Application of Statistical Quality Control Methods in the Manufacture of Mechanical Parts of Induction Motors
Abstract Generated abstract
This paper examines quality control practices in electrical machine manufacturing, focusing on mechanical parts and assemblies of asynchronous motors. It critiques complete inspection and existing sampling inspection as costly or ineffective when not based on risk analysis, and uses measurements of motor components to show that some process variations exceed tolerance fields. The paper recommends introducing statistical quality control, including quantitative control charts and single-stage attribute sampling, while emphasizing the need to align equipment capability with tolerances. It also proposes a generalized probabilistic criterion for assessing manufacturing technology quality, allowing comparison of processes across enterprises producing identical products.
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APPLICATION OF STATISTICAL METHODS OF QUALITY CONTROL IN THE MANUFACTURE OF MECHANICAL PARTS OF ASYNCHRONOUS MOTORS
A. D. NEMTSEV
(Presented by the seminar of the Department of Electrical Machines and General Electrical Engineering)
A fundamental improvement in the quality of manufactured products is one of the most urgent tasks currently facing industry. The solution of this task requires a reconsideration of the established, outdated methods of organizing quality control.
The purpose of the present work is to analyze the control methods used at electrical-machine-building plants, to consider the possibility of applying methods of statistical control in the manufacture of mechanical parts of asynchronous motors, and to develop criteria for assessing the quality of the technology.
At present, two types of control are used at electrical-machine-building plants: complete and sampling control.
Complete control consists in subjecting 100% of the parts to inspection. It is used after operations with unstable technological processes and after operations that are of decisive importance in the further processing of products. Thus, for example, in all end shields the diameter and finish of the bearing bore are checked, as is the diameter of the free end of the shaft when there is no grinding after the pressing-on of the stack.
When complete control is applied, it is assumed that it fully guarantees the consumer against receiving defective products. However, this control has serious shortcomings, for example, the high costs associated with carrying it out. In addition, it also has shortcomings of a technical nature, which consist in the fact that inspectors checking the same product day after day develop symptoms of fatigue and a certain “blindness to defects.” Therefore some defects remain unnoticed.
Sampling control is understood as a method of control in which, during the production process, a certain number of products is systematically selected, in accordance with a previously drawn-up plan, for checking their quality. If 100% control performs only a protective function, then sampling control pursues two objectives:
a) ensuring product quality in the course of production itself, and
b) acceptance control of products manufactured as a result of this process. This method usually consists either of a percentage sampling inspection of products or of a purely random inspection, the frequency of which depends on the number of defects encountered. Both of these methods may be recognized as effective if the risk associated with the application of these methods has first been analyzed.
methods for the supplier and the consumer. This assessment of risk is what distinguishes the scientific method of sampling inspection from “checking at random.”
Sampling inspection that records the results of some individual operation serves as a basis for readjusting machines, and sometimes also signals the need to improve the qualifications of the people who operate the processes.
The sampling-inspection methods used at electrical-machine-building enterprises do not solve any of the stated tasks, since the assigned volumes and frequency of selecting the products to be inspected are carried out without a comprehensive analysis of risk. Therefore, the unjustified conduct of sampling inspection does not help improve the quality of the output and is ineffective. This is confirmed by studies carried out as follows. Motors rejected during control tests for certain parameters were dismantled, and the diameters of the rotor and shield, and the runout of the rotor and shield, were measured. The measurement results showed that a large number of parts have parameters whose values fall outside the tolerance limits. To change the existing situation, it is necessary to introduce sampling statistical control, which is already being successfully used at enterprises of the machine-building and radio-electronic industries in the USSR and in many branches of industry abroad.
When introducing statistical methods of control, it is necessary to know the capabilities of the technological equipment. In this connection, studies were carried out whose results make it possible to judge the actual scatter of parameters in the manufacture of parts and assemblies of asynchronous motors. The measurement results for A02-32-4 motors are presented in Figs. 1, 2, 3, 4, in which the tolerance limits are indicated by vertical lines.
An analysis of the technological processes for manufacturing parts of asynchronous motors showed that at present there are large scatters in some parameters.
Fig. 1. Distribution of deviations from the nominal diameter of rotors
Fig. 2. Distribution of deviations from the nominal diameter of the end-shield rabbet
Fig. 3. Distribution of rotor surface runout relative to bearing journals
Fig. 4. Distribution of shield runout
Effective regulation of the quality of technological processes is possible when the accuracy coefficient is \(K_{\mathrm{T}} = 0.75 \div 0.85\) [1]. In the histograms presented, the scatter of the parameters is greater than the tolerance field, and therefore \(K_{\mathrm{T}} > 1\). Consequently, when introducing statistical methods of control, it is necessary either to increase the tolerance field or to improve the technological processes so that the capabilities of the given equipment correspond to the established tolerances.
As was stated earlier, the principal aim of applying statistical methods of control in industry is the regulation and management of production processes. Regulation of production processes may be carried out both by quantitative and by qualitative characteristics.
For regulating a process by quantitative characteristics, double control charts are most often used; for example, an “\(\bar{x}\)—R” chart, into which the mean values \(\bar{x}\) and ranges \(R\) of samples are entered [3, 4]. The \(\bar{x}\)-chart reflects the course of the process between samples, while the R-chart reflects the position within samples.
To keep control charts, it is necessary to carry out a preliminary analysis, the purpose of which is to find the central line and the control limits.
The calculation of the control limits for the \(\bar{x}\)-chart is performed according to the expression [2]
\[ K_{\mathrm{n.v.}\bar{x}} = \bar{x} \pm \frac{z_{\alpha}\cdot \sigma}{\sqrt{n}}, \tag{1} \]
where
\[ \begin{aligned} \bar{x} &\text{ — mean value of the measured parameter;}\\ z_{\alpha} &\text{ — coefficient, determined from Table 36 [2];}\\ \sigma &\text{ — root-mean-square deviation;}\\ n &\text{ — sample size.} \end{aligned} \]
The control limits for the R-chart are determined according to the expression [2]
\[ K_{\mathrm{B}} = D \cdot \overline{R}, \tag{2} \]
\[ \overline{R} = \frac{1}{k}\sum_{i=1}^{k} R_i, \tag{3} \]
where
\(k\) is the number of samples;
\[ R_i = x_{i\max} - x_{i\min}, \tag{4} \]
where
\(x_{i\max}\), \(x_{i\min}\) are, respectively, the maximum and minimum values of the measured parameter in the sample.
The coefficient \(D\) is determined from [2, 3].
Regulation of the production process by means of control charts for quantitative characteristics requires the preparation of a control chart for each quality characteristic under consideration for the given part, which often deprives the processed data of clarity and is impracticable directly at the machine tool.
Therefore, in practice, for individual characteristics of a product it is checked whether they lie within the tolerance limits or fall outside these limits. This type of sampling inspection is used primarily in those cases when a large number of products is inspected and when it is more expedient to carry out this inspection using measuring devices that establish the presence or absence of qualitative characteristics (of the “go–no-go” type).
The principal methods of inspection by qualitative characteristics include single-stage, double-stage, and sequential inspection. Single-stage acceptance inspection has several advantages: it is easier to plan, and the sample sizes remain constant. The greatest amount of information concerning the quality of product lots can be obtained, since the data of each sample can be entered on a control chart. In addition, if single-stage inspection has not been mastered, it is difficult to master double-stage or sequential inspection immediately.
Possessing these advantages, this type of inspection may be applied at electrical-machine-building plants in monitoring the quality of parts and assemblies after machining.
The most important parameters of single-stage inspection are the sample size \(n\) and the acceptance number \(c\). They are determined by the requirements imposed on the quality of lots as a result of an agreement between the customer and the manufacturer.
When 100% inspection is replaced by sampling inspection, errors of two kinds occur.
An error of the first kind occurs when the customer rejects, on the basis of sampling inspection, an acceptable lot with a low percentage of defects, because the selected sample contained more defective products than allowed by the acceptance number \(c\). The probability of such an erroneous decision is called the producer’s risk.
On the other hand, a lot with a high percentage of defects may be accepted if the sample taken from it contains no defective products at all or contains only a small number of defective products. A false decision to accept quality constitutes an error of the second kind. The probability of such a decision is called the customer’s risk.
A rational inspection plan must be drawn up so that the probabilities of accepting erroneous decisions are as small as possible.
In assessing the quality of the products manufactured, it is necessary to take into account the manufacturing technology. Its consideration must be quantitative and ob-
effective. Such accounting can be carried out with the aid of a generalized criterion of the quality of the technology.
Taking into account that, in the manufacture of asynchronous motors, a large number of random factors operate, it is expedient to adopt as the generalized criterion of the quality of the technology the probability that, in the assembled product, all parts and assemblies satisfy the technical specifications:
\[ P_T = P\left[X^{(1)}\right]\times P\left[X^{(2)}\right]\times \ldots \times P\left[X^{(j)}\right]\times \ldots \times P\left[X^{(m)}\right] = \prod_{j=1}^{m} P\left[X^{(j)}\right], \tag{5} \]
where
\[ \prod_{j=1}^{m} P\left[X^{(j)}\right] \]
is the probability that all parameters of the \(j\)-th assembly or part satisfy the technical specifications.
\[ P\left[X^{(j)}\right] = P\left[x_1^{(j)}\right]\times P\left[x_2^{(j)}\right]\times \ldots \times P\left[x_i^{(j)}\right]\times \ldots \times P\left[x_{ni}^{(j)}\right], \tag{6} \]
where
\(P\left[x_i^{(j)}\right]\) is the probability that the \(i\)-th parameter of the \(j\)-th assembly or part satisfies the technical specifications.
The method of determining the probability \(P\left[x_i^{(j)}\right]\) depends on which attributes—qualitative or quantitative—are used in monitoring the parameter. When monitoring by qualitative attributes,
\[ P\left[x_i^{(j)}\right] = 1 - \frac{n_q}{n} = 1 - \bar{p}, \tag{7} \]
where
\(n\) is the total number of measurements,
\(n_q\) is the number of measurements for which the parameter values do not conform to the technical specifications.
When monitoring by quantitative attributes,
\[ P\left[x_i^{(j)}\right] = \int_{x_n}^{x_v} f(x)\,dx, \tag{8} \]
where \(x_v\) and \(x_n\) are, respectively, the upper and lower permissible values of the parameter being monitored.
\(f(x)\) is the density of the distribution of the parameter being monitored.
Conclusions
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An analysis of existing control methods has been given, and recommendations have been developed for the application of statistical control methods in the manufacture of mechanical parts for large-scale production of asynchronous motors.
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A criterion is proposed for assessing the quality of manufacturing technology, which makes it possible to compare the technological processes of different enterprises producing identical products.
Literature
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O. G. Lisitskii et al. On the first standards for statistical methods of product quality management. Reliability and Quality Control. 4, 1970.
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R. Storm. Probability Theory. Mathematical Statistics. Statistical Quality Control. Moscow, 1970.
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B. Hansen. Quality Control, “Progress,” 1968.
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E. Shindovskii, O. Shyurts. Statistical Methods of Production Control. “Mir,” 1969.