# Nonlinear Vibrations of Carbon Nanotubes with Thermal-Electro-Mechanical Coupling
## 3 Results  
### Primary Resonance of SWCNTs  
When \(\Omega\) is close to \(\omega_{j}\) and there is no internal resonance between the two modes, vibrations of the two model are uncoupled. Therefore, the coupling term of Equation (7) can be removed [28] and the equation is rewritten as:  
\[\ddot{\eta}_{j}+c_{j}\ddot{\eta}_{j}+k_{j1}\eta_{j}+k_{j3}\eta_{j}^{3}=F_{j} \cos\Omega t,\ j=1,2. \tag{9}\]  
For the study of primary resonance, we introduce \(\varepsilon\) for perturbation, where \(\varepsilon\) is a small parameter. Here, \(\varepsilon=0.01\) is used, and Equation (9) is rewritten as  
\[\ddot{\eta}_{j}+\omega_{j}^{2}\eta_{j}+2\varepsilon\overline{\dot{\varepsilon }}\dot{\eta}_{j}+\varepsilon\overline{\dot{k}}_{j3}\eta_{1}^{3}=\varepsilon f _{j}\cos\omega\overline{\dot{t}},\ j=1,2. \tag{10}\]  
Here, \(\omega_{j}^{2}=k_{j1}\), and Equation (10) will be solved by the multi-scale method [32]. We expand \(\eta_{i}\) as follows:  
\[\eta_{j}=\eta_{j0}(T_{0},T_{1})+\varepsilon\eta_{j1}(T_{0},T_{1})+\cdots. \tag{11}\]  
In Equation (11), \(T_{0}=\varepsilon^{0}t\), \(T_{1}=\varepsilon t\), and here, it is  
\[\frac{\mathrm{d}}{dt}=D_{0}+\varepsilon D_{1},\ \frac{d^{2}}{dt^{2}}=D_{0}^{2}+2 \varepsilon D_{0}D_{1}+\varepsilon^{2}D_{1}^{2}. \tag{12}\]  
Substituting Equations (11) and (12) into Equation (10), and making the coefficients of the same order \(\varepsilon\) equal, we obtain [32]  
\[\varepsilon^{0}:\ D_{0}^{2}\eta_{j0}+\omega_{j}^{2}\eta_{j0}=0. \tag{13}\]  
\[\varepsilon:\ D_{0}^{2}\eta_{j1}+\omega_{j}^{2}\eta_{j1}=-2D_{0}D_{2}\eta_{j0 }-2\overline{\dot{\varepsilon}}_{j}D_{0}\eta_{j0}-\overline{\dot{k}}_{j3}\eta _{j0}^{3}+\frac{1}{2}f_{j}\cos(\Omega T_{0}). \tag{14}\]  
Let the solution of the equation of order \(\varepsilon^{0}\) be:  
\[\eta_{j0}=A(T_{1})\exp\left(i\omega_{j}T_{0}\right)+cc. \tag{15}\]  
where \(cc\) is the complex conjugate terms of the previous terms and \(A_{1}\) is a real function of \(T_{1}\). Substituting Equation (15) into Equation (14), we obtain  
\[ \begin{array}{ll}D_{0}^{2}\eta_{j1}+\omega_{j}^{2}\eta_{j1}&=\left[-2i\omega _{j}(A^{\prime}+\overline{\dot{\varepsilon}}_{j}A)-3\overline{\dot{k}}_{j3}A^ {2}\overline{A}\right]\exp\left(i\omega_{j}T_{0}\right)\\ &-\overline{\dot{k}}_{j3}A^{3}\exp\left(3i\omega_{j}T_{0}\right)+\frac{1}{2}f_ {j}\exp[i(\Omega T_{0})]+cc.\end{array}  \tag{16}\]  
where the apostrophe represents the derivative with respect to \(T_{1}\). The dimensionless detuning parameter \(\sigma\) is introduced to make the excitation frequency \(\Omega=\omega_{j}+\varepsilon\sigma\). We eliminate the secular term and obtain:  
\[2i\omega_{j}\big{(}A^{\prime}+\overline{\dot{\varepsilon}}_{j}A\big{)}+3 \overline{\dot{k}}_{j3}A^{2}\overline{A}=\frac{1}{2}f_{j}\exp\left(i\sigma T_{ j}\right). \tag{17}\]  
Introducing a polar form for \(A\), we obtain:  
\[A=\frac{1}{2}a\exp(i\beta). \tag{18}\]where \(a\) and \(\beta\) are real functions of \(T_{1}\). Substituting Equation (18) into Equation (17), and separating the real part and imaginary part, we obtain:  
\[\left\{ \begin{array}{l}a^{\prime}=-\overline{c}_{j}a+\frac{1}{2\omega_{j}}f_{j }\sin\gamma\\ a\beta^{\prime}=\frac{3}{8}\frac{\overline{c}_{j}\beta}{\omega_{j}}a^{2}- \frac{1}{2\omega_{j}}f_{j}\cos\gamma.\end{array} \right. \tag{19}\]  
where \(\gamma=\sigma T_{1}-\beta\). When \(a^{\prime}=\beta^{\prime}=0\), the steady state motion appears; therefore:  
\[\left\{ \begin{array}{l}0=-\overline{c}_{j}a+\frac{1}{2\omega_{1}}f_{j}\sin \gamma\\ a\sigma=\frac{3}{8}\frac{\overline{c}_{j}\beta}{\omega_{j}}a^{3}-\frac{1}{2 \omega_{j}}f_{j}\cos\gamma.\end{array} \right. \tag{20}\]  
From Equation (20), we obtain  
\[\left[\overline{c}_{j}^{2}+\left(\sigma-\frac{3}{8}\frac{\overline{k}_{j}}{ \omega_{j}}a^{2}\right)^{2}\right]a_{j}^{2}=\frac{f_{j}^{2}}{4\omega_{j}^{2}}. \tag{21}\]  
Equation (21) indicates that the response amplitude \(a\) is a function of \(\sigma\) and the excitation amplitude \(f_{j}\).