## 3 Results and Discussion  
### Natural Frequencies and Modal Shapes of the Stator with and without Windings  
The comparisons of experimental, numerical and analytical output results obtained with modal analysis of the stator stack without windings and with windings are given in Tables 3 and 4, respectively. The results of the experimental measurement of modal natural frequencies identified on the stator stack without windings are displayed in Figure 5a, while the experimental results performed on the stator stack with the presence of the slot windings and end-windings are shown in Figure 5b. The results in Figure 5 show that the natural frequency of the stator with slot winding and end-windings decreased compared to the natural frequencies of the stator stack only. \begin{table}
\begin{tabular}{c c c c c} \hline *** & \(f\) **(Hz)**, \(m\) = 2_ & \(f\) **(Hz)** & \(f\) **[Hz]** & \(f\) **(Hz)**, \(m\) = 0_ \\ \hline Experiments & 500.5 & 695.8 & 1315.3 & / \\ \hline Numerical, FEM & 525 & 754 & 1393 & 5113 \\ \hline Analytical, Equation (11) & 458.7 & / & / & 5066.4 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of experimental, numerical and analytical results for the stator with windings. Figure 4: The prototyped components of the IPM motor, on which the experimental modal analysis was performed: (**a**) stator; (**b**) stator with slot windings and end-windings; (**c**) rotor without magnets and (**d**) rotor with permanent magnets, (**e**) impact hammer and (**f**) accelerometer sensor. The white circles indicate the position of the sensor. The displayed IPM components were prototyped in collaboration with the Slovenian company Kolektor Group d.o.o. The front view of the numerically calculated modal shapes (\(m=2\), \(m=3\), \(m=4\), \(m=8\) and \(m=0\) - pulsating mode) of the 3D numerical model of stator stack without windings is shown in Figure 6. The normalized total deformations are indicated in the color bar in Figure 6 from minimum (blue color) to maximum (red color). The color bar is also valid for Figures 7-10. The results were obtained with the final mesh density consisting of 143,573 finite elements and 250,364 nodes. The 3D numerically calculated modal shapes and their corresponding natural frequencies of the stator with the slot windings and the end-windings are displayed in Figure 7. The results were obtained with the final mesh density consisting of 197,911 finite elements and 323,364 nodes. The comparison results shown in Tables 3 and 4 indicate a good agreement between the experimental, numerical and analytical results, which is particularly important for the most critical modal frequency \(m=2\), at which the highest amplitude is expected. The results shown in Table 4 indicate that the analytical solution given by Equation 11 can be used for a rapid preliminary calculation of the most critical modal frequency (\(m=2\)) for the IPM stator with the presence of the slot windings and end windings, with 12% difference with respect to the numerical results and 8% difference with respect to the experimental measurements. However, it should be emphasized that the analytical results (Equation 11) can estimate the natural frequencies only for the winding fill factors \(k_{\rm cu}\) within the range 0.35-0.45. For fill factors outside this range, the equation gives erroneous results, since it neglects the Young's modulus of the windings. In order to achieve accurate results of the electrical machine stator natural frequencies with windings with a given fill factor \(k_{\rm cu}\), the equivalent Young's modulus of composed materials (conductor and insulation) as a function of the fill factor has to be taken into account \(E(k_{\rm cu})\). Moreover, in certain electrical machines, the stiffness of the slot winding and end-windings are not equal. According to the mechanical properties reported in the literature for the concentrated windings (such as switched reluctance machines), the [PERSON]'s modulus of both slot windings and end-windings is equal.