## 3 Results  
The amplitude-frequency characteristics for an industrial centrifuge with parameters \(m=24\) kg, \(M=25\) kg, \(e=0.001\) m, \(c_{0}=1.1\times 10^{7}\) kg/s\({}^{2}\), \(c_{1}=0.87\times 10^{11}\) kg/m\({}^{2}\)s\({}^{2}\), \(c_{2}=3.26\times 10^{5}\) kg/s\({}^{2}\), \(\chi_{0}=6.59\) kg/s, \(\chi=4200\) kg/s were obtained by the following methods: from the algebraic system of Equation (13) using the [PERSON]'s method (Figures 3 and 4, the blue curve), from differential Equation (8) using the 4 th order Runge-Kutta-Fehlberg method (Figures 3 and 4, the dotted red curve), from differential Equation (8) at \(c_{1}=0\) by the 4 th order Runge-Kutta-Fehlberg method (Figures 3 and 4, the black curve) and compared with the amplitude-frequency characteristic obtained from the algebraic system of Equation (23) derived by the method of Jacobi elliptic functions (Figures 3 and 4, the green curve). As a result of calculations, three resonant regions were determined for the rotor and the foundation, the presence of which, as will be shown below, is due to the nonlinearity of the supports and the mobility of the foundation. The curve of the amplitude-frequency characteristic in the first resonant region is typical of linear oscillations and is observed for all cases at \(\omega_{f}=0.051\). When calculating the amplitudes by the [PERSON] method (Figures 3 and 4, the blue curve), the deviation from the results of the numerical method (Figures 3 and 4, the dotted red curve) in the case of the rotor was 2.8%, and in the case of the foundation it was 9.1%. In cases of calculation  
The amplitude-frequency characteristic curve for the second resonant region is typical of nonlinear oscillations with a rigid characteristic. Nonlinear oscillations and bifurcation modes in this case are due to nonlinearity of the restoring forces of the elastic supports. In this case, the amplitude breakdown for each method occurs at different frequencies with different amplitudes. For example, for the 4 th order Runge-Kutta-Fehlberg numerical method, the resonance was observed at \(\omega_{f}\) = 1.33, in the case of the Van der Pol method, the deviation from the amplitude of the numerical method for the rotor was 17.6%, for the foundation 33.87%, while the resonance occurred at \(\omega_{f}\) = 1.25. In the case of Jacobi elliptic functions, this breakdown can be observed at \(\omega_{f}\) = 1.3, the deviation from the amplitudes for the rotor is 8.66%, and for the foundation it is 31.7%. As can be seen in the latter case, the deviations of the values of the resonant frequency and amplitude from the numerical method are much smaller. Amplitude values are presented in Table 2.  
Nonlinear systems are characterized by anharmonicity, i.e., the presence of ultraharmonic and subharmonic resonances that are multiples of the frequency of the main resonance. For the linear case, the main resonance occurs at \(\omega_{f}\) = 1. In our case, the main resonance region, due to nonlinearity, is observed at \(1\leq\omega_{f}\leq 1.33\). In order to determine additional resonances, the amplitude-frequency characteristics of the corresponding linear case (Figures 3 and 4, the black curve), the amplitude-frequency characteristics obtained by the numerical method (Figures 3 and 4, the red curve), and the amplitude-frequency characteristics of the analytical by the Van [PERSON] method (Figures 3 and 4, the blue curve) were constructed. Since the second resonance, located to the right of the main one at \(\omega_{f}\) = 1, \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Second Resonant** & **Rotor’s** & **Foundation’s** \\  & **Frequency-\(\omega_{f}\)** & **Amplitude-\(f_{1}\)** & **Amplitude-\(f_{2}\)** \\ \hline Numerical Solution & & & \\ (Nonlinear Case) & 1.333 & 11.556 & 1.333 \\ \hline [PERSON]’s method & 1.256 & 14.022 & 2.016 \\ \hline Jacobi Elliptic & & & \\ Function method & 1.301 & 12.554 & 1.954 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Amplitude values at the second resonance for different methods.