Quantitative Assessment of the Quality of Manufacturing Technology for Induction Motors

O. P. Muravlev, A. D. Nemtsev

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

One of the conditions for improving induction motors is a well-developed technology for their production. However, there is no objective quantitative method for assessing the technology used in the production of induction motors.

This article proposes a method for the quantitative assessment of the quality of the manufacturing technology of induction electric motors, which makes it possible to compare the technological processes of various electrical-machine-building plants and to evaluate the effectiveness of changes in technology and the use of new materials.

Let us consider the possibility of obtaining a quantitative criterion for assessing the technology of manufacturing induction motors. Each assembly unit or part of an electric motor can be represented by a vector with a certain number of elements:

\[ X^{(1)}=(x_1^{(1)}, x_2^{(1)}, \ldots, x_i^{(1)}, \ldots, x_{n1}^{(1)}); \]

\[ X^{(2)}=(x_1^{(2)}, x_2^{(2)}, \ldots, x_i^{(2)}, \ldots, x_{n2}^{(2)}); \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \]

\[ X^{(j)}=(x_1^{(j)}, x_2^{(j)}, \ldots, x_i^{(j)}, \ldots, x_{nj}^{(j)}); \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \]

\[ X^{(m)}=(x_1^{(m)}, x_2^{(m)}, \ldots, x_i^{(m)}, \ldots, x_{nm}^{(m)}); \]

where \(j=1,2,\ldots,m\) is the number of assembly units and parts under consideration,
\(i=1,2,\ldots,n_j\) is the number of parameters considered for each assembly unit or part.

In this system of notation, each element of a vector characterizes some parameter. The parameters may be dimensions, the mutual arrangement of surfaces, etc., which substantially affect the operation of the machine.

Taking into account that in the production of induction motors a large number of random factors act, it is expedient to take as the quantitative criterion for assessing the technology the probability that, in an assembled motor, all assembly units and parts satisfy the requirements of the technical conditions:

\[ P_T=P[X^{(1)}]\times P[X^{(2)}]\times \cdots \times P[X^{(j)}]\times \cdots \times P[X^{(m)}] =\prod_{j=1}^{m} P[X^{(j)}], \tag{1} \]

where \(P[X^{(j)}]\) is the probability that all parameters of the \(j\)-th assembly or part satisfy the requirements of the technical specifications.

\[ P[X^{(j)}] = P(x_1^{(j)}) \times P(x_2^{(j)}) \times \cdots \times P(x_i^{(j)}) \times \cdots \times P(x_{n_j}^{(j)}), \tag{2} \]

where \(P(x_i^{(j)})\) is the probability that the \(i\)-th parameter of the \(j\)-th assembly satisfies the technical specifications.

The following were selected as the principal parameters.

1. Bearing shield:
   \(x_1^{(1)}\) — rabbet diameter, \(x_2^{(1)}\) — rabbet runout.

2. Rotor assembly:
   \(x_1^{(2)}\) — rotor core diameter, \(x_2^{(2)}\) — rotor runout relative to the bearing journals.

3. Stator:
   \(x_1^{(3)}\) — runout of the housing rabbet relative to the stator bore.

4. Stator winding:
   \(x_1^{(4)}\) — winding manufacturing defects (turn-to-turn, frame, and phase short circuits).

The method for determining the probability \(P(x_i^{(j)})\) depends on the attributes by which the given parameter is inspected.

In inspection by qualitative attributes,

\[ P(x_i^{(j)}) = 1 - \frac{n_g}{n} = 1 - \bar p, \tag{3} \]

where \(n\) is the total number of measurements,
\(n_g\) is the number of measurements for which the parameter values do not comply with the technical specifications.

Taking into account that the fraction of defective items in the sample has a binomial distribution law [1], the upper \(P_{\mathrm{v}}(x_i^{(j)})\) and lower \(P_{\mathrm{n}}(x_i^{(j)})\) probability values can be determined using the following formulas:

\[ P_{\mathrm{v}}(x_i^{(j)}) = 1 - (\bar p - 3\sigma_p), \tag{4} \]

where

\[ \sigma_p = \frac{\sqrt{\bar p(1-\bar p)}}{\sqrt{n}}. \tag{5} \]

If \(\bar p - 3\sigma_p < 0\), one should take \(P(x_i^{(j)}) = 1\),

\[ P_{\mathrm{n}}(x_i^{(j)}) = 1 - (\bar p + 3\sigma_p). \tag{6} \]

In inspection by quantitative attributes,

\[ P(x_i^{(j)}) = \int_{x_{\mathrm{n}}}^{x_{\mathrm{v}}} f(x)\,dx, \tag{7} \]

where \(x_{\mathrm{v}}\) and \(x_{\mathrm{n}}\) are the upper and lower values of the controlled parameter;
\(f(x)\) is the distribution density of the controlled parameter, which characterizes the law of distribution of values.

For the normal distribution law most frequently encountered in practice, the distribution density has the form

\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\bar x)^2}{2\sigma^2}}, \tag{8} \]

where \(\bar x\) is the mean value of the parameter,
\(\sigma\) is the standard deviation.

The standard error of the standard deviation \([1, 2]\)

\[ \sigma_s = \frac{\sigma}{\sqrt{2n}}, \tag{9} \]

where \(n\) is the total number of measurements.

Then the lower \(\sigma_{\mathrm{l}}\) and upper \(\sigma_{\mathrm{u}}\) estimates of the standard deviation will be equal to

\[ \sigma_{\mathrm{l}} = \sigma - 3\sigma_s, \tag{10} \]

\[ \sigma_{\mathrm{u}} = \sigma + 3\sigma_s. \tag{11} \]

On the basis of expressions (7), (8), (10), (11), we obtain the upper \(P_{\mathrm{u}}\!\left(x_i^{(j)}\right)\) and lower \(P_{\mathrm{l}}\!\left(x_i^{(j)}\right)\) values of the probability of defect-free manufacture of a part with respect to the parameter under consideration

\[ P_{\mathrm{u}}\!\left(x_i^{(j)}\right) = \frac{1}{\sigma_{\mathrm{l}}\sqrt{2\pi}} \int_{x_{\mathrm{l}}}^{x_{\mathrm{u}}} e^{-\frac{(x-\overline{x})^2}{2\sigma_{\mathrm{l}}^2}}\,dx, \tag{12} \]

\[ P_{\mathrm{l}}\!\left(x_i^{(j)}\right) = \frac{1}{\sigma_{\mathrm{u}}\sqrt{2\pi}} \int_{x_{\mathrm{l}}}^{x_{\mathrm{u}}} e^{-\frac{(x-\overline{x})^2}{2\sigma_{\mathrm{u}}^2}}\,dx. \tag{13} \]

Expressions (4), (6), (12), (13) make it possible to obtain the upper \(P_{\mathrm{t,u}}\) and lower \(P_{\mathrm{t,l}}\) values of the criterion for assessing the quality of the technology.

\[ P_{\mathrm{t,u}} = \prod_{j=1}^{m} P_{\mathrm{u}}\!\left[X^{(j)}\right]. \tag{14} \]

\[ P_{\mathrm{t,l}} = \prod_{j=1}^{m} P_{\mathrm{l}}\!\left[X^{(j)}\right]. \tag{15} \]

Example of calculation

Let us consider the calculation of the probability of defect-free manufacture of asynchronous motors of the fourth frame size under the existing manufacturing technology and inspection system. The initial data and the results of the calculation are presented in Table 1.

When calculating the probability of defect-free manufacture of the stator winding, the number of assembled machines submitted for inspection tests was taken as \(n\), and the number of machines rejected after inspection tests for turn-to-turn, frame, or phase-to-phase short circuits was taken as \(n_{\mathrm{d}}\).

From the results of Table 1, the lower \(P_{\mathrm{t,l}}\), upper \(P_{\mathrm{t,u}}\), and mean \(P_{\mathrm{t}}\) values of the probabilities of defect-free manufacture of the machine were calculated.

\[ P_{\mathrm{t}} = \prod_{j=1}^{m} P\!\left[X^{(j)}\right] = 0.5679 \cdot 0.875 \cdot 0.9828 = 0.487; \]

\[ P_{\mathrm{t,l}} = \prod_{j=1}^{m} P_{\mathrm{l}}\!\left[X^{(j)}\right] = 0.55 \cdot 0.81 \cdot 0.973 \cdot 0.742 \cdot 0.874 \cdot 0.9826 = 0.276; \]

\[ P_{\mathrm{t,u}} = \prod_{j=1}^{m} P_{\mathrm{u}}\!\left[X^{(j)}\right] = 0.7289 \cdot 0.99 \cdot 0.998 \cdot 1 \cdot 1 \cdot 0.983 = 0.7084. \]

Table 1

It is evident from the calculation that the level of manufacturing technology for motors of the fourth frame size is rather low.

Conclusions

1. The proposed quantitative assessment of the quality of the technology has physical meaning and makes it possible to compare the technological processes of different enterprises producing identical products, and to evaluate the effectiveness of changes in technology and the use of new materials.

2. By calculating the individual probabilities $P[X(i)]$ for assemblies and parts, it is possible to determine the “weakest” point in the manufacturing technology of induction motors.

3. In determining $P_{\mathrm{t}}$, the effectiveness of the inspection system existing at the enterprise is assessed: if the value of $P_{\mathrm{t}}$ is close to unity, then all is well; if not, the enterprise is either producing motors with defects or has to reject finished machines.

References

1. B. Hansen. Quality Control. Moscow, Progress, 1968.

2. A. K. Mitropolsky. Technique of Statistical Computations. Moscow, Fizmatgiz, 1961.

3. G. Korn, T. Korn. Mathematical Handbook for Scientists and Engineers. Moscow, Nauka, 1970.