Unified Engineering Method for Calculating the Main Types of Single-Turn Pressure Gauge Springs
Abstract Generated abstract
This paper develops a unified engineering calculation method for single-turn Bourdon pressure-gauge springs with constant wall thickness and plano-oval, oval, or figure-eight-shaped cross sections. It shows that these practically important profiles can be described by common geometric parameters, enabling shared formulas for spring unwinding angle, maximum equivalent stress, tractive moment, and cavity volume change, with shape-dependent coefficients. The authors indicate that these coefficients may be computed digitally for large design studies or obtained from nomograms for smaller calculations. The proposed formulas are presented as a second approximation relative to a known energy solution, and comparison with experimental bending-angle data suggests discrepancies within the reported experimental error for such springs.
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Unified Engineering Method for Calculating the Main Types of Single-Turn Pressure Gauge Springs
M. P. Shumsky, O. S. Zinkevich, Yu. P. Minchenkov
(Presented by the scientific-technical seminar of the Department of Electrical Equipment)
The number of different cross-sectional forms of Bourdon pressure-gauge springs used in instruments or described in the specialized literature is quite large. For most of these forms, an engineering method for calculating strength and sensitivity has not been developed. If one confines oneself to considering springs with constant wall thickness, then springs with plano-oval, oval, and figure-eight-shaped cross sections can be distinguished. These three types of springs are of the greatest practical interest for the following reasons: 1) the manufacturing technology for them is sufficiently simple and has been mastered by industry; 2) most of the springs produced belong to one of the listed types; 3) by changing the parameters that determine the cross-sectional form, an enormous number of design variants can be obtained.
The three listed types of springs admit a unified analytical description and, consequently, a unified calculation method can be developed for them. Indeed, the principal geometric relationships are identical for oval and figure-eight-shaped (Fig. 1) cross sections. Taking as the independent parameters determining the cross-sectional form the ratio of the axes
\[ \frac{B}{A}=b \]
and the ratio of the radius of rounding at the ends of the major axis of the section to this axis
\[ \frac{R_1}{A}=r_1, \]
we find.
Fig. 1
\[ r_2=\frac{R_2}{A}=\frac{1-2r_1+b}{2(b-r_1)}, \qquad \alpha_0=\frac{\pi}{2}-2\operatorname{arctg}\frac{b-r_1}{1-r_1}. \tag{1} \]
By varying the parameters \(b\) and \(r_1\) and calculating \(r_2\) and \(\alpha_0\) from formulas (1), one can obtain any oval, flat-oval, or figure-eight-shaped profile.
The Cartesian coordinates of a point of the contour on the segment with radius \(R_1\) are expressed by the formulas:
\[ X=R_1(1-\cos\alpha), \qquad y=R_1\sin\alpha, \tag{2} \]
and on the segment with radius \(R_2\), by the formulas:
\[ X=A-R_2\cos\alpha, \qquad y=R_2\sin\alpha-(R_2-B). \tag{3} \]
In [1] a theory is proposed that is suitable for calculating springs with an arbitrary cross-sectional shape. Application of this theory to springs whose cross-sectional shape is specified by equations (2), (3) leads to the following calculation formulas:
- Relative angle of unwinding of the spring:
\[ \frac{\Delta\gamma}{\gamma} = \frac{pA^3}{D}\, \frac{C_1}{1+\dfrac{C}{\nu}}, \tag{4} \]
where
\[ D=\frac{EH^3}{12(1-\mu^2)}, \qquad \nu=\frac{R_0^2H^2}{12A^4(1-\mu^2)}. \]
- Maximum equivalent stress at the ends of the major axis of the section according to the energy theory of strength:
\[ \sigma_{\mathrm{eqv}} = 0.89\,\frac{pKA^2}{H^2}\,C_2\,\frac{\nu+C_3}{\nu+C}. \tag{5} \]
- Tractive moment
\[ M=pR_0A^2\,\frac{C_4}{\nu+C_5}. \tag{6} \]
- Change in the volume of the spring cavity
\[ \Delta V=\frac{pR_0A^5\gamma}{D}\, \frac{C_6}{1+\dfrac{C}{\nu}}. \tag{7} \]
In these formulas: \(p\) is the measured pressure \((\mathrm{kg}/\mathrm{cm}^2)\), \(A\) is the major semiaxis of the section \((\mathrm{cm})\), \(R_0\) is the radius of the central axis of the spring \((\mathrm{cm})\), \(H\) is the wall thickness \((\mathrm{cm})\), \(\mu, E\) are Poisson’s ratio and Young’s modulus, and \(C_i\) are coefficients depending on the shape of the section.
The coefficient \(K\) in formula (5) takes into account the stress concentration on the rounded portion of the spring. A table of values of this coefficient is given in [2].
The formulas for calculating the coefficient \(C_i\) from the given parameters \(b\) and \(r_1\) are cumbersome and cannot be presented because of lack of space. There are two possible ways to use these formulas. In cases where the number of variants under study is large, it is advisable to carry out the calculations on an electronic digital computer. This makes it possible to reduce the time expenditure, avoid errors, and prevent loss of accuracy. When calculating a small number of variants, one should use nomograms (Figs. 2–3), which make it possible to determine the coefficients \(C_i\) entering into the calculation formulas (4)—(7) from
specified values of the parameters \(b\) and \(r_1\). The nomograms were obtained as a result of processing the formulas for calculating the coefficients \(C_i\) on an electronic digital computer.
Fig. 2
The theoretical analysis of the error of the solution [1], on the basis of which the engineering formulas recommended in this work were obtained, shows that, in relation to the known energy solution of V. I. Feodos’ev [3], it may be regarded as a second approximation. Stresses and displacements are determined with the same accu-
by the essence. A comparison of the experimental and calculated values of the relative angle of spring bending (measurement of this quantity is not associated
Fig. 3
with serious difficulties) shows that the discrepancies do not exceed the experimental error, which, according to V. I. Feodos’ev [4], for pressure-gauge springs is 20–25 percent.
REFERENCES
- M. P. Shumskii, Calculation of pressure-gauge springs. Izv. vuzov SSSR—Priborostroenie, vol. VII, No. 6, 1964.
- L. E. Andreeva, Elastic Elements of Instruments. Mashgiz, 1960.
- V. I. Feodos’ev, Elastic Elements of Precision Instrument Engineering. Oborongiz, 1949.
- V. I. Feodos’ev, Calculation of Thin-Walled Bourdon Tubes of Elliptical Cross Section by the Energy Method. Oborongiz, 1940.