## 2 Problem Statement
#### Equations of Motion's Solution
Therefore, geometrically elliptic functions can be represented as:  
\[\mathfrak{cn}(u,k)=\frac{x}{a},\mathfrak{sn}(u,k)=\frac{y}{b},\mathfrak{dn}(u,k )=\frac{r}{a}, \tag{14}\]  
where \(b=1\), and:  
\[u=\int\limits_{A}^{B}rd\varphi=\int\limits_{A}^{B}\frac{d\varphi}{\sqrt{1-k^{ 2}\sin^{2}\varphi}},k=\sqrt{1-\frac{1}{a^{2}}}, \tag{15}\]  
where  
\[\varphi=amp(u,k)=\int\limits_{0}^{u}\mathfrak{dn}\big{(}u^{\prime},k\big{)} du^{\prime}, \tag{16}\]  
here \(amp(u)\) is an inverse function of the elliptic integral of the first kind:  
\[u=F(\varphi,k)=\int\limits_{0}^{\varphi}\frac{d\xi}{\sqrt{1-k^{2}\sin^{2} \xi}}. \tag{17}\]  
Taking into account the above expressions (14)-(17), we represent in the form (18) the main relations for elliptic functions used in the following sections in the form:  
Figure 2: Geometrical representation of the oscillatory motion of the center of inertia of the rotor and the foundation: (**a**)—Linear Case, (**b**)—Nonlinear Case. \[ \begin{split}\text{cn}^{2}(u,k)+\text{sn}^{2}(u,k)&=1, \text{dn}^{2}(u,k)+k^{2}\text{sn}^{2}(u,k)=1,\text{dn}(u,k)=\sqrt{1-k^{2}\text{ sn}^{2}(u,k)},\\ \frac{d}{du}\text{sn}(u,k)&=\text{cn}(u,k)\text{dn} (u,k),\frac{d}{du}\text{cn}(u,k)=-\text{sn}(u,k)\text{dn}(u,k),\\ \frac{d}{du}\text{dn}(u,k)&=-k^{2}\text{sn}(u,k) \text{cn}(u,k).\end{split}  \tag{18}\]  
For a more accurate determination of the amplitudes and resonant frequencies of forced oscillations, we use the Jacobi elliptic functions. To solve Equation (10) by the elliptic function method, we assume that the perturbing force \(\cos(\eta\tau)\) is a special case of the elliptic cosine \(\text{cn}(\omega_{f}\tau k)\), i.e., we assume that \(\eta\equiv\omega_{f}\), and for \(k=0\), \(\text{cn}(\omega/\tau,0)=\cos(\eta\tau)\). Taking these statements into account, we rewrite system (10) in the following general form:  
\[ \begin{split}\ddot{f}_{1}+2\zeta_{1}\dot{f}_{1}+\left(f_{1}-f_{2 }\right)+\varepsilon\left(f_{1}-f_{2}\right)^{3}=\omega_{f}^{2}\text{cn}( \omega_{f}\tau,k),\\ \ddot{f}_{2}+2\mu\zeta_{2}\dot{f}_{2}-\mu\left(f_{1}-f_{2}\right) -\mu\varepsilon\left(f_{1}-f_{2}\right)^{3}+\mu\lambda f_{2}=0,\end{split}  \tag{19}\]  
where \(\omega_{f}\) is a circular frequency of the perturbing force, \(k\) is the elliptic modulus, and \(\text{cn}(\omega_{f}\tau k)\) and \(\text{sn}(\omega_{f}\tau k)\) are the elliptic cosine and sine. The solution to Equation (19), as damping forces are present in the system, will be sought in the following form:  
\[ \begin{split} f_{1}(\tau)&=A_{1}\text{cn}(\omega _{f}\tau,k)+B_{1}\text{sn}(\omega_{f}\tau,k),\\ f_{2}(\tau)&=A_{2}\text{cn}(\omega_{f}\tau,k)+B_{2} \text{sn}(\omega_{f}\tau,k),\end{split}  \tag{20}\]  
Taking into account the main relations of elliptic functions, we obtain:  
\[ \begin{split}\frac{d}{d\tau}f_{i}&=\frac{df_{i}}{du }\frac{du}{du}=-\omega_{f}A_{i}\text{sndn}+\omega_{f}B_{i}\text{cndn},\\ \frac{d^{2}}{d\tau^{2}}f_{i}&=-\omega_{f}^{2}A_{i} \big{(}1-2k^{2}\text{sn}^{2}\big{)}-\omega_{f}^{2}B_{i}\big{(}1-k^{2}+2k^{2} \text{cn}^{2}\big{)},\end{split}  \tag{21}\]  
where \(i=1\), \(2\); \(\text{cn}\equiv\text{cn}(\omega_{f}\tau k)\), \(\text{sn}\equiv\text{sn}(\omega_{f}\tau k)\), \(\text{dn}\equiv\text{dn}(\omega_{f}\tau k)\). Substituting the solution in the form (20) into system (19), and, taking into account (21), we obtain:  
\[ \begin{split}\text{cn}\Big{(}\omega_{f}^{2}\big{(}2k^{2}-1\big{)} A_{1}&+(A_{1}-A_{2})+3\varepsilon(A_{1}-A_{2})(B_{1}-B_{2})^{2}- \omega_{f}^{2}+2\omega_{f}\zeta_{1}B_{1}H_{1}\Big{)}\\ &+\text{sn}\Big{(}\omega_{f}^{2}\big{(}k^{2}-1\big{)}B_{1}+(B_{1 }-B_{2})+3\varepsilon(B_{1}-B_{2})^{3}-2\omega_{f}\zeta_{1}A_{1}H_{1}\\ &+H_{3}\Big{(}\varepsilon\Big{(}3(A_{1}-A_{2})^{2}-(B_{1}-B_{2}) ^{2}\Big{)}(B_{1}-B_{2})-2\omega_{f}^{2}k^{2}B_{1}\Big{)}\Big{)}\\ &+\text{cn}^{3}\big{(}(A_{1}-A_{2})^{3}-2\omega_{f}^{2}k^{2}A_{1}- 3\varepsilon(A_{1}-A_{2})(B_{1}-B_{2})^{2}\big{)}=0,\end{split}  \tag{22}\]  
\[