Determination of the Most Influential Design and Technological Parameters on the Stability of the Technical Characteristics of Asynchronous Roller-Table Motors
Abstract Generated abstract
This paper examines which design and technological parameters most affect the stability of technical characteristics in AR-series asynchronous roller-table motors. Using accuracy theory, equivalent-circuit relationships, and matrix calculations of influence coefficients for 38 motor frame sizes, the authors estimate how input-parameter scatter contributes to deviations in starting torque, maximum torque, starting current, efficiency, power factor, dynamic constant, and slip. The analysis indicates that most output variation is associated with the rotor-cage material resistivity, air gap, steel and mechanical losses, and conditional heat-transfer coefficient, while parameters such as wire diameter and core lengths have comparatively small effects. The results identify priority areas for improving manufacturing processes to achieve more stable motor characteristics.
Full Text
Vol. 301 1975
Determination of the Most Strongly Influencing Design-and-Technological Parameters on the Stability of the Technical Characteristics of Asynchronous Roller-Table Motors
V. V. Dneprovsky, S. A. Shelekhov
(Presented by the joint scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The quality of asynchronous roller-table motors of the AR series is characterized by ensuring the accuracy of the following output parameters: \(M_{\mathrm{p}}\)—starting torque; \(M_{\mathrm{m}}\)—maximum torque; \(I_{\mathrm{k}}\)—starting current; \(\eta\)—efficiency; \(\cos \varphi\)—power factor; \(D\)—dynamic constant; \(S\)—slip.
The accuracy of the output parameters is determined by deviations of the design-and-technological input parameters. The scatter of the output parameters depends, to varying degrees, on the scatter of the input parameters. As such parameters we considered the following: air gap—\(\delta\); lengths of the stator and rotor cores—\(l_1\) and \(l_2\); specific resistance of the rotor cage material—\(\rho_{\mathrm{spl}}\) (for an aluminum-magnesium alloy); wire diameter, mean turn length, and number of turns of the stator winding—\(d_{\mathrm{pr}}\), \(l_{\mathrm{m}}\), and \(W_1\); the sum of losses in steel and mechanical losses—\(p_{\mathrm{st}} + p_{\mathrm{mech}}\); and the conditional heat-transfer coefficient—\(\alpha^*\).
By \(\alpha^*\) is meant the conditional heat-transfer coefficient from the heat-dissipating surface of the active steel (the outer surface of the stator core and the sum of the end surfaces of the stator and rotor), under the condition of the permissible temperature rise of the stator winding. Conditional heat-transfer coefficients are used when checking AR-series motors for heating under complex operating modes, which are often encountered in the operation of rolling mills.
Determining the causes of the scatter of all input parameters and carrying out measures to refine—and, if necessary, change—technological processes in order to reduce the scatter of each input parameter requires extensive and labor-intensive work. Therefore, in order to carry out effective measures to ensure the accuracy of the output characteristics, it is necessary to identify the most strongly influencing input parameters.
The purpose of the present work is to establish the input parameters that most strongly influence the scatter of the output parameters.
We solved this problem on the basis of the general theory of accuracy and the equations relating the input and output parameters.
The relationship between the input \(x_j\) and output \(y_i\) parameters of electrical machines may be represented in general form by the equation
\[ y_i = \varphi_i(x_1, \ldots, x_j, \ldots, x_m), \quad i = 1,2,\ldots,l,\ m). \tag{1} \]
The system of equations relating the tolerances for the output paramet-
parameters with tolerances on the input parameters by the probabilistic method in accordance with [1] is written as
\[ \delta y_i^2=\sum_1^m C_{ij}^2\delta x_j^2, \tag{2} \]
where \(\delta y_i, \delta x_j\) are half the scatter field, respectively, for the output and input parameters;
\(C_{ij}\) is the coefficient of influence of the deviation of the \(j\)-th input parameter on the scatter of the \(i\)-th output parameter.
In the presence of simple analytical dependences of type (1), the calculation of the influence coefficients is performed from the expression
\[ C_{ij}=\frac{\partial y_i}{\partial x_j}\cdot \frac{x_j}{y_i}. \tag{3} \]
For asynchronous motors, analytical expressions of type (1) for most output parameters are complex functions. In addition, the technical literature uses mainly analytical equations relating the output parameters to the parameters of the equivalent circuit, and equations relating the parameters of the equivalent circuit to the input parameters
\[ z_k=\varphi_k(x_1,\ldots,x_j,\ldots,x_m),\ (k=1,2,\ldots,n;\ j=1,2,\ldots,m), \tag{4} \]
where \(z_k\) is the corresponding parameter of the equivalent circuit.
In this case, when the output parameters \(y_1,\ldots,y_i,\ldots,y_l\) are functions of the parameters of the equivalent circuit \(z_1,\ldots,z_k,\ldots,z_n\), and the latter, in turn, are functions of the input parameters \(x_1,\ldots,x_j,\ldots,x_m\), it is advisable to determine the influence coefficients \(C_{ij}\) using the matrix form of notation [2].
To determine the input parameters that strongly influence the scatter of the output parameters, it is necessary to have a matrix of mean values of the coefficients \(C_{ij}\) for all motor frame sizes and averaged accuracy characteristics \(\delta x_j\) of the technological processes adopted in motor manufacture. In this case, the strongly influencing input parameters, determined on the basis of the accuracy equations (2), will reflect the overall picture in the production of this series of motors.
The present work was carried out as applied to a series of asynchronous roller-table electric motors manufactured by the Sibelektromotor plant.
The matrix of mean values of the influence coefficients \(C_{ij}\) was determined as follows. On the basis of expression (3), analytical equations for the interrelation of the output parameters with the parameters of the equivalent circuit, and analytical dependences of type (4), the coefficients of influence of the equivalent-circuit parameters on the output parameters—the matrix “A”—and the coefficients of influence of the input parameters on the equivalent-circuit parameters—the matrix “B”—were calculated.
The elements of matrices “A” and “B” were calculated for 38 frame sizes of roller-table motors of the basic design at 380 V, 50 Hz. By multiplying matrices “A” and “B” for each motor frame size, matrices “C” were calculated for the coefficients of influence of deviations of the input parameters on the scatter of the output parameters.
After statistical processing of the corresponding influence coefficients \(C_{ij}\) for all motor frame sizes, a matrix of mean values of the influence coefficients for motors of the AR series was obtained.
The averaged accuracy characteristics of the technological processes were calculated from the scatters of the input parameters determined from the results of standard tests and specially conducted experimen-
Table 1
| \(j\) | Input parameter \(x_j\) | \(M_p\), \(i=1\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) | \(M_k\), \(i=2\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) | \(I_k\), \(i=3\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) | \(\eta\), \(i=4\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) | \(\cos\varphi\), \(i=5\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) | \(D\), \(i=6\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) | \(S\), \(i=7\) \(\left(C_{1i}^{2}\delta_{x_j}^{2}\right)\) |
|---|---|---|---|---|---|---|---|---|
| 1 | \(w_1\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | \(d_{\mathrm{pr}}\) | 0,885 | 0,839 | 0,221 | 0,196 | 0,073 | 1,556 | 0,157 |
| 3 | \(l_m\) | 1,426 | 1,208 | 0,357 | 0,140 | 0,046 | 1,184 | 0,094 |
| 4 | \(\rho_{\mathrm{st.p}}\) | 9,437 | 0 | 65,338 | 9,915 | 1,858 | 45,158 | 536,15 |
| 5 | \(\delta\) | 116,96 | 79,747 | 29,241 | 12,848 | 72,734 | 9,934 | 0,645 |
| 6 | \(l_1\) | 0,486 | 0,332 | 0,121 | 0,090 | 0,502 | 0,078 | 0,002 |
| 7 | \(l_2\) | 0,063 | 0,105 | 0,334 | 0,022 | 0,003 | 0,074 | 1,262 |
| 8 | \(\rho_{\mathrm{st}}+\rho_{\mathrm{mech}}\) | — | — | — | 6,351 | 14,370 | 1,696 | — |
| 9 | \(\alpha\) | — | — | — | — | — | 278,160 | — |
| — | \(\sum C_{1i}^{2}\delta_{x_j}^{2}\) | 129,26 | 82,231 | 95,612 | 29,562 | 89,586 | 337,84 | 538,30 |
Table 2
| Parameter | Fractional contribution | \(M_{\mathrm{p}}\) \(i=1\) |
\(M_{\mathrm{m}}\) \(i=2\) |
\(I_{\mathrm{k}}\) \(i=3\) |
\(\eta\) \(i=4\) |
\(\cos \varphi\) \(i=5\) |
\(D\) \(i=6\) |
\(S\) \(i=7\) |
|---|---|---|---|---|---|---|---|---|
| \(\rho_{\mathrm{st}}\) | \(C_{14}^{2}\delta_{x4}^{2}/\sum C_{1j}^{2}\delta_{xj}^{2},\ \%\) | 7.3 | — | 68.3 | 33.5 | 2.1 | 13.4 | 99.6 |
| \(\delta\) | \(C_{15}^{2}\delta_{x5}^{2}/\sum C_{1j}^{2}\delta_{xj}^{2},\ \%\) | 90.5 | 97.0 | 30.6 | 43.5 | 81.2 | 2.9 | 0.1 |
| \(p_{\mathrm{st}}+p_{\mathrm{mech}}\) | \(C_{18}^{2}\delta_{x8}^{2}/\sum C_{1j}^{2}\delta_{xj}^{2},\ \%\) | — | — | — | 21.5 | 16.0 | 0.5 | — |
| \(\alpha\) | \(C_{19}^{2}\delta_{x9}^{2}/\sum C_{1j}^{2}\delta_{xj}^{2},\ \%\) | — | — | — | — | — | 82.3 | — |
| \(d_{\mathrm{pr}},\ l_{1},\ l_{2}\) | \(\dfrac{C_{12}^{2}\delta_{x2}^{2}+C_{13}^{2}\delta_{x3}^{2}+C_{16}^{2}\delta_{x6}^{2}+C_{17}^{2}\delta_{x7}^{2}}{\sum C_{1j}^{2}\delta_{xj}^{2}},\ \%\) | 2.2 | 3.0 | 1.1 | 1.5 | 0.7 | 0.9 | 0.3 |
... [3] and measurements carried out under conditions of mass production of AR-series motors.
Table 1 gives the component contributions \(C_{ij}^{2}\delta_{xj}^{2}\) for the output parameters considered, calculated from the mean values of the coefficients \(C_{ij}\) and the mean values of the spreads of the input parameters \(\delta_{xj}\) for the entire series of roller-table motors.
Table 2 gives the fractional contributions of the individual components of the input parameters to the total deviation for each output parameter. These tables show that the principal factors affecting the spread of the output parameters of asynchronous roller-table electric motors are deviations in the specific resistance of the rotor-cage material, the air gap, the mechanical and total losses in the steel, and the conditional heat-transfer coefficient. More than 95% of the total deviation of each output parameter is due to the spread of precisely these input parameters, which affect the deviations of the individual technical characteristics in different ways:
— the spread of the values \(M_{\text{p}}\) and \(M_{\text{m}}\) is determined by the instability of the air gap by more than 90%;
— the spread of \(\eta\) and \(\cos\varphi\) is determined by deviations of \(\delta\), \(p_{\text{st}} + p_{\text{mech}}\), and \(\rho_{\text{spl}}\) by 99%;
— the spread of the dynamic constant is determined mainly by the instability of the conditional heat-transfer coefficient and of the specific resistance of the rotor-cage material;
— the spread of slip is completely determined by the instability of the specific resistance of the alloy.
Identification and study of the most strongly influential input parameters make it possible to outline the most effective ways of carrying out measures to improve technological processes in order to obtain the required accuracy of the output parameters.
REFERENCES
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V. V. Dneprovskii, O. P. Muravlev. Influence of design and technological factors on the specific electrical resistance of rotor cages in asynchronous motors for roller tables. Electrical Engineering Industry, series “Technology of Electrical-Engineering Production,” Moscow, OVNIIZM, issue 5–6 (38–39), 1972.