## 2 Materials and Methods
### Numerical Modeling
The natural frequencies of the analyzed system were calculated by setting the right hand side of the equation to zero, hence the excitation force \(\{F\}=0\). The \([C_{\text{gyr}}]\) denotes the contribution of the Coriolis effects or the damping matrix contribution due to the gyroscopic effect, which imposes the change in stiffness of the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**The Motor Component** & **Material** & \begin{tabular}{c} **Mass Density** \\ \(\rho\) (**lkg/m\({}^{3}\)**)** \\ \end{tabular} & \begin{tabular}{c} **Young’s** \\ **Modulus (Pa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} \\ \hline \hline \begin{tabular}{c} Stator \\ \end{tabular} & M270-35A & 7650 & \(2\times 10^{11}\) & 0.3 \\ \hline \begin{tabular}{c} Rotor \\ \end{tabular} & M270-35A & 7650 & \(2\times 10^{11}\) & 0.3 \\ \hline \begin{tabular}{c} Permanent Magnets \\ \end{tabular} & NdFeB & 7500 & \(1.6\times 10^{11}\) & 0.24 \\ \hline \begin{tabular}{c} Shaft \\ \end{tabular} & \begin{tabular}{c} Steel [Ansys Built-in \\ Library] \\ \end{tabular} & 7850 & \(2\times 10^{11}\) & 0.3 \\ \hline \begin{tabular}{c} The Slot Windings and the \\ End-Windings \\ \end{tabular} & \begin{tabular}{c} Equivalent Mechanical \\ Properties of the Cooper and \\ the Insulation [9] \\ \end{tabular} & 4373 & \(4.97\times 10^{8}\) & 0.343 \\ \hline \
\begin{tabular}{c} Motor Frame/Housing, \\ Back and Front End Cover \\ \end{tabular} &
\begin{tabular}{c} Aluminum Alloy [Ansys \\ Built-in Library] \\ \end{tabular} & 2770 & \(7.1\times 10^{10}\) & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The mechanical material properties of the IPM components. rotating structure, which is reflected into the split of the rotational vibration modes into the forward whirl mode (FW) and backward whirl mode (BW) [16]. The IPM rotor mechanical assembly for which the numerical rotor dynamic analysis was performed is shown in Figure 2. The analyzed IPM rotor assembly with the shaft and bearings is shown in Figure 2a, where the model of the bearing system is implemented in ANSYS as a mechanical connection type defined to the selected shaft locations. The bearings connection type allows for definition of the stiffness and the damping properties of bearings. The equivalent lumped circuit mechanical model of the bearing system (i.e., the spring-damper bearing model) implemented in the ANSYS environment is illustrated in Figure 2b, where \(k_{11}\), \(k_{12}\), \(k_{21}\), \(k_{22}\) are the stiffness coefficients and \(c_{11}\), \(c_{12}\), \(c_{21}\), \(c_{22}\) are the damping coefficients describing the modeled characteristics of the bearings. Figure 2c shows the numerically modeled IPM rotor configuration with shaft, bearings and the load. The output results of the rotational dynamic analysis in ANSYS Mechanical are provided as numerically computed values of eigenvalues \(f\) (Hz) (i.e., natural frequencies, also referred to as whirl frequencies) as a function of the rotational speed \(\Omega\) (rpm) of the modeled rotating structure (i.e., \(f(\Omega)\)). This function \(f(\Omega)\) is referred to as Campbell diagram (named also whirl speed map). The critical speed can be determined from the Campbell diagram at the intersection points of the \(f(\Omega)\) function with the synchronous spin speed line: \(f=\Omega/60\) (i.e., referred to as 1x critical speed) [16; 18; 22]. The corresponding modal shapes (i.e., lateral and torsional modal shapes, rigid-cylindrical and conical, bending and/or coupled modal shapes) of the rotating structure for each of the whirl frequencies and rotational speeds is also provided in the post processing database. Based on the functional dependency \(f(\Omega)\), the profiles are displayed in the Campbell diagram and the nature of the corresponding modal shapes and the crucial rotational dynamic properties of the modeled rotating structure can be detected.