Methodology for Studying Load Conditions of Induction Motors in Universal Metal-Cutting Machines

Yu. M. Bashaurov, E. K. Strelbitskii

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

The need to study the operating conditions of electric drives of universal metal-cutting machines is due to the fact that these machines, which account for the largest share in the country’s stock of metal-cutting equipment, have the lowest relative average motor loads. The main reason for the low average loads is the exponential distribution law, which is characterized by a large variance [2].

It has been established that load graphs are random functions of time, formed as a result of the action of a number of random factors.

Random processes that proceed approximately uniformly in time and have the form of continuous random oscillations about some mean value are called stationary. On any time interval of a stationary process, identical characteristics can be obtained.

It is precisely such processes that are found when investigating the loads of predominantly similar metal-cutting equipment.

The conditions for stationarity of a random process are the constancy of the mean value \(\overline{p}_2\) and of the variance \(D_{p_2}\), which is the case here [2]:

\[ \overline{p}_2 = \text{const}, \tag{1} \]

\[ D_{p_2} = \text{const}, \tag{2} \]

where \(P_2\) is the power at the motor shaft.

However, requirement (1) is not essential, since from the random function \(p_2(t)\) one can always pass to the centered random function \(\overset{\circ}{p}_2(t)\), for which the mathematical expectation is identically equal to zero and, consequently, condition (1) is satisfied. Thus, if the load process under consideration is nonstationary only because of a variable mathematical expectation (which in fact occurs on certain individual intervals of the process under consideration), this does not prevent its being studied as stationary.

Since load processes do not contain initial nonstationary stages, i.e., so-called transient processes, the correlation functions do not depend on the position of the first argument on the time axis, but only on the interval \(\tau\) between the first and second arguments. In this case the only essential condition that a stationary random function must satisfy is fulfilled:

\[ K_{p_2}(t, t+\tau) = R_{p_2}(\tau), \tag{3} \]

since condition (2) at \(\tau=0\) is a special case of condition (3).

Analysis of the autocorrelation functions obtained on universal machine tools shows that the correlation relationship between power values decreases as the interval between them increases and, at \(\tau=50\div100\) min, decays to zero. This means that the random processes under consideration possess the ergodic property

\[ \lim_{\tau\to\infty} R_p(\tau)=0 . \tag{4} \]

This circumstance considerably simplifies data processing, since the aggregate of independent realizations of loads on individual machine tools may be regarded as one single realization of the load process \(T\), and consequently it permits ordinary averaging of power values over time.

When condition (4) is fulfilled, the variance of the estimate of the mathematical expectation tends to zero [3].

\[ \lim_{T\to\infty} D(\tilde p_2)=0, \tag{5} \]

i.e., such an estimate is consistent and unbiased. It is known that if we have no additional information about the properties of the random function \(p_2(t)\) (apart from its stationarity), then this estimate is also the most efficient of all linear estimates [5].

This article presents a methodology for studying the load conditions of universal metal-cutting machines, developed on the basis of survey data from screw-cutting lathes of the most widespread model 1K62, using methods for the study of stationary random processes.

The application of these methods in the present investigations makes it possible to obtain estimates of load parameters with a prescribed accuracy and to carry out studies of motor heating under random-load conditions. The methodology was developed on the basis of the following principles of the experimenter: what must be obtained from the experiment, with what accuracy it is desirable to obtain these results, and what means may be used for data processing.

The following characteristics of the random process are to be determined:

1. The mean value of the power on the shaft \(\tilde p_2\).
2. The probability density of the loads \(f(\tilde p_2)\).
3. The correlation function \(R_{p2}(\tau)\).

Since the values obtained are only estimates of the characteristics of the general population, it is necessary to prescribe in advance a satisfactory accuracy for the estimate of the mean. In doing so, it should be borne in mind that the volume of measurements increases inversely proportionally to the variance of the estimate of the mean [4].

The task consists in creating a methodology that would minimize the costs of the experiment and the time required to process the experimental results, i.e., would establish the optimal length of the realization of the process \(T_{\text{opt.}}\) and the optimal sampling interval \(\Delta_{\text{opt.}}\) to ensure a satisfactory variance of the estimate of the mean.

It should be noted that \(T\) and \(\Delta\) depend not only on the prescribed accuracy \(\sigma^2(\overline{P}_2)\), but also on the correlation function \(R_{p2}(\tau)\), which is unknown before the experiment; therefore, in practice, to obtain the required variances of the estimates, one must resort to the method of successive approximations.

The basic requirements for the methodology are as follows:

1. Obtaining the necessary information in the investigation at the lowest cost.

1. The use of simple, convenient, and rapid methods of data processing in order to obtain sufficiently accurate results.
2. The use of the latest mathematical methods and electronic computers.
3. The possibility of applying it in the investigation of loads for any metal-cutting equipment.

An important point in studying these processes is the choice of recording equipment. To record load regimes, self-recording wattmeters N 383 were used. Since the process under study is rapidly varying, the chart-tape feed speed was chosen to be the maximum, 5400 mm/hour. The values of the power \(p_1\) consumed by the motor, with a sampling interval \(\Delta\) equal to 45 sec, were read by means of a special device in which a mechanical unit was used for advancing the chart tape of this same type of instrument. The tape is moved by turning the knob of a rotary switch (with position locking), connected by a coupling to the reducer of the rewinding mechanism. By changing the switch board and the gears of the reducer, it is possible to vary the interval between the read-off power values from 0.03 sec to \(\infty\).

Since processing data from investigations of this kind under modern conditions can be carried out only on an electronic computer, with discrete data input the estimate of the mathematical expectation for the finite sequence \(p_{2i}\) is the quantity

\[ \widetilde{P}_2=\frac{\sum_{i=1}^{N} P_{2i}}{N}, \tag{6} \]

where \(N\) is the number of points in the realization.

The process is recorded at equal time intervals

\[ \Delta=\frac{T}{N-1}. \tag{7} \]

The rate of recording the process \(\Delta\) must be specified in advance, proceeding from the duration of recording the load process on one machine (the duration of a partial realization), the memory capacity of the electronic computer, etc.

The correctness of the choice of \(\Delta\) is checked after obtaining the normalized correlation function of the process under study \(\rho(\tau)\) by solving the equation

\[ 1{,}6+5/18\,\rho(\Delta)-4/9\,\rho\left(\frac{\Delta}{2}\right)=0 \tag{8} \]

with respect to \(\Delta\) [5]. The solution of equation (8) is

\[ \Delta_{\mathrm{opt}}=\frac{1{,}022}{a}, \tag{9} \]

where \(a\) is the parameter of the correlation function of the form \(\rho(\tau)=e^{-a\tau}\).

The limited memory of electronic computers makes it necessary to process the information in several stages. However, this is not a drawback of the method, since, along with obtaining estimates of the average level of operation over the entire process under study, it is necessary to investigate the heaviest load regimes (in terms of power and time) and their influence on the heating of motors.

The length of the realization \(T\) is determined by the specified accuracy of the estimate of the mean \(D(\widetilde{P}_2)\) and by the correlation function of the process \(R_{p_2}=D_{p_2}e^{-a\tau}\).

The content of the realization \(T\), i.e., the selection of machines by branches of industry, plants, shops, etc., is entirely determined by the researcher’s objectives.

The accuracy of the estimate of the mathematical expectation \(\widetilde{P}_2\) is characterized by the variance

\[ \sigma_{\widetilde{P}_2}^{2}=M(\widetilde{P}_2-\overline{P}_2)^2. \tag{10} \]

The choice of the required accuracy of the estimate \(\widetilde{P}_2\) is determined by the experimenter’s objectives. A two-sigma confidence interval equal to 1–2 percent of the rated power may be considered satisfactory. In the present case, for an AO2-52-4 motor having a rated power of 10 kW, the variance is

\[ \sigma_{\widetilde{P}_2}^{2}=(0.25 \div 1.00)\cdot 10^{-2}\ \text{kW}^{2}, \]

and the mean load, at a 95-percent confidence probability, is

\[ \overline{P}_2=\widetilde{P}_2\pm(0.1 \div 0.2)\ \text{kW}. \]

The accuracy of the estimate obtained is checked from the experimental results. The variance of the arithmetic-mean estimate obtained from formula (6) is equal to [4]

\[ D(\widetilde{P}_2)=\frac{D_{p_2}}{N} \left\{ 1+\frac{2}{N}\, \frac{e^{-\alpha\Delta}\left[N(1-e^{-\alpha\Delta})-(1-e^{-N\alpha\Delta})\right]} {(1-e^{-\alpha\Delta})^{2}} \right\}, \tag{10a} \]

where \(D_{p_2}\) is the variance of the random process.

The total duration of the recorded load conditions on 1K62 machines was \(1.4\cdot10^{6}\) sec \((N=3.1\cdot10^{4})\). As a result of statistical processing of the array of numbers on an M-20 digital computer, two moments (the mean and variance), a histogram, and the correlation function of the power at the shaft \(P_2\) were obtained.

Conversion of the values \(P_1\) into the values \(P_2\) was carried out on the digital computer. For this purpose, from the results of standard tests of the AO2-52-4 motor, the dependence

\[ P_2=\varphi(P_1), \tag{11} \]

was calculated; it is approximated sufficiently well by a second-degree polynomial. The coefficient of the polynomials was calculated by the method of least squares:

\[ P_2=-0.445+0.976P_1-0.004P_1^{2}. \tag{12} \]

The approximation error does not exceed 0.5 percent. The histogram of the distribution of \(P_2\) (Table 1) is well fitted by the exponential

\[ f(P_2)=0.735\exp(-0.735P_2) \tag{13} \]

with mean value \(\widetilde{P}_2=1.37\ \text{kW}\) and variance \(D_{p_2}=1.80\ \text{kW}^{2}\).

Since no-load travel limiters are installed on the machines under investigation, the no-load power values obtained on the diagram tapes as a result of faulty and unadjusted limiter relays were not taken into account.

The calculation of the mean values of the duty factor and the power factor in the given load conditions is performed by the methods presented in [1].

The parameter of the correlation function \(\alpha\), according to partial realizations, varies from 0.1 to 0.001. The smaller value corresponds to the heaviest conditions, whose share in the total length of the realization \(T\) is no more than 8–10 percent. Since the optimal realization length and the sampling interval \(\Delta\) depend on \(\alpha\), we shall formulate a procedure for their preliminary determination.

The correlation function of the load is of interest not only as an auxiliary characteristic of the process under study for obtaining the required estimate of the mean. It is necessary in the study

Table 1

\(N = 31110,\)

where \(I_i\) is the designation of the \(i\)-th interval, kW;
\(m_i\) is the number of points in the given interval;
\(P_i^*\) are the observed frequencies;
\(P_i\) are the theoretical frequencies.

heating of motors under random loading conditions, namely: for determining the scatter of temperatures from the scatter of loads.

The heating process of a motor is characterized by several heating time constants. In particular, for motors of the unified AO2 series the minimum time constant is 300 sec, and the maximum \(\tau_{\mathrm{h}\max}\) is about an hour. It may be assumed that, for \(\dfrac{1}{\alpha} \ll 0.5\tau_{\mathrm{h}\min}\), the presence of an autocorrelation relationship does not lead to any substantial scatter of temperatures. Therefore, correlation functions should be investigated only for those processes for which \(\dfrac{1}{\alpha} > 0.5\tau_{\mathrm{h}\min}\).

Taking into account that expression (10a), for a long realization length, tends to the expression

\[ D(\tilde P_2)=\frac{2D_{p_2}}{\alpha T} \tag{14} \]

and, for laws close to the exponential law, \(D_{p_2}\approx \tilde P_2^{\,2}\), one can determine the minimum realization length

\[ T_{\min}\approx \frac{2\cdot \tilde P_2^{\,2}}{\alpha\cdot D(\tilde P_2)}. \tag{15} \]

The value of \(\tilde P_2\) may be specified preliminarily. According to data from investigations of machine-tool loads [2], \(\tilde P_2=(0.2\div 0.4)P_{\mathrm{n}}\).

Taking the foregoing into account, it is recommended that studies of the random load process of universal metal-cutting equipment be carried out in the following order:

1. Specify the required accuracy of estimating the mean \(D(\tilde P_2)\).

2. Preliminarily specify the values \(\tilde P_2\) and \(\alpha_{\max}=\dfrac{2}{\tau_{\mathrm{h}\min}}\).

3. Using formulas (9) and (15), determine \(\Delta_{\mathrm{opt}}\) and \(T_{\min}\).

4. From the experimental results, determine \(R_{p2}(\tau)\), \(\tilde P_2\), and \(D_{p2}\).

5. Substitute the values obtained in item 4 into formula (15). If

obtained \(T < T_{\min}\), then the characteristics determined in item 4 are final.

1. If \(T > T_{\min}\), then the recording of the process should be continued until the total recording time is equal to the calculated value. The discretization step is refined for each newly obtained \(\alpha\). The length of the realization obtained in this way is representative for the majority of the investigated modes.

If it is necessary to determine the parameters of the most severe modes and their weight in the total length of the realization of the process, one should be guided by the least favorable correlation function, which, according to our studies, has \(\alpha = 0.001\).

REFERENCES

1. Yu. M. Bashagurov. Calculation of the mean values of k.p.d. and of the power factor of induction electric motors under random loading conditions, Izvestiya TPI, No. 190, 1968.

2. Yu. M. Bashagurov, E. K. Strelbitskii. Allowance for load conditions in the design of induction motors. Article in the present collection.

3. A. A. Sveshnikov. Applied Methods of the Theory of Random Functions, “Nauka,” 1968.

4. S. Ya. Vilenkin. Statistical Methods for the Study of Automatic Control Systems, “Sov. radio,” 1967.

5. U. Grenander. Random Processes and Statistical Inference, IL, 1961.