Impact of rotor eccentricity on electromagnetic vibration and noise of permanent magnet synchronous motor

2.1. Analytical derivation

In the healthy motor, the radial force on the inner surface of the stator can be expressed in the form of Fourier series and is given by:

Here, $\theta$ is the mechanical angle, $t$ is the time, $r$ is the space order of force. $A_{r,i}$, $f_{r,i}$ and ${\phi }_{r,i}$ are the amplitude, frequency and phase angle of the force harmonic, respectively. $s_{r,i}$ denotes the rotating direction of the harmonic, being 1 for forward rotating and –1 for backward rotating.

Eccentricity correction factor (ECF) is employed here to derive the force harmonics under the eccentricity. ECF of SE and DE can respectively be expressed as [7]:

Here, $f_r$ is the rotational frequency of the rotor, ${\epsilon }_s=e_s/g^{\text{'}}$, ${\epsilon }_d=e_d/g^{\text{'}}$. $g^{\text{'}}$ is the effective air-gap length, which is equal to $g+h_m/{\mu }_r$ and $g$ in surface-mounted PMSM (SPMSM) and interior PMSM (IPMSM), respectively. $h_m$ is the thickness of PM, ${\mu }_r$ is the relative permeability of PM, and $g$ is the air-gap length. Due to the fact that SPMSM has a larger effective air-gap length, more distortion of the field occurs in IPMSM than SPMSM under the same eccentricity length. In other words, IPMSM is more sensitive to the eccentricity than SPMSM.

According to Eq. (1) and (2), the radial force considering SE is obtained and is given by:

Similarly, based on Eq. (1) and (3), the radial force considering DE is obtained and is expressed as:

It is found in Eq. (4) and (5) that both SE and DE have three effects on electromagnetic force. Due to the fact that the amplitudes of ${\epsilon }_s$ and ${\epsilon }_d$ are small, especially in SPMSM, the second effect is the most important because of its relatively large amplitude. With the help of Eq. (4) and (5), both of the space order and frequency of the additional force harmonics from eccentricity are clear. However, when the amplitude of the additional force harmonics is concerned, these two classical equations would not be precise enough, which is discussed in the next part.

2.2. FE analysis

A PMSM with 6 poles and 9 slots is used in this paper to investigate the effect of the eccentricity. The main parameters of the motor are listed in Table 1. Fig. 2 shows the 2-D electromagnetic model. The motor is studied under the rated working condition. The rotational speed is 3600 rpm and the load torque is 2.7 N∙m.

Fig. 22-D electromagnetic FE model

FE method is employed to calculate the radial force under the healthy case, 0.2 mm SE and 0.2 mm DE, respectively. The results are shown in Fig. 3, where $\alpha$ means the rotational angle. It is shown that SE only changes the spatial distribution of force while DE changes both of the spatial and temporal distribution. Since the electromagnetic force varies in both space and time, the commonly used 1-D FFT is not able to acquire the space order and frequency at the same time. However, these two aspects of the force harmonic determine the spatial mode and the frequency of the vibration and noise it causes. Moreover, many researches have concluded that lower space harmonics produce higher vibration and noise [2, 5, 6]. Hence, not only the frequency but also the space order of the force harmonic should be obtained simultaneously to analyze the vibration and noise caused by this harmonic. To achieve this, 2-D FFT instead of 1-D FFT should be adopted.

Fig. 3 gives the results of 2-D FFT under different types of eccentricity. The sign of the frequency denotes the rotational direction of the force harmonic. It is noted that the 0th space force harmonics have no rotational direction. Thus, the sign of the 0th space harmonics shown in Fig. 3 makes no sense. During the 2-D FFT, a certain 0th space harmonic is divided into two parts, which are equal in the amplitude but opposite in the frequency.

In healthy PMSM with the double layer winding, the space order of the force is $k\times GCD\left(2p,Q_s\right)$ and the force frequency order is $k\times 2p$[16]. $GCD$ is the greatest common divisor and $k$ denotes integer in this paper. Thus, for the motor studied in this paper, which has 6 poles and 9 slots, the space order and frequency order of force are 3$k$ and 6$k$ under the healthy case, respectively. It is found in Fig. 3(b) that force harmonics with space order $\left(3k\pm 1\right)$ are induced under SE but the frequency orders of the harmonics are still 6$k$. However, additional force frequency orders are yielded when DE occurs. So, it is possible to judge the existence of DE from the vibration or noise signal according to such unique feature of frequency.

Fig. 3Calculated radial force (left) and corresponding harmonics (right): a) healthy, b) 0.2 mm SE, c) 0.2 mm DE

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For the sake of clarity, the force harmonic with the space order $r$ and the frequency order $s_{r,i}{f}_O$ is abbreviated as $P_r\left(r,s_{r,i}{f}_O\right)$. When only the second effect of the eccentricity is considered, the force harmonics induced by SE and DE are $P_r\left(r\pm 1,s_{r,i}{f}_O\right)$ and $P_r\left(r\pm 1,s_{r,i}{f}_O\pm 1\right)$, respectively. It is interesting to compare the amplitude of the force harmonic having space order $\left(r+1\right)$ with that having space order $\left(r-1\right)$. When the eccentricity occurs, force harmonics with the 2nd and 4th space order are derived from a certain original 3rd space harmonic. The derived force harmonics from the original 3rd space harmonics are listed in Table 2. It is found that the comparison between the amplitude of the 2nd space harmonic and that of the 4th space harmonic depends on the rotational direction of the original 3rd space harmonic. When the 3rd space harmonic rotates forward, the derived 2nd space harmonic is smaller than the 4th space harmonic. Inversely, when the 3rd space harmonic rotates backward, the derived 2nd space harmonic is larger than the 4th space harmonic. However, according to Eq. (4) and (5), the force harmonic having space order $\left(r+1\right)$ is equal to that having space order $\left(r-1\right)$, which is contradictory with the above conclusion drawn from Table 2. This is because the commonly used ECF only corrects the amplitude of the flux density according to the change of air-gap length but ignores the conversion of coordinates to consider the misalignment of rotor axis and stator axis [17]. Hence, ECF can be used to discuss the space and frequency order of the force harmonics induced by eccentricity, but it is not able to analyze the amplitude of these harmonics quantificationally.

The effect of the eccentricity on the 0th space harmonics is a special case. When eccentricity occurs, two 1st space harmonics with different rotational directions are derived from a certain 0th space harmonic except $P_r\left(\text{0th, 0th}\right)$, as shown in Table 3. The amplitudes of these two harmonics are different and also depend on the rotational direction. The harmonic rotating forward has a larger amplitude than that rotating backward.

3.1. Modeling of electromagnetic vibration and noise

A multiphysics model is proposed in this paper to predict the vibration and noise under eccentricity. The structural model and acoustic model are shown in Fig. 4. In the structural model, which has been validated by modal test [14]. Some stator modes considering the constraints of the test bed are shown in Fig. 5. To observe modal shapes more clearly, the winding is hidden. The acoustic model has been used to calculate the noise under the healthy case, and the calculated result is validated by the noise test in [14].

Fig. 4Vibration and noise model: a) structural model, b) acoustic model

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Fig. 5Stator modes: a) 1267 Hz, b) 2460 Hz, c) 3259 Hz, d) 5056 Hz

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3.2. Vibration and noise under different eccentricity types

Firstly, the 2-D electromagnetic model shown in Fig. 2 is used to calculate the force under the eccentricity. The force can be extended to the 3-D space under the assumption that the force distributes uniformly along the axial direction. Then, the force calculated by the electromagnetic model is applied to the structural model. Finally, the structural vibration and the acoustic noise are calculated by using mode superposition method and acoustic BEM, respectively. The proposed multiphysics model considers the spatial characteristics of the force on teeth surface with the help of the nodal force transfer method in [14] and [16]. So, the spatial distribution changes of the electromagnetic force due to the eccentricity are accurately taken into account. The vibration and noise prediction is performed in LMS Virtual.lab.

Fig. 6 gives the vibration and noise under the healthy case, 0.2 mm SE and 0.2 mm DE. It is shown that great changes can be found when the eccentricity occurs.

Fig. 6Vibration and noise comparison under different eccentricity types: a) vibration, b) noise

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Under the healthy case, the highest noise peak appears at 2520 Hz. Fig. 7(a) shows the spatial mode of vibration/noise at this frequency. Three spatial periods can be found, which indicates that the vibration/noise at this frequency is induced by the 3rd space harmonic with the same frequency, i.e. $P_r\left(\text{3rd,}\text{–42nd}\right)$. Although this harmonic has a small amplitude, it is still able to yield a considerable vibration/noise peak because it is close to the 3rd circumferential mode at 2460 Hz.

When SE occurs, the dominant vibration and noise peak appears at 1440 Hz. The spatial mode of the vibration and noise at 1440 Hz is shown in Fig. 7(b). Two spatial periods can be observed, which explains that the vibration/noise peak at 1440 Hz is induced by $P_r\left(\text{2nd, –24th}\right)$ rather than $P_r\left(\text{3rd, –24th}\right)$. Since the force harmonic $P_r\left(\text{2nd, –24th}\right)$ induced by SE is close to the 2nd circumferential mode at 1267 Hz, a significant vibration/noise peak is found at 1440 Hz due to the occurrence of resonance. When DE occurs, the dominant vibration and noise peak transfers to 1500 Hz. Two spatial periods are found in the spatial mode of the vibration/noise at 1500 Hz, as shown in Fig. 7(c). Similarly, this illustrates that the vibration and noise at 1500 Hz are attributed to $P_r\left(\text{2nd, –25th}\right)$.

From the above analysis, it is concluded that the dominant vibration and noise peak transfers to the frequency band around the lower modal frequency when the eccentricity occurs, because force harmonics with lower space orders are induced. In the motor studied in this paper, additional 2nd space harmonics are induced by eccentricity, the vibration and noise peaks around the 2nd modal frequency stand out.

Fig. 7Spatial mode of the vibration and noise at the peak frequency (on the left is vibration and on the right is noise): a) healthy motor at 2520 Hz, b) 0.2 mm SE at 1440 Hz, c) 0.2 mm DE at 1500 Hz

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When DE occurs, additional vibration/noise harmonics with frequency orders $\left(6k\pm 1\right)$ are induced. Some significant harmonics are listed in Table 4. By comparing the $\left(6k+1\right)\text{th}$ harmonics with the $\left(6k-1\right)\text{th}$ harmonics, it is concluded that the former has a larger amplitude than the latter under the same value of $k$. This can be well explained by the amplitude and the space order of the responsible force harmonics, which are also listed in Table 4.

From Table 2 it is found that $P_r\left(\text{1st, –17th}\right)$ is smaller than $P_r\left(\text{1st, 19th}\right)$, which results in a smaller 17thorder vibration/noise than 19thorder vibration/noise. Because $P_r\left(\text{2nd, –25th}\right)$ has a lower space order and larger amplitude than $P_r\left(\text{4th, –23rd}\right)$, the 25th order vibration/noise is much higher than the 23rd order vibration/noise. In the same way, the difference between the 41st and 43rd order vibration/noise can also be explained. Although $P_r\left(\text{2nd, 29th}\right)$ has a lower space order than $P_r\left(\text{4th, 31st}\right)$, the amplitude of the former force harmonic is quite lager than the latter one. Finally, $P_r\left(\text{2nd, 29th}\right)$ induces a larger vibration/noise than $P_r\left(\text{4th, 31st}\right)$.

3.3. Vibration and noise under different eccentricity lengths

The vibration/noise and the overall SPL under different eccentricity levels are shown in Fig. 8. Since $P_r\left(\text{2nd, –24th}\right)$ from SE and $P_r\left(\text{2nd, –25th}\right)$ from DE increase almost linearly with the eccentricity length, as shown in Fig. 9, the vibration peak at 1440 Hz under SE and that at 1500 Hz under DE also increase linearly. However, the noise peak at 1440 Hz under SE and that at 1500 Hz under DE are more sensitive to the variation of eccentricity length when the eccentricity level is low. It is noted that although the amplitude of $P_r\left(\text{2nd, –24th}\right)$ from SE is almost equal to that of $P_r\left(\text{2nd, –25th}\right)$ from DE, the former harmonic yields a larger vibration and noise, because it is closer to the 2nd modal frequency.

Fig. 8The relation variation between vibration/noise and eccentricity length: a) vibration peak, b) noise

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Fig. 9The relation variation between force amplitude and eccentricity length

The noise level increases greatly when eccentricity occurs. Fig. 8(b) shows the overall SPL under 0.4 mm SE and 0.4 mm DE increase by 14 dB(A) and 11 dB(A), respectively. This enormous effect is attributed to the lower space force harmonics induced by the eccentricity. Hence, any type of eccentricity should be avoided as far as possible to mitigate the motor noise.