## 2 Materials and Methods
### Analytical Analysis
#### 2.2.2 Analytical Solutions for the Stator with Windings  
In order to analytically estimate the influence of the presence of the slot winding and end windings on the circumferential natural frequencies of the IPM stator, the analytical formula developed by [PERSON], [PERSON] and [PERSON] (2017), which takes into account the windings, stator teeth, effects of shear, and rotary inertia, was used, as defined by Equation (11):  
\[f_{m\geq 2}\ =\ \frac{m\left(m^{2}-1\right)a\ f_{0}}{\sqrt{\left(\left(m^{2}+1 \right)\ +\ a^{2}\ \left(m^{2}-1\right)\left(4\ m^{2}+\ m^{2}\ \frac{d}{w}+3\right)\right)}} \tag{11}\]  
where \(m\) is the circumferential mode number, and the coefficient \(a\) relates to the thickness and the mean radius \(T_{\rm sy}\) and \(R_{\rm m}\) of the stator yoke, respectively, according to Equation (12):  
\[a=\frac{1}{2\ \sqrt{3}}\ +\ \frac{T_{sy}}{R_{m}}. \tag{12}\]  
The \(f_{0}\) is the frequency of the pulsation vibration mode (\(m=0\)), which relates to the Young's modulus of elasticity of the stator \(E\), density of the stator material \(\rho_{\rm s}\), the mean radius of the stator yoke \(R_{\rm m}\) and the factor \(w\) according to t Equation (13):  
\[f_{0}\ =\ \frac{1}{2\ \pi\ R_{m}}\ \sqrt{\frac{E}{\rho_{\rm s}\ w}}. \tag{13}\]  
The factor \(w\) in Equations (11) and (13) denotes the weight of the system defined as:  
\[w=1+\ \frac{w_{w}\ +\ w_{i}\ +w_{p}}{w_{y}} \tag{14}\]  
where \(w_{w}\), \(w_{i}\), \(w_{\rm p}\), and \(w_{\rm y}\) denote the total weights of winding in the slots and end-windings, insulation, stator teeth and stator yoke, respectively. In order to express the modal frequency \(f_{m=2}\) and the pulsation frequency \(f_{0}\) as a function of space winding factor \(k_{\rm cu}\) (i.e., \(f_{m=2}\ (k_{cu})\) and \(f_{0}\ (k_{cu})\)), Equation (4) should be introduced into Equation (13) and Equation (5) into Equation (14), which yields:  
\[w=1+\ \frac{V_{wi}\left(\rho_{\rm{\alpha}}\ \ k_{cu}+\rho_{\rm{\beta}ns}\ (1-k_{cu})\right)+w_{p}}{w_{y}} \tag{15}\]  
Finally, the factor \(d\) in Equation (11) is defined as follows:  
\[d=1+\ \frac{91\ N_{sp}\ A_{sp}\ l_{sp}^{2}\left(w_{w}\ +\ w_{i}\ +w_{p}\ \right)}{L_{st}\ R_{m}\ w_{p}\ T_{sy}}\left(\frac{1}{3}+\ \frac{T_{sy}}{2\ h_{sy}}+ \left(\frac{T_{sy}}{2\ h_{sy}}\right)^{2}\right) \tag{16}\]where \(N_{\text{sp}}\), \(A_{\text{sp}}\), \(h_{\text{sp}}\) and \(L_{st}\) are the number of stator poles, the circumferential cross-sectional area of each stator tooth, the height of the stator pole and the length of the stator stack, respectively. In addition to the circumferential natural frequencies, the bending natural frequencies should also be investigated when designing electrical machines. For the stator, the importance of the bending frequency depends on the ratio between the diameter and the length of the stator [7,17]. The analytical solution for calculation of the fundamental natural frequency (\(f_{n}\)) of a solid bar under a free vibration condition is defined as follows:  
\[f_{n}=\frac{a_{n}}{2\pi}\ \sqrt{\frac{E\,I}{\mu_{L}\,L_{st}^{4}}}, \tag{17}\]  
where \(E\) is Young's modulus, \(I\) is inertia, \(\mu_{L}\) is mass per unit length, \(L_{st}\) is axial length and \(a_{n}\) is a numerical constant [17]. According to Equation (17), the bending frequency of a simple solid bar structure depends on the stiffness as well as on the axial length of the bar \(L_{st}\). 2.3 Analytical Calculation of Rotor Critical Speeds, Whirl Natural Frequencies and Torsional Natural Frequencies  
The analytical rotor dynamic analysis is presented for simplified rotating mechanical structures as illustrated in Figure 3 for: a long symmetrical rotor-shaft mounted on identical flexible isotropic bearings system for the whirl natural vibration study with gyroscopic effect (Figure 3a) and a two-disc rotor system for torsional vibration study (Figure 3b).