## 3 Results and Discussion
### Results of the Rotordynamic Analysis
Such an example is the speed 7256.6 rpm, as shown in Table 11, which could not be canceled by increasing the stiffness of the bearings system. However, this speed of our model is probably not critical for the real system, since it represents one of the numerical solutions for the no load situation. Similarly, the numerical solution for the rotor with the load appear at the first critical speed value 3790.4 rpm (Table 11), which can be attributed to the non-uniform shape of the shaft. In order to investigate the critical speeds that may occur in a mechanically loaded IPM rotor system with bearings (Figure 2c), a load with 2.33 times higher moment of inertia \(I_{\text{p2}}\) [kg\(\cdot\)m\({}^{2}\)] compared to the moment of inertia of the IPM rotor was added to the rotor shaft (i.e., \(I_{\text{p2}}\) = 0.0112 kg\(\cdot\)m\({}^{2}\), \(I_{\text{p1}}\) = 0.0048 kg\(\cdot\)m\({}^{2}\) and the inertia ratio \(I_{\text{p2}}\)/\(I_{\text{p1}}\) was 2.33). It is well known that the higher the inertia ratio the higher the risk for a suboptimal system performance in terms of resonance. Therefore, such a load was selected in order to investigate one of the most critical scenarios due to the high difference between the two moments of inertia, since it may result in severe consequences such as premature fatigue, shaft bending or crack of the shaft due to resonance with the natural torsional frequencies. The Campbell diagram for the analyzed rotor assembly with the load for four different bearing stiffnesses is shown in Figure 15. The numerical results were obtained with the final mesh density consisting of 132,604 finite elements and 243,733 nodes. The numerical rotor dynamic simulations identified a larger number of critical speeds at lower motor speeds when the bearings with lower stiffness were used (Figure 15a-c), while for the super precision SKF bearings, only the three critical speeds were identified: 48,774 rpm, 49,683 rpm and 50,393 rpm. The operational speed range of the IPM machine analyzed in this study is defined with the baseline/rated operational speed value of 4000 rpm (at the border between the constant torque and the constant power) up to the maximum speed value of 10,000 rpm (i.e., in the field weakening region). The numerical results shown in Figure 15 indicate that the critical speeds can be shifted out from the operational range of the machine, using bearings with a higher stiffness. It should be also noted that the frequencies shown in Figure 15 are significantly lower compared to the natural frequencies shown in Figures 9 and 10. The load condition analysis showed that the first torsional frequency appeared at a very high critical speed of 48,774 rpm (which corresponds to the critical natural torsional frequency 812 Hz), and cannot be cancelled or shifted toward higher speeds with stiffer bearings, as shown in Figure 15a-d and Table 11. It should be noted that this torsional speed can also be roughly predicted by using the analytical solution given by Equation 27 (Figure 14a). For the less critical load operational regimes (i.e., for the regimes when the rotor is loaded with the loads with lower polar mass moment of inertia), the critical speeds of the loaded rotor system will further increase (above 48,774 rpm) according to Equation (27), as demonstrated also in Figure 14. It should be noted that the calculated critical torsional speed is significantly higher than the operational speed of the real IPM motor. The last two critical speeds of 49,683 rpm and 50,393 rpm represent the forward and backward critical speeds, which may result in the bending natural frequencies (i.e., the bending rotational modal shapes). These results also indicate that the strong bifurcation shown in Figure 15a-c that occurs due to the presence of the rotor (due to the gyroscopic effect) in the proximity of 900 Hz can be reduced with bearings with a higher stiffness.