## 5 OAFM for Nonlinear Vibration of CNTRC  
If we introduce the notations:  
\[m_{i}=\frac{\overline{m}_{i}}{m_{1}},\ i=1,2;\ m_{j}=\frac{\overline{m}_{j}}{m_{ 1}},\ j=3,4;\ a_{i}=\frac{\overline{a}_{i}}{m_{1}},\ i=1,2;\ b_{i}=\frac{ \overline{b}_{i}}{m_{1}},\ i=3,4,5,7;\ b_{6}=\frac{\overline{b}_{6}}{m_{4}}, \ a_{k}=\frac{\overline{a}_{k}}{m_{4}},\ k=3,4;\ X_{0}=\overline{f} \tag{76}\]  
then the nonlinear differential Equations (66)-(68) can be rewritten in the equivalent forms:  
\[\ddot{\mathrm{T}}_{1}+\omega_{1}^{2}\mathrm{T}_{1}+\mathrm{m}_{2}\ddot{ \mathrm{T}}_{2}+(a_{1}-\omega_{1}^{2})\mathrm{T}_{1}+\mathrm{a}_{2}\mathrm{T }_{2}+\mathrm{b}_{3}T_{1}^{3}+\mathrm{b}_{4}T_{1}^{2}\mathrm{T}_{2}+\mathrm{b }_{5}T_{1}^{5}+\mathrm{b}_{7}T_{1}^{7}=X_{0} \tag{77}\]  
\[\ddot{\mathrm{T}}_{2}+\omega_{2}^{2}\mathrm{T}_{2}+\mathrm{m}_{3}\ddot{ \mathrm{T}}_{1}+(a_{4}-\omega_{2}^{2})\mathrm{T}_{2}+\mathrm{a}_{3}\mathrm{T }_{1}+\mathrm{b}_{6}T_{1}^{3}=0 \tag{78}\]  
in which \(\varpi_{1}\) and \(\varpi_{2}\) are unknown frequencies of \(T_{1}\) and \(T_{2}\), respectively. The linear and nonlinear operators of Equations (77) and (78) are, respectively:  
\[L_{1}(\mathrm{T}_{1})=\ddot{\mathrm{T}}_{1}+\omega_{1}^{2}\mathrm{T}_{1};\ N_{1}(\mathrm{T}_{1})=(a_{1}-\omega_{1}^{2} )\mathrm{T}_{1}+\mathrm{a}_{2}\mathrm{T}_{2}+\mathrm{b}_{3}T_{1}^{3}+\mathrm{ b}_{4}T_{1}^{2}\mathrm{T}_{2}+\mathrm{b}_{5}T_{1}^{5}+\mathrm{b}_{7}T_{1}^{7}+ \mathrm{m}_{2}\ddot{\mathrm{T}}_{2}-X_{0} \tag{79}\]  
\[L_{2}(\mathrm{T}_{2})=\ddot{\mathrm{T}}_{2}+\omega_{2}^{2}\mathrm{T}_{2};\ N_{2}( \mathrm{T}_{2})=(a_{4}-\omega_{2}^{2})\mathrm{T}_{2}+\mathrm{a}_{3}\mathrm{T }_{1}+\mathrm{b}_{6}T_{1}^{3}+\mathrm{m}_{3}\ddot{\mathrm{T}}_{1} \tag{80}\]  
with the initial conditions:  
\[T_{1}(0)=A,\ \dot{\mathrm{T}}_{1}(0)=0,\ T_{2}(0)=B,\ \dot{\mathrm{T}}_{2}(0)=0 \tag{81}\]  
The approximate solutions of Equations (77) and (78) are, respectively:  
\[\overline{T}_{1}(t)=T_{10}(t)+T_{11}(t,C_{1},C_{2},\ldots C_{n-1}),\ \overline{T}_{2}(t)=T_{20}(t)+T_{21}(t,C_{n},C_{n+1},\ldots C_{s}) \tag{82}\]  
The initial approximations \(T_{10}\) and \(T_{20}\) are obtained from the linear equations:  
\[\ddot{\mathrm{T}}_{10}+\omega_{1}^{2}\mathrm{T}_{10}=0,\ T_{10}(0)=A,\ \dot{ \mathrm{T}}_{10}(0)=0 \tag{83}\]  
\[\ddot{\mathrm{T}}_{20}+\omega_{2}^{2}\mathrm{T}_{20}=0,T_{20}(0)=\mathrm{B}, \dot{\mathrm{T}}_{20}(0)=0 \tag{84}\]  
whose solutions are, respectively:  
\[\mathrm{T}_{10}(\mathrm{t})=\mathrm{Acos}\omega_{1}\mathrm{t},\mathrm{T}_{20} (\mathrm{t})=\mathrm{Bcos}\omega_{2}\mathrm{t} \tag{85}\]  
The expression of the nonlinear operator given by Equation (79) calculated for the values given by Equation (85) becomes:  
\[ \begin{array}{l}N_{1}(\mathrm{T}_{10},\mathrm{T}_{20})=(a_{1}-\omega_{1}^{2 })\mathrm{Acos}\omega_{1}\mathrm{t}+\frac{1}{4}b_{3}A^{3}(cos3\omega_{1} \mathrm{t}+3\mathrm{cos}\omega_{1}\mathrm{t})+\frac{1}{16}b_{5}A^{5}(cos5 \omega_{1}\mathrm{t}+5\mathrm{cos}3\omega_{1}\mathrm{t}+10 cos\omega_{1} \mathrm{t})\\ \qquad\qquad\qquad\qquad\qquad+\frac{1}{16}b_{7}A^{7}(cos7\omega_{1}\mathrm{t }+7\mathrm{cos}5\mathrm{t}+21 cos3\omega_{1}\mathrm{t}+35 cos\omega_{1} \mathrm{t})+\cdots\end{array}  \tag{86}\]or  
\[N_{1}(T_{10},T_{20})=\alpha_{1}\text{cos}\omega_{1}t+\alpha_{3}\text{cos}3 \omega_{1}t+\alpha_{5}\text{cos}5\omega_{1}t+\alpha_{7}\text{cos}7\omega_{1}t+\cdots \tag{87}\]  
where  
\[\alpha_{1}=(a_{1}-\omega_{1}^{2})A+\frac{3}{4}b_{3}A^{3}+\frac{5}{8}b_{5}A^{5}+ \frac{35}{64}b_{7}A^{7};\;\alpha_{3}=\frac{1}{4}b_{3}A^{3}+\frac{5}{8}b_{5}A^{5 }+\frac{21}{64}b_{7}A^{7};\;\alpha_{5}=\frac{1}{16}b_{5}A^{5}+\frac{7}{64}b_{7 }A^{7}; \tag{88}\]  
The first approximation \(T_{11}(t,C_{i})\) can be obtained from Equation (75):  
\[ \begin{split}\ddot{T}_{11}+\omega_{1}^{2}T_{11}+8\omega_{1}^{2} C_{1}\text{cos}3\omega_{1}t+24\omega_{1}^{2}C_{2}\text{cos}5\omega_{1}t+48 \omega_{1}^{2}C_{3}\text{cos}7\omega_{1}t+\\ 8\omega_{1}^{2}C_{4}\text{sin}3\omega_{1}t+24\omega_{1}^{2}\text{ sin}5\omega_{1}t,T_{11}(0)=\dot{T}_{11}(0)=0\end{split}