## 2 Materials and Methods
Using the continuity Equation (7) from relation Equation (6), taking into account Equations (20) and (21), we obtain the inhomogeneous Euler equation, solving which we obtain an expression for the component \(u\) of the fluid particle velocity in the form:  
\[u=\bigg{[}C_{3}+\frac{C_{4}}{r^{2}}+\frac{i}{\alpha^{2}r}z_{1}(\alpha r)\bigg{]} e^{i(\sigma t-\phi)} \tag{23}\]  
Taking Equation (23) into account, from the continuity equation, we find an expression for the component \(v\) of the fluid particle velocity:  
\[u=\bigg{[}C_{3}-\frac{C_{4}}{r^{2}}-\frac{i}{\alpha^{2}r}z_{1}(\alpha r)+\frac {iz_{0}(\alpha r)}{\alpha}\bigg{]}e^{i(\sigma t-\phi)} \tag{24}\]  
Now using Equations (19), (23) and (24) from the second equation of system Equation (17), we obtain an expression for \(P^{\prime}\):  
\[P^{\prime}=-i\rho r\bigg{[}(2\Omega_{0}+\sigma-ib_{m})C_{3}+\frac{2\Omega_{0} -\sigma+ib_{m}}{r^{2}}C_{4}-\frac{(2\Omega_{0}-\sigma+ib_{m})\
u}{(\sigma-ib_ {m})r}z_{1}(\alpha r)-\frac{\
u}{r}z_{1}(\alpha r)+iB\omega^{2}\bigg{]}e^{i( \sigma t-\phi)} \tag{25}\]  
The integration constants \(C_{1},C_{2},C_{3}\) and \(C_{4}\) are determined using the boundary conditions Equations (8)-(11). Taking into account Equations (18), (24) and (25) from Equations (2) and (3), we find the complex expression for the reaction force of a weakly conductive viscous fluid in the complex plane:  
\[F_{r}=m_{L}\Omega_{0}^{2}Ae^{i\Omega_{0}t}+m_{L}\omega^{2}B\Phi e^{i\omega t} \tag{26}\]  
where \(F_{r}=F_{x}+iF_{y}\), \(m_{L}=\pi\rho R^{2}h\) is the mass of fluid required to completely fill the rotor cavity:  
\[\Phi=\frac{\sigma\
u}{\Delta_{0}}\bigg{\{}f_{12}\bigg{[}\alpha I_{0}(\alpha R )-\frac{I_{1}(\alpha R)}{R}\bigg{]}-f_{11}\bigg{[}\alpha N_{0}(\alpha R)- \frac{N_{1}(\alpha R)}{R}\bigg{]}+1\bigg{\}} \tag{27}\]  
where  
\[ \begin{array}{l}\Delta_{0}=f_{11}f_{14}-f_{12}f_{13}\\ f_{11}=\frac{2R^{2}\alpha}{r_{0}^{2}}I_{0}(\alpha R)-\frac{4\
u}{r_{0}^{2}}I_ {1}(\alpha R)+\Big{(}\frac{4}{r_{0}}+\frac{i(\sigma-ib_{m})}{\
u}r_{0}\Big{)} I_{1}(\alpha r_{0})-2\alpha I_{0}(\alpha r_{0})\\ f_{13}=\frac{\
u\alpha}{2(\sigma-ib_{m})}\bigg{[}\omega^{2}-i\sigma b_{m}-q^{2 }\bigg{(}\Omega_{0}^{2}-\sigma^{2}+2\Omega_{0}(\sigma-ib_{m})+\frac{4\
u i \sigma}{r_{0}^{2}}\bigg{)}\bigg{]}I_{0}(\alpha R)+\\ +\frac{\
u R}{(\sigma-ib_{m})\
u_{0}^{2}}\bigg{(}\Omega_{0}^{2}+2\Omega_{0} \sigma-\sigma^{2}+i\sigma b_{m}+\frac{4\
u i\sigma}{r_{0}^{2}}\bigg{)}I_{1}( \alpha R)-\\ -\frac{\
u}{(\sigma-ib_{m})\
u_{0}}\bigg{(}\Omega_{0}^{2}+2\Omega_{0}\sigma+ \frac{4\
u i\sigma}{r_{0}^{2}}\bigg{)}I_{1}(\alpha r_{0})-\frac{2\
u i\sigma} {\alpha r_{0}^{2}}I_{0}(\alpha r_{0})\end{array}  \tag{28}\]  
Replacing the Bessel functions in the expressions for \(f_{11}\) and \(f_{13}\) by the Neumann functions of the same order, we obtain expressions for the functions \(f_{12}\) and \(f_{14}\). The quantity \(q=\frac{R}{r_{0}}\) characterizes the degree of filling of the rotor cavity with liquid. Now, when the fluid reaction force \(F_{r}\) is known, we can determine the amplitude of forced oscillations and self-oscillations of the system [8; 37; 38; 39; 40]. Taking into account the identity of the first equation to the second, and the third to the fourth in system Equation (1), we will further consider a system of the form:  
\[ \begin{array}{l}m\ddot{x}+2c_{0}(x-x_{2})+2c_{1}(x-x_{2})^{3}+\dot{x}\dot{x}=m \epsilon\Omega_{0}^{2}\cos\Omega_{0}t+F_{x}\\ M\ddot{x}_{2}+2c_{2}x_{2}-2c_{0}(x-x_{2})-2c_{1}(x-x_{2})^{3}+\chi_{0}\dot{x}_ {2}=0\end{array}  \tag{29}\]  
Substituting the real parts from Equations (15), (16) and (26) into Equation (29), using the method of the imaging function, and making dimensionless each term of system Equation (29), we obtain a system of nonlinear algebraic equations for unknown dimensionless amplitudes \(a\), \(b\), \(c\) and \(d\). We obtain:  
\[ \begin{array}{l}l_{11}a+l_{13}c=\mu s^{2}\\ l_{12}b+l_{14}a=0\\ l_{13}c-(a-c)-\eta(a-c)\Big{[}(a-c)^{2}+2(b-d)^{2}\Big{]}=0\\ l_{14}d-(b-d)-\eta(b-d)\Big{[}(b-d)^{2}+2(a-c)^{2}\Big{]}=0\end{array}  \tag{30}\]  
where  
\[