Vibration Enhancement or Weakening Effect Caused by Permanent Magnet Synchronous Motor Radial and Tangential Force Formed by Tooth Harmonics

Abstract

This paper investigates the vibration enhancement or weakening effect caused by permanent magnet synchronous motor radial and tangential force formed by tooth harmonics. First, the analytical expressions of an air gap magnetic field are deduced based on a permanent magnet synchronous motor model. Then, the Maxwell stress tensor method is employed to calculate the radial and tangential force density produced by permanent magnet magnetomotive force harmonics and tooth harmonics. Moreover, the spatial phase difference between the minimum non-zero spatial order radial and tangential force waves under loading operation are also obtained. It is followed by stator vibration deformation induced by radial and tangential force waves, and the vibration enhancement or weakening effect is discussed. This study shows that the tangential force wave has a significant influence on the vibration performance similar to the radial force wave. At specific frequencies, superposition of the radial and tangential force waves can intensify the vibration while weakening each other to reduce the vibration at other specific frequencies. Numerical simulation and a vibration measurement experiment of the prototype motor were carried out to confirm the proposed theory, which can guide motor designers in selecting an appropriate pole and slot combination to apply the weakening effect between radial and tangential force waves and improve PMSM vibration performance.

1. Introduction

With the development of electrification technology, permanent magnet synchronous motors (PMSMs) are used in servo control systems, electric vehicles, industrial automatization, etc. due to benefits, including high efficiency, energy saving, diverse control methods, etc. The vibration performance of PMSMs has also been increasingly considered in recent years [1,2,3]. As the dominating vibration exciting source, research investigating electromagnetic force is significant.

The vibration characteristics caused by the radial electromagnetic force wave have been studied extensively [4,5,6]. In order to fulfil the objectives of low vibration and noise, decreasing the radial electromagnetic force is a common approach [7]. Using an innovative method of controlling the radial force, multiple sector permanent magnet machine vibration is suppressed [8]. In addition, the structural optimal design can weaken the radial force wave [9]. Optimizing the topology by employing electromagnetic structural coupled analysis was carried out to suppress vibration at a specific frequency [10]. An unconventional stator tip arc shape was proposed to weaken the radial electromagnetic force density distribution and vibration displacement [11]. To reduce the stator vibration amplitude under a specified load for dual-redundancy PMSM, the minimum width of the small teeth is adopted to achieve the lowest vibration velocity under a specified load [12].

Besides, the tangential electromagnetic force wave is also essential to the vibraton of PMSMs [13,14,15]. To decrease the electromagnetic vibration produced by magnetic force, multiple-objective optimization of the pole and slot structure parameters is employed by adopting the adaptive weighted particle swarm optimization (AWPSO) algorithm [16]. In [17], the influence of the tangential force on the vibration characteristics of fractional slot concentrated winding PMSMs is discussed, finding that tangential force can also contribute significantly to vibration.

In most previous studies, the vibration characteristics induced by tangential force wave have often been ignored. However, in fact, research has shown that the tangential force wave is an indispensable exciting source [17]. Moreover, relevant literature investigating the effect of the spatial phase of radial and tangential force waves on PMSMs’ vibration does not exist. This spatial phase difference between the radial and tangential force waves can lead to vibration enhancement or a weakening effect. In addition, numerous studies have shown that the minimum non-zero spatial order force wave has a significant influence on vibration behavior and tooth harmonics is one significant factor when the motor is under loading conditions [18]. For these reasons, it is necessary to analyze the vibration enhancement or weakening effect caused by PMSM’s radial and tangential force formed by tooth harmonics.

This paper is organized as follows: firstly, an in-depth study of the air gap magnetic field is outlined, including analytical expressions. It is followed by investigation of the radial and tangential force density produced by permanent magnet magnetomotive force (MMF) harmonics and tooth harmonics. Then, the spatial phase difference between the minimum non-zero spatial order radial and tangential force waves under loading operations are inferred and stator vibration deformation induced by a force wave is simulated. Finally, numerical simulation and a prototype test are adopted to verify the proposed theory. The results show that the tangential force wave also has a significant impact on the vibration performance similar to the radial force wave. With the superposition or weakening of radial and tangential force waves, PMSM vibration is intensified or reduced at specific frequencies. By taking advantage of this vibration weakening effect, the designer can choose the optimal pole and slot combination to ameliorate PMSM vibration performance.

2. Materials and Methods

In order to investigate the air gap magnetic field, analysis of the air gap magnetic flux density is a critical point. In this section, the air gap magnetic field produced by permanent magnet and the armature reaction are studied. After adding the above results, the air gap magnetic flux density can be acquired.

2.1. Air Gap Magnetic Field Produced by Permanent Magnet

A simplified PMSM model schematic is shown in Figure 1. $\theta$ is the model circumferential angle, and the specified permanent magnet center line is defined as the initial position.

For the convenience of problem analysis, the three following hypotheses are proposed: (1) the magnetic permeability of the stator iron core is infinite; (2) the stator iron core is slotless; and (3) the permanent magnet magnetic permeability is the same as the vacuum. Due to the air gap magnetic field being periodic and symmetrical, the radial and tangential air gap magnetic flux density can be written as:

$$B_{pm\_r}(\theta ,t)={\displaystyle \sum _{n=1,3,5\dots }{B}_{pm\_rn}\cos n(p\theta -\omega t)},$$

(1)

$$B_{pm\_\theta }(\theta ,t)={\displaystyle \sum _{n=1,3,5\dots }{B}_{pm\_\theta n}\sin n(p\theta -\omega t)},$$

(2)

where n represents the harmonic order, p is the number of pole pairs, and ω is the electrical angular frequency. For the same parameter n, $B_{pm\_rn}$ and $B_{pm\_\theta n}$ have the same sign.

2.2. Air Gap Magnetic Field Produced by Armature Reaction

In this hypothetical model, the stator iron core is slotless. Hence, the radial air gap magnetic field produced by three-phase windings can be written as [19]:

$$\begin{gathered} B_{\mathrm{arm}\_r}(\theta ,t)={\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }{B}_{\mathrm{arm}\_r\mu \upsilon }}\sin (\upsilon \theta -\mu \omega t-{\theta }_{\mu }+\pi )} \\ ={\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }{I}_{\mu }\frac{3{\mu }_{0}W}{\pi \delta \upsilon }{K}_{dp\upsilon }{F}_{\upsilon }\sin (\upsilon \theta -\mu \omega t-{\theta }_{\mu }+\pi )}}, \end{gathered}$$

(3)

where I_μ is the harmonic current, μ is the temporal order, μ₀ is the magnetic permeability in vacuum, W represents the number of series turns per phase, δ is the effective airgap length, υ is the spatial order, K_dpυ is the winding factor, and F_υ is a function of the radius and spatial order. Similarly, the tangential air gap magnetic field produced by three-phase windings can be derived as [20]:

$$B_{\mathrm{arm}\_\theta }(\theta ,t)={\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }{B}_{\mathrm{arm}\_\theta \mu \upsilon }}\sin (\upsilon \theta -\mu \omega t-{\theta }_{\mu }+\pi \pm \frac{\pi }{2})},$$

(4)

when υ > 0, “$\pm$” takes a plus sign, υ < 0, “$\pm$” takes a minus sign. For the same parameter μ and υ, $B_{\mathrm{arm}\_r\mu \upsilon }$ and $B_{\mathrm{arm}\_\theta \mu \upsilon }$ have the same sign.

2.3. Spatial Phase Difference between the Minimum Non-Zero Spatial Order Radial and Tangential Force Density under Loading Operation

Maxwell stress theory illustrates the relationship between the air gap magnetic flux density and electromagnetic force density, which can be deduced as:

$$f_r=\frac{1}{2{\mu }_0}[B_r{(\theta ,t)}^2-B_{\theta }{(\theta ,t)}^2],$$

(5)

$$f_{\tau }=\frac{1}{{\mu }_0}{B}_r(\theta ,t)B_{\theta }(\theta ,t).$$

(6)

When the motor is under loading operation, the radial and tangential force density produced by the permanent magnet MMF harmonics and tooth harmonics is the main component of the minimum non-zero spatial order force density. Moreover, the slotless motor model is accurate enough for the force density analysis [20]. Thus, this part of the radial and tangential force density is expressed as:

(1) υ > 0

$$\begin{gathered} f_r={\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_\theta n}{B}_{\mathrm{arm}\_\theta \mu \upsilon }-B_{pm\_rn}{B}_{\mathrm{arm}\_r\mu \upsilon }}{2{\mu }_0}}}}\times \sin [(np+\upsilon )\theta -(n+\mu )\omega t-{\theta }_{\mu }] \\ +{\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_\theta n}{B}_{\mathrm{arm}\_\theta \mu \upsilon }+B_{pm\_rn}{B}_{\mathrm{arm}\_r\mu \upsilon }}{2{\mu }_0}}}}\times \sin [(np-\upsilon )\theta -(n-\mu )\omega t+{\theta }_{\mu }], \end{gathered}$$

(7)

$$\begin{gathered} f_{\tau }={\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_\theta n}{B}_{\mathrm{arm}\_r\mu \upsilon }-B_{pm\_rn}{B}_{\mathrm{arm}\_\theta \mu \upsilon }}{2{\mu }_0}}}}\times \cos [(np+\upsilon )\theta -(n+\mu )\omega t-{\theta }_{\mu }] \\ +{\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_\theta n}{B}_{\mathrm{arm}\_r\mu \upsilon }+B_{pm\_rn}{B}_{\mathrm{arm}\_\theta \mu \upsilon }}{2{\mu }_0}}}}\times \cos [(np-\upsilon )\theta -(n-\mu )\omega t+{\theta }_{\mu }-\pi ]. \end{gathered}$$

(8)

(2) υ < 0

$$\begin{gathered} f_r={\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_rn}{B}_{\mathrm{arm}\_r\mu \upsilon }-B_{pm\_\theta n}{B}_{\mathrm{arm}\_\theta \mu \upsilon }}{2{\mu }_0}}}}\times \sin [(np-\upsilon )\theta -(n-\mu )\omega t+{\theta }_{\mu }] \\ +{\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_rn}{B}_{\mathrm{arm}\_r\mu \upsilon }+B_{pm\_\theta n}{B}_{\mathrm{arm}\_\theta \mu \upsilon }}{2{\mu }_0}}}}\times \sin [(np+\upsilon )\theta -(n+\mu )\omega t-{\theta }_{\mu }+\pi ], \end{gathered}$$

(9)

$$\begin{gathered} f_{\tau }={\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_\theta n}{B}_{\mathrm{arm}\_r\mu \upsilon }-B_{pm\_rn}{B}_{\mathrm{arm}\_\theta \mu \upsilon }}{2{\mu }_0}}}}\times \cos [(np-\upsilon )\theta -(n-\mu )\omega t+{\theta }_{\mu }+\pi ] \\ +{\displaystyle \sum _n{\displaystyle \sum _{\mu }{\displaystyle \sum _{\upsilon }\frac{B_{pm\_\theta n}{B}_{\mathrm{arm}\_r\mu \upsilon }+B_{pm\_rn}{B}_{\mathrm{arm}\_\theta \mu \upsilon }}{2{\mu }_0}}}}\times \cos [(np+\upsilon )\theta -(n+\mu )\omega t-{\theta }_{\mu }]. \end{gathered}$$

(10)

It is important to note that the minimum non-zero spatial order force density is the focus of this study. The minimum non-zero spatial order is the greatest common divisor (GCD) of the number of poles and slots, i.e., GCD(2p, z), where z represents the number of slots. When the motor is using fractional slot winding:

$$\mathrm{GCD}\left(2p,z\right)=\frac{2p}{d},$$

(11)

where d is the denominator of slots per pole per phase.

For Formulas (7) and (8), when υ > 0, the spatial harmonic order np + υ cannot be equal to the minimum non-zero spatial order GCD(2p, z). Likewise, when υ < 0, the spatial harmonic order np − υ cannot be equal to the minimum non-zero spatial order GCD(2p, z). Thus, the spatial phase difference between the minimum non-zero spatial order radial and tangential force density under loading operation is as follows: when υ > 0 and np – υ > 0 or υ < 0 and np + υ > 0, the minimum non-zero spatial order tangential force density lags the radial force density 90° spatial phase; when υ > 0 and np − υ < 0 or υ < 0 and np + υ < 0, the minimum non-zero spatial order tangential force density advances the radial force density 90° spatial phase.

2.4. Stator Vibration Deformation Caused by Radial and Tangential Force Waves

A 10-pole 12-slot PMSM was utilized for finite element analysis (FEA). The main structural parameters of the motor are given in

Table 1. Main structural parameters of the motor.

Items	Value	Items	Value
Stator outer diameter	125 mm	Air gap length	1.2 mm
Stator inner diameter	75 mm	PM thickness	3.5 mm
Casing diameter	140 mm	Stack length	106 mm

. The unit radial and tangential force waves were applied on the tooth tip surfaces and the stator vibration deformation in the same initial phase is shown in Figure 2.

As shown in Figure 2, regardless of whether the rotation direction of the unit force wave is clockwise or counterclockwise, the stator vibration deformation induced by the unit tangential force wave lags the 90°spatial phase induced by the unit radial force wave. Based on the discussion about the spatial phase difference between the minimum non-zero spatial order radial and tangential force density under loading operations, when the minimum non-zero spatial order tangential force density lags the radial force density 90° spatial phase, the stator vibration deformation generated by the radial and tangential force waves weakens each wave (see Figure 3a). Moreover, when the minimum non-zero spatial order tangential force density advances the radial force density 90° spatial phase, the stator vibration deformation generated by the radial and tangential force waves enhances each wave (see Figure 3b).

For investigation of the air gap magnetic field, analysis of the air gap magnetic flux density is one critical point. In this section, the air gap magnetic field produced by a permanent magnet and the armature reaction are studied. After adding the above results, the air gap magnetic flux density can be acquired.

The tooth harmonic spatial order υ is kz + p, where k is a non-zero integer. According to the previous conclusion, when the minimum non-zero spatial order tangential force density lags the radial force density 90° spatial phase, $np\pm \upsilon =\mathrm{GCD}\left(2p,z\right)$; thus, the force density temporal order u can be derived as:

$$u=n\pm 1=\frac{2k^{\prime }mN+2}{d},$$

(12)

where $k^{\prime }$ is a positive integer, m is the number of motor phases, and N is the numerator of the slots per pole per phase. When $u=(2k^{\prime }mN+2)/d$, the stator vibration deformation generated by the radial and tangential force waves weakens each wave. Analogously, when the minimum non-zero spatial order tangential force density advances the radial force density 90° spatial phase, $np\pm \upsilon =-\mathrm{GCD}\left(2p,z\right)$; thus, the force density temporal order u can be deduced as:

$$u=n\pm 1=\frac{2k^{\prime }mN-2}{d},$$

(13)

Specifically, the stator vibration deformation generated by the radial and tangential force waves enhances each wave.

3. Results

Based on the above 10-pole 12-slot PMSM, a 14-pole 12-slot PMSM was simulated for contrastive analysis of its vibration characteristics. The main structural parameters and output characteristics of the motor are the same as the previous prototype. In order to compare the spatial phase of the minimum non-zero spatial order radial and tangential force density under loading operations, electromagnetic FEA was employed, and the results are shown in

Table 2. Spatial phase of the minimum non-zero spatial order force density under loading operation of a 10-pole 12-slot PMSM (unit: °).

	u	2f	10f	14f	22f	26f
Force Type
Radial force		−137.5	95.8	121.4	−41.9	−52.5
Tangential force		−52.1	27.3	−143.6	−134.7	46.6
Spatial phase difference		85.4	−68.5	95.0	−92.8	99.1

and

Table 3. Spatial phase of the minimum non-zero spatial order force density under loading operation of a 14-pole 12-slot PMSM (unit: °).

	u	2f	10f	14f	22f	26f
Force Type
Radial force		−166.3	67.3	−89.5	−89.4	141.8
Tangential force		95.9	156.5	−161.1	−5.9	51.5
Spatial phase difference		−97.8	89.2	−71.6	83.5	−90.3

. The spatial phase differences in the table are the tangential force wave spatial phase subtracted from the radial force wave spatial phase.

For these two motors, the force density temporal order u is identical; however, the mutual effect of the minimum non-zero spatial order radial and tangential force waves is different. According to Formulas (12) and (13), when the 10-pole 12-slot motor is used, the stator vibration deformation generated by the radial and tangential force waves is reduced at 10f, 22f and increased at 2f, 14f, 26f. However, when the 14-pole 12-slot motor is used, in fact, the reverse is true. Because the influence of the slots and other winding MMF harmonics are ignored, the spatial phase difference between the minimum non-zero spatial order radial and tangential force density under loading operation is not accurate for the 90° spatial phase. Despite this, the interaction principle of the minimum non-zero spatial order radial and tangential force waves formed by tooth harmonics under loading condition is still correct.

To further study the impact of the stator vibration characteristics caused by the minimum non-zero spatial order radial and tangential force density formed by tooth harmonics, 3-D harmonic response analysis was carried out and the model is shown as Figure 4. A, B, and C are the code names of the three measuring points respectively. Points A and C are situated at the top and the horizontal point of the casing in the middle section plane, respectively, and point B is at the midpoint of the arc formed by points A and C. The average of the accelerations by these three points is shown in Figure 5.

As shown in Figure 5a, when the force density temporal order u is 2f, 14f, and 26f, the acceleration is produced by the enhancement effect of the minimum non-zero spatial order radial and tangential force waves. In contrast, when the 14-pole 12-slot motor is used (see Figure 5b), the acceleration is formed by the weakening effect at the above frequencies. These phenomena confirm the proposed theory. Based on the simulated acceleration spectra, the vibration acceleration level of the two motors was calculated, as shown in

Table 4. Vibration acceleration level of the minimum non-zero spatial order force density under loading operation of a 10-pole 12-slot PMSM.

	u	2f	10f	14f	22f	26f
Force Type
All force		119.19 dB	103.86 dB	105.07 dB	90.45 dB	100.63 dB
Only radial force		115.99 dB	108.41 dB	99.40 dB	100.98 dB	96.44 dB
Only tangential force		108.92 dB	111.24 dB	98.80 dB	103.16 dB	92.61 dB
Enhancement or weakening effect		+2.75%	−4.20%	+5.71%	−10.43%	+4.34%

and

Table 5. Vibration acceleration level of the minimum non-zero spatial order force density under loading operation of a 14-pole 12-slot PMSM.

	u	2f	10f	14f	22f	26f
Force Type
All force		105.90 dB	110.89 dB	98.07 dB	108.97 dB	87.38 dB
Only radial force		119.76 dB	105.43 dB	98.78 dB	104.66 dB	94.13 dB
Only tangential force		118.59 dB	106.81 dB	102.30 dB	101.03 dB	91.62 dB
Enhancement or weakening effect		−11.57%	+5.17%	−0.72%	+4.11%	−7.17%

, respectively. The vibration acceleration level can be calculated by the following equation:

$$L_a=20\lg \frac{a_e}{a_0},$$

(14)

where a_e represents the RMS value of vibration acceleration, and a₀ is the reference value of vibration acceleration, a₀ = 1 × 10⁻⁶ m/s².

The percentage of enhancement or weakening of the tangential force wave to the radial force wave is also listed, and “+” or “−” are used to represent the enhancement or weakening effect. Obviously, the influence of the tangential force wave on the vibration performance cannot be ignored, which is an important excitation source of a radial force wave. By taking advantage of this vibration-weakening effect induced by the tangential force wave, the vibration acceleration level is decreased, and the vibration performance is improved.

Furthermore, a vibration measurement platform was constructed and the 10-pole 12-slot PMSM prototype was tested (see Figure 6). The 10-pole 12-slot prototype motor was installed on an L-shaped bracket and connected to one side of the torque sensor through the coupling. The brake was jointed to the other side of the torque sensor, whose operating range is 0 to 6000 rpm and 0 to ± 110 Nm. The maximum torque of the brake is 200 Nm. A, B, and C are the code names of the three measuring points respectively. The locations of the three vibration acceleration sensors are the same as the previous 3-D harmonic response analysis model. Besides, the operating range of the vibration acceleration sensor is 0.5Hz to 10kHz, and the sensitivity is 10.2 mV/(m/s²), i.e., 100mV/g. The average of the accelerations by these three points is shown in Figure 7. When the force density temporal order $u=(2k^{\prime }mN\pm 2)/d$, i.e., 2f, 10f, 14f, 22f, 26f, and 34f, the experimental results are close to the simulation. The vibration enhancement or weakening effect caused by the PMSM radial and tangential force formed by tooth harmonics is confirmed at these frequencies. Due to the influence of the harmonic current and torque ripple, a difference is observed between the experiment and simulation at 6f, 12f, and 18f, but this does not affect the correctness of the previous conclusion.

4. Discussion

In this study, the vibration enhancement or weakening effect caused by PMSM’s radial and tangential force formed by tooth harmonics was investigated, which was verified by numerical simulation and a prototype test. The spatial phase difference between the minimum non-zero spatial order radial and tangential force density under loading operation is discussed and the stator vibration deformation produced by radial and tangential electromagnetic force waves was simulated. When the rotation direction of the force wave is positive and the force wave spatial order np − υ > 0 or the rotation direction of force wave is negative and the force wave spatial order np + υ > 0, the minimum non-zero spatial order tangential force density lags the radial force density 90° spatial phase; when the rotation direction of force wave is positive and the force wave spatial order np − υ < 0 or the rotation direction of the force wave is negative and the force wave spatial order np + υ < 0, the minimum non-zero spatial order tangential force density advances the radial force density 90° spatial phase. In addition, regardless of whether the rotation direction of force wave is forward or reverse, the stator vibration deformation induced by the unit tangential force wave lags the 90° spatial phase induced by the unit radial force wave.

Based on the above analysis, it can be concluded that when the motor is under loading conditions, the stator vibration deformation generated by the minimum non-zero spatial order radial and tangential force waves enhances each wave when the force shows temporal order $u=(2k^{\prime }mN-2)/d$. On the contrary, when temporal order $u=(2k^{\prime }mN+2)/d$, the stator vibration deformation generated by the minimum non-zero spatial order radial and tangential force waves weaken each wave. For the two motors in this study, when the pole and slot combination is 10-pole 12-slot, the enhancement effect of the minimum non-zero spatial order radial and tangential waves intensifies the vibration at 2f, 14f, and 26f. The weakening effect of the minimum non-zero spatial order radial and tangential waves reduces the vibration at 10f and 22f. The results of the 14-pole 12-slot PMSM are the opposite of the results of the 10-pole 12-slot PMSM. Apparently, the proposed theory can guide designers when estimating the acceleration spectra at specific frequencies. Moreover, this conclusion is also helpful for selecting an appropriate pole and slot combination to improve PMSMs’ vibration performance.

5. Conclusions

Different from most previous studies, this study emphasized the importance of the tangential force wave to vibration performance once again. The tangential force wave is a significant factor that cannot be ignored when evaluating the vibration characteristics of PMSMs. Superposition of the radial and tangential force waves can cause a vibration enhancement effect and aggravate vibration performance. In contrast, the suppression of radial and tangential force waves can induce a vibration-weakening effect to improve the vibration performance. This phenomenon will make people pay more attention to the study of the tangential force wave. Besides, people can take advantage of this vibration-weakening effect to optimize the vibration characteristics of the PMSM.

Since the fractional slot PMSM is a special case of an integer slot PMSM, the same conclusion is applicable to the integer slot PMSM. Thus, in future research, the correctness of the proposed theory confirmed by both simulation and an experiment will be the focus of the study. Furthermore, this conclusion is in good agreement with the simulation and experimentation when the motor was under loading conditions. However, when the motor was running with no load or a light load, the radial and tangential force waves formed by tooth harmonics were smaller, and the electromagnetic force generated by permanent magnet MMF harmonics became the main vibration exciting source; thus, the performance of motor under this operation remains to be studied.