## 2 Materials and Methods
\begin{array}{l}l_{11}=-\mu_{1}s^{2}+ips;\;l_{12}=-(\mu+\mu_{L}\Phi)\tau^{2} +ip\tau;\;l_{13}=\big{(}g_{0}-s^{2}+ip_{0}s\big{)};\;l_{14}=\big{(}g_{0}-\tau^{ 2}+ip_{0}\tau\big{)}.\\ n_{0}^{2}=\frac{2c_{0}}{\dot{M}};\;n_{1}=\frac{2c_{1}}{\dot{M}};\;n_{2}^{2}= \frac{2c_{2}}{\dot{M}};\;\mu=\frac{m_{1}}{\dot{M}};\;\mu_{L}=\frac{m_{L}}{ \dot{M}};\;m_{L}=\pi\rho R^{2}H;\;\mu_{1}=\mu+\mu_{L};\;p=\frac{\chi}{Mn_{0}}; \\ p_{0}=\frac{\chi_{0}}{Mn_{0}};\;g_{0}=\frac{\chi_{0}}{c_{0}};\;\eta=\frac{3\mu n _{1}e^{2}}{4\pi n_{0}^{2}};\;a=\frac{A}{e};\;b=\frac{b}{e};\\ c=\frac{c}{e};\;d=\frac{D}{e}\end{array}  \tag{31}\]  
where \(\tau=\frac{\omega}{n_{0}}\) and \(s=\frac{\Omega_{0}}{n_{0}}\) are the dimensionless frequency of natural vibrations of the system and the angular velocity of the rotor, respectively. From the formulas that determine the unknowns \(a\), \(b\), \(c\) and \(d\), it is obvious that the amplitudes of the forced and natural oscillations of the system depend on the natural frequency of the system and they are interdependent [41]. This is one of the specific features of a nonlinear system. From formulas Equations (15) and (16), it follows that one more frequency is superimposed on the forced oscillations of the system, i.e., the system makes oscillations consisting of the sum of two harmonic oscillations [42]. In this case, the system performs a processional motion that differs from a circular precession [43]. From the first and second equations of system Equation (30), we have:  
\[c=\beta_{13}+\beta_{11}a\]  
\[a-c=a(1-\beta_{11})-\beta_{13} \tag{32}\]  
\[d=-\frac{l_{12}}{l_{14}}b \tag{33}\]  
\[b-d=b\bigg{(}1+\frac{l_{12}}{l_{14}}\bigg{)} \tag{34}\]  
\[b^{2}=\lambda_{2}a^{2}+\lambda_{1}a+\lambda_{0} \tag{35}\]  
where  
\[ \begin{array}{l}\beta_{11}=-\frac{l_{11}}{l_{13}};\;\beta_{13}=\frac{\mu s^{ 2}}{l_{13}};\;\beta_{10}=-\frac{(l_{12}l_{14}+l_{12}+l_{14})}{\eta(l_{12}+l_{14 })};\;\lambda_{0}=-\frac{2l_{14}^{2}(1-\beta_{11})^{2}}{(l_{12}+l_{14})^{2}}; \\ \lambda_{1}=-\frac{4l_{14}^{2}(1-\beta_{11})l_{13}}{(l_{12}+l_{14})^{2}};\; \lambda_{2}=(\beta_{10}-2\beta_{13})\frac{l_{14}^{2}}{(l_{12}+l_{14})^{2}} \end{array} \]Now, from the third equation of system Equation (30), taking into account Equations (32), (33), and (35), we obtain an equation of the third power with respect to \(a\):  
\[a^{3}+\delta_{2}a^{2}+\delta_{1}a+\delta_{0}=0 \tag{36}\]  
\[ \begin{array}{l}\delta_{2}=-\frac{3\mu^{2}}{\beta_{12}};\;\delta_{1}=-\frac{ \frac{1_{11}\beta_{3}^{3}+\beta_{15}\mu^{2}}{\beta_{12}\beta_{14}}+\frac{ \beta_{16}}{\beta_{14}};\;\delta_{0}=\frac{\mu^{2}\left(\frac{\beta_{13}^{3}- \beta_{16}}{\beta_{12}\beta_{14}}\right)}{\beta_{12}\beta_{14}};\\ \beta_{12}=l_{11}+l_{13};\\ \beta_{14}=3\eta\beta_{12}^{2};\;\beta_{15}=-6\eta\beta_{12}\mu s^{2};\;\beta_ {16}=3\eta\mu^{2}s^{4}-2\eta\beta_{10}l_{13}^{2}-l_{13}^{2}\end{array}  \tag{37}\]  
The cubic Equation (36) is solved analytically by the Cardano method and numerically by the Newton-Raphson method. The coefficients of Equation (36) are, in the general case, complex (when friction forces are taken into account). If friction forces are neglected, i.e., at \(p=p_{0}=0\), the coefficients of Equation (36) become real [44]. In general, Equation (36) has three real roots. Taking into account Equation (37), from the solution of Equation (36), the unknown dimensionless amplitude of forced oscillations of the rotor \(a\) is found, then from Equations (32), (33) and (35), the dimensionless amplitude of forced oscillations of the foundation \(c\) is determined, as well as the dimensionless amplitudes of self-oscillations of the rotor \(b\) and the foundation \(d\). As can be seen from Equations (31)-(36), each value of the amplitude of the forced oscillations of the rotor \(a\) corresponds to three values of the amplitude of the forced oscillations of the foundation \(c\) and self-oscillations of the rotor and foundation \(b\) and \(d\).