## 4 3-D FE Analysis of Vibration Characteristics  
In this chapter, the vibration characteristics are calculated using a 3-D finite element method to verify the experimental results. Figure 10 shows a model of the experimental motor. First, the flux distribution in the motor and electromagnetic force at the stator were calculated using the 3-D FE method. The analysis model is 1/2 of the model in the axial direction, because there is no symmetric condition in the circumferential direction. The numbers of nodes and elements used were 132,813 and 225,758, respectively, as shown in Figure 11a. The magnetic flux density was calculated by the 3-D nonlinear FEM with the measured stator current as the input values. Then, the electromagnetic force was calculated using Maxwell's stress tensor. Figure 11b shows the finite element mesh of the cross-sectional region in the 3-D model for the calculation of eigenvalues and vibration characteristics. In Figure 11b, the aluminum case is taken into account. Natural frequencies were obtained by measuring the transfer function of the stator core. A PV08A piezoelectric accelerometer was attached to the top of the stator and was connected to one channel of a UV-06 charge amplifier. A PH-51 impulse hammer was connected to the other channel. The charge amplifier was connected to a SA-01A4 signal analyzer and then to a PC where software for the SA-01A4 was installed. The transfer function was measured by hammering the stator surface. Table 2 shows the six lowest measured natural frequencies compared with the calculated ones. The relative error of the natural frequencies was in the range of 10% through 15%. Figure 10: Experimental IPMSM. Figure 9: Characteristics of the 12 th vibration component relative to load torque: (**a**) radial demagnetization motor; (**b**) axial demagnetization motor. Figure 12 shows the FFT analysis of the electromagnetic force at a stator tooth calculated by 3-D FEM, where Figure 12a,b shows the force in the radial direction and the circumferential direction, respectively. Here, the measured stator current was used for the calculation by the 3-D FEM. It was found that the calculated electromagnetic force in the radial direction at a stator tooth was 80, 160, and 480 Hz, namely, the 2 nd, 4 th, and 12 th harmonic components of the stator current. The electromagnetic force in the circumferential direction was 80, 160, 480, 520, and 960 Hz, namely, the 2 nd, 4 th, 12 th, 13 th, and 24 th harmonic components. Using the harmonic components of the electromagnetic force as the exciting force, the vibration characteristics were calculated by 3-D FEM. Figure 13 shows the harmonic components of vibration calculated by 3-D FEM for no load and for 70% load. It was found that the vibration at 70% load was larger than that at no load, and the vibration amplitudes at 80 Hz and 160 Hz were not large. Table 3 shows the amplitude of vibration at the main frequencies and at eigenvalues. By comparing the calculated vibrations shown in Figure 13 and the measured ones shown in Figures 7 and 8, we observed that the calculated amplitudes of vibration were about 2 times larger than the measured ones. We believe that the error between the measured and the calculated vibrations is due to the calculated model. In the measurement, the motor was connected to a torque meter and a hysteresis brake on a motor pedestal. However, the authors calculated the vibration of only the motor, without including the pedestal and the torque meter, because it is difficult for 3-D FEM to calculate the whole measurement system. However, the qualitative characteristics were the same: We found that the vibration had peak values at the 440, 480, 520, 920, 960, and 1000 Hz components and that the vibration was small at 80 and 160 Hz and the eigenvalues. The amplitude of vibration at heavy load was larger than that at no load, and the amplitude of vibration emitted from the motor with radial demagnetization was larger than that with axial demagnetization.