Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 190　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1968

Calculation of Magnetic Vibrations and Noise of a Cross-Field EMA

Yu. V. Gorst, P. T. Malyshev

(Recommended by the seminar of the departments of electrical machines and general electrical engineering)

The problems of calculating the expected level of vibration and noise when designing electrical machines are becoming increasingly important in connection with the constantly rising requirements imposed on modern products with respect to their vibroacoustic parameters. This is especially important in the design of low-noise electrical machines, since in the design process it is desirable at least approximately to determine by calculation the expected levels of their vibration and noise. The overall noise level can be estimated according to the method proposed in [1]. However, among the many components in the spectra of vibration and noise that determine the overall levels, vibrations and noise of magnetic origin are of interest, since they are often dominant and occur in almost any electrical machine.

A number of works are devoted to the computational determination of magnetic vibrations and noise of electrical machines, for example [2, 5, 7] for asynchronous electric motors and [3, 4, 8] for direct-current machines. In connection with the rapid development of automation, modern electrical-machine engineering faces the problem of creating low-noise electric-machine amplifiers. Since in an EMA, as in a number of other electrical machines, the components of vibration and noise of magnetic origin are determinative in the overall levels, the problem of their computational determination in the design process arises first of all.

According to the methods for calculating magnetic noise set forth in the technical literature, an asynchronous electric motor is regarded as a spherical radiator, and a direct-current machine as a cylindrical radiator of sound waves.

The calculation of the sound-radiation power of a spherical surface, depending on the order of oscillations, is carried out using the formulas [2, 7]:

\[ \text{for } i = 0 \qquad P_{zv}=1.03 \cdot 10^{-3} \cdot R_c^2 f_v^2 U_{vr}^2 \frac{(kR_c)^2}{1+(kR_c)^2}; \]

\[ \text{for } i = 1 \qquad P_{zv}=0.34 \cdot 10^{-3} \cdot R_c^2 f_v^2 U_{vr}^2 \frac{(kR_c)^4}{4+(kR_c)^4}; \tag{1} \]

\[ \text{for } i = 2 \qquad P_{zv}=1.08 \cdot 10^{-3} \cdot R_c^2 f_v^2 U_{vr}^2 \frac{(kR_c)^2} {\left[4-\frac{9}{(kR_c)^2}\right]^2+ \left[kR_c-\frac{9}{kR_c}\right]^2}, \]

where

$R_c$ is the mean radius of the elastic axial line of the stator ring;

$f_v$ is the frequency of the exciting radial forces;

\[ k=\frac{2\pi}{\lambda_{\mathrm{sv}}} \]

is the wave number;

\[ \lambda_{\mathrm{sv}}=\frac{v_{\mathrm{sv}}}{f_v} \]

is the sound wavelength;

$v_{\mathrm{sv}}$ is the speed of sound in air;

$U_{\mathrm{vr}}$ is the magnitude of the amplitude of the dynamic rotating deformation of the stator:

\[ U_{\mathrm{vr}}=0.0543\, \frac{C_p F_{\mathrm{ud}} f_c} {\sqrt{f_c+2.5(f_v-f_c)^2}}, \tag{2} \]

where

$f_c$ is the frequency of the natural vibrations of the stator;

\[ C_p=6\cdot 10^{-2}\frac{R_c^4}{b_c^3} \]

is the coefficient determining the stiffness of a solid stator ring under a static load periodically distributed over its circumference (per unit length);

$b_c$ is the thickness of the stator-ring back;

$F_{\mathrm{ud}}=0.407\cdot 10^{-7} B_\delta^2$ is the specific force causing stator vibrations and acting on $1\ \mathrm{cm}^2$ of the area of its bore.

The value of the square of the induction in the air gap $(B_\delta^2)$ is determined from the expression given in [7], and the function $f(kR_c)$, which depends on the ratio of the radiator dimensions to the length of the radiated wave for $i=2$, is given there as well, in Fig. 7-1.

The sound-radiation power of a cylindrical surface is determined from the expression [4, 8]:

\[ P_{\mathrm{sv}}=\frac{1}{2}r_a x_0^2, \tag{3} \]

where

$x_0$ is the vibration velocity on the machine frame;

$r_a$ is the acoustic resistance of the frame in air, which can be determined by the formula [8]:

\[ r_a=\pi R_c l_c \rho v_{\mathrm{sv}}\varphi_1, \tag{4} \]

$l_c$ is the length of the machine frame;

$\rho v_{\mathrm{sv}}=41.3$ is the specific acoustic resistance of air.

The function $\varphi_1$ depends on the effective radius of the cylinder $\dfrac{2\pi R_c}{\lambda_{\mathrm{sv}}}$ and the order of vibration, and can be determined from Fig. 6 [4].

The amplitude of the flexural vibrations of the stator of an asynchronous electric motor as a function of the order of vibration is calculated by the formula [5]:

\[ A_i= \frac{P_i\, i} {\sqrt{\left(1-\frac{f_v^2}{f_c^2}\right)^2+n^2\frac{f_v^2}{f_c^2}}}\,\chi_\mu, \tag{5} \]

where

$P_i$ is the amplitude of the specific radial forces produced by the interaction of rotor and stator tooth harmonics;

\(P_i\) is the compliance of the stator ring under the action of forces of the \(i\)-th order;

\(n_t\) is a coefficient allowing for internal friction in the metal;

\(\chi_\mu(\chi_\nu)\) is the skew coefficient of a slot on the rotor (stator).

For a direct-current machine, the amplitude of the radial flexural vibrations of the frame is determined by formula [3]:

\[ A_i = \frac{i z R_c^4 B_\delta^2 \cdot 10^{-5}} {4\pi^2 \cdot 9.81\, E I_x} \cdot \frac{ \left[ p^2 k - \frac{r+p^2(R_c-t)}{R_c} S_k \right] \frac{\sin \pi C}{\pi C} }{ \left[ (i^2+1)\frac{\mu R_c^4}{E I_x}(2\pi f_{\mathrm{B}})^2 - i^2(i^2-1) \right] }, \tag{6} \]

where

\(z\) is the number of armature slots;

\(E\) is the modulus of elasticity of the frame material;

\(I_x\) is the moment of inertia of the transverse section of the frame with respect to an axis perpendicular to the plane of bending;

\(p\) is the number of pairs of main poles;

\(r\) is the radius of the pole bore;

\(\mu\) is the mass of the frame together with the main and interpoles per unit length of the neutral line;

\(C\) is the skew of the armature slots or pole, in fractions of the armature slot pitch;

\(R_k, S_k\) are calculation coefficients determined, according to formulas [3], as functions of the ratios of tooth width and slot pitch, slot width and air-gap length, and the air-gap coefficient;

\(i\) is the order of vibration.

The question of which of the two methods for calculating magnetic vibrations and noise more correctly reflects the picture occurring in a cross-field EMU has so far not been addressed in the technical literature. The electromechanical amplifier differs from other types of electrical machines in a number of design features. Being, as a rule, a direct-current machine, the cross-field EMU has a stator with poorly expressed poles. In its construction it is analogous to the stator of an asynchronous electric motor. In addition, from the design standpoint EMUs are manufactured in two versions: unit-type and single-frame (with a built-in motor). These features leave their mark on the method for calculating magnetic vibrations and noise. Obviously, as applied to a cross-field EMU, there can be no unambiguous solution of this question. This proposition was confirmed in calculating the levels of magnetic vibrations and noise of mass-produced electromechanical amplifiers.

Let us consider separately EMUs in unit-type and single-frame versions. In an electromechanical amplifier of unit-type design (the drive motor is made separately), the stator core stack differs little from the stator core stack of an asynchronous motor. In its linear dimensions the EMU fits, with small deviations, into a sphere. It should be noted that in this case the dimensions of the machine are, as a rule, small in comparison with the wavelengths of the radiated magnetic noise. Consequently, a cross-field EMU of unit-type design may, in a first approximation, be regarded as a spherical sound source, and the calculation of its sound-radiation power as a function of the order of vibration may be carried out by formulas (1).

For the same reasons, in all probability, the calculation of the amplitude of flexural vibrations on the EMU frame may be performed by the method developed for asynchronous electric motors, i.e., by formula (5).

When calculating the natural vibration frequencies (\(f_c\)) of the stator ring of a cross-field electric machine amplifier, its compliance (\(\Pi_1\)), and the stiffness coefficient (\(C_n\)), it is most expedient to be guided by the recommendations and formulas set forth in [5, 7]. However, owing to certain features of the design of the stator ring of the cross-field electric machine amplifier, the problem of determining the above-listed parameters cannot be considered solved. And although in the present work we proceed from the assumption “that the partial vibration frequencies of the frame and of the stator steel stack differ significantly from one another, and the coupling coefficient of the system is the greatest; therefore the natural frequencies of the system are close to the partial ones” [5], nevertheless this also requires further refinement.

In a cross-field electric machine amplifier of single-frame construction, a characteristic feature is that its length proves to be considerably greater (by a factor of \(2.5 \div 3\)) than its diameter. Therefore, despite the fact that, for equal power of cross-field electric machine amplifiers of both constructions, the source of magnetic vibrations and noise remains unchanged, the dimensions and shape of the radiating surface of a single-frame cross-field electric machine amplifier differ substantially from those of a unit-type cross-field electric machine amplifier. In this connection, it is evidently most correct to represent the single-frame cross-field electric machine amplifier as a cylindrical sound source and to calculate the magnetic noise by formula (3). The stated assumptions were verified by experiments. Figure 1 presents the directivity characteristic of sound radiation for a unit-type cross-field electric machine amplifier, and Fig. 2—for the single-frame construction.

For calculating the vibration velocity on the frame of a cross-field electric machine amplifier we shall use

Fig. 1. Directivity characteristic of the sound radiation of a unit-type cross-field electric machine amplifier:
a) in the plane passing through the shaft of the cross-field electric machine amplifier;
b) in the plane perpendicular to the shaft of the cross-field electric machine amplifier.

\(I — 0.5\ \text{m}\quad II — 1\ \text{m}\)

Fig. 2. Directivity characteristic of the acoustic radiation of a single-frame EMU:
a) in the plane passing through the EMU shaft; b) in the plane perpendicular to the EMU shaft

\(I - 0.5\text{ m};\ II - 1\text{ m}\)

by the method of electromechanical analogy, set forth in [4, 6]. In Fig. 3 the mechanical system stator—EMU frame and the electrical circuit equivalent to it are presented. Having determined the total resistance of the mechanical system, we find the vibration velocity on the EMU frame. Knowing the value of the vibration velocity \((\dot{x}_0)\), from formula (3) we calculate the power of the acoustic radiation.

In calculating magnetic vibrations on the frame of a single-frame EMU, at first glance it would seem possible to use the method adopted for the aggregate design, with the only difference that the mass of the system increases and, consequently, the level of magnetic vibrations decreases. However, comparison of the results of calculation and experiment shows the untenability of this assumption. The reason for the discrepancy between the results of calculation and experiments evidently lies in the fact that the stiffness of the EMU stator does not have a substantial influence on the stiffness of the entire stator—frame system, since the length of the frame considerably (by 3–4 times) exceeds the length of the stator.

Apparently, in this case the most acceptable method is that of calculating magnetic vibrations for direct-current machines and, consequently, the amplitude of radial flexural vibrations of the EMU transverse-field frame may be determined by formula (6).

Knowing the amplitude of vibrations on the frame, we can determine the magnitude of the effective vibrational acceleration.

Fig. 3. EMA stator–frame system; a) mechanical system; b) equivalent electrical circuit

\[ a = A_1 \cdot (2\pi f_B)^2 . \tag{7} \]

In engineering practice, vibrational acceleration is usually evaluated not in absolute units, but in relative units—decibels. The level of vibrational acceleration is found from the expression

\[ L = 20 \lg \frac{a}{a_0}, \ \text{dB}, \tag{8} \]

where

\[ a_0 = 3 \cdot 10^{-4}\ \text{m/sec}^2 \]

is a constant corresponding to the zero level and taken as the reference origin.

From the known power of sound radiation it is possible to determine the magnitude of the sound pressure produced by the radiator in the room [7]:

\[ P_1 = 40.8 \sqrt{\frac{P_{\text{sound}}}{\sum \alpha S}}, \ \text{N/m}^2, \tag{9} \]

where

\[ \sum \alpha S \]

is the total sound absorption in the room in sabins;

\[ \alpha \]

is the sound absorption coefficient;

\[ S \]

is the area of the room surfaces.

The sound-pressure level in relative logarithmic units is equal to [7]:

\[ L = 20 \lg \frac{P_1}{P_0}, \ \text{dB}, \tag{10} \]

or

\[ L = 20 \lg 2 \cdot 10^6 \sqrt{\frac{P_{\text{sound}}}{\sum \alpha S}}, \ \text{dB}, \]

where

\[ P_0 = 2.04 \cdot 10^{-5}\ \text{N/m}^2 \]

is the sound pressure at the threshold of audibility.

Measurement of the sound-pressure level is usually carried out at some specified distance from the housing of the electrical machine. For this reason it becomes necessary to determine by calculation the sound-pressure level as a function of the distance from the sound source. According to [4, 9], sound pressure is inversely proportional to the distance from the sound source. Consequently, the sound-pressure level is

\[ L = 20 \lg 2 \cdot 10^6 \sqrt{\frac{P_{\mathrm{sv}}}{4\pi (R_c+d)^2 \sum \alpha S}}, \tag{11} \]

where

\(d\) is the distance from the microphone to the surface of the electrical machine.

The experimental verification was carried out on specially manufactured electric-machine amplifiers in single-housing and aggregate designs. To avoid noise from the drive motor, in both cases a vibration- and noise-insulated drive described in [10] was used. Measurement and analysis of the frequency composition of vibrations and noise were carried out with an IVPSh-type vibrometer, an Sh-63 sound-level meter, and a PF-1 type half-octave filter. The tooth frequency of the experimental EMAs, both for the single-housing and for the aggregate design, is \(f_v = 1280\) Hz.

The results of calculating the levels of magnetic vibrations and noise and the experimental data are given in the table (for \(d = 0.5\) m).

Table 1

It is evident from the table that the discrepancy between the calculated results and the experimental data lies within the error limits of the measuring instruments (1–2 dB). It should also be noted that the experimental data given in the table were obtained after mathematical processing of the results of 10–15 measurements for each point.

Conclusions

1. The calculation of magnetic vibrations and noise of a cross-field EMA must be carried out with allowance for its design.

2. As a result of analyzing the existing methods for calculating magnetic vibrations and noise, it has been established that, for an EMA of aggregate design, the method developed for asynchronous electric motors is acceptable, while for a single-housing design the method adopted for direct-current machines is acceptable. The calculation of magnetic vibrations and noise of an EMA of aggregate design should be performed using formulas (1), (5), and that of a single-housing design using formulas (3) and (6).

References

1. E. R. Kucher, N. I. Murkes. Overall noise level of self-ventilated electrical machines. Proceedings of VNIIEM, vol. 20, Energiya Publishing House, 1965.

1. E. R. Kucher. On the foundations of the method for calculating the magnetic noise of a three-phase squirrel-cage asynchronous electric motor. Vestnik elektropromyshlennosti, No. 12, 1956.

2. N. I. Ermolin. Magnetic noise of DC machines. Izvestiya LETI, 1955, issue 28.

3. I. G. Shubov. Determination of the magnetic-noise level of DC machines by the method of electromechanical analogy. Elektrichestvo, No. 4, 1958.

4. Yu. P. Rylov. Investigation of vibrations of asynchronous motors caused by magnetic forces. Vestnik elektropromyshlennosti, 1959, No. 5.

5. V. V. Furduev. Acoustic Fundamentals of Broadcasting. Svyazizdat, 1960.

6. B. B. Voronetskii, E. R. Kucher. Magnetic Noise of Asynchronous Electric Motors. Gosenergoizdat, 1957.

7. Schmidt K. Gehäuseschwingungen von Gleichstrommaschinen als Ursache magnetische Geräusches, A. F. E., 1938, B. 32, H. 8.

8. V. K. Iofe. Electroacoustics. Svyazizdat, 1954.

9. Yu. V. Gorst, P. T. Maltsev, A. T. Molodykh. Influence of load on the vibration and noise of low-power DC electrical machines. Izvestiya TPI, vol. 172, 1967.