## 3 Results and Discussion
### Natural Frequencies and Modal Shapes of the Stator with and without Windings
On the other hand, the studies on the asynchronous machine windings (i.e., distributed windings), the [PERSON]'s modulus of the end-windings (which typically overhang) is higher compared to the [PERSON]'s modulus of the windings in the slots, due to the additional stiffening process of the end-windings [6,9]. Figure 5: Experimental results of natural frequencies measured for: **a**) stator stack without windings and **b**) stator stack with the slot windings and end-windings. The \(E(k_{\rm cu})\) and the \(\rho_{\rm eq}(k_{\rm cu})\) of the slot and end windings can be successfully taken into account by means of numerical FEM modeling. Therefore, in order to investigate the dependency of the second mode shape natural frequency on the fill factor \(f_{m=2}(k_{\rm cu})\), the numerical model presented in this study with the simplified slot windings and end-windings (Figure 1e) was used by taking into account the Young's modulus and mass density of windings as a function of fill factor \(k_{\rm cu}\) (i.e., (\(E(k_{\rm cu})\) according to Equation 4 and \(\rho_{\rm eq}(k_{\rm cu})\) according to Equation 5). A sensitivity analysis was also performed in order to investigate the dependence of the \(E(k_{\rm cu})\) on the fill factor \(f_{m=2}(k_{\rm cu})\). Figure 6: Front view of the 3D numerical model of the stator stack; calculated natural frequencies and their corresponding modal shapes: (**a**) \(m=2\), (**b**) \(m=3\), (**c**) \(m=4\), in (**d**) \(m=8\) and (**e**) \(m=0\) (pulsating mode). The normalized total deformations are indicated from minimum (blue color) to maximum (red color). Figure 7: Numerically calculated modal shapes with the model of stator with slot and end-windings at the natural frequencies: (**a**) f = 525 Hz (\(m=2\)), (**b**) 754 Hz, (**c**) f = 1393 Hz, (**d**) f = 1762 Hz and (**e**) f = 5113 Hz (pulsating modal shape \(m=0\)). identify the \(E(k_{\rm cu})\) of the real IPM stator, since, in this study, the \(E\) of slot windings and end-windings were not directly measured (as stated before, the material properties were taken from the literature [9]). For this purpose, the numerical calculations were done for different \(k_{\rm cu}\) values (within the range from 0.35 to 0.8) and different \(k_{\rm cu}\) ratios of the end-windings were compared to the \(k_{\rm cu}\) of the slot windings (from 0.5\(\cdot\)\(k_{\rm cu}\) to 2\(\cdot\)\(k_{\rm cu}\)) (Figure 11). The \(E(k_{\rm cu})\) was identified by comparison of the numerical results with experimental measurements (Figure 5b). In Figure 11, three different cases of mechanical properties of the slot windings versus end-windings are given: (1) slot windings and end-windings with equal \(k_{\rm cu}\); (2) slot windings with \(k_{\rm cu}\) and end-windings with 2 \(k_{\rm cu}\); and (3) slot windings with \(k_{\rm cu}\) and end-windings with 0.5 \(k_{\rm cu}\). It should be noted that the results of the sensitivity analysis indicate that best agreement/matching between the numerical and experimental results can be obtained when the fill factor of the end-windings is slightly lower (i.e., around 0.75\(\cdot\)\(k_{\rm cu}\)) compared to the \(k_{\rm cu}\) of the slot winding (Figure 11). However, the equal \(k_{\rm cu}\) was kept in both slot and end windings (in the model defined as \(k_{\rm cu}=0.41\)) in order to simplify the numerical model as much as possible, since, in this case, a small difference between the experimental and numerical results was obtained: (i.e., 4.7%, Table 4, \(m=2\), numerical model: \(f=525\) Hz vs. experimental measurements \(f=500.5\) Hz). Based on this, the modeling approach presented in this study, which proposes a simplified geometry of the slot winding and end-winding as a solid object (Figure 1e) with equivalent mechanical properties of the composite material, was successfully validated with the experimental measurements.