## 2 Materials and Methods
### Numerical Modeling  
The numerical modal analysis and the rotor dynamic analysis were performed using ANSYS Mechanical software (ANSYS 2019 R1, ANSYS, Inc., Canonsburg, USA), which is based on the finite element method (FEM). First, the static modal analysis for the free-fee vibration condition was performed in order to investigate the structural mechanical vibration characteristics, such as natural frequencies and the modal shapes of the IPM motor components. The numerical modal analysis in the ANSYS software is executed by numerically solving the general system of equations of motion (Equation 1):  
\[[M]\{u\}+[C]\{\dot{u}\}+[K]\{u\}=\{F\} \tag{1}\]  
where \([M]\), \([C]\) and \([K]\) are matrices of mass, damping and stiffness, respectively, while the vector \([u]\) stands for the unknown modal displacements resulting in the modal shapes to be calculated. The vector \(\{F\}\) represents the external force for modeling the forced mechanical vibration condition (e.g., electromagnetic force). In our study, the \([C]\) and \(\{F\}\) were set to zero since the mechanical structures are analyzed according to the unforced and un-damped mechanical vibration conditions, as follows:  
\[[M]\{u\}+[K]\{u\}=\{0\} \tag{2}\]  
Therefore, by considering the zero damping and zero external excitation force, the ANSYS program numerically solves the eigenvalue problem for the free vibration condition according to the following equation:  
\[\left\{[K]-\left(\omega\right)^{2}[M]\right\}\{u\}=\{0\} \tag{3}\]  
The output solution provides the natural angular frequencies \(\omega\) (in rad/sec) and the corresponding modal shapes resulting from the calculated modal displacement \(\{u\}\). The natural frequencies \(f\) in Hz are then calculated as \(f=\omega/(2\cdot\pi)\). The eigenvectors \(\{u\}\) thus represent the mode shapes of the analyzed IPM mechanical structure when vibrating at the natural frequency \(f\). In addition to the geometry of the mechanical structure to be analyzed, the key input material parameters needed for the static numerical analysis are: [PERSON]'s modulus \(E\) (Pa), [PERSON]'s ratio and mass density \(\rho\) (kg/m\({}^{3}\)) of the materials composing the modeled structure. In our study, the 3D geometries of the investigated IPM components were preprocessed using the Ansys Space Claim environment in order to describe the geometry of the real IPM motor configurations as accurately as possible. Figure 1 depicts the geometry of the following IPM components/assemblies that were modelled: the motor assembly with the housing (Figure 1a), stator stack with the rotor and the shaft, rotor with magnets and the shaft (Figure 1c), the shaft only (Figure 1d), the stator stack with slot windings and end-windings (Figure 1e), and the stator lamination (Figure 1f). It should be noted that one of the goals of this study was to investigate whether an accurate and fast mechanical vibration analysis can also be done with adequately simplified geometrical and mechanical material properties of the real IPM windings. A simplified stator geometry model was therefore built in the form of solid bodies of slot and end-windings (as shown in Figure 1e) with equivalent mechanical material properties of composite materials assigned based on the experimental and numerical data found in the literature [9]. The accuracy of the modal analysis results was controlled by comparing the numerical results to the experimental measurements performed on the real IPM stator with the distributed windings. This allowed us to validate the simplified numerical model with equivalent geometrical and material properties (Figure 1e), with the goal to reduce the time needed for numerical calculations while still preserving the accuracy of the numerical results. However, one should bear in mind that for the precise magnetic study and/or coupled multi-physics analysis, a more complex numerical model with a more detailed description of the IPM distributed windings (its detailed geometrical and material properties) should be built.