Exciting Force and Vibration Analysis of Stator Permanent Magnet Synchronous Motor

This paper presents the detail analysis of the radial force and bending moment responsible for vibration in the stator permanent magnet synchronous motors (SPMSM). Firstly, the air gap flux density distribution and radial force density distribution are derived in detail. In addition, the relationships between these two vibrations and pole width, pole number and slot number are analyzed. The multi-probe mode-included test method of vibration acceleration is proposed and the experimental results on a6-pole/36-slot SPMSM match well with the simulation results. 0 Introduction With the advancement of the electrical vehicles such as hybrid-electric and electric vehicles, improving the noise and vibration characteristics of rotating electric machines is becoming important consideration issues for the development of vehicles[1–4]. Electrical vehicles make use of permanent magnet synchronous traction motors for their high torque density and efficiency compared with other types of motors[5–9]. The stator permanent magnet synchronous motors (SPMSM) have good advantages and are widely applied in the industry. Few papers reported to focus on the vibration characteristics of this type of motor[10ߝ12]. So, this paper investigates the vibration mode and frequencies in the stator permanent magnet synchronous motor, which is shown in Fig.1. 1The magnetic field and exciting force 1.1 analysis of Magnetic field When ignoring magnetic saturation, the flux density distribution in the air gap under no load can be expressed as$\mathrm{b}(\theta, \mathrm{t}) = \pm ({B_{\delta}} {\Lambda _{0}}+ \sum {B_{\delta}} {\Lambda _{k}} {\mathrm {coskZ}} (\theta- {\omega _{{\mathrm {r}}}}\mathrm{t})) \theta \epsilon \left[- \alpha \pi /(2\mathrm{p}) + \mathrm{j}2 \pi /(2\mathrm{p})\alpha \pi /(2\mathrm{p}) + \mathrm{j}2 \pi /(2\mathrm{p}) \right]$(1) Where Z is the number of rotor slots, ${\Lambda_{0}}$ and ${\Lambda_{k}}$ are the amplitude of average and k-th harmonic magnetic permeability. $\alpha$ is the pole arc coefficient. When $\mathrm{j=2}\mathrm{m}$,$\mathrm{m=0}$,1,2,3…, the sign in the equation takes a positive value; When $\mathrm{j=2} \mathrm{m+1}$, the sign is negative. 1.2 Force analysis The force on ferromagnetic substance in the electromagnetic field can be described by Maxwell stress tensor method. Radial force wave generated by magnetic field is${\mathrm{p}_{n}}=1 / (2 {\mu _{0}}) \left\{ {(B_{\delta}} {\Lambda_{0}})^{2}+ \sum \left[{B_{\delta}} {\Lambda _{k}} {\mathrm {coskZ}} (\theta- {\omega _{r}}\mathrm{t})\right] ^{2}+ 2 {B_{\delta}} ^{2} {\Lambda_{0}}{\Lambda _{k}} {\mathrm {coskZ}} (\theta- {\omega _{r}}\mathrm{t})\right\}$(2) The third term indicates radial force which changes with time generated by interaction of the not time-varying magnetic field and the periodic magnetic field, which causes a considerable vibration. The vibration frequencies are the integral multiples of the product of the slot number and the rotation frequency. 2 Two vibration sources In the motor, the radial force ${P_{n}}$ on one pole can be equivalent to one equivalent pulsating force ${P_{np}}$ varied with time on the center line of the pole and equivalent bending moment ${m_{np}}$ varied with time on the pole. The general expression of pulsating force is ${P_{np}}({\mathrm{t}}) =2 {\mathrm{l}_{p}} {\mathrm{RB}_{\delta}} ^{2}{\Lambda_{0}}{\Lambda _{1}} \sin ({\alpha_{1}}\pi /(2\mathrm{p}))\cos$(jZ $2 \pi /(2\mathrm{p})- \mathrm{Z}{\omega _{r}}\mathrm{t})/ (\mathrm{Z}{\mu _{0}})(3)$ Where $R$ is the radius of the inner surface of the pole, ${\alpha_{1}}$ is pole arc width expressed by rotor number. Because the uneven distribution of the force density, the corresponding bending moment is ${m_{np}}(\mathrm{t}) = {\mathrm{l}_{p}} {{\mathrm{R}^{2B}}_{\delta} ^{2}}{\Lambda_{0}}{\Lambda_{1}} \cos ({\alpha_{1}}\pi) \sin (\alpha \pi /(2\mathrm{p}))\sin$(jZ $2 \pi /(2\mathrm{p})- \mathrm{Z}{\omega _{r}}\mathrm{t})/ (\mathrm{Z}{\mu _{0}})(4)$ From the equation(3) and (4), we can know that: If $2 \pi \mathrm{Z} /(2\mathrm{p}) =2 \mathrm{n}\pi$, n is integral, the radial forces and the bending moment on the two adjacent poles are in the same phase. When ${\alpha_{1}}$ is $n$, radial force reaches minimum, bending moment reaches maximum, and the mode number equal to the pole number. When ${\alpha_{1}}$ is $n+1 /2$, radial force reaches maximum, bending moment reaches minimum, and the vibration mode equal to 0. If $2 \pi \mathrm{Z} /(2\mathrm{p}) =(2 \mathrm{n+1}) \pi$, the phases of the forces on the two adjacent poles are opposite, and so do the bending moment. When ${\alpha_{1}}$ is $n$, radial force reaches minimum, bending moment reaches maximum, and the mode number equal to the pole pairs. When ${\alpha_{1}}$ is $n+1 /2$, radial force reaches maximum, bending moment reaches minimum. The vibration mode equal to pole pairs. 3 simulation analysis In the simulation, a harmonic response analysis has been performed by using the mode-superposition method. the motor is constrained by elastic support through the end face. A damping ratios of 4.4% extracted from the modal test is used. The harmonic forces are projected on the structure mesh of the stator in order to calculate the deformation, and the vibration mode. 4 Experiment test In order to verify the validity of analytical model studied in this paper, the Operational Deflection Shapes experiment is performed. The designed motor with 6-pole, 36-slot, ${\alpha_{1}} -5$ is hanged on the bracket by elastic rope, and seven accelerometer sensors are evenly attached on the stator in the circumferential direction. The motor run at 1500rpm. Fig.2(a) compares the measured acceleration with those FEM at five frequencies associated with slot frequency. It clearly reveals that the acceleration components at the harmonic frequencies of 36fr, 72fr, 108fr, 144fr, etc.. Vibration shape at slot frequency is measured seen in Fig.2(b). It can be seen clearly that the vibration mode at slot frequency is six order, consistent with the analytical results. 5 Conclusions In this paper, a detailed analysis of vibration in stator permanent magnet synchronous motor (SPMSM) is presented. Some conclusions can be summarized from the paper: The frequencies of the exciting radial force and vibration of the SPMSMs mainly are integral multiples of the product of the slot number and the rotation frequency, which is different from the frequencies related about the pole number in the rotor permanent magnet motors. The vibration mode is decided by the radial force, which causes the pulsating mode, and the bending moment, which causes the bending mode.