Determination of Correlation Dependencies for Simplified Calculation of the Coefficients of Influence of Technological Errors on the Technical Characteristics of Roller Table Motors

DETERMINATION OF CORRELATION DEPENDENCIES FOR SIMPLIFIED CALCULATION OF THE COEFFICIENTS OF INFLUENCE OF TECHNOLOGICAL ERRORS ON THE TECHNICAL CHARACTERISTICS OF ROLLER-TABLE MOTORS

V. V. Dneprovskii

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

For calculating tolerances on output parameters at the design stage, it is necessary to determine the coefficients of the equations for the principal errors and the technological errors in the input parameters. Since most coefficients of influence of changes in input parameters on deviations of output parameters have a large variability over the standard sizes of electric motors, there naturally arises the need to determine them for individual standard sizes of machines. This is important in developing intermediate modifications, which is characteristic of roller-table motors. The previously considered method [1] for determining the coefficients of influence \(C_{ij}\) is complicated, since the analytical formulas for their determination are cumbersome.

When determining the coefficients of influence for the standard sizes of motors of interest to us, a simpler way of determining them is necessary; therefore, of particular interest is finding simple regression equations relating the coefficients of influence \(C_{ij}\) to the basic characteristics of the motor: the machine volume—\(D_a^2l\), the rated power—\(P_n\), and the number of poles—\(2p\). Thus, from the results of calculating each coefficient \(C_{ij}\) for all standard sizes of the series, it is necessary to find regression equations of the form

\[ \hat{y}_u = b_0 + b_1(D_a^2l) + b_2P_n + b_3 2p,\quad (u = 1,2,\ldots,n), \tag{1} \]

where \(\hat{y}_u\) is the corresponding coefficient of influence.

Finding the regression coefficients \(b_0, b_1, b_2, b_3\) of equation (1) is the usual problem of regression analysis, whose solution is based on minimizing the sum of squares of deviations between experimental and calculated values according to equation (1). The regression coefficients were computed on the “Promin” computer. The data for calculating the coefficients of influence by equation (1) are given in Table 1. For the obtained regression equations a statistical analysis was carried out. As a quantity characterizing the contribution of \(k\) regression coefficients to equation (1), containing \(k+1\) terms, the multiple correlation coefficient is introduced:

\[ R = \sqrt{ \frac{S_k}{ \displaystyle \sum_{u=1}^{n}(y_u-\bar{y})^2 } }, \tag{2} \]

where

\(S_k\) is the sum of squares relating to the \(k\) regression coefficients;

\(y_u\) is the value of the influence coefficient obtained from the formulas [1] for each motor type size;

\(\overline{y}\) is the mean value of the influence coefficients for all type sizes, given in matrix \(A\) [1].

All intermediate calculations connected with the determination of \(R\) were carried out according to the procedure of [2].

The multiple correlation coefficient \(R\) can be interpreted as a measure of the linear relationship between the influence coefficients \(C_{ij}\) and the independent variables \(D_a^2 l\), \(P_n\), \(2p\). Its value may vary from zero to \(+1\). If the contribution made by the regression coefficients is zero, then \(S_k = 0\) and \(R = 0\); but if the regression equation completely describes the experimental results, then \(R = 1\).

The significance of the multiple correlation coefficient is determined by the \(F\)-ratio

\[ F = \frac{s_k^2}{s_R^2}, \tag{3} \]

where

\[ s_k^2 = \frac{S_k}{f_k}; \]

\[ f_k = k = 3; \]

\[ s_R^2 = \frac{S_R}{f_R} \]
is the residual variance;

\[ f_R = n - k - 1 = 34; \]

\(S_R\) is the residual sum of squares.

If the computed value \(F > F_T\) for the selected significance level, then \(S_k\) and \(R\) are significant, and one may speak of the significance of the regression equation as a whole.

In our case, for the 5% significance level, with \(f_k = 3\) for the larger variance and \(f_R = 34\) for the smaller variance, from [3] we find \(F_T = 2.88\). Table 1 gives the computed \(F\)-ratio.

As is evident from the table, \(F > F_T\); consequently, the multiple correlation coefficient for the influence coefficients calculated by equation (1) proved to be significant.

A necessary task in the statistical analysis of regression equations is also to test the hypothesis of the adequacy of the obtained equations to the initial calculated data.

If the variance \(s^2\{y\}\), characterizing the experimental error, is known, then the \(F\)-ratio can be found as

\[ F = \frac{s_R^2}{s^2\{y\}}. \tag{4} \]

Since calculated values of the motor parameters were used in determining the influence coefficients, unambiguous values (without scatter) of the influence coefficients were obtained for each type size, \(s^2\{y\} = 0\). Here one may proceed as follows. Considering equation (1) adequate, one can determine the maximum error in determining the influence coefficients from the obtained equations. If the variance of this error is less than the variance of the influence coefficients over the entire series, then the regression equations can be used to calculate the influence coefficients for various type sizes of motors.

Table 1

The calculation procedure is as follows:

1. Specifying the significance level, from the number of degrees of freedom \(f_R\) for the residual variance \(s_R^2\) and from the number of degrees of freedom \(f_{s(y)}\), we find the \(F_T\)-ratio (tabulated). The number of degrees of freedom for the error variance \(s^2\{y\}\) must be taken as equal to infinity, as for the general population.

2. We determine

\[ s^2\{y\} = \frac{s_R^2}{F_T}. \]

In our case, for a 5% significance level we have

\[ f_R = n - k - 1 = 38 - 3 - 1 = 34;\quad F_T = 1.46;\quad s^2[y] = 0.685s_R^2. \]

The error variance proves to be smaller than the residual variance and, consequently, the regression equations may be used to calculate the influence coefficients. The absolute magnitude of the error \(s\{y\}\) is given in Table 1.

Thus, using the mathematical apparatus of regression analysis, simple regression dependences of the influence coefficients \(C_{ij}\) on the principal characteristics of the motor frame size have been obtained, which makes it possible to significantly simplify the calculation of the coefficients of influence of the input parameters on the output parameters.

REFERENCES

1. V. V. Dneprovskii, O. P. Muravlev. Evaluation of the sensitivity of the technical characteristics of roller-table motors to technological errors. Present collection.

2. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. “Nauka,” Moscow, 1965.

3. N. V. Smirnov, I. V. Dunin-Barkovskii. A Course in Probability Theory and Mathematical Statistics for Technical Applications. 2nd ed., “Nauka,” Moscow, 1965.