**2.4. Calculation of natural frequencies**

First, we calculate the natural frequencies for the stator core only of the M-model motor, whose mechanical properties include mass density of 7,850kg/m3, Young's modulus of 2.1× 1010N/m2 and Poisson's ratio of 0.3. Table 3 shows the comparison of the calculated natural frequencies with the measured ones. It shows a good agreement between the measured values and the calculated ones. In this calculation, we use 18,811 finite element nodes. If we calculate the natural frequencies with a rough mesh, they become higher values. Fig. 5 shows the modes of stator due to each harmonic. The natural frequencies of 1,369 and 1,425Hz have mode 2, and 3,446 and 3,926Hz have mode 3.



(b) 1,545Hz

**Figure 6.** Natural vibration modes for stator with winding.

(a) 587Hz

**3.1. Analysis method of radial magnetic force** 

constant. Then, the government equations are as follows,

large, that is, close to the measured one.

**3. Radial magnetic force** 

As the natural frequencies around 1,200Hz are generated from rotor. Three smallest natural frequencies except around 1,200Hz are shown in Table 4 as well as the measured ones. It is shown that the calculated natural frequencies are a little smaller than the measured ones. This is because we calculate the space factor of winding composed of the enameled wires only. If the insulation films and vanish are taken into account, the space factor is larger. Fig. 7 shows the lowest natural frequency by changing the space factor. Therefore, if the insulation films and vanish are taken into account, the smallest natural frequency becomes

(c) 2,739Hz

The simulation of the electromagnetic force is implemented by using a 2D non-linear finite element method considering the rotor current coupled with voltage equations. As we consider the force and vibration at a steady state, the rotating speed is assumed to be

**Table 3.** Comparison of the calculated natural frequencies with the measured ones for the stator core only.

**Figure 5.** Natural vibration modes for stator core only.

Next, we calculate the natural frequencies of the stator with winding, where the space factor of winding is chosen to be 0.43 by considering the enameled wires. Three lowest natural frequencies and the natural vibration modes are shown in Table 4 and Fig. 6. The natural frequencies of 587, 1,545 and 2,739Hz have mode 2, 3 and 4, respectively.


**Table 4.** Comparison of the calculated natural frequencies with the measured ones for the stator with winding.

**Figure 6.** Natural vibration modes for stator with winding.

As the natural frequencies around 1,200Hz are generated from rotor. Three smallest natural frequencies except around 1,200Hz are shown in Table 4 as well as the measured ones. It is shown that the calculated natural frequencies are a little smaller than the measured ones. This is because we calculate the space factor of winding composed of the enameled wires only. If the insulation films and vanish are taken into account, the space factor is larger. Fig. 7 shows the lowest natural frequency by changing the space factor. Therefore, if the insulation films and vanish are taken into account, the smallest natural frequency becomes large, that is, close to the measured one.
