Modal analysis of magnetic vibration of induction motor used in household appliances

Abstract

The paper deals with analysis of vibrational behavior of induction motor using the combined approach linking the finite elements method (FEM) applied to modelling of magnetic and mechanical phenomena in the motor with its modal description in spectral domain. Results of investigations presented in this paper proved that relatively small rotor axis misalignment against stator leads to visible vibration if forces resulted from this effect have frequency close to rotor bending resonance. Paper contains a rigorous description of all steps in analysis with a special attention paid to details of numerical approach both in FEM and in spectral consideration. Results of theoretical investigations were confirmed by measurements.

Keywords

Induction motor vibration finite elements spectral analysis

1. Introduction

The induction motors are widely used in many household devices like laundry machines, dryers and others. Nowadays they have the three-phase supply from a power converter enabling the smooth control in a wide range of an angular velocity. Their application inside devices having relatively large and thin housing in connection with the possibility of non-centric shaft load makes them a potential source of vibration and accompanying acoustic pollution.

Vibration and acoustic noise emitted by induction motors has been studied almost one hundred years. Starting from pioneering Jordan work [1] and its followers [2–4] dozens of papers were afterwards written containing different approaches to modelling, measurements and other designing problems related to these machines [5–7]. The substantial development of numerical modelling techniques both in electromagnetic and mechanical domains caused that finite elements approach is the basic tool for the solution of vibration of electromechanical devices [8,9].

Excessive vibration is mainly mitigated by the correct choice of the number of stator and rotor slots relative to the number of pole pairs required together with their mutual skew. Such rules are well known and they may be found in [2,3]. For medium and high power motors there is no problem to find these numbers, the eventual problem comes from rather low values of resonant frequencies of the stator structure which may be too close to frequencies of exciting forces. The oppose situation arises in small machines where the stator resonances appear quite far in the frequency spectrum but the available slot numbers are very limited due to small size of these motors. Up to some extent the vibration may be reduced using the appropriate control strategy [10] or applying the carefully designed damper windings connected with the resonance circuit [11]. The magnetic stress in the air gap is assumed to be the input quantity forcing vibrations, however, magnetostriction effects present in stator core may also contribute [13,15]. Some results concerning other types of AC machines can be directly applied when vibration of induction motors are of interest [12,14].

The paper presents combined analysis of vibration caused by magnetic forces in three-phase induction motor of power 1.5 kW supplied from network and inverter. The finite elements technology is used in magnetic and vibrational calculus by means of commercial software (Magnet and ANSYS). The results are obtained with the modal approach specified for magnetic forces definition and also for calculations of the vibration response.

2. Motor description

Analyzed motor is intended for a laundry machine what explains a specific shape of its housing consisting of two separate bearing plates connected with a pair of handles. The stator is equipped in a single-layer winding, the rotor has an aluminum cage with the slot skew of 1.15 stator slot pitch. The outlook of the motor is shown in Fig. 1 and basic data are given in Table 1.

Fig. 1.

Outlook of model geometry of analyzed motor.

Table 1

Motor data

Parameter	Value
Torque [Nm]	6.8
Pole pair number	2
Velocity range [rpm]	300–10 500
Shaft height [mm]	90
Stator core outer diameter [mm]	119
Stator core length [mm]	120
Stator slot number	36
Rotor slot number	30
Air gap [mm]	0.25

3. Electromagnetic analysis

The non-linear 2D solution of the magnetic field distribution in terms of magnetic vector potential in the cross-section of the motor was obtained in two ways. Firstly, the harmonic solver was applied at no-load state and forced voltage on terminals of the motor. Secondly, the time-stepping with motion for the same conditions was used. The computing time was, of course, very different. The time stepping required about 40 min but the nonlinear harmonic approach took few seconds only. Results observed on the single field plot were quite similar, e.g. magnitude of flux density in stator tooth was 1.65 T for time stepping and 1.7 T for the harmonic solution. Also the instantaneous pictures of the flux density distribution along the air gap in both cases were comparable. The exemplary results are displayed in Fig. 2. The main difference between these approaches concerns the spectral content of currents induced in the rotor bars. The time harmonic solution by definition gives the single frequency only resulting from the input value of the rotor slip. When the time stepping connected with the rotor motion is solved the additional time changes of the rotor flux linkage appear caused by the variable permeance of the air gap which enrich significantly the time profile of the rotor currents. Exemplary distribution of the rotor current within one period of the supply is shown in Fig. 3.

Fig. 2.

Instantaneous flux density distribution in motor cross-section at no-load.

Fig. 3.

Rotor bar current versus time at no-load resulting from 2D time stepping with motion.

The analyzed motor has a small air gap, therefore, the contribution of the permeance harmonics in the 2D solution is substantial. Their reduction in a real motor is obtained by the rotor slot skew which has no chance to be applied in a 2D model. It is possible to build the 3D model having this property but the computing time of the 3D time-stepping would be unreasonable. Therefore, the simplified algorithm using the modal skew factor 𝜉_sr was applied which is given by $\begin{eqnarray}\displaystyle {\xi}_{sr}=sinc\left(r{\displaystyle \frac{{\alpha}_{s}}{2}}\right) & & \displaystyle\end{eqnarray}$ (1) where r is mode number and 𝛼_s denote the angular skew of the rotor core. The smoothing of the flux density distribution along the air gap by the skew factor requires the decomposition of this distribution into Fourier components. All components multiplied by their skew factor may be used afterward to synthesize back the averaged space profile of the flux density. The input data and the Fourier transform are different for both approaches. In the case of a time-harmonic solution, we have the vector B (r =1… N) where N denotes the number of samples of the radial component of the flux density extracted in the airgap. The one-dimensional Fourier transform is applied here. For the time stepping results we have the array B (r, q) where q = (1… M) is the number of the time steps. In such a case the two-dimensional transform must be used. In both cases, we operate in a complex number domain. The block diagram of the above algorithm is presented in Fig. 4. The flux density spectrum obtained from the harmonic solution does not contain information on the frequency of its particular components. Up to some extent, it is possible to add it a’posteriori following the classic permeance harmonics method. Assuming that the fundamental component of MMF H_p(𝛼, t) with p pole pairs is able only to create the flux density field inside the slotted structure of the motor airgap we get the simplified equation of the flux density here as $\begin{eqnarray}\displaystyle B({\alpha},t)=H_{p}\left\{\begin{array}{@{}l@{}}{\lambda}_{0}\exp [+j({\omega}t-p{\alpha})]+\\ {\lambda}_{S}\exp [+j({\omega}t+(N_{S}-p){\alpha})]+\\ {\lambda}_{S}\exp [+j({\omega}t-(N_{S}+p){\alpha})]+\\ {\lambda}_{R}\exp \left[+j\left({\omega}t(1-\displaystyle \frac{N_{R}}{p}(1-s))-(p-N_{R}){\alpha}\right)\right]+\\[12.0pt] {\lambda}_{R}\exp \left[+j\left({\omega}t(1+\displaystyle \frac{N_{R}}{p}(1-s))-(p+N_{R}){\alpha}\right)\right]+\ldots \end{array}\right\}. & & \displaystyle\end{eqnarray}$ (2) Unfortunately, two and more frequencies may possess the same spatial mode that has also happened in the analyzed case. Following that harmonic approach was not used in further considerations. Figure 5 demonstrates the results of the application of the smoothing algorithm and Fig. 6 shows the mode-frequency spectrum of the flux density after smoothing. Besides, the main, non-dispersive wave (r = kp, q = k), k = 1,3,5, … in this spectrum contains the higher frequency components resulting from stator slotting modulation of rotor currents wave and vice versa stator currents wave modulated by rotor slotting. The components having magnitude below 5 mT were removed from this picture assuming this value to be the computational noise level.

Fig. 4.

Smoothing algorithm of flux density distribution in air gap of induction motor.

Fig. 5.

Results of application of smoothing algorithm on radial component of flux density distribution in air gap of induction motor at no-load, centric rotor.

Fig. 6.

Single-sided mode-frequency spectrum of radial component magnitude of flux density distribution in air gap of induction motor at no-load, centric rotor.

Apart the case of centric rotor analysis two types of the possible eccentricity were also investigated. Firstly, the constant misalignment of 60 μm between stator and rotor axes was applied. Secondly, this misalignment was sinusoidally varying in time along chosen axis with the frequency equal to the rotational speed. In both mentioned instances the angular step of rotor rotation was set to 1 degree. The results of calculations having the form of space-time [360 × 360] arrays of flux density vector along the air gap were extracted for further post-processing. The time stepping method has also some limitations connected with the requested computational effort. Number of steps N in time domain results from parameters of DFT analysis given by Shannon theorem $\begin{eqnarray}\displaystyle f_{Ny}={\displaystyle \frac{N}{2}}{\Delta}f. & & \displaystyle\end{eqnarray}$ (3) where f_Ny is the Nyquist frequency and Δf is the frequency resolution. Setting the upper limit of frequency spectrum in motor having p pole pairs to stator slotting harmonic of second order $\begin{eqnarray}\displaystyle f_{Ny}=2{\displaystyle \frac{N_{S}}{p}}f_{supply} & & \displaystyle\end{eqnarray}$ (4) and defining the frequency resolution as the maximum value enabling separation of frequencies caused by stator N_S and rotor N_R slotting at slip s $\begin{eqnarray}\displaystyle {\Delta}f={\displaystyle \frac{sf_{supply}}{2}} & & \displaystyle\end{eqnarray}$ (5) we have the requested value of time steps which is equal to number of independent solutions of nonlinear 2D problem $\begin{eqnarray}\displaystyle N=8{\displaystyle \frac{N_{S}}{sp}}. & & \displaystyle\end{eqnarray}$ (6) In considered motor N_S = 36 and rated slip s = 0.05 what gives N = 2880. Adding the supplementary time necessary for the decay of transient phenomena which was equal to three periods of the supply voltage we get the total number of time steps N = 11 520. This value raises dramatically when the no-load state is of interest. The another problem is related with the requested full periodicity of analyzed signal when the Discrete Fourier Transform (DFT) is used. In other words, it is expected that at the end of the acquisition time the magnetic field and rotor geometry will be exactly in the same place as they were at starting point. In general, it is very difficult and costly to get such a coincidence for arbitrary asynchronous speed of the rotor. Therefore, one must reckon with the power leakage in spectral analysis. The compromise values of the acquisition time T_ac = 100 ms and step number N = 900 were chosen which result with the frequency resolution Δf = 0. 5f_supply = 25 Hz used in all further analyses.

4. Magnetic force calculations

The presence of any eccentricity must trigger the forces F of so-called magnetic pull acting simultaneously on stator and rotor of the machine. It may be calculated from the magnetic stress distribution σ_ij(x, y) on the closed surface S marked out in the air around the stator or rotor. The value of this force equals to (index summation applied) $\begin{eqnarray} \displaystyle F_{i}=\unicode[STIX]{x222F}{}_{S}{\sigma}_{ij}n_{j}dS. & & \displaystyle \end{eqnarray}$ (7) The magnetic stress tensor is given by $\begin{eqnarray}\displaystyle {\sigma}_{ij}={\displaystyle \frac{1}{{\mu}_{0}}}B_{i}B_{j}-{\delta}_{ij}w_{m} & & \displaystyle\end{eqnarray}$ (8) where B_i and B_j are components of flux density vector, w_m is magnetic energy density and δ_ij is Kronecker sign. The instantaneous forces of magnetic pull acting on stator are the same as on rotor but with opposite sign because they are calculated from the same stress field and they differ only with the direction of the outward normal n_j applied to. Their time distributions for variants having introduced eccentricity are presented in Fig. 7. The main component of these signals has the frequency of angular rotation, the high frequency content is mostly related to currents induced in rotor cage by magnetic permeance variation. When the rotor is centric the basic sinusoidal components are absent and higher frequencies are observed only. Vibration analysis is usually not performed in a realistic time-space domain, more information at much shorter computation time can be obtained from spectral considerations. Having space-time arrays of smoothed Cartesian components of flux density $B_{\text{x}}^{\ast }({\alpha}_{n},t_{m}),B_{\text{y}}^{\ast }({\alpha}_{n},t_{m})$ along the airgap it is possible to convert them into cylindrical coordinates $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}B_{r}^{\ast }\\ B_{{\alpha}}^{\ast }\end{array}\right\}=\left[\begin{array}{@{}cc@{}}\cos {\alpha} & \sin {\alpha}\\ -\sin {\alpha} & \cos {\alpha}\end{array}\right]\left\{\begin{array}{@{}c@{}}B_{x}^{\ast }\\ B_{y}^{\ast }\end{array}\right\} & & \displaystyle\end{eqnarray}$ (9) and calculate by means of ((8)) the array σ_rr(𝛼_n, t_m) of the radial stress component. When the magnetic core is not oversaturated the stress distribution may be interpreted as an approximation of the normal surface force F_s present on the inner stator surface (and with negative sign on the rotor as well) which can be given in its modal form as $\begin{eqnarray}\displaystyle {\sigma}_{rr}({\alpha}_{n},t_{m})\cong F_{S}({\alpha}_{n},t_{m})=\displaystyle \mathop{\sum }_{{\mu}=(1\ldots M)\atop {\nu}=(1\ldots N)}F_{S{\mu}{\nu}}\exp [j({\omega}_{{\mu}}t_{m}-{\nu}{\alpha}_{n}+{\varphi}_{{\mu}{\nu}})] & & \displaystyle\end{eqnarray}$ (10) where ω_μ is angular frequency, 𝜈 is the spatial mode number and 𝜙_μ𝜈 is modal phase angle. The each component in ((10)) represents the force wave rotating in the air gap. Applying the 2DFT for the F_S(𝛼_n, t_m) we get the mode-frequency spectrum of the surface force magnitude F_Sμ𝜈. Due to its symmetry it can be presented in a single-sided and centered form where only the positive frequency components are kept such μ ≥ 0 and 𝜈 = [−(N∕2 −1)… 0… N∕2]. The sign of mode 𝜈 informs about direction of rotation – positive in the direction of rotor movement. In accordance with this convention the magnitudes of all components of the resulting array F_Sμ𝜈 for μ > 0 are multiplied by two.

Fig. 7.

Time functions of Cartesian components of magnetic pull in induction motor at no-load, (a) stator versus rotor axes misalignment of 60 μm, (b) rotor axis 60 μm oscillation with frequency 25 Hz along y direction.

From the point of view of affecting the amount of resultant vibration, only the forces of the lowest orders matter. That statement concerns both radial and tangential components of forces applied to stator teeth tips. It should be remembered that magnetic attraction forces on boundary iron/air by definition have the normal component only. The resultant displacement profile of the outer yoke surface is created mostly by the bending deformations of the yoke. This effect in a hollow cylinder arises when the balanced set of moments of forces is applied to its structure. It comes from alternating (in space) radial or tangential magnetic stress integrated over surface of their existence and transmitted through the teeth. The radial stress forms a continuous distribution along the air gap but the tangential stress has a shape of narrow peaks appearing under slot openings only. Besides, the tangential forces are substantial at rated load. The analysis presented here concerns the no-load case what allows to neglect the tangent forces and reduces the force spectrum to orders −12 ≤ 𝜈 ≤ 12. Quantitative assessment of this assumption is given in the next chapter where values of modal compliances of low-order modes are presented.

Fig. 8.

Frequency spectra of spatial rms values of magnetic forces in induction motor at no-load, (a) centric rotor (b) stator versus rotor axes misalignment of 60 μm, (c) rotor axis 60 μm oscillation with frequency 25 Hz along y direction.

Fig. 9.

Modal spectra of magnetic forces having frequency 900 Hz in induction motor at no-load, (a) centric rotor (b) stator versus rotor axes misalignment of 60 μm, (c) rotor axis 60 μm oscillation with frequency 25 Hz along y direction.

For illustration purposes the frequency spectrum of spatial rms values F_Sμ was calculated following the Parseval theorem as $\begin{eqnarray}\displaystyle F_{S{\mu}}=\sqrt{{\displaystyle \frac{1}{2}}\mathop{\sum }_{{\nu}}(F_{S{\mu}{\nu}})^{2}}. & & \displaystyle\end{eqnarray}$ (11) The resulting frequency spectra for F_Sμ in considered cases are displayed in Fig. 8. They consist of three main parts. At low end there are components invoked by the main, non-dispersive wave of flux density. Then occur groups related to first and second order of stator and rotor slotting perturbations. All spectra contain also the visible power leakage around dominated components caused by the lack of full periodicity in analyzed time signals. The value of rotational component at 25 Hz is the main difference between force spectra for the centric rotor case and two variants of eccentricity. The remaining parts of these diagrams look similar. Nevertheless, there are also some more subtle differences related to the content of modal spectrum. The example is presented in Fig. 9 where motor with axial misalignment has force modes of first order at frequency significantly higher than rotational one. Amplitudes of modes 𝜈 = ±1 for this motor are not equal each other what means that magnetic pull will have both oscillating and rotating part. Note that their magnitudes are about 5 ×10⁻⁴ times smaller than fundamental wave at 100 Hz. The importance of that effect will be detailed in the next chapter.

5. Modal analysis of motor vibration

The cross-section of the mechanical model of analyzed motor and boundary constraints applied in calculations are shown in Fig. 10. Steel parts of the motor (housing, stator core, shaft) are isotropic. The windings and bearing area have a distinct transverse isotropy only. The isotropic rotor material has stiffness of the steel but its density is fictious resulting from the sum of the cage and core mass inserted into real volume. The conversion of the force excitation from magnetic surface forces F_Sμ𝜈 having the mode 𝜈 and frequency number μ into structural discrete domain requires the definition of nodal force in structural mesh. The amplitude of nodal force $\hat{F}_{{\mu}{\nu}}$ for given space-time mode is approximated by $\begin{eqnarray}\displaystyle \hat{F}_{{\mu}{\nu}}=F_{S{\mu}{\nu}}{\displaystyle \frac{{\rm\pi}DL}{N_{S}n_{\mathit{eff}}}}. & & \displaystyle\end{eqnarray}$ (12)D, L are gap diameter and its axial length, N_S is slot number and n_eff represents effective number of nodes on the surface of stator tooth neglecting nonuniformity of nodal force resulting from numerical integration of constant force density over a tooth surface. The value of this number is n_eff = 125. The components of surface forces are waves, so the nodal forces must be defined using complex numbers. The exemplary distributions of nodal forces applied to the stator are presented in Fig. 11.

Fig. 10.

Section of structural model of induction motor.

Fig. 11.

Nodes of structural mesh on inner stator surface and instantaneous distributions of forces having mode 𝜈 = 1 and 𝜈 = 4.

Some comments are necessary on forces applied to rotor. When the mode number 𝜈 > 1 the overall sum of nodal forces applied to both the stator and rotor amounts to zero. The stator yoke having the form of a relatively thin hollow cylinder is deformed by bending moments but the rotor has no holes inside and it is compressed/elongated only. Therefore, in view of difference between bending and compression stiffness one may assume that rotor deformation is negligible. Completely different conditions exist when 𝜈 = 1. The magnetic pull applied to the rotor is balanced by reaction forces occurring in the support area situated on the stator and transmitted to the rotor via bearings. Now the elastic shaft may be bended what means that rotor core and cage move following the shaft deformation in a form of a rigid body. It allows to represent the rotor pull by the single radial nodal force counteracting the distributed stator pull and having the same rotation. The value of this force can be obtained from ((12)) setting N_Sn_eff = 1. The dynamic behavior of the motor depends on values of natural frequencies and is tightly connected with the shape of natural modes. Figure 12 presents chosen natural modes having low frequencies and simultaneously exhibiting vibration in the plane perpendicular to the rotation axis. The modes containing first order bending deformation of the rotor belong to the frequency range 800–1000 Hz, the lowest stator bending mode 𝜈 = 2 has the frequency f = 2555 Hz. The natural mode of order 𝜈 = 4 which dominates in the force spectrum was not detected below 6000 Hz.

Fig. 12.

Chosen natural modes of induction motor, shaded map shows modulus of displacement.

All material properties in the model are constant, therefore, it is possible to calculate separately for the each mode the vibration response of the motor structure assuming the unit value of the nodal force magnitude. Such a treatment is often met [16] and gives the value of modal compliance C_μ𝜈 relating the magnitude of the outer displacement of the motor surface to the magnitude of nodal force. Values of the compliance defined in this way for the few lowest mode numbers and frequencies of interest are given in Table 2.

Table 2

Modal compliances

Mode number 𝜈	Compliance C_μ𝜈 [μm/N]
	100 Hz	900 Hz
0	0.21	0.21
1	24.6	71.0
2	2.0	6.5
4	0.27	0.31
8	0.023	0.024

Dynamic modal displacement C_μ𝜈 patterns are now easy to get. Having the mode-frequency spectra of surface forces F_Sμ𝜈 and motor compliance C_μ𝜈 it is possible to calculate values of C_μ𝜈 by simple extension of ((12)) $\begin{eqnarray} \displaystyle u_{{\mu}{\nu}}=\hat{F}_{{\mu}{\nu}}C_{{\mu}{\nu}}=F_{S{\mu}{\nu}}{\displaystyle \frac{{\pi}DL}{N_{S}n_{\textit{eff}}}}C_{{\mu}{\nu}}. & & \displaystyle \end{eqnarray}$ (13) The mode numbers 𝜈 of the force field existing in the spectrum are 0, 1, 4, 8. The components of the first order F_Sμ1 in the force field matter most following the values of compliances given in Table 2. Therefore, the spectral position of the resonance for these components indicated by the compliance C_μ1 spectrum shows where the dominated vibration may appear. It is visualized in Fig. 13. It is clear that only forces around 900 Hz may be the origin of main components in vibration spectrum with a single exception of fundamental force component having the doubled frequency of the supply and the mode order 𝜈 = 4 which amplitude is about 10⁵ Pa – see Fig. 8. The exemplary shapes of dynamic displacement fields are plotted in Fig. 14.

Fig. 13.

Frequency spectra of the first mode of surface forces $F_{S\mu1}$ and compliance C_μ1 for motor with misalignment of stator versus rotor axes.

Fig. 14.

Chosen instantaneous shapes of forced vibration of induction motor, shaded map shows modulus of displacement.

6. Experimental verification

Vibration of investigated motor was measured by the set of piezoelectric accelerometers mounted on the stator circumference as it is shown in Fig. 15. The simultaneous acquisition of signals from all gauges enabled separation of instantaneous shapes of vibration for particular frequencies. The shapes of two components having frequency 100 Hz and 899 Hz are also displayed in this figure. The first one with the doubled network frequency forms the rotating wave of the fourth order. Its irregular picture (different shapes of positive and negative half-waves) results from the number of accelerometers because twelve gauges cannot accurately map the shape of wave having four periods in space. The second one presents the almost pure oscillation of the stator core along some axis. Its frequency f^′ exactly matches the expression $\begin{eqnarray}\displaystyle f^{\prime }=N_{S}f_{supply}(1-s_{0}) & & \displaystyle\end{eqnarray}$ (14) where s₀ is the no-load slip. It corresponds with the frequency of cage currents presented in Fig. 3.

Fig. 15.

View of test stand for vibration measurements of induction motor with instantaneous shapes of radial vibration field.

Fig. 16.

Measured and calculated acceleration spectra (rms) of outer surface of induction motor at no-load.

The emitted acoustic pressure is proportional to the acceleration patterns on outer surface. The measured spectrum of acceleration is shown in Fig. 16 where few values coming from calculations at important frequencies were also added. Commenting differences between theoretical and experimental values we should keep in mind that spectral components of flux density in the air gap responsible for forces around 900 Hz are of order of 10 mT. Besides, the change of modal compliance in 1 or 2 percent may cause the changes of vibration magnitudes in this area in amount of dozens percent because of the closeness between excitation and resonance frequencies. The small frequency shift of Δf between calculated and measured spectral components of forces resulting from stator slotting may be caused by the incomplete decay of transient effects in rotor currents. Also the value of eccentricity in this analysis equal to 24% of the air gap size was arbitrary.

7. Final conclusions

Results of investigations presented in this paper proved that relatively small rotor axis misalignment against stator leads to visible vibration if forces resulted from this effect have frequency close to rotor bending resonance. Probably it is not possible to avoid the eccentricity in a mass production of electric motors. The given construction of the motor has so-called first and second critical rotor frequencies at known points of the spectrum. Therefore, the choice of stator and rotor slot numbers should also to take them into account. This is particularly important for motors designed for drives with a wide range of a variable speed.

Footnotes

Acknowledgements

The research was granted in part by The European Regional Development Fund no. POIR.04.01.04-00-0002/16-00.

References

Jordan

, Geräuscharme Elektromotoren, Verlag W. Girardet, Essen, Germany, 1950.

Heller

and Hamata

, Harmonic Field Effects in Induction Machines, Academia, Prague, 1977.

Yang

S.J.

, Low Noise Electrical Motors, Clarendon Press, Oxford, 1981.

Timar

P.L.

, Noise and Vibration on Electrical Machines, Elsevier, New York, 1998.

Verma

S.P.

Williams

and Singal

R.K.

, Vibration of long and short laminated stators of electrical machines, theory, experimental models, procedure and setup, Journal of Sound and Vibration129 (1989), 1–13.

Vandevelde

Gyselinck

J.J.C.

Bokose

and Melkebeek

J.A.A.

, Vibrations of magnetic origin of switched reluctance motors, COMPEL22(4) (2003), 1009–1020.

Wang

and Lai

J.C.S.

, Vibration analysis of an induction motor, Journal of Sound and Vibration224(4) (1999), 733–756.

Mori

and Ishikawa

, Force and vibration analysis of induction motors, IEEE Transactions on Magnetics41(5) (2005), 1948–1951.

Schlensok

Ch.

van der Giet

Herranz Gracia

van Riesen

and Hameyer

, Structure-dynamic analysis of an induction machine depending on stator–housing coupling, IEEE Transactions on Industry Applications44(3) (2008), 753–759.

10.

Ch.

and Xiao

, Unbalanced vibration control strategy of bearingless induction motor based on inverse system decoupling, International Journal of Applied Electromagnetics and Mechanics51(4) (2016), 455–469.

11.

Bauw

Cassoret

Romary

and Ninet

, Damper winding for noise and vibration reduction of induction machine under sinusoidal conditions, International Journal of Applied Electromagnetics and Mechanics57(S1) (2018), 13–21.

12.

Witczak

Kubiak

and Swiniarski

, Vibration of permanent magnet motor with concentrated windings caused by rotor eccentricity, International Journal of Applied Electromagnetics and Mechanics57(2) (2018), 193–203.

13.

Witczak

, Vibration analysis of induction motors caused by magnetic forces, Lodz University of Technology, Scientific Notebooks No 725, 1995, Lodz, p. 348 (in Polish).

14.

Wang

Xia

Long

and Howe

, Radial force density and vibration characteristics of modular permanent magnet brushless AC machine, in: IET Proc. on Electric Power Applications, Vol. 153, 2006, pp. 793–801.6.

15.

Sathyan

Aydin

Lehikoinen

Belahcen

Vaimann

and Kataja

, Influence of magnetic forces and magnetostriction on the vibration behavior of an induction motor, International Journal of Applied Electromagnetics and Mechanics59(3) (2019), 825–834.

16.

Boesing

and De Doncker

R.W.

, Exploring a vibration synthesis process for the acoustic characterization of electric drives, IEEE Transactions on Industry Applications48(1) (2011), 70–78.