ASSESSMENT OF THE DURABILITY OF P-SERIES MACHINES

I. A. Volkomirskii, A. N. Nimirovskii, A. Ya. Tsirulik

(Presented by the scientific seminar of the Department of Electrical Machines)

Tomsk Polytechnic Institute, together with the research institute at the Elektromashina plant, is conducting studies of the reliability and durability of P-series direct-current machines. For this purpose, life tests of machines have been carried out for a number of years at the research institute of the Elektromashina plant. Machines of various sizes are being tested, with armature windings made of wire of the grades PETV, PSDKT, and PETKSO.

Table 1 gives information on the operating time of each machine, the grade of the armature-winding wire, and the cause of failure, where the letters o. ya. denote failure due to a short circuit in the armature winding, k.—due to impermissibly severe sparking at the commutator, z. k.—a short circuit in the commutator, and o. k.—a break in the connection of an armature-winding section with the commutator.

Table 1

Very low reliability was shown by machines with an armature winding made of PETKSO wire. Of the six machines, four failed within the first 6000 hours of operation. This is also consistent with the operating experience of such machines. Since the plant at present produces mainly machines with armature windings made of PETV and PSDKT wire, the processing of the test results will be carried out for these machines.

The small number of machines that underwent testing and the low density of failures per unit time do not make it possible, from the experimental data, to obtain sufficiently reliable information on the failure rate \(\lambda\). It is recommended that this parameter be calculated from experimental data in the case when, over a sufficiently small time interval, there are at least ten failures [1]. Therefore, the values of the probability of failure-free operation \(P(t)\) were calculated for various instants of time, and dependences \(P(t)\) were constructed.

According to GOST 13377-67, the probability of failure-free operation is calculated by the formula

\[ P(t) \simeq \frac{N(t)}{N}, \tag{1} \]

where

\[ \begin{aligned} N &\text{ — the number of items placed on test,}\\ N(t) &\text{ — the number of items remaining operational up to the end of the operating time } t. \end{aligned} \]

By this formula \(P(t)\) for the time instant \(t\) can be calculated in the case where all the machines remaining operational have an operating time not less than \(t\). As can be seen from Table 1, for \(t > 10000\) hours some operational machines have an operating time less than the selected fixed values of \(t\). These machines provide additional information on reliability, and excluding them from the number of machines \(N\) will lead to an underestimation of reliability. The test data for such machines can be taken into account if, when calculating \(P(t)\) for the time instant \(t\), the machines that have not failed but have operated less than \(t\) are included with a weighting coefficient proportional to their operating time. We proceed from the following.

For a test plan of type \([N, B, t]\), under which \(N\) items are tested for a time \(t\), with failed items replaced by new ones, the failure rate is determined by the formula [1]

\[ \lambda = \frac{n}{N \cdot t}, \tag{2} \]

where \(n\) is the number of failed items.

For a plan of type \([N, B, t]\), in which failed items were not restored,

\[ \lambda = \frac{n}{S_{\text{n}}}, \tag{3} \]

where \(S_{\text{n}}\) is the total operating time of all items, determined by the formula

\[ S_{\text{n}} = \sum_{k=1}^{n} \tau_k + \sum_{i=1}^{m} t_i + t(N - n - m), \tag{4} \]

where

\[ \begin{aligned} m &\text{ — the number of items having operating time } t_i < t,\\ \tau_k &\text{ — the operating time of the } k\text{-th item among those that failed.} \end{aligned} \]

Transform expression (3) to the form

\[ \lambda = \frac{n}{t \cdot \dfrac{S_{\text{n}}}{t}} = \frac{n}{N_{\text{e}} \cdot t}. \tag{5} \]

Here \(N_{\mathrm{e}}\) is a certain equivalent number of tested machines, determined on the basis of (4) by the formula

\[ N_{\mathrm{e}}=\frac{1}{t}\sum_{k=1}^{n}\tau_k+\frac{1}{t}\sum_{i=1}^{m}t_i+(N-n-m), \tag{6} \]

where

\[ (N-n-m) \]

is the number of serviceable products with operating time not less than \(t\);

\[ \frac{1}{t}\sum_{i=1}^{m}t_i \]

is the equivalent number of serviceable products having operating time less than \(t\);

\[ \frac{1}{t}\sum_{k=1}^{n}\tau_k \]

is the equivalent number of products that failed in the time interval \(0\)–\(t\).

This quantity is less than the actual number \(n\) of failed products. It is advisable to take the true number of failed products into account in the calculation, so as not to overstate the reliability indices. Then the equivalent number of products tested over time \(t\), in the case when some of the serviceable products have operating time less than \(t\), should be determined by the formula

\[ N_{\mathrm{e}}(t)=n+\frac{1}{t}\sum_{i=1}^{m}t_i+(N-n-m). \tag{7} \]

Then the probability of failure-free operation for any time instant \(t\) is determined by the formula

\[ P(t)\cong \frac{N_{\mathrm{e}}(t)-n}{N_{\mathrm{e}}(t)}. \tag{8} \]

Using formulas (7) and (8), values of \(P(t)\) were calculated for different time instants. Figure 1 shows the dependences \(P(t)\) for machines with screw-type insulation of the armature winding of type PETV (curve 1), PSDKT (curve 2), and for all machines together (curve 3).

Fig. 1. Function of the probability of failure-free operation of machines of the P series, frames 1–6

In Fig. 2 the dependences \(P(t)\) are given for armature windings of both grades of winding wire and for all machines together. In determining these dependences, failures due to causes not associated with winding failure were not taken into account. On semilogarithmic paper, the statistical dependences \(P(t)\) in all cases are well smoothed by a straight line, which indicates an exponential law of reliability of P-series machines, of the form

\[ P(t)=\exp[-\lambda t]. \tag{9} \]

Fig. 2. Probability-of-failure-free-operation function of the armature winding of series P machines of frame sizes 1–6

For an exponential law of reliability, the failure rate \(\lambda\) is constant, and this indicates the absence of the previously assumed run-in period for series P machines of frame sizes 1–6. In the range of 0–30,000 hours of operation, no increase in the failure rate is observed, which indicates the absence of noticeable aging of the insulation of the armature windings and wear of other units of the machine; the failure pattern of the machines corresponds to a pattern of instantaneous failures from the standpoint of the mathematical description of the process. Run-in and wear-out failures are absent. The service life of series P machines of frame sizes 1–6, defined according to GOST 13377-67 as the operating time up to the limiting condition that precludes further operation of the machines, is at least 30 thousand hours.

The smoothing of the experimental dependences was performed by the method of least squares. From the curves obtained, the failure rates of the machines and, separately, of the armature windings were determined; their values are given in Table 2.

Table 2

The confidence limits for the failure rate were determined at a 95% confidence probability by the formulas:

\[ \lambda_{\mathrm{B}}=\frac{\lambda}{r_3}, \qquad \lambda_{\mathrm{H}}=\frac{\lambda}{r_1}. \tag{10} \]

The coefficients \(r_1\) and \(r_3\) are determined from Table 11.2 [2].

Although in the range of 0–30,000 hours of operation there are no wear-out failures, the mean service life (50% service life) of series P machines is small. It is 16,500 hours for machines with an armature winding made of PETV wire and 21,000 hours for machines with an armature winding made of PSDKT wire. On average for all machines it is 18,500 hours. The decrease in the probability of failure-free operation during machine operation occurs rapidly. The 80-percent service life of the machines does not exceed 6,000 hours. The main causes of machine failures are failures of the armature winding (75%), unsoldering of the connections of the armature winding with the commutator or short-circuiting in the col-

the commutator (12.5%) and the appearance of heavy sparking at the commutator (12.5%). The mean operating time of the armature windings is 20,300 hours, although wear phenomena are not observed even up to 30,000 hours. An increase in the reliability of DC machines of the P series, in frame sizes 1–6, can be achieved primarily by increasing the reliability of the armature winding, and also by improving the quality of soldering at the points where the winding is connected to the commutator.

REFERENCES

1. B. V. Gnedenko, Yu. K. Belyaev, A. D. Solov’ev. Mathematical Methods in Reliability Theory. Moscow, “Nauka,” 1965.

2. Ya. B. Shor, F. I. Kuz’min. Tables for the Analysis and Control of Reliability. Moscow, “Sov. radio,” 1968.