OPTIMAL DESIGN OF SERIES OF MANOMETRIC SPRINGS USING AN ELECTRONIC DIGITAL COMPUTER

M. P. Shumskii, Yu. P. Minchenkov, O. S. Zinkevich

(Presented by the scientific-technical seminar of the Department of Electrical Equipment)

The metrological characteristics of a manometric spring depend on a large number of parameters: the wall thickness \(H\), the radius \(R_0\) of curvature of the spring axis, the size \(A\) of the major semiaxis of the cross section, the shape of the cross section \(y(x)\), the central angle \(\gamma\), the modulus of elasticity \(E\), and Poisson’s ratio \(\mu\). The number of possible design solutions corresponding to different values of these parameters is large, and in order to select, from all possible design variants, the variant that best satisfies the specified conditions, certain criteria are necessary.

As a rule, the designer knows the limiting value of the measured pressure \(p\) and the overall dimensions of the spring, i.e., the radius \(R_0\) of the central axis and the major semiaxis of the section \(A\). We shall require that, when the upper limit of the measured pressure \(p\) is reached, the strength condition not be violated

\[ \sigma_{\mathrm{eq}} \leq [\sigma]. \tag{1} \]

All variants that do not satisfy this condition must be excluded from further consideration.

The remaining springs are suitable for measuring pressure in the specified range, but they have different values of the driving torque \(M\), the relative change of the central angle \(\dfrac{\Delta\gamma}{\gamma}\), the maximum stress \(\sigma_{\mathrm{eq}}\), and the change in cavity volume \(\Delta V\). Depending on the additional requirements imposed on the spring, the principles of further selection may be different. Suppose that the displacement \(\lambda\) of the end of the spring, by means of a mechanism with transmission ratio \(i\), must be converted into displacement of the pointer. Usually one transmission mechanism is used with a series of springs measuring different pressures. Therefore it is necessary to require that the spring being designed satisfy this mechanism. Assuming, for simplicity, that the value of the central angle \(\gamma\) is specified*, we obtain the following condition: at maximum pressure \(p\), the condition

\[ \left(\frac{\Delta\gamma}{\gamma}\right)_{\mathrm{nom}} - \Delta \leq \frac{\Delta\gamma}{\gamma} \leq \left(\frac{\Delta\gamma}{\gamma}\right)_{\mathrm{nom}} + \Delta, \tag{2} \]

must be satisfied.

* The question of choosing the optimal value of the central angle of the spring is considered in [1].

where

\[ \left(\frac{\Delta\gamma}{\gamma}\right)_{\mathrm{nom}} \]

is the nominal value of the relative angle of unbending, corresponding to the specified mechanism and scale;

\[ \pm \Delta \]

is the tolerance for the deviation of the value \(\dfrac{\Delta\gamma}{\gamma}\) from the mean value.

Discarding the variants that do not satisfy condition [2], we obtain a group of variants differing in the values of the tractive moment \(M\), the maximum stress \(\sigma\), and the change in volume \(\Delta V\). Let us suppose that no requirements are imposed on the measuring spring with respect to the magnitude of the change in volume \(\Delta V\). In this case it remains to analyze the influence on the quality of the instrument of the quantities \(M\) and \(\sigma\).

The transmission mechanism offers resistance to the displacement of the end of the spring. This resistance is caused by the tension of the hair spring and by friction. The tension of the hair spring leads to a slight change in the stiffness of the system, which may be neglected. Since, in serial and mass production of instruments, individual springs are manufactured with deviations of their characteristics from the nominal values, provision is usually made for the possibility of adjusting the transmission ratio of the instrument mechanism. If the travel of the spring end exceeds the nominal value, the transmission ratio must be reduced so as to preserve the specified angle of rotation of the pointer, and conversely. Thus, between the travel \(\lambda\) of the spring end and the transmission ratio \(j\) of the mechanism there is an inversely proportional relationship

\[ j \cdot \lambda = j_0 \cdot \lambda_0, \]

where \(j_0\) and \(\lambda_0\) are the nominal values of the transmission ratio and sensitivity.

Let us denote by \(\Delta M\) the friction moment in the mechanism, reduced to the pointer axis;

\(\varphi\) is the angle of rotation of the pointer corresponding to the upper limit of the measured pressure.

The tractive moment developed by the spring, after reduction to the pointer axis, will have the value \(\dfrac{M}{j}\). The presence of the friction moment \(\Delta M\) causes a change in the angle of rotation of the pointer by the amount \(\Delta\varphi\). We may write

\[ \frac{\Delta\varphi}{\varphi} = \frac{\Delta M}{\dfrac{M}{j}} = j_0\lambda_0\,\frac{\Delta M}{M\lambda}. \tag{3} \]

The ratio \(\Delta\varphi/\varphi\) is the relative error of the pointer rotation angle caused by friction. It follows from the formula obtained that the relative error caused by friction will be the smaller, the greater the product \(M\lambda\).

For a given value of the central angle \(\gamma\) and radius of the spring’s central axis \(R_0\), the travel \(\lambda\) is proportional to the relative angle of unbending. In order that the error caused by friction in the mechanism not exceed the permissible value, the following condition must be satisfied:

\[ M \cdot \frac{\Delta\gamma}{\gamma} \geq \left( M_{\mathrm{t}} \cdot \frac{\Delta\gamma}{\gamma} \right)_{\mathrm{perm}} . \tag{4} \]

The variants remaining after condition (4) has been fulfilled can be evaluated on the basis of the following considerations: 1) the greater the product

$M \cdot \Delta\gamma/\gamma$, the more accurate the instrument; 2) the lower the stress, the longer the service life and the greater the accuracy of the instrument.

Thus, we arrive at the condition

\[ \frac{M \cdot \dfrac{\Delta\gamma}{\gamma}}{\sigma_{\mathrm{eqv}}} = \left( \frac{M \cdot \dfrac{\Delta\gamma}{\gamma}}{\sigma_{\mathrm{eqv}}} \right)_{\max}. \tag{5} \]

We shall consider as optimal a spring design satisfying all the requirements listed above. The scheme for finding the optimal design variant must change as the requirements imposed on the spring change. Thus, for springs operating according to the principle of force compensation, the need for condition (2) disappears. It must be replaced by the following:

\[ M_{\mathrm{opt}}-\delta \leq M_T \leq M_{\mathrm{opt}}+\delta . \tag{6} \]

For instruments whose error depends on the change in the volume of the spring cavity, $\Delta V$, one of the principal conditions in spring design will be that the ratio $\dfrac{\Delta\gamma}{\gamma}/\Delta V$ or $M/\Delta V$ attain a maximum value (depending on whether the measurement is made according to a kinematic scheme or according to a force-compensation scheme). Thus, the scheme for finding the optimal design variant of a spring, first, contains a large number of conditions and, second, must be modified depending on the intended purpose of the spring.

In light of the above, the inadequacy is evident, for solving the problem of optimal design, of the universal quality criterion for manometric springs proposed by G. I. Tyzhnov [2] and used by B. N. Vasil’ev [3] for a comparative evaluation of springs with different cross-sectional shapes. According to [2], the “quality” of a spring is the higher, the greater the quantity

\[ \frac{\Delta\gamma}{\gamma}\, \frac{E}{(1-\mu^2)\cdot \max \sigma_{\mathrm{eqv}}}. \]

L. E. Andreeva [4] constructed, for a flat-oval-section spring, a series of nomograms that make it possible to select the spring parameters according to the requirements specified for it. However, as the cross-sectional shape becomes more complicated, the number of parameters that must be taken into account in constructing the nomograms increases; consequently, the number of nomograms increases, and their use becomes rather ineffective. These difficulties can be overcome with the aid of electronic digital computers. The availability of a standard program for optimal design makes it possible to carry out the design in a short time, to take into account with the required accuracy the influence of various factors, and to eliminate errors associated with the individual characteristics of the designer.

Several concrete problems of optimal design were solved according to the proposed scheme. As possible cross-sectional shapes, the oval, flat-oval, and figure-eight shapes were selected. The program for calculating the relative angle of unbending $\dfrac{\Delta\gamma}{\gamma}$, the maximum equivalent stress $\sigma_{\mathrm{eqv}}$, and the tractive moment $M_T$ was compiled on the basis of the solution of the problem of a spring with an arbitrary cross-sectional shape, presented in [5].

The first problem is formulated as follows.

It is required to determine the optimal values of the geometric parameters of a series of springs intended for measuring pressure from $0.6\ \mathrm{kg/cm^2}$ to $100\ \mathrm{kg/cm^2}$ (Table 1).

The cross-sectional shape is oval or figure-eight. The radius of the spring centerline is specified as $R_0 = 55\ \mathrm{mm}$ and the major semiaxis of the section

Table 1

$A = 17$ mm. Material: Br. BNT-1.7; modulus of elasticity of the material
$E = 1.28 \cdot 10^6$ kg/cm$^2$. The geometric parameters of the section may take the following discrete values:

\[ b=b_n=0.08n,\quad n=1,2,3,\ldots,6; \]

\[ r=r_{n+1}=\frac{r_n}{1.18+0.02n},\quad r_1=0.50,\quad n=1,2,3,\ldots,14; \]

\[ |b-r|\geq 0.1; \]

\[ H=0.1\cdot n,\quad n=1,2,3,\ldots,25. \]

The total number of variants of the section shape is 2100.

Selection procedure.

For the variant under consideration, the stresses and the travel $\lambda(p_i)$ are calculated $(i=1,2,\ldots,12)$.

If $\sigma(p_i)\leq[\sigma]$, $0.8\lambda_i\leq\lambda(p_i)\leq1.2\lambda_i$, then the variant is assigned to the $i$-th group. Variants that do not fall into any of the 12 groups are rejected. For the remaining variants the quantity is calculated

\[ X=M_{\mathrm{T}}\cdot \frac{\lambda}{\sigma}. \]

In each group, 10 variants having the largest value of $X$ are selected.

Part of the calculation results for $[\sigma]=3430$ kg/cm$^2$ is given in Table 2. For each pressure step (with the exception of $p=0.6$ and $p=1$), 10 variants were selected, arranged in decreasing order of the quantity

\[ \frac{M_{\mathrm{T}}\cdot\lambda}{\sigma}. \]

At pressures $p=0.6$ and $p=1$, the number of variants satisfying the specified conditions proved to be less than ten.

In the region of low pressures (from 0.6 kg/cm$^2$ to 10 kg/cm$^2$), the optimal designs selected on the electronic digital computer have an oval-shaped section, and the radius $R_1$ of rounding at the ends of the major axis of the section is the smaller, the lower the measured pressure. Beginning with a pressure of 16 kg/cm$^2$, the optimal designs have a figure-eight-shaped section, and the minor semiaxis $B$ of the section decreases as the measured pressure increases. Ideally flat-oval springs were excluded from consideration. Analysis of the results shows that section shapes close to flat-oval correspond to pressures of 10 kg/cm$^2$ and 16 kg/cm$^2$.

The results of optimal design on the electronic digital computer confirm the established view that oval-section springs are most suitable for measuring low pressures, while figure-eight-section springs are most suitable for measuring high pressures. Flat-oval springs, which have become widely used because of their manufacturability, and also because of the absence of clear recommendations on the range of application of particular section shapes, in the overwhelming majority of cases did not withstand comparison with oval and figure-eight-

different. In this connection it should be noted that the conclusion drawn in [3], according to which “springs with a plane-oval cross section have a number of advantages in comparison with springs of other types of profiles

Table 2

profiles (it provides the greatest travel at the nominal pressure and a comparatively large tractive moment and serviceability-

Table 3

...“fatigue life”), requires specification. In the example of optimal design considered, this conclusion was confirmed in 2 cases (for pressures of 16 kg/cm² and 10 kg/cm²) and was not confirmed in 10 cases (for all other pressures from 0.6 kg/cm² to 100 kg/cm²).

The selection criteria in the problem considered, in the pressure range from 0.6 kg/cm² to 6 kg/cm², approximately correspond to the conditions adopted by the Smolensk branch of NIITeplopribor in developing the standard for springs with an internal diameter \(D_{\mathrm{in}} = 100\) mm made of dispersion-hardening alloy Br. BNT-1.7. Table 3 gives, for comparison, the principal calculated data of springs included in the draft standard and of springs selected according to the specified program on an electronic digital computer.

Design using an electronic digital computer on the basis of the theory set forth in the present work has the following advantages:

1. The springs selected on the computer, while satisfying all the stated requirements, have a large strength margin. If, for springs from the draft standard, the highest calculated stress reaches 7076 kg/cm², then for springs selected on the computer it is 5760 kg/cm². With some relaxation of the requirements, springs can be selected for which the maximum stress value does not exceed 3440 kg/cm².

2. The requirements on the value of serviceability are satisfied in all cases, whereas springs for measuring pressures of 4 kg/cm² and 6 kg/cm² from the draft standard have calculated serviceability values below the prescribed value.

3. The requirements on the value of the relative unbending angle are satisfied with high accuracy, whereas the calculated values of the relative unbending angle for springs from the draft standard deviate considerably from the specified values, both upward and downward.

4. High productivity. In the problem considered, 2100 variants were calculated and about 120 were selected (10 best variants for each of the 12 levels of measured pressure); this required 8 minutes of machine time on the M-20 computer. As may be concluded from the comparison given above of the results of machine and conventional design, the variants selected by the machine will require minimal expenditure for experimental refinement.

REFERENCES

1. V. I. Kopytov, M. P. Shumskii. On the question of the rational length of a Bourdon spring. Izv. Tomsk Polytechnic Institute, vol. 133, 1965.

2. G. I. Tyzhnov. Deformations and stresses in tubular manometric springs. Author’s abstract of dissertation, Tomsk Polytechnic Institute, 1964.

3. B. N. Vasil’ev. Calculation of thin-walled Bourdon manometric springs. Author’s abstract of dissertation, LIIZhT, L., 1966.

4. L. E. Andreeva. Elastic Elements of Instruments. Mashgiz, 1960.

5. M. P. Shumskii. Calculation of manometric springs. Izv. vuzov SSSR.—Instrument Engineering, No. 5, 1966.