Calculation of Tolerances for Technological Deviations Affecting the Commutation of DC Machines and Prediction of Commutation Failures

E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik

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

There are three methods for calculating tolerances: the maximum–minimum method, the “quadratic summation” method, and the probabilistic method. The first method, now the most widely used in factories, gives values of the errors of the output parameter that are overestimated by 1.5–10 times, while the second gives values underestimated by approximately 6 times. In the present work the probabilistic method is used as the most theoretically justified and the most accurate. The probabilistic method is based on the linear theory of accuracy. However, its direct application in the present case is impossible for the following reasons. First, nominal values are not specified for the commutator parameter \(\sigma_{\Delta}\)—the root-mean-square deviation of the step changes in the levels of the bars—and for the output quantity, the sparking level \(I\). Therefore their mean values must be determined from the tolerance equation together with their variances. Second, the dependence of sparking on technological parameters is substantially nonlinear.

In deriving the tolerance equation, we start from the sparking model [1]. It is necessary to ensure such tolerances for the input parameters of the sparking model \(x_i\) that the probability that the sparking \(I\) exceeds the critical level \(I_{\mathrm{cr}}\) is equal to a specified value \(Q\). We shall regard the distribution of the sparking levels of machines in a series as normal, which has an adequate justification (the limit theorem of probability theory). Then the probability \(Q\) can be determined from the formula

\[ Q = 1 - F\left[\frac{I_{\mathrm{cr}}-\overline{I}}{\sqrt{D(I)}}\right] = 1 - F[Z], \tag{1} \]

where \(F[Z]\) is the integral function of the normal distribution. Each value of \(Q\) corresponds to a definite value of \(Z\). To calculate \(Z\), it is necessary to determine the mean value \(\overline{I}\) and the variance \(D(I)\) of the sparking of the machines in the series. The dependence of sparking on technological factors was found by us in the form of a second-degree polynomial [1]:

\[ I(x_1, x_2, \ldots, x_n) = b_0 + \sum^n b_i \cdot x_i + \sum^n b_{ii} \cdot x_i^2 + \sum_{j<i}^{n} a_{ij} \cdot x_i \cdot x_j . \tag{2} \]

The values of the parameters \(x_i\) are random owing to random technological deviations. The mean value and variance of the sparking are found from the formulas:

\[ \bar I = \int_{-\infty}^{+\infty}\cdots\int_{-\infty}^{+\infty} I(x_1,x_2,\ldots,x_n)\cdot f_1(x_1)\cdot f_2(x_2)\cdots f_n(x_n)\cdot dx_1\cdots dx_n; \tag{3} \]

\[ D(I)= \int_{-\infty}^{+\infty}\cdots\int_{-\infty}^{+\infty} I^2(x_1,x_2,\ldots,x_n)\cdot f_1(x_1)\cdots f_n(x_n)\cdot dx_1\cdots dx_n-\bar I . \tag{4} \]

Here

\[ f_i(x_i)=\frac{1}{\sqrt{2\pi\cdot \sigma(x_i)}}\cdot \exp\left[-\frac{(x_i-\bar x_i)^2}{2\sigma^2(x_i)}\right] \]

are the normal distribution densities of the input parameters \(x_i\). Carrying out the integration, we obtain:

\[ \bar I=b_0+\sum_i^n b_i\bar x_i+\sum_i^n b_{ii}\bar x_i^2+ \sum_{i<j}^n b_{ij}\bar x_i\cdot \bar x_j+ \]

\[ +\sum_i^n b_{ii}\cdot D(x_i)=I(\bar X)+\sum_i^n b_{ii}\cdot D(x_i); \tag{5} \]

\[ D(I)=\sum_i^n\left[b_i+2b_{ii}\bar x_i+\sum_{i\ne j}^n b_{ij}\bar x_j\right]^2\cdot D(x_i)+ \]

\[ +2\sum_i^n b_{ii}^2\cdot D^2(x_i)+\sum_{i<j}^n b_{ij}^2\cdot D(x_i)\cdot D(x_j). \tag{6} \]

Expressions (5), (6), and (1) make it possible to solve the direct problem of the theory of accuracy: from known tolerances on the output parameters \(x_i\), to calculate the scatter of sparking of a series of machines and the probability of rejection due to sparking. With symmetric tolerance fields, between half the width of the tolerance field \(\delta_i\) and the variance \(D(x_i)\) there is the relation

\[ \delta_i=3\cdot\sqrt{D(x_i)}. \tag{7} \]

The mean values \(\bar x_i\) and variances \(D(x_i)\) of the output parameters can also be determined from statistical data on the distribution of the parameter values for a given level of technology. Knowing \(\bar x_i\) and \(D(x_i)\), one can calculate \(\bar I\) and \(D(I)\), determine the argument \(Z\) of the Laplace function, find the value of the function from tables [3], and calculate the probability that the sparking of a series of machines will exceed the permissible level \(I_{\mathrm{cr}}\).

Of greatest practical interest is the solution of the inverse problem of the theory of accuracy: calculation of tolerances on the input parameters for given \(I_{\mathrm{cr}}\) and permissible probability of rejection \(Q\). To derive the tolerance equation, on the basis of expression (1) we write the equation

\[ [I_{\mathrm{cr}}-\bar I]^2=Z^2\cdot D(I) \tag{8} \]

and substitute into this equation the values of \(\bar I\) and \(D(I)\) from expressions (5) and (6). After transformations we obtain the tolerance equation:

\[ [I_{\mathrm{cr}}-I(\bar X)]^2= \sum_i^n a_i\cdot D(x_i)+ \sum_i^n a_{ii}\cdot D^2(x_i)+ \sum_{i<j}^n a_{ij}\cdot D(x_i)\cdot D(x_j) \tag{9} \]

The coefficients of the equation are determined by the formulas:

\[ a_i=Z^2\cdot\left[b_i+2b_{ii}\bar x_i+ \sum_{j\ne i}^n b_{ij}\cdot \bar x_j\right]^2 +2[I_{\mathrm{cr}}-I(\bar x)]\cdot b_{ii}; \]

\[ a_{ii}=b_{ii}^2\cdot(2z^2-1); \qquad a_{ij}=z^2\cdot b_{ij}^2-2b_{ii}-b_{ij}. \tag{10} \]

The coefficients \(b_i,\ b_{ii},\ b_{ij}\) are the coefficients of the mathematical model of sparking [1]. The transition from variances to tolerances is carried out by formula (7). We have found the empirical dependence

\[ D(x_3)=0.0827\cdot x_3^2, \tag{11} \]

which makes it possible to consider equation (9) in the axes \(D(x_1)\), \(D(x_2)\), and \(\bar{x}_3\).

Equation (9) defines the boundaries of the tolerance region. Figure 1 presents the solution of the tolerance equation for machines of the third frame size of series P, making it possible to select tolerances for the brush contact pressure \(P\), the commutator parameter \(\sigma_\Delta\), and the optimum excitation current \(I_{\mathrm{p}}\). As shown in [1], the current \(I_{\mathrm{p}}\) is a generalized parameter characterizing deviations of the principal geometrical parameters of the magnetic circuit.

Fig. 1. Tolerance system for machines of the third frame size of series P at permissible sparking of \(1\frac14\) points and a rejection probability of 0.0027

Having assigned the tolerance for \(I_{\mathrm{p}}\) from the curves in Fig. 1, one can calculate the tolerances for the geometrical parameters of the magnetic system.

The tolerance equation was derived as applied to a sparking model specified by a second-degree polynomial. To calculate tolerances it is necessary to know the coefficients of the sparking model; therefore, for each type of machine, a sparking model must first be constructed. For geometrically similar types and sizes with nearly equal armature rotational speeds, it is possible to construct a tolerance system for the type and size for which a sparking model has been found.

The essence of the recalculation consists in changing the scale along the axis \(\delta I_{\mathrm{p}}\) in the ratio

\[ \frac{I_{\mathrm{p}}''}{I_{\mathrm{p}}'}, \]

where \(I_{\mathrm{p}}'\) is the value of \(I_{\mathrm{p}}\) for the type size \((\,'\,)\) for which the tolerance system is known, leading to the appearance of first sparking; \(I_{\mathrm{p}}''\) is the value of \(I_{\mathrm{p}}\) for the type size \((\,''\,)\) for which it is necessary to construct a tolerance system leading to the same sparking.

An important problem is the prediction of gradual machine failures due to sparking, by which is meant the gradual departure of the brush sparking level in service beyond the boundary of the critical value \(I_{\mathrm{cr}}\). The fact that the sparking process belongs to random processes of the Markov type, and the high adequacy of the mathematical sparking model obtained, make it possible to use it for predicting gradual failures.

If the input parameters are substituted into the sparking model (2) in the form

\[ x_{it}=x_{i0}+k_i t, \tag{12} \]

where \(x_{i0}\) are random initial values, and \(k_i\) are random rates of change of the input parameters in operation, then a time coordinate will appear in the sparking model, making it possible to use the model to predict the development of sparking in operation.

The probability of failure-free operation of the commutator and brushes, i.e., the probability that the sparking level remains within the tolerance region \(D\) during time \(t\), can be determined from the expression

\[ P\{D,t\}=\int_{(D)} \ldots \int f(X_t,t)\cdot dx_{1t}\cdot dx_{2t}\ldots dx_{nt}, \tag{13} \]

where \(f(X_t,t)\) is the \(n\)-dimensional density of the normal distribution of the input parameters as a function of time. The boundaries of the region \(D\) are specified by expression (2), if \(I_{\mathrm{cr}}\) is substituted into the left-hand side. This expression is nonlinear; therefore the integral (13) cannot be solved in general form. It is advisable to use the numerical Monte Carlo method and solve the problem on a digital computer.

The solution procedure consists in repeatedly determining the sparking level by formula (2) for nonrandom values of the parameters \(x_{i0}\) and \(k_i\), chosen at random from populations with the corresponding distribution laws. The probability of failure-free commutation over time \(t\) is equal to

\[ P\{t\}=\frac{N_p}{N}, \tag{14} \]

where

\[ \begin{aligned} N_p&\text{ is the number of successful outcomes, when } I(X_t,t)<I_{\mathrm{cr}},\\ N&\text{ is the total number of cases (statistical trials).} \end{aligned} \]

To calculate reliability one may also use the tolerance-calculation algorithms set forth above, if the input parameters are written as functions of time (12). The probability of failure-free operation of the commutator and brushes is equal to

\[ P\{t\}=F\left[\frac{I_{\mathrm{cr}}-\bar I_t}{\sqrt{D(I_t)}}\right]=F[Z(t)]. \tag{15} \]

The mean value \(\bar I_t\) and the variance \(D(I_t)\) of sparking as functions of operating time are determined by formulas (3) and (4). In this case the parameters \(x_{i0}\) and \(k_i\) are random. The following expressions are obtained:

\[ \bar I=b_0+\sum_i^n b_i x_{it}+\sum_i^n b_{ii}x_{it}^2+\sum_{i<j}^n b_{ij}x_{it}x_{jt}+ \]

\[ +\sum_i^n b_{ii}D(x_{i0})+t^2\sum_i^n b_{ii}D(k_i). \tag{16} \]

\[ D(I_t)=\sum_i^n\left[b_i+2b_{ii}x_{it}+\sum_{j\ne i}^n b_{ij}x_{jt}\right]^2\cdot \left[D(x_{i0})+t^2D(k_i)\right]+ \]

\[ +2\sum_i^n b_{ii}^2\cdot\left[D^2(x_{i0})+t^4D^2(k_i)\right] +\sum_{i<j}^n b_{ij}^2D(x_{i0})D(x_{j0})+ \]

\[ +4t^2\sum_i^n b_{ii}^2D(x_{i0})D(k_i) +t^2\sum_{i\ne j}^n b_{ij}D(x_{i0})D(k_j)+ \]

\[ +t^4\sum_{i<j}^n b_{ij}^2D(k_i)D(k_j). \tag{17} \]

Having calculated, for any time \(t\), \(\overline{I_t}\) and \(D(I_t)\), one can determine the argument \(Z(t)\) of the Laplace function \(F[Z]\) and, from tables of this function, find the probability of failure-free operation of the commutation system.

The parameters \(x_{i0}\) and \(D(x_{i0})\) must be known either from statistical data on the scatter of the input parameters for the given level of technology, or may be determined from the known tolerances for these parameters. The parameters \(k_i\) and \(D(k_i)\) are determined from statistical data on the rates of change of the parameters during operation. For machines of frame sizes 1–3 of series П with an armature rotation speed of 1500 rpm, based on the results of service-life tests, we determined the rates of change of the input parameters of the sparking model and calculated the values of the probability of failure-free operation for various values of operating time (Table 1).

Table 1

In the developed reliability model of the commutation system, the operating conditions are methodologically simply taken into account through the rates \(k_i\) of change of the parameters during operation.

The reliability model also makes it possible to calculate tolerances for the input parameters, taking into account a specified operating time.

The developed algorithms for calculating tolerances and commutation reliability are valid for any number of input parameters of the sparking model. It is necessary only to know the coefficients of the sparking model. For specific machines these coefficients are obtained by the method developed in [2].

REFERENCES

1. E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik. Influence of technological deviations on brush sparking in DC machines. Izvestiya TPI, present collection.

2. E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik. Application of the method of mathematical statistics to the study of commutation. Izvestiya TPI, vol. 160, 1966.

3. L. N. Bol’shov, N. V. Smirnov. Tables of Mathematical Statistics, Nauka, Moscow, 1965.