Analysis of the Effect of the Skewed Rotor on Induction Motor Vibration

Abstract

Induction motors have a simple structure, have low manufacturing costs and are widely used. However, various vibration effects with mechanical or electromagnetic origins are also very common. To analyze the impact of rotor skewing on electromagnetic vibrations in induction motors, this paper investigated the skew factor of skewed rotor slots and proposes an electromagnetic force wave analysis method. The method aimed to optimize the skew angle parameters for vibration amplitude reduction, with its effectiveness verified through simulations and experiments. Taking a 7.5 kW four-pole induction motor with 36 stator slots and 28 rotor slots as the research object, the suppression law of different skew parameters on force waves generated by stator harmonics was obtained. Results show that when the rotor is skewed by an angle equivalent to three stator teeth pitch, electromagnetic forces of different orders are attenuated by approximately 5% on average. Physical rotors with skew angles of 0°, 10°, 12.8°, 14°, and 20° were manufactured for experimental validation, while considering the influence of rotor skewing on starting torque and maximum torque. The study concludes that the amplitude of tooth harmonics varies with the skew coefficient, consistent with the skew factor analysis. By analyzing motor vibration with the skew coefficient, the amplitude relationship of electromagnetic vibration under different optimization parameters can be determined, thereby selecting reasonable skew parameters for rotor optimization.

1. Introduction

As the core power unit of industrial production systems, induction motors occupy an important position in the global manufacturing industry. However, electromagnetic vibration noise of motors caused by eccentricity and of other origins is a very common occurrence. In the magnetic field of the motor, all parts of the motor undergo different degrees of deformation due to electromagnetic forces, and the amount of deformation varies with time, which leads to macroscopic electromagnetic vibration. Reference [1] demonstrates a methodology for vibration analysis of typical mechanical and electromagnetic problems in the field of vibration motor diagnostics. Reference [2] reveals the effect of voltage interharmonics on cage induction motors, where simple harmonics may lead to torque pulsations and cause torque resonance. The findings of interharmonic malpractice in seven motors rated at 3 kW–5.6 MW are presented.

An effective way to reduce electromagnetic vibration in induction motors is to use a skewed rotor. Early studies on skewed rotors focused on the weakening effect of rotor skew parameters on the harmonic air gap magnetic field, and the variation of the air gap magnetic field when the rotor is skewed was determined by using analytical and finite element methods [3,4]. According to the analysis of the air gap magnetic field in the references [5,6,7], when the rotor deflection distance is small, the effect on the fundamental magnetic field of the motor is small, but the effect on the harmonic magnetic field is large, especially on the magnetic torque and the permeability tooth harmonics. Reference [8] gives an expression for the saturation factor of the leakage reactance when the rotor of an induction motor is skewed, and [9] shows by a two-dimensional time-stepping finite element method that rotor skewing significantly attenuates the toothed harmonic component of the induced counter electromotive force.

Reference [10] analyzed different skew angles of switched reluctance motor rotor and their effects on key electromagnetic parameters, which resulted in a 56.2% reduction in radial force at 5% skew angle. References [11,12] reduced the cogging torque by changing the rotor structure of electric vehicle motors, such as skewed pole splitting of the rotor and applying wedge-shaped skewing. Reference [13] investigated and analyzed the torque pulsation of a servomotor by changing the rotor topology and reduced the torque pulsation, which was 3.75%, to about 2.08%.

Due to the difficulty of process manufacturing, the motor rotor cannot be tilted by too large a number of degrees, and as the rotor oblique groove inclination can only be within a certain range to reduce the tooth oblique wave content, so [14,15] analyzed the double oblique rotor on the vibration and noise effects. Double-sloping and a new type of sloping slot can make the rotor sloping slot inclination angle larger, so as to reduce the vibration and noise of the motor more effectively.

Practical and theoretical analyses show that for different types of motors, such as small and medium-sized induction motors, permanent magnet synchronous motors, and switched reluctance motors, choosing the appropriate rotor slant slot angle can also effectively reduce vibration and noise [16,17,18,19,20].

Previously, scholars analyzed the damping effect of sloping slots on motor vibration by analyzing the weakening effect of different types of motors using different rotor sloping slots on harmonics. The advantage of this method is that it is more convenient to analyze the harmonics and torque pulsations that generate force waves, but the effect of the skew slot on the electromagnetic excitation force cannot be obtained. In this paper, the skew coefficient of the rotor is investigated, and the degree of weakening of the electromagnetic excitation force is directly analyzed by the skew coefficient, taking a 7.5 kW motor as an example. In this way, the optimum angle of the skew slot was found to reduce the electromagnetic vibration of the motor. The optimum angle for reducing the electromagnetic vibration of the motor was found, and the validity of the theoretical simulation verified by the electromagnetic–mechanical excitation separation test bench for rotors with tilt angles of 10°, 12.8°, 14°, and 20°, respectively. The skew coefficient can be substituted into different rotor types of induction motors for analysis. The stages of this study and evaluation are detailed in the following sections.

2. Skew Factor

A single-slewed rotor refers to a rotor conductor that is offset by a certain angle in the axial direction, which causes both the fundamental and harmonics of the stator to be weakened. In order to calculate the weakening effect, a skewed coefficient needs to be added in addition to the potential distribution coefficient and pitch coefficient. Figure 1 shows a simplified schematic diagram of a single skewed rotor. In the rotor structure of induction motors, the role of the end rings is more significant during motor starting or speed change. The theoretical model of this study focuses on the analysis of electromagnetic vibration under steady state operation (slip rate s < 2%), where the rotor current frequency is very low, the resistance of the end rings dominates the impedance characteristics, and the regulation of the magnetic field distribution in the air gap is negligible. Thus the end ring portion of an ordinary aluminum rotor is omitted from the figure.

The main parameters of the motor used in this paper are shown in the following

Table 1. Parameters of motor.

Rate Power	7.5 kW	Rate Voltage	380 V
Stator slots	36	Rotor slots	28
Pole pairs	2	Skewed	Different
Air gap length	0.5 mm	Core length	160 mm
Stator diameter	210 mm	Rotor diameter	135 mm

[21]:

The skew factor is calculated from the skew angle β where α is the small angular difference between neighboring guide bars and c is the spacing between two neighboring guide bars. The physical model shown in Figure 1 can be regarded as an infinite series of infinitesimal segments (n→∞), with each segment angular displacement being dθ and satisfying β = ∫dθ. In the discrete model, if the number of segments is n, the angle of each segment α = β/n, which strictly satisfies nα = β when n→∞, where β is the entire conductor skew section of the radian, i.e., skewness.

Using the synthesis method of electric potential in distributed conductors, the skewing factor of the fundamental wave can be obtained [8,22], as seen in Equation (1):

$$k_{sk1}=\underset{\begin{array}{l}\alpha \to 0\\ n\alpha \to \beta \end{array}}{\lim }{\displaystyle \frac{\sin \frac{n\alpha }{2}}{n\sin \frac{\alpha }{2}}}={\displaystyle \frac{\sin \frac{\beta }{2}}{\frac{\beta }{2}}}$$

(1)

where λ is the rotor skewed number of stator slots.

For the convenience of calculation, the number of skew stator slots is used to represent, then proceeding as follows:

$$k_{sk1}={\displaystyle \frac{\sin \frac{\lambda {\pi }^{2}D}{2S_1\tau }}{\frac{\lambda {\pi }^{2}D}{2S_1\tau }}}$$

(2)

where D is the stator inner diameter, S₁ is the number of stator slots, τ is the pitch.

For the skew factor of the vth harmonic, β is replace in the above formula with υβ, so the skew factor for the vth harmonic is the following:

$$b_{skv}={\displaystyle \frac{\sin \frac{v\beta }{2}}{\frac{v\beta }{2}}}={\displaystyle \frac{\sin v\frac{\lambda {\pi }^{2}D}{2S_1\tau }}{v\frac{\lambda {\pi }^{2}D}{2S_1\tau }}}$$

(3)

where b_skv is the rotor skew factor for the vth harmonic wave.

Therefore, to eliminate the vth harmonic, it is only necessary to make the skew factor of this harmonic equal to zero, but since harmonics of different orders exist simultaneously, it is necessary to consider them in a comprehensive way and to adopt an appropriate skew distance for processing. Based on the skew coefficients of the fundamental and harmonics, the relationship between the skew distance and the magnetomotive force (MMF) coefficients of each order is shown in Figure 2.

As shown in Figure 2, the fundamental or harmonic coefficients can reach bskv = 0 when the rotor skew angle is at certain specific values. Since the fundamental wave is the main component that generates the torque output in the motor, the skew distance should be minimized to ensure that the fundamental coefficient is large enough. However, in order to reduce the content of the harmonic waves, the skew distance should be selected at the position where the harmonic wave coefficient is 0 as much as possible. At the same time, as shown in the figure, when the skew distance is 1 or 2 stator teeth, the fundamental wave coefficients are bsk1 = 0.9949 and bsk1 = 0.9798, respectively, and the coefficients of the 17th and 19th harmonics are bsk17 = 0.0585, bsk19 = 0.05236, bsk17 = 0.05764, and bsk19 = 0.0515, respectively. The amplitude of the tooth harmonics decreases significantly. The coefficients of the 11th and 13th harmonics change from bsk11 = 0.4895 and bsk13 = 0.3316 to bsk11 = −0.1674 and bsk13 = −0.217, respectively. Therefore, when only the 17th and 19th harmonics weakening is considered, the skew angle can be selected as 1 stator slot distance, and when the 11th and 13th harmonics weakening is considered, 2 stator slot distances are more optimal. When the skew angle is selected between 1.3 and 1.5 stator slot distance, the fundamental coefficient is bsk1 = 0.99, and it has a good effect on the weakening of the 13th harmonic coefficient approaching 0.

3. Radial Force Wave with Skewed Rotor

After analyzing the skew factor, the radial force wave of the rotor skew is analyzed, and Figure 3 gives the spatial model of the rotor skew.

The stator harmonic magnetic field can be expressed as follows:

$$b_{\mu }=B_{\mu }\cos (\mu px-{\omega }_{1}t-{\phi }_{\mu })$$

(4)

where B_μ is the magnitude of the μ stator harmonic magnetic field and μ is the number of stator harmonics.

The harmonic magnetic field generated by the rotor fundamental current is the following:

$$b_v=B_v\cos \left(vpx-{\omega }_{v}t-{\phi }_v-v{\displaystyle \frac{b_{sk}}{R}}{\displaystyle \frac{Y}{l}}\right)$$

(5)

where B_v is the magnitude of the v rotor harmonic magnetic field and v is the number of rotor harmonics.

The above formulas of Equations (4) and (5) obtain the radial electromagnetic force wave generated by the interaction of the two magnetic fields as below:

$$\begin{array}{ll}{p}_r& ={\displaystyle \frac{2b_{\mu }{b}_v}{2{\mu }_0}}\\ & ={\displaystyle \frac{1}{2{\mu }_0}}{B}_{\mu }{B}_v\cos \left[\left(\mu \pm v\right)px-({\omega }_v\pm {\omega }_1)t-\left({\phi }_{\mu }\pm {\phi }_v\right)-v{\displaystyle \frac{b_{sk}}{R}}{\displaystyle \frac{Y}{l}}\right]\\ & =P_r\cos \left(rx-{\omega }_{r}t-{\phi }_r-v{r}_s{\displaystyle \frac{Y}{l}}\right)\end{array}$$

(6)

in the formula $P_r={\displaystyle \frac{B_{\mu }{B}_v}{2{\mu }_0}}$, $r=\mu \pm v$, ${\omega }_r={\omega }_v\pm {\omega }_1$, ${\phi }_r={\phi }_{\mu }\pm {\phi }_v$, $r_s={\displaystyle \frac{b_{sk}}{R}}$ where P_r is the rth order force wave amplitude, φ_0r, φ_μz, φ_vz are the phase angles of the different harmonic magnetic fields, ω₁, ω_v are the angular velocities of different harmonic magnetic fields.

It can be seen from Equation (3) that the distribution amplitude of the radial force wave varies in the axial direction due to the rotor being skewed.

To further analyze the influence of the oblique slot on the 0-order axial vibration, the average radial electromagnetic excitation force is obtained by integrating the longitudinal direction of the iron core as follows:

$$\begin{array}{ll}{p}_a& ={\displaystyle \frac{1}{l}}{\displaystyle {\int }_{-l/2}^{l/2}{P}_{n}dY}\\ & ={\displaystyle \frac{1}{l}}{\displaystyle {\int }_{-l/2}^{l/2}{P}_r\cos \left(rx-{\omega }_{n}t-{\phi }_n-v{r}_s\frac{Z}{l}\right)dY}\\ & =P_r\cos \left(rx-{\omega }_{n}t-{\phi }_n\right)\sin \left({\displaystyle \frac{v{r}_s}{2}}\right){\displaystyle \frac{2}{v{r}_s}}\\ & =P_r{K}_{skv}\cos \left(rx-{\omega }_{n}t-{\phi }_n\right)\end{array}$$

(7)

The expression for $K_{skv}$ is $K_{skv}={\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\nu }{\mathrm{r}}_{\mathrm{s}}/2\right)}{\mathsf{\nu }{\mathrm{r}}_{\mathrm{s}}/2}}$.

It can be seen that K_skv agrees formally with the skew coefficient in Equation (1), so K_skv is the skew coefficient described above. Therefore, when the slot is tilted, each order of the magnetomotive force acting on the stator and the skew coefficient also satisfies the relationship equation shown in Figure 2.

4. Electromagnetic Excitation Force Under Tooth Skew

To further analyze the effect of skew on vibration, it is necessary to analyze the electromagnetic excitation force on the tooth. Since the force wave derived in the previous section cannot be directly extended to a three-dimensional plane, the stator iron core must be extended along the circumferential direction. Therefore, the analysis is carried out in the Cartesian coordinate system. The Y direction is the motor rotation axis, i.e., the longitudinal coordinate of the iron core, and the X direction is the tangential coordinate of the circumferential direction along the expanded plane of the stator. γ₁ is the axial inclination angle of the stator slot, t₁ is the tooth pitch on the surface of the inner hole of the stator and l is the effective length of the core. The spatial calculation model of the motor stator for the force wave acting on the tooth is shown in Figure 4.

The force density of the radial force wave skewed along the inner hole of the stator and the rotation axis is given by the following formula:

$$p_r=P\mathrm{c}\mathrm{o}\mathrm{s}\left(r{\displaystyle \frac{2x}{D}}-\nu {\displaystyle \frac{2y}{D}}-{\omega }_{r}t-{\phi }_r\right)$$

(8)

In order to calculate the electromagnetic excitation force on the stator side-teeth, the inclination of the cogging slot in Figure 4 is projected onto the XOY plane, and Figure 5 shows the schematic diagram of the projection, and the relationship between the various parameters can be obtained. The number of stator teeth is from S = 0 to S = λ, where S₁ is the number of stator teeth, γ₁ is the axial inclination angle of the stator slot, which is determined by the inclination distance and the iron core length, which is tanγ₁ = bck₁/l. Among them, bck₁ is the arc length of the skewed slot distance, and t₁ is the tooth pitch on the surface of the inner hole of the stator, which is t = πD/S1; x₁ = ytanγ₁ + λt₁−t₁/2, x₁ = ytanγ₁ + λt₁ + t₁/2, λ is the number of the tooth.

The excitation force acting on the rigid teeth can be determined by integrating the force wave density along a certain stator pitch surface. The force uniformly distributed along the tooth section of No. λ is transmitted to the tooth root according to the force wave expressed by Equation (9):

$$\begin{array}{l}{p}_{\lambda }^1=\int \int P\cos \left(r\frac{2x}{D}-v\frac{2y}{D}-{\omega }_{r}t-{\phi }_r\right)dxdy\\ ={\int }_0^y{\int }_{x_1}^{x_2}P\cos \left(r\frac{2x}{D}-v\frac{2y}{D}-{\omega }_{r}t-{\phi }_r\right)dxdy\end{array}$$

(9)

After rearranging the above formula, we obtain the following equation:

$$p_{\lambda }^1=P_z\cos \left(\left({\displaystyle \frac{r\tan {\gamma }_1}{D}}-{\displaystyle \frac{v}{D}}\right)l+{\displaystyle \frac{2r}{D}}\lambda t_1-{\omega }_{r}t-{\phi }_r\right)$$

(10)

$$P_z=Pl{t}_1{\xi }_r{\xi }_{zck}/2$$

(11)

$${\xi }_r=\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\pi r}{S_1}}\right)/{\displaystyle \frac{\pi r}{S_1}}$$

(12)

$${\xi }_{zck}={\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\left(\frac{r\mathrm{t}\mathrm{a}\mathrm{n}{\gamma }_1}{D}-\frac{\nu }{D}\right)l\right)}{\left(\frac{r\mathrm{t}\mathrm{a}\mathrm{n}{\gamma }_1}{D}-\frac{\nu }{D}\right)l}}$$

(13)

where ξ_r is the vibration amplitude coefficient, ξ_zck is the slant factor, and l is the rotor axial length.

Although the above equations are obtained by tilting the stator teeth, the essence of both stator tooth tilting and rotor slant slotting is to introduce a spatial phase difference through axial geometric offsets that attenuate a specific number of magnetic field harmonics. The form of the phase difference equation is identical for both, with only the parameters replaced with rotor-related values. Therefore, the same applies to the case of rotor slant slots. It can be seen from Equation (7) that the coefficient of the excitation force transmitted by the electromagnetic force wave through the teeth is similar to the coefficient of the skewed, which is a function continuously oscillating through the zero point.

After optimizing the parameters of the skew, when the excitation force transmitted to the teeth is zero, the appropriate skewed slot distance is obtained. The skew slot distance can be calculated using the following formula:

$$b_{ck1}={\displaystyle \frac{\pi D}{r}}k+{\displaystyle \frac{vl}{r}}$$

(14)

In, k = 1, 2, 3, … where b_ck is the distance the rotor is skewed in the circumferential direction.

According to Equation (14), there are two parts to consider when considering the rotor slot. If the distance of the rotor slant slot is larger, a larger reduction of the magnetic moment wave will be produced, as shown in Figure 2, which will affect the performance of the motor. Due to the limitations of motor processing and manufacturing techniques, it is difficult to produce a large, skewed angle in the iron core of small and medium-sized motors. Even if it can be produced, there are many defects. Therefore, the maximum slant slot distance of five stator slots is selected in the simulation.

Figure 6 shows the variation of the slope coefficient for a 36–28 motor rotor with skew slot spacing from 1–5 stator teeth distance. The slope coefficient directly reflects the change in amplitude of the electromagnetic excitation force, and from this coefficient, it can be seen that when the rotor is tilted at a certain angle, the higher-order electromagnetic forces are slightly weakened. At the same time, when the tilt angle is greater than one stator tooth, only specific electromagnetic forces higher than a certain order are weakened to within 15%. When the pitch angle is three stator teeth apart, it is optimal, with each order of force being weakened by an average of approximately 5%.

After obtaining the rotor skew coefficient for induction motors with different parameters, the relationship between electromagnetic excitation force and skew coefficient can be obtained by Equation (7). Thus, the optimized skew slot parameters to reduce the electromagnetic excitation force are calculated, and the rotor tilt distance in the circumferential direction can be obtained by combining with Equation (14).

5. Experiment

Vibration tests were performed for rotors with skew angles of 0°, 10°, 12.8°, 14°, and 20°. The rotor with 0° skew is a straight skewed rotor and is given in the literature [21]. Figure 7 shows the plots for rotors with other parameters.

As the skewed rotor reduces the torque performance of the induction motor, the torque performance of the skewed rotor was tested with different parameters. The maximum torque and the locked rotor torque of the motor for different rotors were tested. The test equipment is shown in Figure 8.

The test results for straight slot rotors and different skewed parameter slot rotors are shown in Figure 9. The comparison of maximum torque and stall torque in the figure shows that the actual data trend of the decrease in skewed slot torque is basically consistent with the results of the simulation analysis in the previous text. The reason why the torque at a rotor skewed slot angle of 12.8° is smaller than that at 14° is mainly due to the fact that the skewed slot parameters of these two rotors are similar—the manufacturer ensures the skewed slot parameters through molds, so there is some error, but the overall trend has not changed. If the skewed slot distance increases, it will lead to a decrease in the starting and overload capability of the motor. Therefore, before adopting the skewed slot method to reduce vibration, it is necessary to consider whether the change in maximum torque can meet the performance requirements of the motor.

In order to compare better the reduction of tooth harmonics by straight slot and skewed rotors, the air gap magnetic density data are analyzed to further obtain the variation law of the magnetic density, as shown in Figure 10.

From the comparison of the time-domain data of the air gap magnetic flux density of the straight groove rotor and the chute rotor, it can be seen that the air gap magnetic flux density waveform of the straight groove rotor contains more harmonics in the no-load state, while the air gap magnetic flux density of the chute rotor contains fewer harmonics, and the harmonic content in the gap magnetic flux density waveform is significantly reduced. Through the comparison of Fourier transform in the frequency domain, it can be seen that the amplitude of the harmonic component of the tooth is weakened due to the skew. Under no-load conditions, the time-domain waveform of the air gap flux density contains more harmonics. After the Fourier transform, comparing the harmonic amplitude of the tooth with different deflection grooves, it can be seen that the amplitude of the tooth harmonics does not decrease with the increase of the deflection groove, but changes with the change of the deflection groove coefficient. This is consistent with the results of the skew coefficient analysis.

In order to verify the damping effect of the skewed rotor on vibration, vibration tests were carried out on rotors with different skewed rotors on an electromagnetic–mechanical excitation separation test rig. The specific structure of the test rig is shown in Figure 11, and its detailed parameters can be found in [21].

In the diagram, X, Y, and Z represent the axial, circumferential radial, and tangential directions of the motor, respectively. Vibration acceleration sensors are installed at points P1 to P7, which are located at the bottom of the non-shaft end cover, the top of the non-shaft end cover, the top of the stator, the stator foot (front and back), the base plate, the top of the shaft end cover, and the bottom of the shaft end cover, in order.

Taking the test results of three different rotors with different inclined slots together, we can observe the overall electromagnetic vibration amplitude and the trend of the different measurement points in three directions. Meanwhile, as can be seen from the amplitude comparison in Figure 12, the overall amplitude of the casing decreases significantly when the rotor is tilted. The damping effect on the electromagnetic vibration of the stator top and stator foot is the largest, when the tilt is 12.8°; the vibration amplitude in the radial direction of the stator top is 106 dB, and the vibration amplitude of the stator foot is 105.3 dB.

From the vibration data of the sensor on the motor casing in Figure 13, it can be seen that the electromagnetic vibration of the motor in the transverse and longitudinal directions (i.e., the radial and tangential directions in the circumferential direction) decreases with the increase in the degree of skew, which is consistent with the results of the previous analysis of the skew factor on the force and vibration amplitude of the motor teeth. The axial direction increases with the increase in the degree of skew. This is because in the case of a single skewed, since the skewed will generate an axial force in the axial direction with the increase of the skewed degree, the proportion of the axial force increases, and the vibration increases accordingly. This is without comparing the axial vibration.

According to previous analysis, the optimum vibration skew slot of the motor is three stator teeth, i.e., the angle of the skew slot is 30°. However, it is difficult to produce such a large skew angle using conventional motor technology. In order to verify the optimum parameters of the skew vibration, the double-skew and four-skew slot methods can be used to increase the skew angle accordingly. The double-skew slot divides the whole rotor core into two symmetrical single skew slot rotors along the axial direction, and the double skew can double the skew angle of the single skew. The four-skew slot method can achieve a skew angle four times that of the single-skew method.

6. Conclusions

This paper described an analytical method for a skewed rotor induction motor to analyze the magnetic flux density distribution and calculate the electrical magnetic force. By converting the single-slot, double-slot, and four-slot rotor transformations into slot skew parameters, the effect of slot skew on the electromagnetic vibration damping of the motor was validated for all parameters. The described analytical method was verified by simulation results and experimental tests. The conclusions are as follows:

Motor vibration decreases as the skew slot angle increases, but vibration changes occur when a certain skew slot angle is reached. It can be assumed that there is an optimized skew slot angle within the permissible parameter range that can effectively reduce the electromagnetic vibration of the motor.

The variation analysis of skew coefficients in 36–28 motor rotors demonstrates that the skew coefficient directly reflects the amplitude variations of electromagnetic excitation forces. When the rotor is skewed by a specific angle, higher-order electromagnetic forces exhibit slight attenuation. Moreover, when the skew span covers three stator teeth, the forces at each order are weakened by approximately 5% on average, indicating the optimal condition. By analyzing motor vibrations through the skew coefficient, the amplitude relationships of electromagnetic vibrations under different optimization parameters can be determined, thereby enabling the selection of appropriate optimization parameters.

Experimental analyses and tests on rotors with skew angles of 0°, 10°, 12.8°, 14°, and 20° reveal that, the tooth harmonic amplitude varies with the skew factor, which is consistent with the theoretical analysis. Regardless of the skew configuration, the skew angle remains the predominant factor affecting electromagnetic vibrations in motors. As the skew angle increases, both the starting torque and the maximum torque of the motor gradually decrease. Consequently, when optimizing motor vibrations using skewed rotors, it is essential to evaluate comprehensively the interplay between motor performance and vibration characteristics and select suitable skew angle parameters.