## 2 Materials and Methods
### Analytical Analysis
#### 2.2.2 Analytical Solutions for the Stator with Windings
For the long symmetrical rotor mounted on identical flexible isotropic bearings with the stiffness \(k=k_{11}=k_{22}\), (Figure 3a), the analytical solutions for the natural whirl frequencies and critical rotational speed considering the gyroscopic effect can be derived based on the equations of motion approach, i.e., force and moment equations for the translational and rotational movement, respectively [20, 21, 22]. Therefore, the analytical solution for the natural frequencies of the rotating rotor on the shaft with the bearing system considering the harmonic motion in four degrees of freedom (4 DOF) (Figure 3a), can be written as follows:  
\[\left[ \begin{array}{cccc}m_{sh}\ f^{2}+2\ k&0&0&0\\ 0&m_{sh}\ f^{2}+2\ k&0&0\\ 0&0&I_{d}\ f^{2}+0.5\ k\,L^{2}&I_{p}\ \Omega\,\
u\\ 0&0&-I_{p}\Omega\,\
u&I_{d}\ f^{2}+0.5\ k\,L^{2}\end{array} \right]\left\{ \begin{array}{c}X_{transDOF}\\ Y_{transDOF}\\ Y_{rotDOF}\\ Y_{rotDOF}\end{array} \right\}\ =\ \left\{ \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \right\} \tag{18}\]  
where \(f\) is the natural whirl frequency, \(m_{sh}\) is the mass of the shaft, \(k\) is the dynamic parameter (the stiffness) of the isotropic bearings, \(\Omega\) is the rotational speed, \(X_{transDOF}\), \(Y_{transDOF}\), \(X_{rotDOF}\), \(Y_{rotDOF}\) are displacement in the x direction, displacement in the y direction, rotation around the x axis, and rotation around the y axis, respectively (as depicted in Figure 3a), and \(I_{\text{p}}\) and \(I_{\text{d}}\) are the cylinder polar moment of inertia and its diameter moment of inertia, respectively. Figure 3: Illustration of: (**a**) a long symmetrical rotor mounted on identical flexible isotropic bearings and (**b**) a two-disc rotor system for torsional vibration study. The natural frequencies are than calculated from the determinant of the matrix of Equation (18), which yields:  
\[\left(m_{sh}\;f^{2}+2\;k\right)^{2}\left\{\left(l_{d}\;f^{2}+0.5\;k\;L^{2}\right) ^{2}+\;\left(l_{p}\;\Omega\;v\right)^{2}\right\}=0. \tag{19}\]  
By solving Equation (19), four natural frequencies are obtained. The first two (Equation (20)) are pure translational speeds (which result in cylindrical modal shapes) and do not depend on the rotational speed of the rotor (no influence of the gyroscopic effect). This frequency therefore does not split and it is typical for symmetric and perfectly centered rotational systems, describing pure translational rotational motion (i.e., resulting in the cylindrical rotational modal shape) [20; 21]. \[f_{1}=f_{2}=\frac{1}{2\pi}\sqrt{\frac{2\;k}{m_{sh}}} \tag{20}\]  
The third and the fourth natural whirl frequency equations are affected by the gyroscopic moment, which typically reflects the dependency of the natural frequency of the rotating mechanical system on the rotational speed of the system. Based on the theory of the rotor dynamics, the critical rotational speeds can appear in the same direction as the direction of the rotor rotation (i.e., forward whirl direction FW) and in the opposite direction compared to the rotation direction of the rotor (i.e., backward whirl direction BW). The value and the direction of the rotation depend on the configuration and the mechanical material properties of the rotor, shaft and the bearings system. Thus, the whirl frequencies typically split into two functional dependencies indicating the forward whirl motion (FW) and backward whirl motion (BW) due to the gyroscopic effect. Analytically computed eigenvalues as a function of the shaft's rotation speed given in Equation 21 and Equation 22 can be used to draw the Campbell diagram \(f(\Omega)\).