## 5 Results and Discussion
\begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{\(\mathbf{B}\), \(\mathbf{\gamma}\) = 13.8} & \begin{tabular}{l} **First Critical Speed** \\ **Amplitude Value** \\ \end{tabular} & \begin{tabular}{l} **Second Critical** \\ **Speed** \\ **Amplitude Value** \\ \end{tabular} & \begin{tabular}{l} **Third Critical Speed** \\ **Amplitude Value** \\ \end{tabular} \\ \hline \(b_{m}=0\) & 0.1068045741, \(\tau=0.16\) & 0.090720226, \(\tau=0.78\) & 0.345717672, \(\tau=1.36\) \\ \(b_{m}=0.5\) & 0.106796702, \(\tau=0.16\) & 0.090718, \(\tau=0.78\) & 0.345709, \(\tau=1.36\) \\ \(b_{m}=5\) & 0.106725474, \(\tau=0.16\) & 0.09065, \(\tau=0.78\) & 0.341007, \(\tau=1.36\) \\ \(b_{m}=25\) & 0.10640083, \(\tau=0.16\) & 0.089259, \(\tau=0.78\) & 0.256448, \(\tau=1.36\) \\ \(b_{m}=50\) & 0.105976946, \(\tau=0.16\) & 0.085366, \(\tau=0.78\) & 0.155548, \(\tau=1.36\) \\ \(b_{m}=100\) & 0.105071989, \(\tau=0.16\) & 0.073742909, \(\tau=0.78\) & 0.109144682, \(\tau=1.36\) \\ \(b_{m}=500\) & 0.095853985, \(\tau=0.16\) & 0.039189072, \(\tau=0.79\) & 0.03976382, \(\tau=1.43\) \\ \(b_{m}=5000\) & 0.054819732, \(\tau=0.15\) & 0.032983197, \(\tau=1\) & – \\ \hline \hline \multirow{2}{*}{\(\mathbf{D}\), \(\mathbf{\gamma}\) = 13.8} & \begin{tabular}{l} **First Critical Speed** \\ **Amplitude Value** \\ \end{tabular} & \
\begin{tabular}{l} **Second Critical** \\ **Speed** \\ **Amplitude Value** \\ \end{tabular} &
\begin{tabular}{l} **Third Critical Speed** \\ **Amplitude Value** \\ \end{tabular} \\ \hline \(b_{m}=0\) & 0.066385927, \(\tau=0.16\) & 0.04010122, \(\tau=0.78\) & 0.453737506, \(\tau=1.36\) \\ \(b_{m}=0.5\) & 0.066382727, \(\tau=0.16\) & 0.040109872, \(\tau=0.78\) & 0.45372321, \(\tau=1.36\) \\ \(b_{m}=5\) & 0.066353881, \(\tau=0.16\) & 0.040087202, \(\tau=0.78\) & 0.447400839, \(\tau=1.36\) \\ \(b_{m}=25\) & 0.066224805, \(\tau=0.16\) & 0.045069061, \(\tau=0.79\) & 0.334005995, \(\tau=1.36\) \\ \(b_{m}=50\) & 0.066061617, \(\tau=0.16\) & 0.048328381, \(\tau=0.79\) & 0.198268343, \(\tau=1.36\) \\ \(b_{m}=100\) & 0.065730061, \(\tau=0.16\) & 0.041947113, \(\tau=0.79\) & 0.088761642, \(\tau=1.36\) \\ \(b_{m}=500\) & 0.063037959, \(\tau=0.16\) & 0.01798351, \(\tau=0.79\) & 0.016219871, \(\tau=1.37\) \\ \(b_{m}=5000\) & 0.054819732, \(\tau=0.16\) & 0.014450226, \(\tau=0.85\) & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Critical Speeds. Figure 2: Amplitude–frequency characteristics of natural oscillations of the rotor for \(s=1\) and \(\gamma=13.8\). As the extent of filling increases, as mentioned above, the amplitudes of the second and third critical frequencies increase (see Figures 4 and 5). For example, at \(\gamma\) = 4.56 (\(r_{0}=0.8R\)) and \(b_{m}\) = 0, the maximum values of the rotor amplitudes at the second and third critical frequencies are 3.3-fold and 5.4-fold greater than the values for the same case at \(\gamma\) = 13.8, and are 2.64-fold and 17.45-fold greater than the amplitudes at the first critical frequency, i.e., at \(\tau\) = 0.16. Moreover, an increase in the amount of liquid in the rotor cavity affects the displacement of the second and third critical frequencies [26]. For example, in this case, the second critical frequency occurs a little earlier, compared with the case of \(\gamma\) = 13.8, at \(\tau\) = 0.69, whereas the third critical frequency appears a little later, at \(\tau\) = 1.76. With an increase in the parameter \(b_{m}\), for example, at \(b_{m}\) = 500 and \(b_{m}\) =5000, the oscillation amplitudes at the second and third critical speeds of the rotor and foundation, as expected, are damped up to their complete damping [27], which also indicates a positive effect of the high-frequency magnetic field on the system. The numerical values of the amplitudes for this case are presented in Table 2.