Calculation of Tolerances for Parameters Determining the Eccentricity in an Induction Motor
Abstract Generated abstract
This paper addresses the problem of assigning manufacturing tolerances for induction motor components that determine rotor eccentricity and air-gap nonuniformity, a factor affecting efficiency, vibration, noise, magnetic forces, and reliability. It models eccentricity-related quantities probabilistically, using a Rayleigh-type distribution for misalignment and runout and a normal distribution for rotor diameter and air-gap dimensions, and derives relationships between permissible eccentricity, reject rate, bearing clearance, and tolerances for bearing-shield alignment, frame alignment, and rotor runout. The authors propose several tolerance-allocation procedures, including allocation based on equipment capability, equal influence of parameters, and cost minimization, while noting practical limits in available cost functions. The method is presented as a basis for setting tolerances during motor design and for choosing economically feasible inspection or equipment-improvement measures in existing production.
Full Text
CALCULATION OF TOLERANCES FOR PARAMETERS DETERMINING THE VALUE OF ECCENTRICITY IN AN INDUCTION MOTOR
O. P. MURAVLEV, A. D. NEMTSEV
(Submitted by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
The quality of induction motors is determined to a considerable degree by the uniformity of the air gap. Nonuniformity of the air gap worsens the energy indices and characteristics of induction motors, increases the level of noise and vibration, and gives rise to unbalanced magnetic forces. With a nonuniform air gap, the possibility arises that the rotor will catch on the inner surface of the stator, which leads to a substantial reduction in motor reliability.
Despite the great influence that nonuniformity of the air gap has on the quality and reliability of induction motors, its magnitude at present is practically not regulated. In 1971 OST No. 16.0688.091-71 was adopted, which establishes the permissible nonuniformity of the air gap for induction motors as a function of the number of poles [1]. The eccentricity of the rotor relative to the stator bore is taken as the magnitude of the nonuniformity of the air gap.
The assignment of tolerances for dimensions and for the mutual arrangement of surfaces that affect the magnitude of eccentricity is carried out without taking into account the permissible value of eccentricity. Therefore, in the present article a scientifically substantiated method is proposed for assigning tolerances to the parameters that affect the magnitude of eccentricity, according to the specified value of the permissible eccentricity.
The eccentricity of the rotor relative to the stator is a random quantity. The distribution density of the eccentricity has the form [2]:
\[ f(\varepsilon)=\frac{1}{\sigma^{2}}\varepsilon\cdot e^{-\frac{\varepsilon^{2}}{2\sigma^{2}}}, \tag{1} \]
where \(\varepsilon\) is the magnitude of eccentricity, and \(\sigma\) is the distribution parameter. For this distribution the mean value of the eccentricity is
\[ \bar{\varepsilon}=1.25\sigma. \]
We shall consider the following parameters that affect the eccentricity of the rotor relative to the stator:
\(l_{1}\) — misalignment of the bearing-shield lock relative to the bore for the bearing, mm;
\(l_{2}\) — misalignment of the frame lock relative to the stator bore, mm;
\(l_3\) — runout of the rotor surface relative to the bearing seats, mm;
\(2\delta_p\) — tolerance field for the rotor diameter, mm;
\(\Delta_p\) — displacement of the tolerance field for the rotor diameter relative to the nominal value, mm;
\(\delta\) — air gap, mm;
\(\delta_p\) — radial clearance of the bearing in the assembly, mm.
The quantities \(l_1, l_2, l_3\) are random; their distribution density is described by expression (1). The value of the rotor diameter and the magnitude of the air gap obey the normal distribution law, whose distribution density is [2]
\[ f(x)=\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\bar{x})^2}{2\sigma^2}}, \tag{3} \]
where \(\bar{x}\) is the mean value of the random variable, mm; \(\sigma\) is the standard deviation.
When calculating tolerances for quantities distributed according to the normal law, it is usually assumed that \(\delta_x=3\sigma\). For the air gap, one half of the tolerance field is
\[ \delta_{\delta}=\sqrt{\delta_p^2+\delta_c^2}, \tag{4} \]
where \(\delta_c\) is one half of the tolerance field for the diameter of the stator bore, mm.
The value of the stator-bore diameter for general-purpose asynchronous motors is determined mainly by the diameter of the mandrel on which the core stack is assembled. One half of the tolerance field for the stator-bore diameter is determined by the permissible wear of the mandrel
\[ \delta_c=\frac{\delta_l}{2}, \tag{5} \]
where \(\delta_l\) is the permissible wear of the mandrel, mm. The mean eccentricity of the rotor relative to the stator bore is
\[ \bar{\varepsilon}=\frac{K}{\delta_n}\left(\sqrt{l_1^2+l_2^2}+l_3\right)+\frac{\delta_p}{\delta_n}, \tag{6} \]
where \(\delta_n\) is the nominal value of the air gap, mm; \(K\) is a coefficient depending on the permissible reject level.
This coefficient is determined as follows:
\[ P\left(\varepsilon_i>\varepsilon_{i\mathrm{dop}}\right) =\exp\left(-\frac{\varepsilon_{i\mathrm{dop}}^2}{2\sigma_i^2}\right) =\frac{q}{100}, \tag{7} \]
after taking logarithms we obtain
\[ -\frac{\varepsilon_{i\mathrm{dop}}^2}{2\sigma_i^2} =\ln\frac{q}{100}, \tag{8} \]
whence we find
\[ \sigma_i= \frac{1}{\sqrt{-2\ln\frac{q}{100}}}\cdot \varepsilon_{i\mathrm{dop}}. \tag{9} \]
Taking expression (2) into account, we obtain the dependence relating the mean eccentricity to the permissible value of eccentricity
\[ \bar{\varepsilon}_i =1.25\sigma_i =\frac{1.25}{\sqrt{-2\ln\frac{q}{100}}}\cdot \varepsilon_{i\mathrm{dop}} =K\cdot \varepsilon_{i\mathrm{dop}}. \tag{10} \]
Table 1 gives the values of the coefficient \(K\) for different levels of permissible rejects.
Table 1
| \(q,\ \%\) | 0.3 | 0.5 | 10 | 2.0 | 5.0 | 10.0 |
|---|---|---|---|---|---|---|
| \(K\) | 0.368 | 0.394 | 0.425 | 0.448 | 0.514 | 0.823 |
The quantities \(l_1\) and \(l_2\) form a composition of two laws of distribution of eccentricity and are its static component \(\varepsilon_{\mathrm{st}}\), with mean value
\[ \bar{\varepsilon}_{\mathrm{st}}= \frac{K}{\delta_{\mathrm{n}}}\sqrt{l_1^{\,2}+l_2^{\,2}}. \tag{11} \]
The runout of the rotor surface relative to the bearing seats \(l_3\) forms a rotational eccentricity \(\varepsilon_{\mathrm{vr}}\), with mean value
\[ \bar{\varepsilon}_{\mathrm{vr}}= \frac{K}{\delta_{\mathrm{n}}}\cdot l_3. \tag{12} \]
Let us consider expression (6) in more detail. The radial clearance of the bearing \(\delta_{\mathrm{p}}\) depends on its initial radial clearance, on the magnitude of the interferences when the outer bearing ring is fitted into the end shield and the inner ring onto the shaft. All these quantities are relatively small, and for sizes 1–6 of AO2-series induction motors one may take \(\delta_{\mathrm{p}}=0.01\) mm. It is desirable to make the rotational eccentricity \(\varepsilon_{\mathrm{vr}}\) as small as possible, since during rotation of the rotor the worst case of the relative position of the static and rotational eccentricities will always occur. In addition, from the technological point of view it is easier to ensure the tolerance for rotor runout than for \(l_1\) and \(l_2\).
The values of \(l_1\), \(l_2\), and \(l_3\) can be calculated either from the specified mean value of the rotor eccentricity relative to the stator \(\bar{\varepsilon}\), in which case expression (6) is used, or from the permissible value of the eccentricity
\[ \varepsilon_{\mathrm{dop}}= \frac{1}{\delta_{\mathrm{n}}} \left(\sqrt{l_{1\mathrm{dop}}^{\,2}+l_{2\mathrm{dop}}^{\,2}}+l_{3\mathrm{dop}}\right) +\frac{\delta_{\mathrm{p}}}{K\cdot\delta_{\mathrm{n}}}. \tag{13} \]
In what follows we shall consider calculation by equation (13). Taking \(\delta_{\mathrm{p}}\) as a known quantity, we obtain
\[ \varepsilon'_{\mathrm{dop}} = \varepsilon_{\mathrm{dop}}-\frac{\delta_{\mathrm{p}}}{K\delta_{\mathrm{n}}} = \frac{1}{\delta_{\mathrm{n}}} \left(\sqrt{l_{1\mathrm{dop}}^{\,2}+l_{2\mathrm{dop}}^{\,2}}+l_{3\mathrm{dop}}\right), \tag{14} \]
i.e., we have one equation with three unknowns.
The permissible values for \(l_1\), \(l_2\), and \(l_3\) can be assigned in different ways:
- For \(l_3\), the minimum possible value \(l_{3\mathrm{dop}}\) is adopted, proceeding from the capabilities of the existing equipment; \(l_{1\mathrm{dop}}=l_{2\mathrm{dop}}\) is assumed, and then
\[ l_{1\mathrm{dop}}=l_{2\mathrm{dop}}= \frac{\varepsilon'_{\mathrm{dop}}\cdot\delta_{\mathrm{n}}-l_{3\mathrm{dop}}}{\sqrt{2}}. \tag{15} \]
- The permissible values are determined according to the principle of equal influence on \(\varepsilon'_{\mathrm{dop}}\). In this case
\[ l_{1\mathrm{dop}}=l_{2\mathrm{dop}}= \frac{\sqrt{2}\,\varepsilon'_{\mathrm{dop}}\cdot\delta_{\mathrm{n}}}{3}; \tag{16} \]
\[ l_{3\,\mathrm{adm}}=\frac{\varepsilon'_{\mathrm{adm}}\cdot \delta_{n}}{3}. \]
- In the case of the requirement of minimum cost for ensuring the admissible values \(l_1\), \(l_2\), and \(l_3\), it is necessary to minimize the objective function
\[ C=\sum_{i=1}^{m}\varphi_i(l_{i\,\mathrm{adm}}), \tag{17} \]
where \(\varphi_i(l_{i\,\mathrm{adm}})\) is the dependence of the cost of ensuring the \(i\)-th admissible value on the magnitude of \(l_{i\,\mathrm{adm}}\).
The last method is the most promising, but at present it is difficult to implement because of the absence of the dependences \(\varphi_i(l_{i\,\mathrm{adm}})\).
It is advisable to assign \(l_{1\,\mathrm{adm}}\), \(l_{2\,\mathrm{adm}}\), and \(l_{3\,\mathrm{adm}}\) by the methods considered above when the manufacturing technology for motors is being designed. If induction motors are produced on the existing equipment of a plant, then there are quite definite values of \(l_{i\,\mathrm{adm}}\) that can be ensured on this equipment. If it is necessary to obtain motors with a smaller eccentricity value, then two solutions are possible:
- Modernization or replacement of equipment.
- Rejection of parts with large eccentricity.
The first method usually requires large capital expenditures, obtaining funds for new equipment, and cannot be implemented in a short time. The second method can be used fairly quickly by introducing appropriate inspection with rejection of parts. This method can be used when it is necessary to manufacture a relatively small number of motors with a small admissible eccentricity, for example, in the production of high-precision and export motors. The costs of ensuring admissible eccentricity by this method can be estimated from the expression
\[ C=\sum_{i=1}^{3}\left(c_i\frac{N}{1-q_i}+c_{q_i}\cdot q_i\frac{N}{1-q_i}\right), \tag{18} \]
where \(c_i\) is the cost of inspecting the \(i\)-th part;
\(q_i\) is the reject rate for the \(i\)-th part;
\(c_{q_i}\) is the cost of one rejected \(i\)-th part.
Thus, the method proposed in the present work for calculating eccentricity makes it possible to determine, on a justified basis, tolerances for the dimensions that determine eccentricity at the design stage, and to outline an economically expedient way of ensuring the specified eccentricity for the existing manufacturing technology of induction motors.
REFERENCES
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OST No. 16.0688.091-71. “Three-phase squirrel-cage induction motors for general application with power above 0.12 kW. Methods for monitoring nonuniformity of the air gap.”
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Kh. B. Kordonskii. Application of Probability Theory in Engineering. L., Fizmatgiz, 1963.
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O. P. Muravlev, E. K. Strelbitskii. Calculation of tolerances for parameters of induction motors. Elektrotekhnika, 1968, No. 11.