## 3 Results and Discussion
### Results of the Rotordynamic Analysis  
The comparison of the analytical and numerical rotor dynamic analysis was first performed on a simple rotor system with isotropic bearings and uniform shaft in order to validate the numerical FEM model built using Ansys Mechanics software. Analytically and numerically computed whirl natural frequencies (i.e., eigenvalues) as a function of the shaft's rotation speeds \(f(\Omega)\) are presented in Figure 13 and Table 5.  
Good agreement between the analytical and numerical results was obtained for the solid shafts (modelled as a solid bar with a uniform diameter \(D\)). For example, for the uniform/solid shaft with dimensions: \(D=27\) mm and L = 205.7 mm (which correspond to the mean diameter and length of the real IPM shaft, respectively), the following deviations between the numerical and analytical results was obtained: 1.4% for transversal, 0.26% for rotational FW and 0.16% for rotational BW critical speeds (Table 5). Figure 13a also shows the numerically calculated critical speed for the geometry of the real IPM shaft with non-uniform diameter (red solid line), which is in good agreement with the analytical solution of the uniform IPM shaft approximated with a solid shaft with diameter \(D=27\) mm (as stated above this is the mean diameter value of the non-uniform real IPM shaft with dimensions \(23\) mm \(\leq\) D \(\leq\) 31 mm). The obtained difference between the analytically and numerically calculated critical speeds is as follows: transversal 1.38%, rotational FW 4% and rotational BW 5.4%, which is in line with the  
Figure 12: The results of the experimental modal analysis: (**a**) rotor without magnets and (**b**) rotor with magnets. requirements of the API standards [18]. Such agreement between the analytical (uniform shaft with \(D=27\) mm, Figure 13a) and numerical results (non-uniform real IPM shaft, red lines in Figure 13a) validates the numerical models proposed in this study and confirms the practicality of the analytical solutions, indicating that they can be used for a rapid preliminary prediction of the critical speeds. (The numerical results show that the non-uniform real IPM shaft was obtained with the final mesh density consisting of 14,438 finite elements and 23,993 nodes). Further, the investigation of the influence of the dimensions of the shaft (i.e., different diameters) and the bearing properties (i.e., different bearing stiffnesses) on the critical speeds of the rotating mechanical structure was carried out based on the analytical solutions given with Equation (20), (21) and (22). The comparison of the analytically calculated critical speeds for different diameters of the rotor shaft (Figure 13a) indicates that the critical speeds decrease with the increase of the shaft diameter. The comparison of the analytically calculated critical speeds for different bearing stiffnesses indicates that the higher the bearing stiffness, the higher the critical speed of the shaft (Figure 13b). For the IPM shaft analyzed in this study (with a maximum operational speed of 10,000 rpm), this means that the bearings with the \(k_{11}=k_{22}=10^{6}\) and \(k_{11}=k_{22}=10^{7}\) N/m could be more appropriate compared to the bearings with \(k_{11}=k_{22}=10^{5}\) N/m (Figure 13b). Figure 13: Comparison of Campbell diagrams and critical speeds: (**a**) analytically calculated for different shaft diameters and the numerical results for the real non uniform IPM motor shaft geometry (all the calculations are done for the bearings stiffness \(k_{11}=k_{22}=10^{6}\) N/m) and (**b**) analytically calculated for different properties of bearings (all shafts are L = 207.5 mm long). Figure 14 shows the comparison of the analytical and numerical solutions for a simple two-disc rotor system (illustrated in Figure 3) where the first and second disc represent the rotor and the load, respectively.