METHOD FOR CALCULATING THE DURABILITY CHARACTERISTICS OF COMMUTATORS IN LOW-POWER MACHINES

A. Ya. Tsirulik, V. S. Stukach

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

The durability characteristics are the service life and the gamma-percent service life. Calculation of these characteristics is possible if the realizations of wear are known—that is, the dependences of the commutator state parameter on time and operating conditions. We have carried out studies of the processes of wear caused by commutation sparking of the commutators of machines of the P 1 series, sizes 1–6, as a result of which the character and quantitative characteristics of the wear process were determined [1]. The rate of electric-spark erosion of the commutator bars was determined by the expression

\[ \alpha = 0.426 \cdot 10^{-10} \cdot \frac{n}{(\Sigma l_{\mathrm{sh}} \cdot L)^{0.73}} \cdot \tau_{\mathrm{i}}^{1.46}, \tag{1} \]

where

\(n\) — armature rotational speed, rpm;
\(\Sigma l_{\mathrm{sh}}\) — total length of brushes per bolt, cm;
\(L\) — inductance of the armature-winding section, H;
\(\tau_{\mathrm{i}}\) — duration of the microarc discharge on the bar, μs.

This expression is valid for machines having simple equal-section windings. When passing from one standard size to another in the power range \(0.4\)–\(50\) kW, \(\Sigma l_{\mathrm{sh}}\) and \(L\) vary in such a way that their product remains almost unchanged. Therefore, one may use a simpler expression for \(\alpha\)

\[ \alpha = 2.21 \cdot 10^{-8} \cdot n \cdot \tau_{\mathrm{i}}^{1.46},\ \text{mm/hour}. \tag{2} \]

In [1] it is shown that commutator failure should be taken to mean a state in which the width of the erosion trace \(b\), on at least one of the bars connected with the outermost sections completing commutation of the slot, becomes equal to the bar width \(b_k\), that is, the erosion covers the entire width of the bar. The rate of increase of the trace width is determined by expression (2), and the realization of wear, with a linear dependence on time, has the form

\[ b(t) = \alpha \cdot t,\ \text{mm}. \tag{3} \]

Time to first failure

\[ T=\frac{b_{\mathrm{k}}}{\alpha}. \tag{4} \]

It has different values for different bars of one commutator and for commutators of different specimens of machines of one frame size, since there is scatter in the width of the bars and in the wear rate of the bars. The latter is due to the different intensity of sparking discharges on different bars. Statistical studies of deviations in the width of commutator bars of machines of the P 1—6 frame-size series showed that the coefficient of variation of the bar width does not exceed 3%, which contributes to the variance of \(T\) by no more than 0.96%. Therefore, in what follows we shall regard the bar width as a nonrandom quantity equal to the mean value determined by the formula

\[ b_{\mathrm{k}}=\frac{\pi\cdot D_{\mathrm{k}}}{K}-\overline{\Delta}_{\mathrm{i}}. \tag{5} \]

Here the mean width of the interbar gap \(\overline{\Delta}_{\mathrm{i}}\) must be calculated from the formula

\[ \overline{\Delta}_{\mathrm{i}}=\delta_{\mathrm{i}}+0.155,\ \mathrm{mm}, \tag{6} \]

where

\[ \delta_{\mathrm{i}} \]
is the design thickness of the interbar insulation, mm.

Sparking at the commutator is a stationary random process in which the sparking intensity drifts within a certain range of levels while remaining unchanged on average (this is true for defect-free machines). In individual time intervals sparking may be absent, and in this case there is no electroerosive wear either. This increases the service life of the commutator. If the probability of sparkless operation of the machine at a known mean sparking level is equal to \(\xi\), then the time to failure must be determined by the formula

\[ T=\frac{b_{\mathrm{k}}}{\alpha}\cdot(1+\xi)=\frac{b_{\mathrm{ke}}}{\alpha}. \tag{7} \]

Here \(b_{\mathrm{ke}}=b_{\mathrm{k}}\cdot(1+\xi)\) is the equivalent value of the bar width, introduced to take into account the effect of sparkless commutation in individual time intervals on the commutator life.

The scatter of the parameters of spark discharges on different bars is due to a great variety of causes; as a result, \(\tau_{\mathrm{i}}\) and \(\alpha\) may be considered normally distributed. Then the realization of wear is a normal random function with parameters (mean value and variance)

\[ \overline{b}(t)=\overline{\alpha}\cdot t,\quad D[b(t)]=t^{2}\cdot D[\alpha]. \tag{8} \]

Here the mean value of \(\alpha\) is determined by expression (2) at the mean value \(\overline{\tau}_{\mathrm{i}}\); the variance of \(\alpha\) must be calculated from formula (9) (with allowance for the nonlinear dependence of \(\alpha\) on \(\tau_{\mathrm{i}}\))

\[ D[\alpha]=[10.4\cdot \overline{n}^{2}\cdot \overline{\tau}_{\mathrm{i}}^{-0.92}\cdot D\{\tau_{\mathrm{i}}\}+1.1\cdot \overline{n}^{2}\cdot \overline{\tau}_{\mathrm{i}}^{-1.08}\cdot D^{2}\{\tau_{\mathrm{i}}\}+ \]
\[ +4.88\cdot \overline{\tau}_{\mathrm{i}}^{-2.92}\cdot D[n]+10.4\cdot \overline{\tau}_{\mathrm{i}}^{0.92}\cdot D[\alpha]\cdot D[n]]\cdot 10^{-16}. \tag{9} \]

For the existing relationship between the quantities \(\overline{\tau}_{\mathrm{i}}\) and \(D[\tau_{\mathrm{i}}]\), the second term of the sum in expression (9) is 1.17% of the first,

therefore the term with \(D^{2}\{\tau_i\}\) need not be taken into account [2]. The third and fourth terms of the sum should be taken into account only when the calculation is being made for a batch of machines of one standard size operating, under service conditions, at different speeds. If, however, the calculation is being made for machines operating at nominal speed, then the speed variance \(D\{n\}\) is due only to the scatter of the parameters of the speed characteristic. According to the data of [3], the coefficient of variation of nominal speeds does not exceed 2.37%; therefore the third and fourth terms of the sum in expression (9) may be neglected because of their smallness.

The mean operating time to failure is

\[ \overline{T}=\frac{\overline{b}_{k e}}{\alpha},\ \text{hours}, \tag{10} \]

and the variance of the operating time to failure is

\[ D\{T\}=\frac{\overline{b}_{k e}^{\,2}}{\alpha^{4}}\cdot D\{\alpha\} +2\cdot \frac{\overline{b}_{k e}^{\,2}}{\alpha^{6}}\cdot D^{2}\{\alpha\},\ \text{hours}^{2}. \tag{11} \]

From the normality of the wear realization function it follows that the integral distribution function of the time of failure-free operation obeys Bernstein’s law [4]

\[ F(T)=\Phi\left[\frac{t-\overline{T}}{t\cdot\sqrt{a}}\right], \tag{12} \]

where

\[ a=\frac{D\{\alpha\}}{\overline{\alpha}^{2}}, \]

\(\Phi[X]\) is the Laplace function. It has been tabulated.

After a commutator “failure,” the commutator is turned or polished with a bar, after which the commutator can operate for one more cycle until the next failure. Over the entire service life it is possible to perform \(m\) turnings, where

\[ m=\frac{D_k-D_{k\min}}{2\cdot h_{\mathrm{pr}}} =\frac{h_{\mathrm{izn}}}{h_{\mathrm{pr}}}, \tag{13} \]

where

\(h_{\mathrm{pr}}\) is the turning depth at which all electroerosion recesses on the segments disappear and good quality of the commutator surface is ensured. For machines of frame sizes 1–6, one may take \(h_{\mathrm{pr}}=0.2\ \text{mm}\).

The total cumulative life of the commutator after \(m\) turnings is equal to

\[ T_c=\sum_{j=1}^{m+1}T_j . \tag{14} \]

Here

\[ T_j=\frac{\overline{b}_{k e j}}{\alpha} \tag{15} \]

is the operating time to failure in the \(j\)-th operating cycle between failures; \(\overline{b}_{k e j}\) is the equivalent segment width for the \(j\)-th cycle. The values \(T_j\) and \(b_{k e j}\) may be averaged over all cycles.

Then

\[ \overline{b}_{ke}=\frac{\pi\cdot(D_k+D_{k\min})}{2K}-\overline{\Delta}_i =\overline{b}_{ke}-\frac{\pi\cdot h_{\mathrm{izn}}}{K}, \tag{16} \]

\[ T_c=\sum_{j=1}^{m+1}\frac{\overline{b}_{ke}}{\alpha} =(m+1)\cdot\frac{\overline{b}_{ke}}{\alpha}. \tag{17} \]

The mean value of the total service life of the commutator is equal to

\[ \overline{T}_c=(m+1)\cdot\frac{\overline{b}_{ke}}{\overline{a}}, \ \text{hours}, \tag{18} \]

and the variance of the service life is

\[ D\{T_c\}= \frac{(m+1)^2\cdot\overline{b}_{ke}^{\,2}}{\overline{a}^{\,4}}\cdot D\{a\} + 2\frac{(m+1)^2\cdot\overline{b}_{ke}^{\,2}}{\overline{a}^{\,6}}\cdot D^2\{a\}. \tag{19} \]

The total gamma-percent service life for a given value of the probability of failure-free operation \(\gamma\%\) can be determined from the relation

\[ \gamma=\frac{\gamma\%}{100}=\Phi[X_\gamma], \tag{20} \]

where

\[ X_\gamma=-\frac{t_{c\gamma}-\overline{T}_c}{t_{c\gamma}\cdot\sqrt{a}}. \tag{21} \]

For the specified value of \(\gamma\), from the table of values of the Laplace function \(\Phi[X]\) we find the value of the argument \(X_\gamma\), after which we calculate the total gamma-percent service life \(t_{c\gamma}\) by the formula

\[ t_{c\gamma}=\frac{\overline{T}_c}{X_\gamma\cdot\sqrt{a}+1}. \tag{22} \]

This expression is valid for values \(\gamma\%>50\).

The necessary periodicity of turning operations (grinding) of the commutator, ensuring the specified probability of failure-free operation \(\gamma\), should be calculated by the formula

\[ t_{\mathrm{prot}}=t_\gamma=\frac{\overline{T}}{X_\gamma\cdot\sqrt{a}+1}. \tag{23} \]

Here

\[ \overline{T}=\frac{\overline{b}_{ke}}{\overline{a}}. \tag{24} \]

Example: calculate the durability characteristics of commutators of series P machines of the second frame size at \(n_n=1500\) rev/min. Initial data:
\(\overline{b}_k=2.47\) mm, \(h_{\mathrm{izn}}=3\) mm, \(h_{\mathrm{pr}}=0.2\) mm, \(\overline{\tau}_i=4\) microsec, \(D\{\tau_i\}=1.78\) microsec\(^2\), \(\xi=0.35\), \(K=72\).
Solution: the equivalent value of the mean lamella width according to (7)

\[ \overline{b}_{kэ}=2.47\cdot(1+0.35)=3.33\ \text{mm}. \]

The average rate of advance of the electro-spark erosion trace over the width of the lamella, according to (2),

\[ \overline{\alpha}=2.21\cdot10^{-8}\cdot1500\cdot4^{1.46}=2.51\cdot10^{-4}\ \text{mm/hour}. \]

The variance of the erosion rate, according to (9), for a constant rotational speed,

\[ D[\alpha]=10.4\cdot10^{-16}\cdot1500^2\cdot4^{0.92}\cdot1.78=1.48\cdot10^{-8}\ \text{mm}^2/\text{hour}^2. \]

The average operating time of the commutator until the first failure, according to (10),

\[ \overline{T}=\frac{3.33}{2.51\cdot10^{-4}}=13300\ \text{hours}. \]

The variance of the operating time until the first failure, according to (11),

\[ D[T]=\frac{3.33^2}{2.51^4\cdot10^{-16}}\cdot1.48\cdot10^{-8} +2\cdot\frac{3.33^2}{2.51^6\cdot10^{-24}}\cdot \]

\[ \cdot\,1.48^2\cdot10^{-16}=0.604\cdot10^{-8}\ \text{hours}^2. \]

Parameter \(a\):

\[ a=\frac{1.48\cdot10^{-8}}{2.51^2\cdot10^{-8}}=0.235. \]

The number of possible commutator undercuttings, according to (13),

\[ m=\frac{3}{0.2}=15. \]

The equivalent lamella width averaged over all operating cycles, according to (16),

\[ \overline{b}_{kэ}=3.3-\frac{\pi\cdot3}{72}=3.2\ \text{mm}. \]

The total average service life of the commutator, according to (18),

\[ \overline{T}_{c}=(15+1)\cdot\frac{3.2}{2.51\cdot10^{-4}}=204500\ \text{hours}. \]

Let us determine the 95% service life of the commutator \((\gamma=0.95)\). From the tables of the Laplace function, for the value \(\gamma=\Phi[Х_{\gamma}]=0.95\), we find: \(X_{0.95}=1.65\) [4]. The 95% service life is equal, according to (22), to

\[ t_{c\,0.95}=\frac{204500}{1.65\cdot\sqrt{0.235}+1}=113500\ \text{hours}. \]

The average operating time between failures, according to (24),

\[ \overline{T}=\frac{3.2}{2.51\cdot10^{-4}}=12750\ \text{hours}. \]

The periodicity of commutator turning (grinding) according to (23)

\[ t_{\text{turn}}=\frac{12750}{1.65\cdot\sqrt{0.235}+1}=7080\ \text{hours}. \]

The calculation shows that, with average sparking of \(1\frac{1}{2}\) points, the service life of the commutators is great.

Conclusion

A method has been developed for calculating durability indices of commutators of low-power machines, making it possible to compute the service life of commutators operating at any level of sparking.

References

1. E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik. Wear of commutators of DC machines due to commutation sparking and calculation of commutator durability. Proceedings of TPI, vol. 211, Tomsk, 1970.

2. V. S. Stukach, A. Ya. Tsirulik, V. N. Popov. Statistical characteristics of the sparking process on the commutator in series P machines. Proceedings of TPI. The present collection.

3. I. P. Isaev. Stability of the Characteristics of Electric Locomotives. Moscow, Transzheldorizdat, 1956.

4. I. B. Gertsbar, Kh. B. Kordonskii. Failure Models. Moscow, “Sovetskoe radio,” 1966.