Determining Air Gap Nonuniformity in Induction Motors from Quality Control Data on the Number of Rotor-Stator Rubs
Abstract Generated abstract
The paper addresses the problem of estimating static air gap nonuniformity in induction motors when direct measurement of eccentricity is difficult or unavailable. It proposes a probabilistic statistical method that uses plant test station records on the fraction of machines exhibiting rotor to stator rubbing, together with calculated bearing clearances, air gap tolerances, shaft deformation, and magnetic unbalanced force, to infer the Rayleigh distribution parameter and mean eccentricity for a production batch. The method includes corrections for random air gap variation and discusses how bearing wear can be incorporated to predict future rubbing during operation. A worked example for an A31-4 motor illustrates the calculation of limiting eccentricity, distribution variance, mean eccentricity, and the fraction of machines meeting a specified eccentricity limit.
Full Text
DETERMINATION OF AIR-GAP NONUNIFORMITY IN INDUCTION MOTORS BASED ON QUALITY CONTROL DATA ON THE NUMBER OF ROTOR-TO-STATOR CONTACTS
O. P. Muravlev, E. K. Strelbitskii
(Recommended by the seminar of the Department of Electrical Machines and General Electrical Engineering.)
The air gap of induction motors is not uniform for technological reasons. Moreover, the nonuniformity increases during operation. There are two types of air-gap nonuniformity: static nonuniformity, determined by errors in the dimensional chain, and rotating nonuniformity, caused by the eccentric position of the rotor core relative to the shaft axis.
We shall consider only static nonuniformity (static eccentricity), since the eccentricity of the rotor core relative to the shaft is usually small.
Air-gap nonuniformity adversely affects the characteristics of the motor: the efficiency decreases, losses in the steel increase (by 15–30%) [1], dips in the torque curve increase, acceleration time increases, and current is redistributed in parallel branches. In addition, the possibility arises of the rotor contacting the stator, which leads to high local overheating. All these factors shorten the service life of the insulation and reduce the reliability of motor operation.
The probability of contact during operation increases because of bearing wear. The fatigue life of a bearing is inversely proportional to the cube of the radial forces arising with a nonuniform air gap. But in electrical machines the bearings are usually selected with a large safety margin; bearing wear due to fatigue failure of the material is not the dominant type of failure, and therefore the influence of eccentricity on service life may be neglected.
The calculation of forces, vibrations, and noise arising in the presence of eccentricity has been developed quite fully, but the eccentricity values themselves, on which the calculation is based, are poorly substantiated. The standards allow a limiting eccentricity value of 0.15, while calculations and experimental data show that in a number of cases the eccentricity is considerably greater. Statistics on eccentricities are not kept for all machines; measurement of the air gap is usually carried out with feeler gauges. The design of a number of machines altogether precludes the possibility of measurements by this method, and indirect methods for measuring air-gap nonuniformity have not found wide application.
In this connection, the problem arises of obtaining, by a simple path, information on the mean value and the distribution law of eccentricities without carrying out additional measurements.
In the present work a probabilistic-statistical method is proposed for determining eccentricities from data of the plant test station.
If there is an initial displacement of the shaft axis relative to the stator bore, a magnetic unbalanced force \(P\) arises, directed toward the smallest gap. The force \(P\) causes elastic deformation of the shaft and bearing, takes up the radial clearance of the bearing and the clearance between the outer ring and the shield in the direction of the action of the force, which leads to a further increase of the force. At a certain initial eccentricity, contact of the rotor with the stator may occur.
We shall assume that contact of the rotor with the stator occurs when the equality
\[ e_0 + f_\delta + \delta_{\mathrm{p}} + x + y = \delta, \tag{1} \]
is satisfied, where \(\delta\) is the air gap, cm;
\(e_0\) is the initial displacement of the shaft axis relative to the axis of the stator bore, cm;
\(f_\delta\) is the magnitude of the elastic deformation of the shaft, cm;
\(\delta_{\mathrm{p}}\) is the displacement of the inner ring of the bearing relative to the outer ring under the action of the magnetic unbalanced force, cm;
\(x\) is the radial clearance of the bearing in the assembly, cm;
\(y\) is the clearance between the outer ring of the bearing and the shield, cm.
The random quantities \(x\), \(y\), and \(\delta\) have a normal distribution law with density
\[ f(x)=\frac{1}{\sigma_x\sqrt{2\pi}}\exp\left[-\frac{(x-m_x)^2}{2\sigma_x^2}\right], \tag{2} \]
where \(m_x\) is the mean value of the quantity \(x\);
\(\sigma_x\) is the root-mean-square deviation of the quantity \(x\).
When the normal quantities \(u=\delta-x-y\) are composed, a normal law [2] is again obtained, with parameters
\[ m_u=m_\delta-m_x-m_y, \tag{3} \]
\[ \sigma_u^2=\sigma_x^2+\sigma_y^2+\sigma_\delta^2. \tag{4} \]
The mean values and root-mean-square deviations of the random quantities can be readily found if the type of bearing and its fits in the machine are known. The method of calculating them is given in the example.
Let us find the value \(e_0\) at which contact of the rotor with the stator occurs when all the random quantities have their mean values.
Using formulas for calculating the force of unilateral magnetic attraction [3], the magnitude of elastic deformation [4], and the displacement of the inner ring of the bearing relative to the outer ring [5], one can compose a system of equations for determining \(e_0\):
\[ \left. \begin{aligned} e_0+f_\delta+\delta_{\mathrm{p}}&=m_u,\\ P&=\kappa_1 Dl\,\frac{e_0}{m_\delta},\\ \delta_{\mathrm{p}}&=98\cdot 10^{-6}\sqrt[3]{\frac{P^2}{4d_{\mathrm{sh}}}}, \end{aligned} \right\} \tag{5} \]
where \(D\) and \(l\) are the diameter and length of the rotor, cm;
\(K_1 = 1.5\) for \(2p > 2\) and \(K_1 = 1\) for \(2p = 2\);
\(K\) is the shaft rigidity, kg/cm;
\(d_{\mathrm{ш}}\) is the diameter of the bearing ball, mm;
\(m_\delta\) is the mean value of the air gap, cm.
After simple transformations we obtain the equation for determining
\[ e_0\left(1+\frac{K_1Dl}{\delta K}\right) +98\cdot 10^{-6}\sqrt[3]{\frac{K_1^2D^2l^2e_0^2}{4m_\delta^2d_{\mathrm{ш}}}} = m_{\mathrm{ш}} . \tag{6} \]
For each particular case this equation is solved rather simply by the iteration method.
The distribution density of the eccentricity obeys the Rayleigh law [6]
\[ f(\varepsilon)=\frac{\varepsilon}{D}\exp\left(-\frac{\varepsilon^2}{2D}\right), \tag{7} \]
where \(\varepsilon=\dfrac{e}{m_\delta}\) is the eccentricity;
\(D\) is the variance.
The curve \(f(\varepsilon)\) for \(D=0.1\) is shown in Fig. 1. The area bounded by the curve \(f(\varepsilon)\), the abscissa axis, and the straight line \(\varepsilon_m=1\) is proportional to the number of machines in the assembly of which rotor-to-stator rubbing was detected. By adjustment (replacement or rotation of the end shields), this rubbing is usually eliminated by the assembly fitter. The shaded area in the figure is equal to the fraction of machines \(q\) in which rotor-to-stator rubbing was detected at the test station
Fig. 1.
\[ q=\int_{\varepsilon_0}^{1}\frac{\varepsilon}{D}\,e^{-\frac{\varepsilon^2}{2D}}\,d\varepsilon . \tag{8} \]
The quantity \(q\) is recorded in the documents of the test station. Knowing \(q\) and having preliminarily calculated the eccentricity from equation (6), one can determine the variance \(D\), which uniquely determines the distribution law. The solution of equation (8) can be obtained by a numerical method.
The mean eccentricity value of the machines in the given batch is
\[ \varepsilon_{\mathrm{av}}=\sqrt{\frac{\overline{\pi D}}{2}} . \tag{9} \]
The fraction of machines having an eccentricity less than the specified \(\varepsilon\) is
\[ q' = 1 - e^{-\frac{\varepsilon^2}{2D}} . \tag{10} \]
To facilitate the calculations, Fig. 2 presents a family of curves \(D=f(q)\), calculated by formula (8), for several values of the eccentricities \(\varepsilon_0\).
Fig. 2.
The quantity \(q\) is calculated by formula (8) under the assumption that the air-gap magnitude is unchanged. In reality this is a random quantity having a normal distribution law. Because the curve (Fig. 1) in the neighborhood of the point \(\varepsilon=\varepsilon_0\) is not parallel to the abscissa axis, the additional number of machines that will have rotor-to-stator contact when the air gap decreases is greater than the number of machines in which contact is eliminated when the air gap increases. This must be taken into account by reducing the limiting eccentricity by the amount
\[ \Delta \varepsilon = -\frac{\Delta q}{2f(\varepsilon_0)} , \tag{11} \]
where \(\Delta q=q_1-q_2\) is the increase in the fraction of machines having rotor-to-stator contact because of fluctuations in the value of \(u\);
\[
q_1
\]
is the fraction of machines in which contact appeared because the value \(u\) decreased relative to the mean value;
\[
q_2
\]
is the fraction of machines in which contact was eliminated because the value increased,
\[ q_1=\int_0^{\varepsilon_0}\int_0^\infty \frac{\varepsilon}{\sqrt{2\pi}\,D\sigma} e^{-\frac{\varepsilon^2}{2D}-\frac{\varepsilon^2}{2\sigma^2}} \,d\varepsilon\,d\varepsilon , \tag{12} \]
\[ q_2=\int_{\varepsilon_0}^{1}\int_{0}^{1-\varepsilon_0} \frac{1}{\sqrt{2\pi}\,D\sigma}\, e^{-\frac{\varepsilon^2}{2D}-\frac{\zeta^2}{2\sigma^2}}\, d\varepsilon\,d\zeta . \tag{13} \]
The quantities \(q_1\) and \(q_2\) can be found by a numerical method. If the interval is taken to be \(0.5\sigma\), then
\[ q_1=s\int_{0}^{0.5\sigma} f(\gamma)\,d\gamma +(s+\Delta s)\int_{0.5\sigma}^{\sigma} f(\gamma)\,d\gamma +(s+2\Delta s)\times \]
\[ \times\int_{\sigma}^{1.5\sigma} f(\gamma)\,d\gamma +(s+3\Delta s)\int_{1.5\sigma}^{2\sigma} f(\gamma)\,d(\gamma), \tag{14} \]
\[ S=0.5\sigma\,\frac{f(\varepsilon_0-0.5\sigma)+f(\varepsilon_0)}{2}, \tag{15} \]
\[ \sigma=\frac{\sigma_n}{m_\delta};\qquad \frac{u}{m_\delta}=\gamma, \tag{16} \]
\[ \Delta s=0.5\sigma\,\frac{f(\varepsilon_0-0.5\sigma)-f(\varepsilon_0)}{2}. \tag{17} \]
For
\[ 1-\varepsilon_0 \ge 2\sigma \]
\[ q_2=(S-\Delta S)\int_{0}^{0.5\sigma} f(\gamma)\,d\gamma +(S-2\Delta S)\int_{0.5\sigma}^{\sigma} f(\gamma)\,d\gamma+ \]
\[ +(S-3\Delta S)\int_{\sigma}^{1.5\sigma} f(\gamma)\,d\gamma +(S-4\Delta S)\int_{1.5\sigma}^{2\sigma} f(\gamma)\,d\gamma, \tag{18} \]
\[ \Delta q=q_1-q_2=\Delta S\left[ \int_{0}^{0.5\sigma} f(\gamma)\,d\gamma +3\int_{0.5\sigma}^{\sigma} f(\gamma)\,d\gamma +\int_{\sigma}^{1.5\sigma} f(\gamma)\,d\gamma+ \right. \]
\[ \left. +7\int_{1.5\sigma}^{2\sigma} f(\gamma)\,d\gamma \right]=1.409\,\Delta S. \tag{19} \]
The value of the integrals is determined from tables [7]. If \(1-\varepsilon_0<2\sigma\), for example, \(1-\varepsilon_0=1.2\), then
\[ q_2=(S-\Delta S)\int_{0}^{0.5\sigma} f(\gamma)\,d\gamma +(S-2\Delta S)\int_{0.5\sigma}^{\sigma} f(\gamma)\,d\gamma +\int_{\sigma}^{1.2\sigma} (f(\gamma)\,d\gamma . \tag{20} \]
Similarly, \(q_2\) can be found for any relation between \(\varepsilon_0\) and \(\sigma\).
The calculated correction \(\Delta \varepsilon\) is found with 95% confidence.
When the rotor temperature considerably exceeds the stator temperature, the rotor expands more, which is equivalent to a reduction of the air gap.
During operation, the eccentricity increases because of bearing wear. Bearing wear is a random process with mean rate \(g\), i.e., over time the eccentricity changes according to the law
\[ \varepsilon=\varepsilon_{\mathrm{init}}+gt. \]
The fraction of machines in which rotor-to-stator contact will occur during the time \(\Delta t\) is
\[ \Delta q=\int_{\varepsilon_0-g\Delta t}^{\varepsilon_0}\frac{\varepsilon}{D}\,de-\frac{\varepsilon^2}{2D}. \tag{21} \]
Example of calculation
The calculation was carried out for an A31-4 electric motor having the following data:
\(D=8.9\ \text{cm},\ l=6.4\ \text{cm},\ \delta=0.025^{+0.005}_{+0.002}\ \text{cm},\ K=8.74\cdot10^6\ \text{kg/cm}\), bearings—No. 304. The bearing data were taken from [5].
The root-mean-square deviations were determined under the assumption that the tolerance field \(2\delta_i=6\sigma\).
The reduction of the bearing radial clearance is taken as equal to 0.7 of the interference when fitted on the shaft.
The parameters of the calculated quantities are summarized in the table.
Table
| Name of quantity | Mean | Root-mean-square deviation |
|---|---|---|
| Initial radial clearance | \(m_1=17\) | \(\sigma_1=2.33\) |
| Change in radial clearance due to change in the diameter of the inner ring | \(m_2=-3.5\) | \(\sigma_2=1.16\) |
| Change in radial clearance due to change in shaft diameter | \(m_3=-6.65\) | \(\sigma_3=1.75\) |
| Value of the radial clearance of the assembled bearing | \(m_x=6.85\) | \(\sigma_x=3.14\) |
| Change in the diameter of the outer bearing ring | \(m_4=4.5\) | \(\sigma_4=1.5\) |
| Change in the diameter of the seating hole in the end shield | \(m_5=4.5\) | \(\sigma_5=3.84\) |
| Clearance between the outer bearing ring and the end shield | \(m_y=9\) | \(\sigma_y=4.12\) |
| Air gap | \(m_\delta=2.65\) | \(\sigma_\delta=11.7\) |
\[ m_u=m_\delta-m_x-m_y=2.49\cdot10^{-2}\ \text{cm}, \]
\[ \sigma_u=\sqrt{\sigma_x^2+\sigma_y^2+\sigma_\delta^2}=0.128\cdot10^{-2}\ \text{cm}. \]
The displacement and eccentricity at which contact occurs [6] are
\[ e_0=2.44\cdot10^{-2}\ \text{cm}, \]
\[ \varepsilon_0=\frac{e_0}{m_\delta}=0.921. \]
From the actual \(q\) and \(\varepsilon_0\), using Fig. 2 we find \(D\). For example, for \(q=1\%\), \(D=0.105\).
The mean eccentricity value for this batch of machines is
\[ \varepsilon_{\mathrm{av}}=\sqrt{\frac{\pi D}{2}}=0.406. \]
The correction \(\Delta\varepsilon\) is found from formulas (11, 14–20).
\[ \Delta\varepsilon=0.0017. \]
The value of \(\varepsilon'_0\), taking into account the fluctuations of the above quantities, is
\[ \varepsilon'_0 = \dot{\varepsilon}_0 - \Delta \varepsilon = 0.9193. \]
The fraction of machines having an eccentricity permissible according to the standards, \(\varepsilon \leq 0.15\), is calculated by formula (10):
\[ q' = 0.1034 = 10.3\%. \]
Conclusions
-
The proposed method makes it possible to determine the mean and the distribution law of the eccentricity from test-station data on the number of rotor-to-stator contacts for a given batch of machines; this, in turn, makes it possible to take proper account in calculations of the actual magnitude of the eccentricity.
-
Knowing the rate at which the radial clearance of the bearing increases, one can predict rotor-to-stator contact during operation.
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