## 2 Materials and Methods  
The rotor system rotates on rolling bearings (see Figure 1). In this case, elastic deformations in the rolling bearing occur in the radial and axial directions and are nonlinear [1]. The radial compliance of the bearings is caused by the deformation of the rolling elements and roller ways at the points of contact [2]. Consider a symmetrical vertical rotor of mass \(m\), having a cylindrical cavity of radius \(R\), and a static imbalance \(e\). A cylinder of height \(h\) is partially filled with a weakly conductive viscous liquid. The angular velocity of the rotor (shaft) \(\Omega_{0}=const\pi\) is considered sufficiently large as it is beyond its critical velocity. The equations of the bearing static equilibrium are compiled in accordance with the [PERSON] theory [26; 27; 28]. The rolling bearing has a nonlinear stiffness characteristic of the type:  
\[F_{C}=c_{0}\delta_{r}+c_{1}\delta_{r}^{3}\]  
where \(F_{C}\) is a component of the restoring force in the radial direction, \(\delta_{r}\) is the deformation in the radial direction, \(c_{0}\) and \(c_{1}\) are stiffness coefficients for the linear and cubic terms. The outer ring is rigidly connected to the foundation of mass \(M\), which is mounted on an elastic support, with a linear stiffness coefficient \(c_{2}\). To compile the equation of motion of the system, a fixed coordinate system \(OXY\) is introduced. In an equilibrium state, the geometric center of the shaft (rotor) and the center of gravity of the foundation coincide with the origin of the fixed coordinate system [3]. The coordinates in the displaced position of the center of the shaft (rotor) \(O_{1}\) are denoted by \(x_{1}\) and \(y_{1}\), and the center of gravity of the rotor is denoted by \(x\) and \(y\). The coordinates of the center of gravity of the foundation \(O_{2}\) are denoted by \(x_{2}\) and \(y_{2}\), \(c_{0}\) and \(c_{1}\) are the coefficients of rigidity of the rotor support (rigidity of the rolling bearing), \(\chi\) and \(\chi_{0}\) are the coefficients of external friction [4; 5; 6]. It is assumed that the rotor performs a plane-parallel motion, and there is no rotation of the foundation around the coordinate axes. In this case, the equations of motion of the system are written as:  
\[\left. \begin{array}{c}m\ddot{x}+2c_{0}(x-x_{2})+2c_{1}(x-x_{2})^{3}+\chi \dot{x}=me\Omega_{0}^{2}\cos\Omega_{0}t+F_{x}\\ m\ddot{y}+2c_{0}(y-y_{2})+2c_{1}(y-y_{2})^{3}+\chi\dot{y}=me\Omega_{0}^{2}\sin \Omega_{0}t+F_{y}\\ m\ddot{x}_{2}+2c_{2}x_{2}-2c_{0}(x-x_{2})-2c_{1}(x-x_{2})^{3}+\chi_{0}\dot{x}_{ 2}=0\\ m\ddot{y}_{2}+2c_{2}y_{2}-2c_{0}(x-y_{2})-2c_{1}(y-y_{2})^{3}+\chi_{0}\dot{y}_{ 2}=0\end{array} \right\} \tag{1}\]  
where F\({}_{\rm x}\) and F\({}_{\rm y}\) are components of the fluid reaction force:  
\[F_{x}=Rh\int_{0}^{2\pi}\sigma_{n}\Big{|}_{r=R}\cos(\Omega_{0}t+\phi)d\phi \tag{2}\]  
\[F_{y}=Rh\int_{0}^{2\pi}\sigma_{n}\Big{|}_{r=R}\sin(\Omega_{0}t+\phi)d\phi \tag{3}\]  
where \(h\) is the height of the rotor cavity and \(\sigma_{n}\mid_{r=R}\) is the normal pressure of the viscous fluid on the rotor wall. Figure 1: Scheme of a rotor partially filled with a weakly conductive liquid mounted on an elastic foundation. Excluding the unknown pressure \(P\) from system Equation (5), taking into account Equations (6) and (7), we obtain:  
\[\frac{\partial f}{\partial t}+bf-\
u\Delta f=-\left(u\frac{\partial f}{\partial r }+\frac{\
u}{r}\frac{\partial f}{\partial\phi}\right) \tag{12}\]  
where \(\Delta\) is the Laplace operator in the polar coordinate system. Equation (12) is solved by the method of successive approximations [30]. In the equilibrium position of the rotor and the foundation, i.e., in the absence of oscillations of the system [31], the viscous fluid rotates with the rotor as a single solid body.