## 3 Results and Discussion
### Results of the Rotordynamic Analysis
The calculations were done for the different polar masses of inertia of the load \(I_{\rm p2}\) spanning the range 0.005 kg\(\cdot\)m\({}^{2}\)\(<\)\(I_{\rm p2}\)\(<\) 0.025 kg\(\cdot\)m\({}^{2}\), while the rotor polar mass of inertia was kept constant at \(I_{\rm p1}\) = 0.0048 kg\(\cdot\)m\({}^{2}\) (i.e., equal to the \(I_{\rm p1}\) of the real/prototyped IPM rotor). The observed difference between the analytical solution (Equation 27) and the numerical results was 5% (Figure 14a), which validated the numerical results (the 5% difference complies with API standards required for a torsional system design [18]). The results in Figure 14a also show that the torsional natural frequencies decrease with the increase of the polar mass of inertia of the load \(I_{\rm p2}\). The investigation results shown in Figure 14b indicate that the torsional natural frequency of the shaft can be increased by increasing the shaft diameter and decreasing the shaft length. Finally, the numerical rotordynamic analysis of the real/prototyped IPM rotor configuration (Figure 2) was performed for four different types of bearings in order to investigate the influence of the bearing properties on the rotational critical speeds. The following bearing stiffnesses (i.e., from the soft up to the very stiff bearings) were investigated: k\({}_{11}\) = k\({}_{22}\) = 10\({}^{5}\) N/m, k\({}_{11}\) = k\({}_{22}\) = 10\({}^{6}\) N/m, k\({}_{11}\) = k\({}_{22}\) = 10\({}^{7}\) N/m, and k\({}_{11}\) = k\({}_{22}\) = 8.2 \(\cdot\)10\({}^{8}\) N/m (i.e., super precision SKF bearings NN3006 KTN/SP). The numerical calculations were done for the no-load (FEM model is shown in Figure 2a) and for the mechanically loaded operational condition (FEM model is shown Figure 2c) of the IPM rotor. The total mass and the moment of inertia of the real IPM rotor are 3.8763 kg and 0.0048 kg\(\cdot\)m\({}^{2}\), respectively. The calculated critical speeds are given in Tables 12 and 13 (Appendix A), for no-load and the loaded condition, respectively. The results show the presence of a higher number of critical speeds when the soft bearings are used. Due to the more complex geometry (presence of the rotor \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Critical Speed** & **Transversal \(\Omega\) [(rpm)]** & **FW Rotational \(\Omega\) (rpm)** & **BW Rotational \(\Omega\) (rpm)** \\ \hline Analytical-Uniform IPM & & & \\ Shaft D = 27 mm & 13,990 & 24,400 & 23,780 \\ \hline Numerical-Uniform IPM & & & \\ Shaft D = 27 mm & 13,790 & 24,336 & 23,741 \\ \hline Numerical-Non-Uniform & & & \\ IPM Shaft 23 mm \(\leq\) D \(\leq\) 3 mm (D\({}_{\rm mean}\) = 27 mm) & 14,184 & 23,417 & 22,479 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of analytical and numerical results (IPM shaft with \(k_{11}\) = \(k_{22}\) = 10\({}^{6}\) N/m, \(L\) = 205.7 mm). Figure 14: Dependency of the torsional natural frequencies and critical speeds on the: (**a**) different polar mass of inertia \(I_{\rm p2}\) of the load (0.005 kg\(\cdot\)m\({}^{2}\)\(<\)\(I_{\rm p2}\)\(<\) 0.025 kg \(\cdot\)m\({}^{2}\)) at the constant \(I_{\rm p1}\) of the rotor being \(I_{\rm p1}\) = 0.0048 kg\(\cdot\)m\({}^{2}\) (for the IPM rotor: \(D\) = 31 mm and \(L\) = 80 mm) and (**b**) different lengths and diameters of the shaft. with magnets), a higher number of critical speeds is obtained compared to the critical speed profile of the shaft only. The increase of the bearing stiffness results in shifting of critical speeds towards significantly higher values out from the operational range of the machine. The results show that the rotational natural frequencies and critical speeds strongly depend on the properties of the bearings. Therefore, the critical speeds can be reduced or cancelled with the stiffer bearing system. However, the bearing properties do not affect the torsional frequencies, which strongly depend on the material and geometrical properties of the system.