Vibration of permanent magnet motor with concentrated windings caused by rotor eccentricity

Abstract

The paper presents the combined results of numerical analyses of magnetic vibration introduced by the rotor eccentricity in permanent magnet motor with concentrated windings. The analysis is carried for two possible rotor displacements – uniform shift of its axis against stator and twist of that axis about the gravity point. The investigations were obtained by means of finite elements approach, both in electromagnetic and mechanical domain. The influence of these on magnitude and form of forced vibration is also considered.

Keywords

Permanent magnet motor vibration eccentricity finite elements spectral analysis

1. Introduction

Permanent magnet (PM) motors equipped with concentrated windings are designed for low speed drives having number of pole pairs p usually greater than four. Large number of poles causes that thickness of stator yoke is relatively small and simultaneously, the radius of the air gap is substantial in order to obtain the possibly high torque [1, 2, 14, 16]. These geometric properties lead to rather low mechanical stiffness of the stator and increased rotor mass comparing them with typical machines of similar size. Because the number of stator teeth is close to number of rotor poles, the time-space distribution of magnetic forces is also different [3, 4]. The rotor eccentricity against stator is statistically always present in mass production of electric machines. It generates the additional forces of magnetic origin, which may produce the excessive vibration and noise.

The list of works devoted to vibration of electric machines, including PM ones, is quite long. The earliest [1, 2] gave analytical solutions for simple cases, good for understanding physical rules of their behavior, but mostly not sufficient to get realistic values of forced vibration magnitudes. Nowadays, almost all papers dealing with vibration of electric machines use numerical solutions coming from finite elements technology, both in magnetic and structural domain. Some representative contributions may be found in [3, 4, 5, 12, 15] and also in book [11]. Nevertheless, the question of vibration caused by rotor eccentricity is not fully explained, especially for machines with concentrated windings.

The numerical models presented below were obtained using Magnet 7 system (magnetic fields and forces) and ANSYS 14 software for vibration analysis.

2. Analysis of magnetic field and stress

The paper considers the small power, feet mounted, three phase PM motor (shaft height of 90 mm) having 16 poles and 18 stator slots with double layer winding. The rated torque at velocity 375 rpm amounts to 12 Nm. The motor cross-section is presented in Fig. 1. The neodymium magnets were applied having the magnetic remanence of 1.21 T. All calculations were made using time-stepping of 2D static fields forced by symmetric three phase currents supplied from PWM inverter together with a rotor angular movement with a constant value of an advance angle. The motor has almost sinusoidal back EMF, distortions are below 1%. The finite elements model has 12680 DoF, the computing time of the single step is a few seconds.

Figure 1.

Geometry and performance of PM motor. a. outlook of finite elements model, ferromagnetic parts only; b. calculated phase currents; c. measured phase currents.

Two variants were considered – when rotor is symmetrically placed against stator and when static eccentricity exists having the value of relative (against the air gap size) shift $\varepsilon=$ 0.25. That value is rather high but still possible in manufacturing. The analysis of magnetic forces is based on Maxwell stress tensor $\sigma_{ij}$ , which in 2D solutions for non-magnetic area is given as

$\displaystyle\sigma_{ij}=B_{i}H_{j}-\delta_{ij}w_{m}$ (1)

where $B_{i}$ , $H_{j}$ denote components of flux density and field intensity, respectively, $w_{m}$ is energy density and $\delta_{ij}$ means a Kronecker sign.

Exemplary instantaneous stress distributions for centric case and uniform eccentricity applied in 0y direction (0xy is an arbitrary Cartesian coordinate system in a lamination plane) are shown below.

Figure 2.

Instantaneous space distributions of radial stress $\sigma_{rr}$ in air gap along machine circumference $\alpha_{\textit{geo}}$ , $p=$ 8, concentrated winding. a. centric rotor, no-load; b. uniform eccentricity $\varepsilon=$ 0.25, rated load.

The magnetic force component $F_{i}$ acting on given volume $V$ having the outer surface $S(V)$ holds (index summation applied)

$\displaystyle F_{i}=\oiint_{S(V)}\sigma_{ij}n_{j}dS$ (2)

When the surface $S$ encloses the rotor the Eq. (2) results with one-sided magnetic pull. The time distributions of its Cartesian components are plotted in Fig. 3. The DC component of force, trying to push further the rotor towards the stator, is present in 0y direction only. The alternating components exist however in both directions being simultaneously shifted in phase in $\pi$ /2. Therefore, the resultant one sided force acting on the rotor (and by the momentum conservation on the stator as well) rotates in space. At no-load these AC components are almost sinusoidal, when the load appears the influence of armature reaction becomes visible and some additional harmonics are present.

Figure 3.

Time distributions of one-sided forces acting on rotor in PM motor at relative eccentricity $\varepsilon=$ 0.25.

The particular frequency components can be obtained applying the 2D Fourier Transform to the array of radial stress $\sigma_{rr}(\mu,\nu)$

$\displaystyle S(m,n)=\frac{2}{MN}\mathop{\sum}\limits_{\mu=0}^{M-1}\mathop{% \sum}\limits_{\nu=0}^{N-1}\sigma_{rr}(\mu,\nu)e^{-j2\pi\left({\frac{\mu m}{M}+% \frac{\nu n}{N}}\right)}$ (3)

where $\mu=$ 0 … $M-1$ , $\nu=$ 0 … $N-1$ , denote number of sample in space and time, respectively and $S(m,n)$ is single-sided mode-frequency spectrum, which for modes $m=$ 1, 2 is presented in Fig. 4.

Figure 4.

Frequency spectrum of radial component magnetic stress of 1 ${}^{\text{st}}$ and 2 ${}^{\text{nd}}$ order in space at rated load and uniform eccentricity $\varepsilon=$ 0.25.

In consequence, the frequency spectrum of magnetic pull is

$\displaystyle F\left({1,n}\right)=\pi RLS\left({1,n}\right)$ (4)

where $R$ , $L$ are rotor radius and length. The rotating stress wave in the air gap creates also the pressure $p(m,n)$ , harmonic in time and space, applied in radial direction to the stator teeth tips

$\displaystyle p\left({q,t}\right)=S\left({m,n}\right)\frac{2\pi R}{Q_{s}t_{s}}% e^{j2\pi\left({\frac{nt}{T}-\frac{mq}{Q_{s}}}\right)}$ (5)

Figure 5.

Frequency spectrum of radial component magnetic stress of 1 ${}^{\text{st}}$ order in space at rated load with and without uniform eccentricity.

where $q$ and $Q_{s}$ denote index and overall number of stator teeth, $t_{s}$ is the tip width. The influence of eccentricity on the stress spectrum $S(1,n)$ is shown in Fig. 5. The visible small magnetic pull at centric rotor comes from not fully periodic distribution of phase windings – there are slots with bars of the same phase and with bars of different ones. Nevertheless, the values shown in Fig. 5 should be compared with the mean (in space) value of radial stress, which in given case amounts to about 106 kPa – see Fig. 2b.

All calculations presented so far were done with the usage of 2D model of the machine. The out-of-centric position of the rotor against stator was obtained there by means of a uniform shift between rotor and stator axes but it is not a sole possibility of that effect. We may also consider the twist of the rotor about the axis perpendicular to the axis of rotation, what is schematically shown in Fig. 6. That angle $\alpha_{t}$ is very small, therefore, we can set

$\displaystyle\tan\alpha_{t}\cong\alpha_{t}$ (6)

Figure 6.

One sided force distribution at angular eccentricity between rotor and stator.

Let’s assume that assembly resulting twist angle $\alpha_{t}$ , measured in radians at the ends of rotor, fulfills

$\displaystyle\frac{\alpha_{t}\frac{L}{2}}{\delta}=\varepsilon$ (7)

where $\delta$ is the mean gap size. It means the same shift in radial direction of terminal planes of rotor lamination against stator as it was set in 2D model. Having in mind that cos $\alpha_{t}\approx$ 1 we may predict from Eq. (4) the elementary force $\delta F$ acting in $z$ direction on the rotor part $\delta x$ and applied to the shaft in distance $x$ as

$\displaystyle\delta F\left({x,n}\right)=\pi R\frac{2x}{L}\sigma_{1,n}\delta x$ (8)

Moment of these forces $M(n)$ about $y$ axis is given by

$\displaystyle M\left(n\right)=\mathop{\int}\limits_{-0.5L}^{0.5L}x\delta F% \left({x,n}\right)dx=\frac{\pi RL^{2}}{6}\sigma_{1,n}$ (9)

Inserting geometric dimensions we have $M(0)=$ 6.6 Nm and $M(2)=$ 2.4 Nm. It can be compared with the rated torque which in analyzed motor amounts to 12 Nm. It should be also remembered that stress wave $\sigma_{1,n}(\alpha,t)$ rotates about 0x axis, therefore, the moment $M(n>0)$ does it as well.

3. Magnetic vibration analysis

Computation of the vibration field on the outer surface of the stator was done using the 3D finite elements model. Its outlook is presented in Fig. 7. Material properties of lamination, shaft and frame were assumed to be isotropic but the coils of stator winding have a distinct orthotropy. The Young modulus of the equivalent composite along wires is equal to 60 GPa and in perpendicular direction it amounts to 8 GPa [11]. The numerical model is quite large, it has 13.6 10 ${}^{6}$ DoF, the computing time for modal analysis amounts to 8 hours.

Figure 7.

Overall and partial views of finite elements mesh used in vibration analysis.

Few comments are necessary for the finite elements models of the bearings. It was found [10] that mechanical connection between rotor and stator by means of roller bearings can be modeled using so-called simple support (null displacements and free rotations). When solid elements are used in the finite elements model the nodal rotations do not belong to the set of DoF. Unfortunately, the finite elements technology requires that no part of the model can move as a rigid body. Therefore, the rigid connection between stator and rotor was applied along two circular lines on the shaft surface in bearings planes. This approach neglects the rotation of the shaft’s plane inside bearing making the rotor support a bit stiffer than in reality. Applying the restriction of lack of displacements on one or both ends of the shaft we simultaneously allow the formation of the fictitious free torsional vibration, which must be removed “by hand” from numerical results, what was applied in presented analysis. The application of the huge orthotropy by the small stiffness in angular direction for the bearing material is not a good solution as well, because it makes the resultant matrix of the system ill conditioned.

The equation governing the vibrational behavior of the motor at the steady state written in the matrix form is

$\displaystyle\left({\left[K\right]-\omega^{2}\left[M\right]+j\omega\left[C% \right]}\right)\{\underline{u}\}=\{\underline{F}\}$ (10)

The $K$ , $M$ and $C$ matrices represent stiffness, mass and damping, as usual. The vector of exciting forces ${\{}\underline{F}{\}}$ comes from Eq. (5), if an uniform misalignment between stator and rotor axes exists. When the rotor twist shown in Fig. 6 appears, the additional factor is added covering the variability along the axis of rotation

$\displaystyle p_{Qq}\left({q,t,x}\right)=S\left({m,n}\right)\frac{2\pi R}{Q_{s% }t_{s}}\frac{2x}{L}e^{j2\pi\left({\frac{nt}{T}-\frac{q}{Q_{s}}}\right)}$ (11)

In order to simplify the data exchange between magnetic and vibration calculations the set of points, where the forces ${\{}\underline{F}{\}}$ are applied, was chosen in a direct way. The points $P_{iq}$ are uniformly distributed on the each tooth tip as it is presented in Fig. 8.

Figure 8.

Placement of points $P_{iq}$ where magnetic forces are applied to inner stator surface.

The rotating force wave ${\{}\underline{F}(n,m){\}}$ harmonic in time and space of order ( $n$ , $m$ ) applied in normal direction to teeth tips may be represented as

$\displaystyle\{\underline{F}(n,m)\}=\{\underline{F}(m,P_{iq})\}e^{j\omega_{n}t}$ (12)

where ${\{}\underline{F}(m,P_{iq}){\}}$ is the set of forces given by relation

$\displaystyle\{\underline{F}(m,P_{iq})\}=\{F(P_{i})\}e^{-j2\pi m\frac{q}{Q_{s}}}$ (13)

The value of force applied to selected point $P_{i}$ of a q-th tooth tip results from Eqs (5) or (11), respectively. For eccentricity investigations the unbalanced forces are also applied to the rotor. Because rotor laminations have a high elongation stiffness, the point of application of these forces can be shifted to the rotor axis and their linear density magnitude $f_{1}(n)$ results from Eq. (4) as

$\displaystyle f_{1}\left(n\right)=-S\left({1,n}\right)\pi R$ (14)

The components $f_{1y}$ , $f_{1z}$ are shifted in phase in $\pi$ /2, as explained earlier. When the value of m is greater than one the magnetic radial forces in the rotor are self-compensated and their influence on stator vibration may be neglected. When the rotor twist is analyzed, the Eq. (14) is modified by the factor like in Eq. (11)

$\displaystyle f_{1}\left({n,x}\right)=-S\left({1,n}\right)\pi R\frac{2x}{L}$ (15)

Solution of vibration ${\{}u(\omega){\}}$ is typically represented in the form of modal components series

$\displaystyle\{\underline{u}(\omega)\}=\sum_{k}\frac{\{\psi_{k}\}^{T}\{% \underline{F}\}}{K_{k}}\{\psi_{k}\}\underline{H}_{k}(\omega)e^{j\omega t}$ (16)

where { $\psi_{k}$ } is the modal shape, $K_{k}$ is the modal stiffness and $\underline{H}_{k}$ is the modal frequency response function. The natural mode search in the frequency range up to 2 kHz (doubled frequency of the last meaning component in the force spectrum) showed the presence of seven free modes { $\psi_{k}$ }. The shapes of four of them are presented in Fig. 9. The ability of a given force mode { $\underline{F}(m,P_{iq})$ } to interact with the considered natural mode { $\psi_{k}$ } depends on the value of a dot product representing so-called modal coordinate of the displacement field

$\displaystyle\underline{u}_{k}=\frac{\{\psi_{k}(P_{iq})\}^{T}\{\underline{F}(m% ,P_{iq})\}}{\omega_{k}^{2}}$ (17)

where the complex numbers are related to the space shift only. The modal stiffness $K_{k}$ was replaced in Eq. (17) by the square of natural frequency $\omega_{k}^{2}$ because the natural modes were mass normalized.

Figure 9.

Examples of natural modes of analysed motor (some parts intentionally removed from pictures). a. f $=$ 489 Hz – rigid body rocking of stator and rotor cylinders about axis; b. f $=$ 729 Hz – 2 ${}^{\text{nd}}$ order bending of stator with 1 ${}^{\text{st}}$ order bending of rotor; c. f $=$ 1243 Hz – rigid body twist of stator and rotor cylinders about axis; d. f $=$ 1947 Hz – rigid body rocking of stator with 2 ${}^{\text{nd}}$ order bending of rotor.

The good measure of that relationship it is the correlation factor $\gamma$

$\displaystyle\gamma^{2}=\frac{[\{\psi_{k}(P_{iq})\}^{T}\{\underline{F}(m,P_{iq% })\}]^{2}}{||\psi_{k}(P_{iq})||^{2}||\{\underline{F}(m,P_{iq})\}||^{2}}$ (18)

The diagrams presenting the distribution of correlation for the lowest force modes against natural modes are displayed in Fig. 10.

Figure 10.

Correlation between lowest modes of magnetic forces and free vibrations of stator. a. displacement eccentricity $\varepsilon=$ 25%, b. angular eccentricity $\varepsilon=$ 25%.

When the electric machine is feet mounted it must cause that some of low frequency natural modes have the form of semi-rigid movement or twist of stator and rotor cylinders against the mounting plane. Such a deformation may be excited and amplified by internal forces if they are able to form the resultant force or moment having simultaneously the requested frequency. This is illustrated in Fig. 9a and c, where the modes are of that shape. But the resultant magnetic force and moment of force are calculated integrating the stress on the surface inside the air gap, which is the same for stator and rotor calculations – the sign of outer normal is the only difference. Therefore, the net magnetic force and moment acting on stator and rotor together must equal to zero. It means that natural modes like in Fig. 9a and c cannot be reinforced by magnetic phenomena. They can be dangerous e.g. in the case of the axial misalignment of the motor load. Finally, two natural modes only are able to cooperate with magnetic force waves. Both contain bending deformation of the rotor, the first one with frequency 729 Hz has the deflection of the first order created by the uniform eccentricity and the second one with frequency 1947 Hz has the deflection of second order caused by the angular eccentricity.

The similar coupling between natural modes and force waves of first and second order is the next effect of the presence of forced null displacements in the neighborhood of the mounting plane. It is caused by a specific increase of the stiffness in the place where deformations are forced by boundary conditions. The estimation of forced vibration was obtained in two steps – firstly, the static compliance was calculated for the lowest space modes understood as the ratio of displacement magnitude by the stress magnitude. Secondly, the modal contributions for the given frequency were obtained as it is shown in Eq. (14). The example of such a treatment is presented in Table 1.

Table 1

Calculations of radial acceleration of modal components at frequency 100 Hz, rated load and centric rotor

Mode order m	$-$ 16	$-$ 10	$-$ 4	2	1
Magnetic stress $\sigma_{m2}$ [kPa]	71.2	5.8	4.0	28.4	17.0
Modal compliance $C_{m}$ [nm/kPa]	0.003	0.018	0.8	20.0	110.0
Radial acceleration a [m/s ${}^{2}$ ]	$\sim$ 0	$\sim$ 0	$\sim$ 0	0.22	0.762

We see that fundamental wave of magnetic field with pole pair number $p=$ 8 creates dominating stress wave of order $m=-$ 16, which has a negligible influence on resultant vibration due to very low bending compliance. Visible vibration may appear when the force components are of 1 ${}^{\text{st}}$ or 2 ${}^{\text{nd}}$ order which are introduced by rotor eccentricity together with mechanical asymmetry caused by feet mounting. The vibration level will be amplified if the frequency of forces will be close to the resonant value.

The aforementioned theory was confronted with results of experiment shown in Fig. 11 including the effect of A-weighting commonly used in acoustic considerations.

Figure 11.

Measured spectrum of radial acceleration on the top of stator surface.

The component of frequency ( $n=-$ 16) is insensitive to the rate of eccentricity in opposite to the component having doubled frequency of the supply ( $n=$ 2), which rises more than 4 times. The component of 100 Hz is created mostly by the mode ( $m=$ 1) coming from uniform rotor eccentricity – see Fig. 5. Some amplification may be added by a significant mode ( $m=$ 2) which is produced by the structure of phase windings having 2 belts in analysed motor – see Fig. 2b. That mode is able to excite the deformations of the 1 ${}^{\text{st}}$ order as well, as it was explained above. The 800 Hz component was amplified by the resonance with the natural mode having the frequency of 729 Hz – see Fig. 9b. The bending deflection of the rotor are now caused by vibration of the stator bearing plates because the rotor forces of 2 ${}^{\text{nd}}$ order are self-compensated. This type of vibration will appear regardless of eccentricity size. The agreement between theoretical and measured values looks good – the real level and the shape of eccentricity remains unknown. It should be added that application of the A-weighted logarithmic scale commonly used in vibration and acoustic considerations will suppress the 100 Hz component in 18 dB. This in turn will cause the remaining 800 Hz component to dominate in the spectrum.

All considerations in the paper were done with the assumption of the supply frequency equal to 50 Hz. One may expect a significant increase of vibration when the motor speed will rise to 680 rpm. In such a case the force mode ( $m=$ 2, $n=$ 4) – see Fig. 4, will have the frequency close to the resonant value of 729 Hz.

4. Conclusions

The detailed analysis of the influence of possible rotor eccentricity in PM motor with concentrated windings on its vibration presented in this paper leads to following conclusions:

The close numbers of stator teeth and rotor poles gave the considerable variability of the flux density envelope in the air gap, both at no-load and rated conditions. The applied shift of rotor magnets reduced unwanted average space harmonic content to the low level.

Analysed motor is of a small size, therefore, the number of phase belts was only two, what resulted in significant value of the second order force wave. For bigger motors that wave will be of higher order, what will make it less effective as the vibration source.

The differences between displacement and angular eccentricity were clearly explained. Both lead to rotating force or moment waves, which can be the reason of excessive rotor bending vibration of the first – in the case uniform shift of rotor axis, or second order, when the axes twist is present.

The presence of feet constraints introduces the mechanical asymmetry and it always leads to the increase of vibration.

Footnotes

Acknowledgments

The research presented in this paper was granted by Polish Ministry of Science and Higher Education, project N510 590 440.

References

Cros

and Viarouge

, Synthesis of high performance PM motors with concentrated windings, in IEEE Trans. on Energy Conversion 17(2) (2002), 248–253.

Salminen

Niemela

and Pyrhonen

, Performance analysis of fractional slot wound PM-motors for low speed applications, in: Proc. of IEEE Industry Applications Society Conf., Vol. 2, 2005, pp. 1032–1037.

Magnussen

and Lendenmann

, Parasitic effects in PM machines with concentrated windings, in IEEE Trans. on Industry Applications 43(5) (2007), 1223–1232.

Wang

Xia

and Howe

, Comparison of vibration characteristics of 3-phase permanent magnet machines with concentrated, distributed and modular windings, in: Proc. of Int. Conf. on Electrical Machines, 2006, pp. 131.

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 153(6) (2006), 793–801.

EL-Refaie

Jahns

and Novotny

, Analysis of surface permanent magnet machines with fractional-slot concentrated windings, in IEEE Trans. on Energy Conversion 21(1) (2006).

Araki

Kondou

and Yamagiwa

, Analyses of electromagnetic forces of concentrated winding permanent magnet brushless motors with rotor eccentricity, in: Int. Conf. on Electrical Machines and Systems, 2009, pp. 1–4.

Guemes

J.A.

Iraolagoitia

A.M.

Del Hoyo

J.I.

and Fernandez

, Torque Analysis in Permanent-Magnet Synchronous Motors, A Comparative Study, IEEE Transactions on Energy Conversion 26(1) (March 2011), pp. 55–63.

Mizia

Adamiak

Eastham

A.R.

and Dawson

G.E.

, Finite element force calculation: Comparison of methods for electric machines, IEEE Trans. Magn. 24(1) (Jan. 1988), 447–450.

10.

Swiniarski

Mlotkowski

and Witczak

, Finite elements model of roller bearing for natural frequency analysis of the shaft (in Polish), in: IX Conf. on Finite elements in CAE/CAM, Gizycko, 2005.

11.

Witczak

, Vibration and Acoustics of Electric Machines with Permanent Magnets (in Polish), Wydawnictwa Politechniki Lodzkiej, 2012, ISBN 978-83-7283-464-5.

12.

Witczak

and Swiniarski

, Vibration analysis of permanent magnet motor with concentrated windings, International Journal of Applied Electromagnetics and Mechanics 46(2) (2014), 419–425.

13.

Zhu

Zhao

and Ling

, Comparative investigation of concentrated winding and vernier double-stator permanent-magnet motors, International Journal of Applied Electromagnetics and Mechanics 53(3) (2017), 387–395.

14.

Zhao

and Kwon

, Dual rotor flux switching permanent magnet machines with phase-group concentrated-coil windings for high performance, International Journal of Applied Electromagnetics and Mechanics 52(1–2) (2016), 599–607.

15.

Yamaguchi

Kawase

Sano

Nagai

and Nakamura

, Induced voltage analysis of interior permanent magnet motor taking into account Y- or Δ-connected circuit using 3-D finite element method, International Journal of Applied Electromagnetics and Mechanics 19(1–4) (2004), 69–73.

16.

Koutroulis

C.H.

Kladas

A.G.

and Mamalis

A.G.

, Advanced permanent magnet machine design and construction for wind power generation and traction applications, International Journal of Applied Electromagnetics and Mechanics 13(1–4) (2001/2002), 285–290.