## 2 Problem Statement
#### Equations of Motion's Solution  
Since we are looking for a quasi-periodic solution of Equation (10) and, accordingly, we consider the case with weak nonlinearity, we will use the classic proven Van der Pol method to compare the results that will be obtained using the Jacobi elliptic function method. Since there are damping forces in the system, we represent the functions \(f_{1}(\tau)\) and \(f_{2}(\tau)\) in system (10) in the following form:  
\[ \begin{split}& f_{1}(\tau)=A_{1}(\tau)\cos(\eta\tau)+B_{1}(\tau) \sin(\eta\tau),\\ & f_{2}(\tau)=A_{2}(\tau)\cos(\eta\tau)+B_{2}(\tau)\sin(\eta\tau),\end{split}  \tag{11}\]  
where derivatives \(A_{1}(\tau)\), \(A_{2}(\tau)\), \(B_{1}(\tau)\), and \(B_{2}(\tau)\) are amplitudes slowly changing over the oscillation period. Substituting (11) into the Equation (10) and equating the left and right parts of the equations for the same functions and harmonics, we obtain:  
\[ \begin{split}&\ddot{A}_{1}-\eta^{2}A_{1}+2\dot{B}_{1}\eta+2\zeta_{1 }\Big{(}\dot{A}_{1}+B_{1}\eta\Big{)}+(A_{1}-A_{2})+\frac{3}{4}\varepsilon\Big{(} \big{(}A_{1}-A_{2}\big{)}^{3}+(A_{1}-A_{2})(B_{1}-B_{2}\big{)}^{2}\Big{)}=\eta^{ 2},\\ &\ddot{B}_{1}-\eta^{2}B_{1}-2\dot{A}_{1}\eta+2\zeta_{1}\Big{(} \dot{B}_{1}-A_{1}\eta\Big{)}+(B_{1}-B_{2})+\frac{3}{4}\varepsilon\Big{(} \big{(}B_{1}-B_{2}\big{)}^{3}+(A_{1}-A_{2})^{2}(B_{1}-B_{2}\big{)}\Big{)}=0,\\ &\ddot{A}_{2}-\eta^{2}A_{2}+2\dot{B}_{2}\eta+2\mu\zeta_{2}\Big{(} \dot{A}_{2}+B_{2}\eta\Big{)}-\mu(A_{1}-A_{2})-\frac{3}{4}\mu\varepsilon\Big{(} \big{(}A_{1}-A_{2}\big{)}^{3}+(A_{1}-A_{2})(B_{1}-B_{2}\big{)}^{2}\Big{)}+\mu \lambda A_{2}=0,\\ &\ddot{B}_{2}-\eta^{2}B_{2}-2\dot{A}_{2}\eta+2\mu\zeta_{1}\Big{(} \dot{B}_{2}-A_{2}\eta\Big{)}-\mu(B_{1}-B_{2})-\frac{3}{4}\mu\varepsilon\Big{(} \big{(}B_{1}-B_{2}\big{)}^{3}+(A_{1}-A_{2}\big{)}^{2}(B_{1}-B_{2}\big{)}\Big{)} +\mu\lambda B_{2}=0.\end{split}  \tag{12}\]  
The first and second order derivatives of the amplitudes of the rotor and the foundation are equal to zero, since for the stationarity of the solutions of system (12) it is necessary that \(A_{1}(\tau)\), \(A_{2}(\tau)\), \(B_{1}(\tau)\)), and \(B_{2}(\tau)\) be constant in time, then:\[ \begin{split}&\big{(}1-\eta^{2}\big{)}A_{1}-A_{2}+2\eta\zeta_{1}B_{1}+ \tfrac{3}{4}\varepsilon\Big{(}\big{(}A_{1}-A_{2}\big{)}^{3}+(A_{1}-A_{2})(B_{1} -B_{2})^{2}\Big{)}=\eta^{2},\\ &\big{(}1-\eta^{2}\big{)}B_{1}-B_{2}+2\eta\zeta_{1}A_{1}+\tfrac{3 }{4}\varepsilon\Big{(}\big{(}B_{1}-B_{2}\big{)}^{3}+(A_{1}-A_{2})^{2}(B_{1}-B_ {2})\Big{)}=0,\\ -\mu A_{1}+\big{(}\mu\lambda-\eta^{2}+\mu\big{)}A_{2}+2\mu\eta \zeta_{2}B_{2}-\tfrac{3}{4}\mu\Big{(}\big{(}A_{1}-A_{2}\big{)}^{3}+(A_{1}-A_{2} )(B_{1}-B_{2})^{2}\Big{)}=0,\\ -\mu B_{1}+\big{(}\mu\lambda-\eta^{2}+\mu\big{)}B_{2}+2\mu\eta \zeta_{2}A_{2}-\tfrac{3}{4}\mu\varepsilon\Big{(}\big{(}A_{1}-A_{2}\big{)}^{3}+ (A_{1}-A_{2})(B_{1}-B_{2})^{2}\Big{)}=0.\end{split}  \tag{13}\]  
From the algebraic equations of system (13), by varying the dimensionless frequency \(\eta\), and analytically solving cubic amplitude polynomials for each case, the amplitude-frequency characteristics, shown in figures below, were obtained. As the elliptic functions are periodic functions on the complex plane, they can be considered as a generalized case of trigonometric functions. In the case of stable forced oscillations, the oscillation trajectory of the center of inertia of a linear oscillator can be depicted as a circle (Figure 2, case (a)). In the case of small nonlinearity, the trajectory of the center of inertia of our system, which is a nonlinear oscillator, will describe an ellipse (Figure 2, case (b)).