## 2 Problem Statement
#### Equations of Motion's Solution
\begin{split}\text{cn}\Big{(}\omega_{f}^{2}\big{(}2k^{2}-1\big{)} A_{2}&-\mu\big{(}A_{1}-A_{2}\big{)}-3\mu\big{(}A_{1}-A_{2}\big{)}(B_{1 }-B_{2})^{2}+\mu\lambda A_{2}+2\omega_{f}\zeta_{2}B_{2}H_{2}\Big{)}\\ &+\text{sn}\Big{(}\omega_{f}^{2}\big{(}k^{2}-1\big{)}B_{2}-\mu \big{(}B_{1}-B_{2}\big{)}-\mu\big{(}B_{1}-B_{2}\big{)}^{3}+\mu\lambda B_{2}-2 \mu\omega_{f}\zeta_{2}A_{2}H_{1}\\ &+H_{3}\Big{(}-\mu\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_{2}\Big{)}\Big{)}\\ &+\text{cn}^{3}\big{(}-\mu\varepsilon(A_{1}-A_{2})^{3}-2\omega_{f}^ {2}k^{2}A_{2}+3\mu\varepsilon(A_{1}-A_{2})(B_{1}-B_{2})^{2}\Big{)}=0,\end{split} \]  
where is:\[H_{1}=\frac{1}{\pi}\int\limits_{0}^{2\pi}\text{sn}u\text{dn}u\sin\varphi d \varphi=\frac{1}{\pi}\int\limits_{0}^{4K}\text{sn}^{2}u\text{dn}^{2}udu=\frac{4 }{3k^{2}\pi}\big{(}\big{(}2k^{2}-1\big{)}E+K\big{(}1-k^{2}\big{)}\big{)},\] \[H_{2}=\frac{1}{\pi}\int\limits_{0}^{2\pi}\text{cn}u\text{dn}u\cos \varphi d\varphi=\frac{1}{\pi}\int\limits_{0}^{4K}\text{cn}^{2}u\text{dn}^{2} udu=\frac{4}{3k^{2}\pi}\big{(}\big{(}1+k^{2}\big{)}E-K\big{(}1-k^{2}\big{)}\big{)},\] \[H_{3}=\frac{1}{\pi}\int\limits_{0}^{2\pi}\text{sn}ucn^{2}u\sin \varphi d\varphi=\frac{1}{\pi}\int\limits_{0}^{4K}\text{sn}^{2}ucn^{2}ud\text{ dn}udu=\frac{1}{4},\] \[K=\int\limits_{0}^{4\varphi}\frac{d\varphi}{\sqrt{1-k^{2}\sin^{2 }\varphi}}\approx\frac{\pi}{2}\Big{(}1+\frac{1}{4}k^{2}+\frac{9}{64}k^{4}+O \big{(}k^{6}\big{)}\Big{)},\] \[E=\int\limits_{0}^{\varphi}\sqrt{1-k^{2}\sin^{2}\varphi}d\varphi \approx\frac{\pi}{2}\Big{(}1-\frac{1}{4}k^{2}-\frac{3}{64}k^{4}+O\big{(}k^{6} \big{)}\Big{)}.\]  
Here \(K\equiv K(\varphi,k)\) is a complete elliptic integral of the first kind, whereas \(E\equiv E(\varphi,k)\) is an incomplete elliptic integral of the second kind. Equating the coefficients for the same functions and harmonics of Equation (22), we obtain a system of algebraic equations that enables us to determine the amplitudes of forced vibrations in the form (23):  
\[(A_{1}-A_{2})+\frac{1}{2}\omega_{f}^{2}k^{2}A_{1}-\omega_{f}^{2}A _{1}+\frac{3}{4}\varepsilon(A_{1}-A_{2})\Big{(}(A_{1}-A_{2})^{2}+(B_{1}-B_{2} )^{2}\Big{)}-\omega_{f}^{2}+2\omega_{f}\zeta_{1}B_{1}H_{2}=0,\] \[(B_{1}-B_{2})+\frac{1}{2}\omega_{f}^{2}k^{2}B_{1}-\omega_{f}^{2} B_{1}+\frac{3}{4}\varepsilon(A_{1}-A_{2})\Big{(}(A_{1}-A_{2})^{2}+(B_{1}-B_{2} )^{2}\Big{)}-2\omega_{f}\zeta_{1}A_{1}H_{1}=0,\] \[\varepsilon(A_{1}-A_{2})^{3}-2\omega_{f}^{2}k^{2}A_{1}-3 \varepsilon(A_{1}-A_{2})(B_{1}-B_{2})^{2}=0,\] \[\mu((\lambda+1)A_{2}-A_{1})+\frac{1}{2}\omega_{f}^{2}k^{2}A_{2}- \omega_{f}^{2}A_{2}-\frac{3}{4}\mu\varepsilon(A_{1}-A_{2})\Big{(}(A_{1}-A_{2} )^{2}+(B_{1}-B_{2})^{2}\Big{)}+2\omega_{f}\zeta_{2}B_{2}H_{2}=0,\] \[\mu((\lambda+1)B_{2}-B_{1})+\frac{1}{2}\omega_{f}^{2}k^{2}B_{2}- \omega_{f}^{2}B_{2}-\frac{3}{4}\mu\varepsilon(B_{1}-B_{2})\Big{(}(A_{1}-A_{2} )^{2}+(B_{1}-B_{2})^{2}\Big{)}-2\omega_{f}\zeta_{2}A_{2}H_{1}=0, \tag{23}\] \[H_{2}=\frac{1}{\pi}\int\limits_{0}^{2\pi}\text{cn}u\text{dn}u\cos \varphi d\varphi=\frac{1}{\pi}\int\limits_{0}^{4K}\text{cn}^{2}u\text{dn}^{2} udu=\frac{4}{3k^{2}\pi}\big{(}\big{(}1+k^{2}\big{)}E-K\big{(}1-k^{2}\big{)}\big{)},\]0 \[H_{2}=\frac{1}{\pi}\int\limits_{0}^{2\pi}\text{cn}u\text{dn}u\cos \varphi d\varphi=\frac{1}{\pi}\int\limits_{0}^{4K}\text{cn}^{2}u\text{dn}^{2} udu=\frac{4}{3k^{2}\pi}\big{(}\big{(}1+k^{2}\big{)}E-K\big{(}1-k^{2}\big{)}\big{)},\]1  
By varying the cyclic frequency in the system of Equation (23), the amplitude-frequency characteristics of the forced oscillations of the rotor and foundation were obtained (Figures 3-12). As can be seen, for \(k=0\), the system of Equation (23) degenerates into system (13). Equation (13) can be obtained by such classical approximate methods as the Van der Pol method, the harmonic balance method, etc. This fact is a direct confirmation of the generalization of the method of elliptic functions.