SELECTION OF MANUFACTURING TOLERANCES FOR PRESSURE-GAUGE SPRINGS

V. G. AFONIN, M. P. SHUMSKII

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

The general form of the relationship between tolerances for the geometric parameters of a spring and deviations of the spring characteristics from their nominal values can be taken from the theory of instrument errors [1]. If between the instrument reading \(y\), the measured quantity \(x\), and the parameters \(l_0, m_0, \ldots q_0\) there exists a dependence

\[ y = f(x,\ l,\ m,\ldots q), \]

then small increments of the parameters \(\Delta l, \Delta m, \ldots \Delta q\) will correspond to an increment of the quantity \(y\)

\[ \Delta y = \frac{\partial f}{\partial l}\cdot \Delta l + \frac{\partial f}{\partial m}\Delta m +\ldots+ \frac{\partial f}{\partial q}\Delta q . \]

This equation can be used to establish a justified system of tolerances for the geometric and mechanical parameters of the spring, if the specific form of the dependence \(y=f(x,\ l,\ m,\ldots q)\) between the principal characteristics of the spring and its geometric and mechanical parameters is known.

Thus, for example, the formula obtained by V. I. Feodos’ev [3] for determining the relative angle of unwinding of thin-walled springs of elliptical cross section,

\[ \frac{\Delta\gamma}{\gamma} = p\frac{1-\mu^2}{E}\, \frac{R_0^2}{BH} \left(1-\frac{B^2}{A^2}\right) \cdot \frac{ \alpha\!\left(\frac{B}{A}\right) }{ \beta\!\left(\frac{B}{A}\right)+\frac{R_0^2 A^2}{A^4} } \tag{1} \]

can be used to establish the relationship between the tolerances for the geometric parameters \(A, B, H, R_0\) and the maximum possible deviation of the relative angle of unwinding from its value corresponding to the nominal values of the geometric parameters.

Here \(A, B\) are the major and minor semiaxes of the centerline of the spring cross section;

\(H\) is the wall thickness of the spring;

\(R_0\) is the radius of the axis of the spring channel.

However, since formula (1) is valid only for springs with an elliptically shaped cross section, it can be used for ana-

only such a combination of deviations of the geometrical parameters in which an elliptical section is again obtained. Thus, for specified deviations of the dimensions of the semiaxes of the ellipse \(\Delta A\) and \(\Delta B\), formula (1) can be used only if the deviation of the radius at the ends of the major axis satisfies the equation

\[ R_1+\Delta R_1=\frac{(B+\Delta B)^2}{A+\Delta A}. \tag{2} \]

Meanwhile, the deviation \(\Delta R_1\) depends on random causes and is not governed by equation (2). The formulas for calculating springs with oval, flat-oval, and figure-eight-shaped sections, obtained in [2], can be used to establish the dependence between deviations of the geometrical parameters and the corresponding changes in the quantities: the relative angle \(\dfrac{\Delta\gamma}{\gamma}\) of spring bending, the design stress \(\sigma_{\mathrm{eqv}}\), the tractive moment \(M_t\), and the change in the volume of the strip \(\Delta V\). In this case, it is possible to estimate the influence on the spring characteristics of deviations in the quantities \(A, B, R, H, R_0\), the modulus of elasticity \(E\), and Poisson’s ratio \(\mu\). Since, in deriving the formulas, it was assumed that the section is symmetric with respect to two axes, only those dimensional deviations that do not violate the symmetry of the section can be evaluated. Thus, the formulas obtained do not make it possible to investigate the case when the radii \(R_1\) at the ends of the major axis of the section do not coincide in magnitude. The influence of technological nonuniformity in wall thickness and of the mismatch of the radii \(R_2\) on the outer and inner sides of the section cannot be investigated.

Let us investigate the relation of tolerances on the spring parameters to the change in displacement of the end of springs with a small constant wall thickness.

The dependence of the displacement of the spring end on its parameters is determined by the formula

\[ \lambda = p\cdot \Gamma(\gamma)\, \frac{A\cdot r_0}{E h_0}\, \frac{C_1(b_1 r_1)} {\dfrac{h^2}{12(1-\mu^2)}+\dfrac{C(b_1 r_1)}{r_0^2}}. \]

Here \(\lambda\) is the displacement of the spring end;

\(\Gamma(\gamma)\) is a coefficient depending on the magnitude of the central angle of the active part and on the structural design of the spring end (Fig. 1);

\(b, r, h, r_0\) are dimensionless ratios of the corresponding dimensions to the length of the major semiaxis of the section.

Let us find the relative change in the displacement of the spring end corresponding to specified parameter deviations:

\[ \begin{aligned} \frac{\Delta\lambda}{\lambda} &= \frac{1}{\lambda}\left\{ \frac{\partial\lambda}{\partial A}\cdot \Delta A + \frac{\partial\lambda}{\partial\Gamma}\cdot \frac{d\Gamma}{d\gamma}\cdot \Delta\gamma + \frac{\partial\lambda}{\partial r_0}\cdot \Delta r_0 + \frac{\partial\lambda}{\partial h}\cdot \Delta h +\right.\\ &\quad + \frac{\partial\lambda}{\partial C_1} \left[ \frac{\partial C_1}{\partial b}\cdot \Delta b + \frac{\partial C_1}{\partial r_1}\cdot \Delta r_1 \right] + \frac{\partial\lambda}{\partial C} \left[ \frac{\partial C}{\partial b}\cdot \Delta b + \frac{\partial C}{\partial r_1}\cdot \Delta r_1 \right] +\\ &\quad\left. + \frac{\partial\lambda}{\partial E}\cdot \Delta E + \frac{\partial\lambda}{\partial\mu}\cdot \Delta\mu \right\} = \alpha_A\cdot \frac{\Delta A}{A} + \alpha_B\cdot \frac{\Delta b}{b} + \alpha_r\cdot \frac{\Delta r}{r} +\\ &\quad + \alpha_h\cdot \frac{\Delta h}{h} + \alpha_{r_0}\cdot \frac{\Delta r_0}{r_0} + \alpha_\gamma \frac{\Delta\gamma}{\gamma} + \alpha_E \frac{\Delta E}{E} + \alpha_\mu \frac{\Delta\mu}{\mu}. \tag{3} \end{aligned} \]

Here

\[ \left. \begin{gathered} \alpha_A=1;\quad \alpha_B=b\left[ \frac{\dfrac{\partial C_1}{\partial b}}{C} - \frac{\dfrac{\partial C}{\partial b}}{\nu+C} \right], \quad \text{where }\ \nu=\frac{r_0^2h^2}{12(1-\mu^2)},\\[1.2em] \alpha_{r_1}=r_1\left[ \frac{\dfrac{\partial C_1}{\partial r_1}}{C_1} - \frac{\dfrac{\partial C}{\partial n}}{\nu+C} \right], \quad \alpha_h=-\left[1+\frac{2}{1+\dfrac{C}{\nu}}\right], \quad \alpha_{r_0}=1-\frac{2C}{\nu+C},\\[1.2em] \alpha_\gamma=\frac{\gamma}{\Gamma}\cdot\frac{d\Gamma}{d\gamma}, \quad \alpha_E=-1, \quad \alpha_\mu=-\frac{2}{1+\dfrac{C}{\nu}}\frac{\mu^2}{1-\mu^2} \end{gathered} \right\} \tag{4} \]

\(\Delta A,\ \Delta b,\ (\Delta\gamma),\ \Delta r_0,\ \Delta h,\ \Delta r,\ \Delta E,\ \Delta\mu\) are the specified deviations of the parameters.

Fig. 1.

The partial derivatives \(\dfrac{\partial C}{\partial b},\ \dfrac{\partial C}{\partial r},\ \dfrac{\partial C}{\partial b},\ \dfrac{\partial C}{\partial r_1},\ \dfrac{\partial \Gamma}{\partial \gamma}\) can be calculated on the basis of the formulas and graphs given in [2].

Let certain mastered or designed technological processes correspond to specific values of the relative errors

\[ \frac{\Delta A}{A},\ \frac{\Delta b}{b},\ldots,\ \frac{\Delta\mu}{\mu}, \]

achievable at equal costs. Taking one of the relative errors, for example \(\dfrac{\Delta A}{A}\), as the reference, we compute the coefficients

\[ n_B= \frac{\dfrac{\Delta b}{b}}{\dfrac{\Delta A}{A}}, \quad n_{r_1}= \frac{\dfrac{\Delta r_1}{r_1}}{\dfrac{\Delta A}{A}}, \ldots,\quad n_\mu= \frac{\dfrac{\Delta\mu}{\mu}}{\dfrac{\Delta A}{A}}. \]

The formula for determining the relative error of displacement of the spring end can be represented in the form

\[ \frac{\Delta \lambda}{\lambda} = \pm \frac{\Delta A}{A}\left[|\alpha_A|+n_b|\alpha_b|+\ldots+n_\mu|\alpha_\mu|\right]. \]

Let \(\delta\) be the permissible value of the relative error of the quantity \(\lambda\). In order to ensure the specified tolerance for the quantity \(\lambda\) at minimum cost, it is necessary to establish the following limiting values of the relative errors of the parameters:

\[ \frac{\Delta A}{A} = \frac{\delta}{|\alpha_A|+n_b|\alpha_b|+\ldots+n_\mu|\alpha_\mu|}, \qquad \frac{\Delta b}{b}=n_b\frac{\Delta A}{A},\ldots,\qquad \frac{\Delta \mu}{\mu}=n_\mu\frac{\Delta A}{A}. \tag{5} \]

Since deviations of parameter values from their nominal values are random in character, the probability that the relative error of the quantity \(\lambda\) will reach the limiting value \(\delta\) is very small. If broadening the tolerances for parameter deviations makes it possible to reduce manufacturing costs, such broadening may be allowed on the basis that some portion of the springs will be manufactured with deviations of the quantity \(\lambda\) exceeding the tolerance limits. In this case, the effect achieved must exceed the losses due to rejects.

Fig. 2. \(\lambda_0 \pm \Delta\lambda_0\)—nominal value of the displacement of the spring end and the tolerance established by the designer;
1—region of possible deviations of the true nominal value of the spring-end displacement from the value \(\lambda_0\) established by the designer;
\(\pm \lambda_{0\,\text{calc}}\)—the tolerance from which the tolerances for the parameter values should be determined

Relations (5) were obtained on the assumption that the displacement of the spring end corresponding to the nominal parameter values is known.

In practice it is difficult to establish precisely the magnitude of the displacement \(\lambda\) of the spring end corresponding to the nominal parameter values. It must be taken into account that in practice the nominal value of the spring-end displacement will always be specified with some deviation from the true value; consequently, when selecting permissible parameter deviations, one can count on only part of the tolerance field established for \(\lambda\) (Fig. 2).

Because the dimensions of the spring are specified on working drawings differently than in the theoretical treatment, some parameter deviations that are assumed here to be independent may turn out to be interrelated. This should be taken into account when applying the method considered in specific calculations.

In the same way, one can analyze the influence of deviations in geometrical and mechanical parameters on the magnitudes of maximum stresses, tractive torque, and changes in the cavity volume.

Example. Calculate the tolerances for the principal parameters of a spring for measuring pressure up to \(0.6\ \mathrm{kg/cm^2}\), the dimensions of which are indicated in Fig. 3, based on

from the condition that the deviations of the displacement of the spring end from the nominal value must not exceed \(\delta \pm 20\%\). The modulus of elasticity is \(E=1.28\cdot 10^6\ \text{kg}/\text{cm}^2\), Poisson’s ratio \(\mu=0.32\). The spring under consideration corresponds to the following parameter values: \(A=1.88\ \text{cm}\), \(b=0.194\); \(r_1=0.0425\), \(r_0=2.86\); \(h=0.0106\); \(E=1.28\cdot 10^6\ \text{kg}/\text{cm}^2\); \(\mu=0.32\); \(\gamma=4.2\).

Fig. 3

The values of the coefficients \(C_1\), \(C\), \(\Gamma\), and of the derivatives \(\dfrac{\partial C_1}{\partial b}\), \(\dfrac{\partial C_1}{\partial r_1}\), \(\dfrac{\partial C}{\partial b}\), \(\dfrac{\partial C}{\partial r_1}\), \(\dfrac{\partial \Gamma}{\partial \gamma}\) are determined from the graphs [2]:

\[ C_1=0.185;\qquad C=0.0024;\qquad \Gamma=5.3;\qquad \frac{\partial C_1}{\partial b}\simeq -0.91; \]

\[ \frac{\partial C_1}{\partial r_1}\simeq 0.06;\qquad \frac{\partial C}{\partial b}\simeq -0.01;\qquad \frac{\partial C}{\partial r_1}\simeq 0.03;\qquad \frac{d\Gamma}{d\gamma}\simeq 1.26. \]

Using formulas (4), we compute the values of the coefficients
\(\alpha_A=1\); \(\alpha_b=-0.18\); \(\alpha_{r_1}=-0.5\); \(\alpha_h=-1.07\); \(\alpha_{r_0}=2.93\); \(\alpha_\gamma=1\); \(\alpha_E=-1\); \(\alpha_\mu=-0.007\).

The value of the coefficient \(\alpha_\mu\) is determined under the assumption that the displacement \(\lambda\) of a point on the spring axis corresponding to the end of the active part of the spring is considered. Of practical interest is the displacement of the point at which the mechanism link is fastened, which is also usually located on the spring axis, but is situated at some distance from the end of the active part. Thus, if it is assumed that the position of the link-fastening point is determined by a central angle of \(270^\circ\), while the arc of the active part corresponds to an angle of \(240^\circ\), then with the aid of graph 2 in Fig. 1 we find: \(\alpha_\gamma=0.25\). In general, in order to reduce the influence of the technological error of the central angle \(\gamma\) on the magnitude of the displacement of the spring end, one should choose values of \(\gamma\) corresponding to the flat portion of the graph \(\Gamma(\gamma)\). In this case, for a spring with an elongated tip, smaller values of \(\gamma\) may be taken.

Let us assume that the relative accuracy of the parameters corresponding to established production is characterized by the following coefficients:

\[ n_b=n_h=n_{r_0}=n_\gamma=n_E=n_\mu=1;\qquad n_{r_1}=3. \]

Then the relative deviation of the displacement of the spring end corresponding to the specified positive deviations of the parameters is determined by the formula:

\[ \frac{\Delta \lambda}{\lambda} = \frac{\Delta A}{A} -0.18\frac{\Delta b}{b} -0.5\frac{\Delta r_1}{r_1} -1.07\frac{\Delta h}{h} +2.93\frac{\Delta r}{r_0} +0.25\frac{(\Delta \gamma)^*}{\gamma} - \frac{\Delta E}{E} -0.007\frac{\Delta \mu}{\mu}. \]

From the absolute values and signs of the coefficients on the right-hand side of the formula, one can obtain a clear picture of the effect of parameter deviations on the deviation of the displacement of the end of the spring from its nominal value. Thus, the deviation of the value of the coefficient \(\mu\) from the nominal will have a very small effect on the value of \(\lambda\). The greatest effect on the value of \(\frac{\Delta \lambda}{\lambda}\) is exerted by the errors \(\frac{\Delta r_0}{r_0}\), \(\frac{\Delta h}{h}\), \(\frac{\Delta E}{E}\), and \(\frac{\Delta A}{A}\). From formulas (5) we find

\[ \frac{\Delta A}{A} = \frac{\Delta b}{b} = \frac{\Delta h}{h} = \frac{\Delta R_0}{R_0} = \frac{(\Delta \gamma)^*}{\gamma} = \frac{\Delta E}{E} = 2.5\%, \]

\[ \frac{\Delta r}{r} = 3\cdot \frac{\Delta A}{A} = 7.5\%. \]

Fig. 4.

If it is necessary to reduce the possible deviations of the value of \(\lambda\) from the nominal value, the greatest effect can be achieved by reducing the tolerances on those parameters for which the coefficient \(\alpha\) has a large absolute value. Thus, reducing the permissible relative error of the parameter \(r_0\) by a factor of 2 makes it possible to reduce the value of \(\frac{\Delta \lambda}{\lambda}\) by 18%. The same reduction of the relative error of the parameter \(b\) leads to a reduction of the value of \(\frac{\Delta \lambda}{\lambda}\) by 1%, i.e., is 18 times less effective.

Let the drawing of the spring specify, instead of the parameters \(r_0\) and \(b\), the dimensions indicated in Fig. 4. Assuming, for simplicity, that the parameters \(h\) and \(A\) have their nominal values, let us analyze the relationship between the possible deviations of the parameters \(r_0\) and \(b\). Suppose that the possible deviations from the nominal value for the dimension \(B_{\text{nar}}\) are bounded by the quantity \(A\cdot \Delta b\), and for the dimension \(R\) by the quantity \(A\cdot \Delta r_0\). The relative po-

the error of the parameter \(b\) will be equal to \(\dfrac{\Delta b}{b}\). The relative error of the parameter \(R_0\) will be equal to

\[ \frac{\Delta r_0+\Delta b}{r_0} = \frac{\Delta r_0}{r_0} + \frac{b}{r_0}\cdot\frac{\Delta b}{b}. \]

Substituting into formula (4) the values of the relative errors of the parameters \(b\) and \(R_0\), we find that the coefficient of \(\dfrac{\Delta r_0}{r_0}\) retains its previous value \(\alpha_{r_0}\), while the coefficient of \(\dfrac{\Delta b}{b}\) has the following magnitude:

\[ \alpha'_B=\alpha_B+\frac{b}{r_0}\cdot\alpha_{r_0}; \]

After substituting the numerical values of the quantities, we obtain \(\alpha'_B=0{,}019\), which is substantially smaller than the previous value \(\alpha_0=-0{,}18\). Thus, specifying the parameters \(r_0\) and \(b\) of the spring in accordance with Fig. 4 in the case under consideration leads to a significant reduction in the influence of the error of the parameter \(b\) on the magnitude of the displacement of the end of the spring.

LITERATURE

1. V. A. Bodner, N. I. Fridlender. Aircraft Instruments. Oborongiz, Moscow, 1960.

2. M. P. Shumskii, O. S. Zinkevich, Yu. P. Minchenkov. A unified engineering method for calculating the principal types of single-coil pressure-gauge springs. Proceedings of TPI, vol. 192, 1969.

3. V. I. Feodos’ev. Calculation of thin-walled Bourdon tubes of elliptical cross section by the energy method. Oborongiz, Moscow, 1940.