Study of the Relationship Between the Tolerance of Input Parameters for Unified-Series Induction Motors and the Cost of Ensuring Them
Abstract Generated abstract
This paper addresses the problem of assigning economically optimal tolerances to input parameters of unified-series induction motors while satisfying prescribed tolerances on output characteristics such as efficiency, power factor, and torque. It argues that conventional accuracy-theory methods are insufficient for electrical machines, because tolerances concern both geometric and material parameters, and proposes formulating the task as a mathematical programming problem using influence coefficients to express output-parameter variance constraints. Using factory data and aggregated cost indices, the authors derive approximate cost-tolerance relationships for core length, winding turn length, rotor diameter and air gap, and discuss fixed or externally determined tolerances for winding turns, wire diameter, aluminum resistivity, and rotor-bar integrity. These relationships provide the minimizing function needed for optimal tolerance assignment in induction motor production.
Full Text
STUDY OF THE RELATIONSHIP BETWEEN THE MAGNITUDE OF THE TOLERANCE OF INPUT PARAMETERS FOR UNIFIED-SERIES ASYNCHRONOUS MOTORS AND THE COST OF ENSURING THEM
V. I. ARZAMASKIN, O. P. MURAVLEV, E. K. STRELBITSKY
(Recommended by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering.)
Manufacturing accuracy is a decisive factor in improving the quality of machines and instruments [1]. Ensuring accuracy is a complex problem that must be solved when carrying out design-and-engineering work, when designing technology, and at all stages of production.
Assigning tolerances to individual parameters is a very complex task: it is necessary, on the one hand, to ensure the specified accuracy and, on the other, the minimum cost of manufacturing the machine.
The correct assignment of tolerances is associated with taking into account an enormous number of the most diverse factors and therefore cannot be carried out by means of simple formulas. When choosing tolerances, it is necessary to take into account the nature and condition of the equipment, the type of technological process, the qualifications of the workers, the materials used, etc.
The tolerances for the output parameters of asynchronous electric motors ($\eta$, p.f., $\cos \varphi$, starting and maximum torque, etc.) are established by GOST 183-55. When designing motors, the problem arises of selecting tolerances for the input parameters (air gap, length of the stator and rotor core, etc.), i.e., the inverse problem of the theory of accuracy.
In mechanical engineering and instrument making, where the theory of accuracy is most widely applied, three variants have been developed for solving the inverse problem [2]:
1) all tolerances on the input parameters are assigned equal values,
2) one manufacturing accuracy class is selected for all elements,
3) for all elements except one, tolerances may be selected taking economic requirements into account, while for the last one they are calculated.
In electrical-machine engineering, where tolerances must be assigned not only to geometrical dimensions but also to physical and mechanical parameters, these methods have little applicability. An optimal solution of the problem can be achieved by means of mathematical programming methods.
The problem of mathematical programming may be formulated as follows: there is some quantity (for example, the cost of manufacturing the machine) that is a function of a number of variables. These variables, in turn, are subject to constraints expressed in the form of a system of linear inequalities or equalities.
It is required to find such nonnegative values of the variables, satisfying the corresponding constraints, for which the quantity that is a function of them assumes the least (greatest) value.
The set of nonnegative values of the variables satisfying these conditions constitutes the optimal solution of the problem [3].
Depending on how the constraints and the minimizing function are specified—linear or nonlinear—methods of linear or convex programming are used [4]. Electronic computers are used to solve the problems.
Thus, in order to determine the optimal tolerances for the input parameters of asynchronous electric motors, for given tolerances on the output parameters, it is necessary to formulate the constraints and the minimizing function.
The system of constraints, which are linear for our problem, can be formulated using the coefficients of influence of the variance of the input parameters on the variance of the output parameters. If there are simple relationships between the input and output parameters, the coefficient of influence of the variance of the \(j\)-th input parameter on the variance of the \(i\)-th output parameter \(\alpha_{ij}\) can be determined analytically
\[ \alpha_{ij} = \left[ \frac{\partial f_i(x_1,\ldots,x_j,\ldots)}{\partial x_j} \cdot \frac{x_j}{f_i(x_1,\ldots,x_j,\ldots)} \right]^2, \tag{1} \]
where \(y_i = f_i(x_1,\ldots,x_j,\ldots)\) is the relationship between the output and input parameters.
The application of mathematical programming methods in solving the problem posed is made difficult by the fact that the relationships between the accuracy of the input parameters and the cost of ensuring it have not yet been established.
The purpose of the present work is to establish such relationships for asynchronous electric motors of a unified series.
As input parameters we shall consider the following quantities: the size of the air gap, the lengths of the stator and rotor cores, the length of a winding turn, the number of turns of the stator winding, the wire diameter, the specific resistance of the rotor aluminum, and the number of whole rotor bars.
We assume the distributions of all input quantities to be normal, with parameters \((m_j,\sigma_j)\), where \(m_j\) is the nominal value and \(\sigma_j\) is the standard deviation
\[ \sigma_j = \frac{2\delta_j}{6}, \tag{2} \]
where \(2\delta_j\) is the tolerance range.
The tolerance on the diameter of the stator-winding wire is determined according to GOST 2112-62 for copper wire and GOST 6132-52 for aluminum wire. The tolerance is ensured at the plants producing winding wire, and it may be assumed that, in the manufacture of electrical machines, taking into account the small tension of the wire when it is wound on a form, its value does not change.
Deviation of the number of turns of the stator winding from the nominal value, with automatic stopping of the winding machine, is unlikely, and the amount of rejection for this parameter for low-power asynchronous electric motors is very insignificant (according to data from the Quality Control Department of the Sibelektromotor plant). Therefore the number of turns of the stator winding may be regarded as a constant quantity, i.e. \(\sigma_w = 0\).
The length of the rotor or stator core depends on the accuracy of weighing the packet and on the pressing force. Weighing is performed
with an accuracy of up to one sheet, so that pressure is the determining factor in the formation of tolerances. The cost of ensuring the tolerance for machines of the A3 series, by frame size, may be approximately calculated from formula (3) for \(2\delta_e = 0.25 \ldots 1.0\) mm:
\[ C = 0.324 - 0.23 \delta_e \ \text{rub.} \tag{3} \]
The dependence of cost on the magnitude of the tolerance, as well as the following, was calculated on the basis of factory data using aggregated indices:
\[ C = З_{\text{п}} + Ц_{\text{р}} + О_{\text{р}} + В_{\text{п}} \ \text{rub.}, \tag{4} \]
where \(C\) is the cost of ensuring the tolerance;
\(З_{\text{п}}\) is the basic wage;
\(Ц_{\text{р}}\) is shop expenses;
\(О_{\text{р}}\) is general plant expenses;
\(В_{\text{п}}\) is nonproduction expenses.
The average turn length \(l_m\) depends on the magnitude and stability of the winding-wire tension. The cost as a function of the tolerance within \(2\delta = 0.8 \ldots 4.0\%\) for an AO 32-4 motor may, as a first approximation, be calculated by the formula
\[ C = 0.243 - 0.075\delta_{l_m} \ \text{rub.} \tag{5} \]
The magnitude of the air gap of an asynchronous electric motor is determined by the rotor diameter and the stator bore diameter. We regard the stator bore diameter as a constant quantity. Thus, the scatter in the air-gap magnitude is completely determined by the tolerance on the rotor diameter.
The dependence of machining cost on tolerance for cylindrical surfaces has been studied rather fully [5]. Since this dependence is nonlinear, we approximate it by a broken line.
The cost of ensuring tolerances for AO32-4 rotors is calculated by the following formulas:
\[ \delta = 0.005 \ldots 0.02 \ \text{mm} \quad C = 2.2 - 83.5\delta \ \text{rub.} \tag{6} \]
\[ \delta = 0.02 - 0.10 \ \text{mm} \quad C = 0.62 - 4.8\delta \ \text{rub.} \tag{7} \]
The specific resistance \(\rho_{\mathrm{Al}}\) of the aluminum used for casting the rotors of squirrel-cage asynchronous motors substantially affects the efficiency and starting characteristics of the machine. The value of the specific resistance is determined by impurities, by the presence of gases and nonmetallic inclusions [6, 7], which significantly increase \(\rho_{\mathrm{Al}}\). Refining aluminum makes it possible, to a considerable extent, to eliminate the influence of gas saturation and nonmetallic inclusions; however, it is not possible to exclude it completely, and the increase in resistance remains at the level of 1–3%.
The existing standard for alloys provides for control of the specific resistance of aluminum by chemical composition. In pure aluminum used for casting the rotors of motors of normal design, silicon content of up to 0.3% is permitted, and iron content of up to 0.7%, with total iron and silicon content of up to 0.9%. Such a composition guarantees a specific resistance in the cast rotor of
\[ \rho_{\mathrm{Al}} = 0.0312 \pm 0.0025 \ \text{ohm mm}^2/\text{m}, \text{ i.e. } \pm 8\%. \]
Analysis of the results of measuring the chemical composition of the aluminum used for casting rotors at the Sibelektromotor plant shows that the content of iron and silicon impurities is within the es-
established limits. Therefore, in solving our problem we shall assume that the deviation of the specific resistance of aluminum from the nominal value lies within $\pm 8\%$.
The amount of rejection due to poor casting of rotors is insignificant. Breakage of rotor bars occurs rarely, and such machines are rejected at the testing station. Therefore it may be assumed that the motors produced by the plant have all rotor bars intact, i.e., $\sigma_z = 0$.
Conclusions
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Ensuring the specified accuracy of the output parameters of electrical machines at minimum production cost is a primary task of modern electrical-machine engineering.
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To select the optimal system of tolerances, it is necessary to apply methods of mathematical programming.
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The obtained dependences of the cost of ensuring tolerances on the magnitude of the tolerance make it possible to construct a minimizing function, which is necessary for the optimal assignment of tolerances.
References
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G. G. Putskyn, P. I. Fedotov, E. A. Goncharova. Aluminum alloys for casting rotors. Proceedings of VNIIEM, vol. 6, 1960.
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G. G. Putskyn, E. A. Goncharova. The influence of gas saturation and nonmetallic inclusions on the electrical resistivity of aluminum and its alloys. Proceedings of NII Elektroprom, vol. 2, 1959.