CALCULATION OF A TUBULAR PRESSURE-GAUGE SPRING WITH A UNIVERSAL CROSS-SECTION

G. I. Tyzhnov

(Presented by the scientific seminar of the Department of Strength of Materials)

Tubular pressure-gauge springs are manufactured with various cross-sectional shapes: flattened-oval, oval, elliptical, figure-eight-shaped, etc. At present, calculations of springs have been developed only for two cross-sectional shapes: elliptical and flattened-oval [1, 2].

It is expedient to have a unified calculation for springs of several cross-sections, but for this it is necessary to choose such a calculation scheme whose variation would make it possible to encompass all these cross-sections. As studies have shown, the calculation scheme of the cross-section shown in Fig. 1 is sufficiently convenient and simple. For \(r = 0\) the cross-section becomes a rhombus; for \(r < b\) it approaches an ellipse or oval; for \(r = b\) it has a flattened-oval cross-section; and for \(r > b\) it has a figure-eight-shaped cross-section. The scheme also has the advantage that it closely approximates the cross-section of greatest sensitivity, which has not yet been calculated [1].

In the subsequent solution, V. I. Feodos’ev’s equation [1] for the total potential energy of a tubular spring having a cross-section symmetric with respect to the axes \(xy\) is used:

\[ \mathcal{E} = \frac{2Eh\gamma}{R(1-\mu^2)} \left[ \int_0^{s/4} w^2\,ds - 2\left(\frac{\Delta \gamma}{\gamma}\right) \int_0^{s/4} wy\,ds + \left(\frac{\Delta \gamma}{\gamma}\right)^2 \int_0^{s/4} y^2\,ds + \frac{R^2h^2}{12} \int_0^{s/4}(\Delta x)^2\,ds \right] - pR\gamma\Delta f . \tag{1} \]

Here there are four unknowns: the relative angle of rotation of the end section \(\Delta\gamma/\gamma\); the displacement \(w\) of any point of the centerline of the cross-section contour in the direction of the \(oy\) axis; the change in curvature of the contour \(\Delta x\); and the change in the area of the contour cross-section \(\Delta f\). The quantities of the spring bending radius \(R\), wall thickness \(h\), modulus of elasticity \(E\), Poisson’s ratio \(\mu\), and medium pressure \(p\) are known.

Assuming that the nature of the deformation of the tubular-spring cross-section will be the same as that of a closed contour, one can find the relations:

\[ w=\varphi_1(w_0);\qquad \Delta x=\varphi_2(w_0);\qquad \Delta f=\varphi_3(w_0). \tag{2} \]

After substituting the dependences (2) into equation (1), only two unknowns remain: \(\Delta\gamma/\gamma\) and \(w_0\), where the latter quantity is the deformation of the contour in the direction of the \(oy\) axis at \(x=0\).

Fig. 1.

Equation (1) consists of five terms. The first term depends on the magnitude and nature of the deformation of the contour. Experiments have shown that the nature of the deformation of the contour depends very little on its shape. The experimental data are also confirmed by calculations. In accordance with this, the first assumption is introduced: considering that the nature of the deformations of the contour of any section will be the same as for a rhombus \((r=0)\), the first term of equation (1) is taken to be the same in all cases.

Fig. 2.

After analogous reasoning, the same assumptions are introduced for the second, fourth, and fifth terms. Only the third term is calculated specifically for the given section. The adopted assumptions significantly reduce the amount of computation; as for the accuracy of the solution, it should increase for those sections that approach the rhombic one; for other sections the accuracy of the calculation should be lower.

On the basis of the considerations set forth, using the dependences (2) for the rhombus, we obtain the following expressions for the terms of equation (1).

First term:

\[ \int_{0}^{s/4} w^{2} ds = w_{0}^{2}\frac{a^{9}\sqrt{1+k^{2}}}{24^{2}m_{0}^{2}} (0{,}406-0{,}74k^{2}+0{,}34k^{4}) = w_{0}^{2}\frac{a^{9}\sqrt{1+k^{2}}}{25^{2}m_{0}^{2}}A_{1}. \]

Second term:

\[ 2\left(\frac{\Delta \gamma}{\gamma}\right) \int_{0}^{s/4} wy\,ds = w_{0}\frac{a^{6}}{12m_{0}} \left(\frac{\Delta \gamma}{\gamma}\right) \left\{ kC_{1}\delta \left[ (1-k^{2}) - \frac{2}{3}(1-2k^{2})\delta^{2} - k^{2}\delta^{3} \right] + \right. \]

\[ + \frac{1}{5}(1+\kappa^2)\delta^4 \bigg] - \frac{C_1 C_2 \delta^2}{2} \bigg[ (1-\kappa^2) - (1-2\kappa^2)\delta^2 - 1{,}6\kappa^2\delta^3 + \]
\[ + \frac{1}{3}(1+\kappa^2)\delta^4 \bigg] + \kappa_1 \bigg[ (1-\kappa^2)(1-\delta) - \frac{2}{3}(1-2\kappa^2)(1-\delta^3) - \]
\[ - \kappa^2(1-\delta^4) + \frac{1}{5}(1+\kappa^2)(1-\delta^5) \bigg] \bigg\} = w_0 \frac{a^6}{12m_0}\left(\frac{\Delta\gamma}{\gamma}\right) A_2 . \]

Third term:

\[ \left(\frac{\Delta\gamma}{\gamma}\right)^2 \int_0^{s/4} y^2\,ds = \left(\frac{\Delta\gamma}{\gamma}\right)^2 a^3 \left[ C_1\left(\kappa^2\delta - \kappa C_2\delta^2 + \frac{C_2^2}{3}\delta^3\right) + \frac{\pi}{4}\kappa_1^3 \right] = \left(\frac{\Delta\gamma}{\gamma}\right)^2 a^3 A_3 . \]

Fourth term:

\[ \frac{R^2 h^2}{12}\int_0^{s/4}(\Delta x)^2\,ds = w_0^2\frac{\eta^2 a^9(4-7\kappa^2+4\kappa^4)}{2160\,m_0^2\sqrt{1+\kappa^2}} = w_0^2\frac{\eta^2 a^9 A_4}{2160\,m_0^2\sqrt{1+\kappa^2}} . \]

Fifth term:

\[ pR\gamma\Delta f = pR\gamma w_0\,\frac{a^5}{45m_0}\,A_4 . \]

Here

\[ \kappa=\frac{b}{a},\qquad \kappa_1=\frac{r}{a},\qquad m_0=\frac{a^4}{24}(1-\kappa^2),\qquad \delta=1-\kappa_1, \]

\[ C_1=\sqrt{1+\left(\frac{\kappa-\kappa_1}{1-\kappa_1}\right)^2}, \qquad C_2=\frac{\kappa-\kappa_1}{1-\kappa_1}, \]

\(a,\ b,\ r\) are the dimensions of the section (Fig. 1).

The characteristic \(\eta = Rh/a^2\) is called the principal parameter of the spring. To find the values of \(w_0\) and \(\Delta\gamma/\gamma\) for which the energy attains a minimum, we resort to the Ritz method:

\[ \frac{\partial \mathcal{E}}{\partial w_0}=0,\qquad \frac{\partial \mathcal{E}}{\partial\left(\dfrac{\Delta\gamma}{\gamma}\right)}=0 . \]

After differentiation we obtain two equations:

\[ \frac{4Eh\gamma}{R(1-\mu^2)} \left[ w_0\frac{a^9 A_1\sqrt{1+\kappa^2}}{24^2 m_0^2} - \left(\frac{\Delta\gamma}{\gamma}\right)\frac{a^6 A_2}{24m_0} + \frac{\eta^2 a^9 A_4}{2160\,m_0^2\sqrt{1+\kappa^2}} \right] - \frac{pR\gamma a^5 A_4}{45m_0} =0 . \tag{3} \]

\[ w_0\frac{a^6 A_2}{12m_0} - 2\left(\frac{\Delta\gamma}{\gamma}\right)a^3 A_3 =0 . \tag{4} \]

The simultaneous solution of equations (3) and (4) leads to the desired formula for determining the relative opening angle of the spring:

\[ \frac{\Delta\gamma}{\gamma} = p\,\frac{1-\mu^2}{E}\,\frac{R^2}{bh}\, \frac{(1-\kappa^2)\kappa A_2\sqrt{1+\kappa^2}} {2A_3(1-\kappa^2) \left[ \dfrac{15\sqrt{1+\kappa^2}\left(A_1A_3\sqrt{1+\kappa^2}-A_2^2\right)}{4A_3A_4} +\eta^2 \right]} . \]

By analogy with the formula of V. I. Feodos’ev [1], we introduce the coefficients \(\alpha\) and \(\beta\):

\[ \alpha=\frac{\kappa A_2\sqrt{1+\kappa^2}}{2A_3(1-\kappa^2)}, \]

\[ \beta=\frac{15\sqrt{1+\kappa^2}\left(A_1A_3\sqrt{1+\kappa^2}-A_2^2\right)}{4A_3A_4}, \]

after which the formula for calculating the relative opening angle of the spring takes the following form:

\[ \frac{\Delta\gamma}{\gamma} = p\frac{1-\mu^2}{E}\frac{R^2}{bh} \left(1-\frac{b^2}{a^2}\right) \frac{\alpha}{\beta+\eta^2}. \tag{5} \]

Figure 2 gives graphs of the coefficients \(\alpha\) and \(\beta\), calculated on an electronic machine.

Table

Table 1 compares the results of calculation by our formula with experimental data. As expected, the greatest error is obtained when calculating springs of plano-oval cross section (springs No. 1 and No. 5). For springs No. 3, 4, 7, and 8, whose cross sections approach a rhombic one, the calculation agrees well with experiment.

Conclusions

1. The possibility has been demonstrated of creating a unified calculation for springs of various cross-sectional shapes on the basis of introducing a universal cross section.

2. The proposed version of the calculation scheme for the universal cross section has made it possible, by a unified approach, to calculate springs of several cross sections. In comparison with other variants, for example, those composed of arcs of two and, still more, three circles, this variant is the simplest.

3. The rather arbitrary and crude assumptions adopted at the outset did not have a very strong effect on the accuracy of the calculation. This circumstance gives confidence that the calculation of springs without these assumptions will have good accuracy.

References

1. V. I. Feodos’ev. Elastic Elements of Precision Instrument Engineering. Oborongiz, 1949.

2. L. E. Andreeva. Elastic Elements of Instruments. Mashgiz, 1962.