On the Question of the Rational Length of a Bourdon Spring

V. I. Kopytov, M. P. Shumsky

(Presented by the scientific seminar of the Department of Gyroscopic Instruments and Devices)

The purpose of the present work is to show that a substantial shortening of a pressure-gauge tube (Bourdon spring) is possible while preserving its sensitivity (the displacement of the end of the spring).

At present, Bourdon springs with a central angle \(\gamma^\circ\) (Fig. 1) close to \(270^\circ\) are widely used in pressure gauges. This choice is due to the fact that a further increase in the central angle does not give a noticeable increase in sensitivity, while decreasing the central angle, for example, to \(180^\circ\), i.e., by \(33\%\), reduces the displacement of the end of the spring by \(35\%\).

The dependence between the central angle of the spring and the displacement of the end \(\delta\) can be obtained as follows.

It is known that an increase in pressure in a Bourdon spring leads to an increase in the radius of curvature of the spring axis (Fig. 1). Assuming that the length of the axis is preserved in this process, we obtain the equation for the trajectory of the end of the spring \(B\):
\(r_0\gamma_0 = r\gamma\).

After passing to Cartesian coordinates, the equation of the trajectory will be:

\[ x = -r_0 \frac{\gamma_0}{\gamma}\sin\gamma, \]

\[ y = r_0 \frac{\gamma_0}{\gamma}(1-\cos\gamma). \]

Fig. 1.

A small displacement of the end of the spring along the trajectory may be regarded as a rotation about the instantaneous center of rotation \(P(x_p, y_p)\) [1]. To find the coordinates of the instantaneous center of rotation, one may use formulas (2)

\[ x_p = -r_0 \frac{1-\cos\gamma_0}{\gamma_0}, \qquad y_p = r_0\left(1-\frac{\sin\gamma_0}{\gamma_0}\right). \tag{2} \]

When the radius of the spring \(r_0\) is increased by \(\Delta r\), the end of the spring will rotate through the angle

\[ \alpha=\gamma_0-\gamma=\gamma_0\frac{\Delta r}{r}. \tag{3} \]

The displacement of the end of the spring \(\delta\) is equal to the product of the angle of rotation and the distance \(\rho\) from the instantaneous center to the end of the spring:

\[ \delta=\alpha\cdot\rho=\gamma_0\cdot\frac{\Delta r}{r}\sqrt{(x-x_p)^2+(y-y_p)^2}. \tag{4} \]

Curve 1 (Fig. 2) shows how the displacement of the end of the spring \(\delta\) changes when the central angle \(\gamma_0\) varies from 0 to \(270^\circ\).

Fig. 2.

Let us now determine the displacement of the end of a spring with a central angle of \(270^\circ\), a portion of which from the end, with central angle \(270^\circ-\gamma\), is made rigid (Fig. 3).

The displacement of the end of the spring will be equal to the product of the angle of rotation of section \(C\) and the distance from its instantaneous center of rotation to the end of the spring:

\[ \delta_1=\alpha\cdot\rho_A=\gamma_0\frac{\Delta r}{r}\sqrt{(x_A-x_p)^2+(y_A-y_p)^2}. \tag{5} \]

The dependence of the displacement of the end of this spring on the magnitude of the central angle of its elastic part is shown in Fig. 2 by curve 2.

It is seen from the graph that reducing the central angle of the elastic part of the spring, for example by \(90^\circ\), i.e., by \(1/3\), reduces the displacement of the end of the spring by only 11%.

Furthermore, from formula (5) it follows that the displacement of the end can be increased by increasing \(\rho_A\), but in this case the pulling force will decrease.

To verify the theoretical conclusions, experiments were carried out. The displacements of the end of the spring and of the end of the pointer were measured (Fig. 4).

attached to the spring, as the pressure is increased in equal steps. The measurements were carried out on a projector at a magnification of 50 times.

I have established that, within the accuracy of the experiment, the theoretical and experimental results coincide. In the calculations it was assumed that both ends of the tube, over a length corresponding to a central angle of \(8^\circ\), are rigid [2].

Fig. 3.

Fig. 4.

Conclusions

It is expedient to design the pressure-gauge spring so that the central angle of the elastic element does not exceed \(220 \div 240\).

To preserve sensitivity, the axis of pull of the transmission-multiplying mechanism should be placed not at the end of the shortened elastic element, but at a point on the spring axis with coordinates \(270^\circ \div 300^\circ\).

Reducing the length of the pressure-gauge spring will make the instruments cheaper and will increase resistance to dynamic loads by reducing the weight and dimensions of the elastic part.

References

1. V. I. Feodos'ev. Elastic Elements of Precision Instrumentation. State Publishing House of the Defense Industry, Moscow, 1949.

2. Walter Wuest. Die Bewegungslehre von Röhrenfedern. Leitschrift fur Instrumentenkunde, 63 jahrgang Dezember, 1943.