sv}}\) and \(n_{\max}\) was obtained, presented in generalized form in the table.

In studying the arrangement of the experimental points \((n_{\max}, i_{\mathrm{sv}})\) for the motors of three watch brands, it was observed that \(i_{\mathrm{sv}\,\max}\) varies inversely proportionally to the quantity \(\left(\frac{r}{h}\right)^2\).

Therefore, for approximating the dependence of \(i_{\mathrm{sv}}\) on \(n_{\max}\), the formula adopted was

\[ i_{\mathrm{sv}}=\frac{\beta}{\left(\dfrac{r}{h}\right)^2}\left(1-e^{-\alpha\cdot n_{\max}}\right). \]

The coefficients \(\alpha\) and \(\beta\) were selected by computer using the least-squares method.

Table

Experimental values of \(i_{\mathrm{sv}}, n_{\max}\), averaged over five springs

The smallest root-mean-square deviation of the approximating curves from the experimental points \((n_{\max}, i_{\mathrm{sv}})\) is attained at \(\alpha \approx 0{,}2055\), \(\beta = 1020\). The figure shows the experimental points and the approximating curves.

Fig. 1

\[ i_{\mathrm{sv}}=\frac{1020}{\left(\dfrac{r}{h}\right)^2}\cdot\left(1-e^{-0{,}2055\cdot n_{\max}}\right) \]

for

\[ \frac{r}{h}=15.03 \qquad \text{(the “Slava” watch),} \]

\[ \frac{r}{h}=14.83 \qquad \text{(the “Baltika” watch),} \]

\[ \frac{r}{h}=13.50 \qquad \text{(the “Polet” watch).} \]

The approximation error does not exceed 5%.

REFERENCES

1. T. A. Gevondyan. Spring Motors. Moscow, Oborongiz, 1956.

2. Yu. Ya. Kovylin. On the number of turns of a free spiral spring. In: Collection of Materials of the Jubilee Scientific and Technical Conference Dedicated to the 50th Anniversary of October, Tomsk Polytechnic Institute. Issue II, Tomsk, 1968.