## 3 Results and Discussion
### Modal Analysis of the IPM Rotor Configuration  
In order to investigate the mechanical vibrations of the rotor configuration and its components, such as magnets and the magnetic barriers, the modal analysis of the IPM rotor without magnets and with magnets was experimentally and numerically performed.  
The results of experimental modal measurements are shown in Figure 12, investigated within the frequency range \(0<f<18\) kHz.  
Figure 12a shows the experimentally determined natural frequency for the rotor without magnets (only magnetic air barriers are present), while Figure 12b shows the natural frequencies measured on the IPM rotor with the magnets (the magnets are placed into the magnetic air barrier compartments).  
The numerically calculated natural frequencies and their corresponding modal shapes are shown in Figure 9. The modal shapes at two different frequencies calculated in the proximity of the measured ones are displayed for each configuration. The modal shapes for the IPM rotor without magnets are displayed at the exiting frequencies \(f=9498\) Hz and \(f=10,924\) Hz (Figure 9a, the results were obtained with the final mesh density consisting of 48,791 finite elements and 86,253 nodes). The modal shapes for the IPM rotor with the magnets are displayed at \(f=14,832\) Hz and \(f=15,451\) Hz (Figure 9b; the results were obtained with the final mesh density consisting of 54,901 finite elements and 118,299 nodes).  
From Figure 9 it can be seen that the modal shape strongly depends on the rotor configuration geometry (i.e., number of magnetic flux barriers and consequently the number of magnetic poles). The natural frequencies of the IPM rotor configuration are higher compared to the natural frequencies of the rotor without magnets due to the presence of the magnets (Figure 9b). Based on the results reported here, the IPM configuration with the magnets results in more uniform modal shape at higher frequencies compared to the rotor configuration without magnets (with the magnetic air flux barriers only). The non-uniform modal shape is obtained at the lower natural frequency \(f=9498\) Hz - Figure 9a, while a more uniform modal shape (similar to the second modal shape \(m=2\)) is obtained at the higher frequency \(f=10,924\) Hz.  
As shown in Figure 9b, the deformation of the inner radius of the rotor of IPM will be considerably deformed due to the presence of the magnets. The analysis was therefore also performed for the IPM rotor with the shaft in order to investigate whether the deformation of the rotor with magnet will also result in deformation of the shaft (Figure 10). The presence of the shaft results in a lower natural frequency of the while rotor-shaft assembly. The modal analysis showed that the first bending longitudinal model will occur as \(f=5212\) Hz (Figure 10a), the second bending will occur at \(f=5933\) Hz (Figure 10b) and the modal shape, which will result in the critical deformation of both the rotor and the shaft, occurs at the frequency \(f=18\),\(449\) Hz (Figure 10c). The results shown in Figure 10 were obtained with the final mesh density consisting of 107,958 finite elements and 214,752 nodes.