## 2 Materials and Methods
### Numerical Modeling
The main geometrical parameters and dimensions are summarized in Table 1. The main mechanical material properties of the modelled IPM motor components are listed in Table 2.  
The mechanical material properties of ferromagnetic material M270-35A (Cogent Power Ltd, Surahammar, Sweden) and permanent magnets NdFeB (Arnold Magnetic Technologies AG, Lupfig, Switzerland) listed in Table 2 were taken from the manufacturer data sheets. For the shaft, motor frame, back and front end cover, the built-in material properties from the ANSYS library were assigned to the \begin{table}
\begin{tabular}{c c} \hline \hline
**Stator Length** & **90 mm** \\ \hline Inner Radius of the Stator & 50.7 mm \\ \hline Outer Diameter of the Stator & 90 mm \\ \hline Stator Teeth Length & 29.3 mm \\ \hline Number of Stator Slots & 36 \\ \hline Height of the End Windings & 40 mm \\ \hline Rotor Length & 90 mm \\ \hline Rotor Outer Radius & 50 mm \\ \hline Rotor Inner Radius & 15.5 mm \\ \hline Length of the Shaft & 207.5 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical properties of the IPM components. Figure 1: The 3D geometry of the interior permanent magnet IPM motor: (**a**) stator, rotor and shaft with the housing; (**b**) stator, rotor with magnets and the shaft; (**c**) rotor with magnets and shaft; (**d**) shaft only; (**e**) stator with windings; (**f**) dimensions of the stator lamination sheet. model since they matched the real materials of the IPM motors. All the mechanical material properties for all motor components were considered as homogeneous and isotropic (i.e., core lamination was neglected). The mechanical material properties such as Young's modulus \(E\), mass density \(\rho\) and Poisson's ratio of the slot windings and the end-windings were defined in our model based on previous experimental and numerical studies of similar mechanical structures [9]. More precisely, the mechanical properties of the slot and end-winding needed for the structural modal analysis were determined as the equivalent mechanical properties of the composite materials, which consist of enameled wires, insulation sheets and epoxy resin [9]. The equivalent Young's modulus of the windings, which has previously been experimentally determined as the second order polynomial function of the copper fill factor \(k_{\text{cu}}\), was therefore used, according to Equation (4) [9]:  
\[E(k_{\text{cu}})=0.0004k_{\text{cu}}^{2}+0.0212k_{\text{cu}}+0.694 \tag{4}\]  
where the fill factor \(k_{\text{cu}}\) is the actual cross section of a conductor to the total area of the slot expressed in percent. In general, the fill factor in the conventional machine windings designs spans the range of values \(0.35\leq k_{\text{cu}}\leq 0.8\)[9]. The equivalent mass density \(\rho_{\text{eq}}\) of a composite material can also be easily expressed as a function of fill factor \(k_{\text{cu}}\), according to Equation (5):  
\[\rho_{\text{eq}}(k_{\text{cu}})=\rho_{\text{cu}}\;\;k_{\text{cu}}+\rho_{\text {ins}}\;(1-k_{\text{cu}}) \tag{5}\]  
where \(\rho_{\text{cu}}\) and \(\rho_{\text{ins}}\) stand for mass density of copper and insulation, respectively. Using the numerical model proposed in this study and considering Equations (4) and (5), the investigation of the natural frequency dependency on the fill factor \(k_{\text{cu}}\) (being one of the key parameters in mechanical, magnetic and thermal aspects of the electrical machine design) was carried out. The reference model of slot winding and end-windings was built and analyzed for \(k_{\text{cu}}=41\%\)[9]. The rotor dynamic analysis implemented into the ANSYS Mechanical was performed by numerically solving the equation of motion, which takes into account the gyroscopic effect of the mechanical system, which can be written as follows:  
\[[M][u]+\left([C]+\left[C_{gyr}\right]\right)\{i\}+[K][u]=\{F\} \tag{6}\]  
where \([M]\), \([C]\) and \([K]\) are structural mass, damping and stiffness matrices, respectively.