## 2 Problem Statement  
A symmetrical vertical rotor of mass \(m\), with polar moment of inertia \(J\), having a static imbalance \(e\), rotates on rolling bearings with a constant angular velocity \(\Omega_{0}\). The foundation of mass \(M\) is installed on elastic supports with an equivalent linear stiffness coefficient \(c_{2}\) (Figure 1). It is assumed that the rotor performs a plane-parallel motion, and there is no rotation around the coordinate axes. The radial compliance of bearings occurs due to the deformation of the rolling elements and raceways at the points of contact. Two Cartesian coordinate systems are introduced, fixed--\(Oxyz\) and mobile, rigidly connected with the geometric center of the rotor--\(O_{1}\xi\eta\), where \(\eta\) is the polar axis, and the \(\xi\) axis is arbitrarily drawn through the eccentricity vector of the rotor mass. The motion of the system is considered relative to the fixed coordinate system \(Oxyz\). The coordinates of the geometric center of the rotor are designated as \(O_{1}(x_{1},y_{1})\), and its center of mass as \(O_{s}(x,y)\). The foundation mass center coordinates are denoted as \(O_{2}(x_{2},y_{2})\), \(\chi\) and \(\chi_{0}\) are damping coefficients (Figure 1). The non-linear restoring force of a bearing can be described using the Hertzian contact formula:  
\[F_{c}=C_{b}\delta_{r}^{\frac{3}{2}} \tag{1}\]  
where \(F_{C}\) is the radial force, \(\delta_{r}\) is the radial deformation, \(C_{b}\) is the non-linear stiffness coefficient. For a qualitative analysis of the equations of motion of the system, the restoring force of rolling bearings in the form (1) can be approximated by a power series:  
\[F_{c}=c_{0}\delta_{r}+c_{1}\delta_{r}^{3}, \tag{2}\]  
where \(c_{0}\) and \(c_{1}\) are cubic and linear coefficients of a cubic polynomial, respectively. The kinetic energy of the system is defined as:  
\[T=\frac{1}{2}\Big{(}m\Big{(}\dot{x}^{2}+\dot{y}^{2}\Big{)}+J\Omega_{0}^{2}+M \Big{(}\dot{x}_{2}^{2}+\dot{y}_{2}^{2}\Big{)}\Big{)}. \tag{3}\]  
Considering that:  
\[\delta_{r}^{2}=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}. \tag{4}\]  
The potential energy of the isotropic non-linear elastic field of the system is defined as:  
\[\Pi=\frac{1}{2}\Big{(}c_{0}\Big{(}(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}\Big{)} +c_{2}\Big{(}x_{2}^{2}+y_{2}^{2}\Big{)}\Big{)}+\frac{1}{4}\Big{(}c_{1}\Big{(} (x_{1}-x_{2})^{4}+(y_{1}-y_{2})^{4}\Big{)}\Big{)}. \tag{5}\]  
The energy dissipation function reads:  
\[\Phi=\frac{1}{2}\Big{(}\chi\Big{(}\dot{x}^{2}+\dot{y}^{2}\Big{)}+\chi_{0}\Big{(} \dot{x}_{2}^{2}+\dot{y}_{2}^{2}\Big{)}\Big{)}. \tag{6}\]  
The coordinates of the center of mass of the rotor are given as:  
\[x=x_{1}+e\cos\Omega_{0}t,y=y_{1}+e\sin\Omega_{0}t, \tag{7}\]  
Figure 1: Rotor system on rolling bearings. where \(e\) is the value of the linear displacement of the center of mass of the rotor from its geometric center \(O_{1}O_{s}\), i.e., static imbalance. Using the Lagrange equations of the second kind, we write the equations of motion of the system in the following form:  
\[ \begin{split}& m\ddot{x}_{1}+c_{0}(x_{1}-x_{2})+c_{1}(x_{1}-x_{2})^{ 3}+\dot{x}\dot{x}_{1}=me\Omega_{0}^{2}\cos\Omega_{0}t,\\ & m\ddot{y}_{1}+c_{0}(y_{1}-y_{2})+c_{1}(y_{1}-y_{2})^{3}+\dot{x} \dot{y}_{1}=me\Omega_{0}^{2}\sin\Omega_{0}t,\\ & M\ddot{x}_{2}-c_{0}(x_{1}-x_{2})-c_{1}(x_{1}-x_{2})^{3}+c_{2}x_{ 2}+\chi_{0}\dot{x}_{2}=0,\\ & M\ddot{y}_{2}-c_{0}(y_{1}-y_{2})-c_{1}(y_{1}-y_{2})^{3}+c_{2}y_{ 2}+\chi_{0}\dot{y}_{2}=0,\\ & x_{1}(0)=e,x_{2}(0)=0.1e,y_{1}(0)=0,y_{2}(0)=0,\\ &\dot{x}_{1}(0)=0,\dot{x}_{2}(0)=0,\dot{y}_{1}(0)=0,\dot{y}_{2}( 0)=0.\end{split}  \tag{8}\]  
Equation (8) describes joint movement of the movable foundation and the rotor with static imbalance installed in elastic supports with a nonlinear characteristic. In the case of a fixed foundation, Equation (8) will consist of two Duffing equations describing the equation of motion of a nonlinear Jeffcot rotor, which have been studied in sufficient detail. We introduce the following dimensionless parameters:  
\[