# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 5 Cable Motion-Induced Resonances
### Primary Resonance  
First- and second-order solutions of the hierarchical systems of equations, Equations (10) to (12), are given by Equations (13) and (14), respectively. The latter are then substituted into the \(e^{3}\)-order problem Equation (12) and the singularity generated by the secular terms can be removed by enforcing the solvability condition. To this end, after introducing the general solution of the adjoint homogeneous problem (i.e., Equation (16)), the orthogonality between (16) and the inhomogeneous term (right-end side) in Equation (12) is enforced as: \(\frac{\omega_{\rm{m}}}{2\pi}\int_{0}^{2\pi\omega_{1}}{\bf x}_{0}^{*}{\bf\,}^{ \top}{\bf r}_{3}\,dt_{0}=0\), where:  
Figure 5: Coefficients of the inertial (\(c_{i2}\) and \(c_{i3}\)), velocity-proportional (\(c_{d2}\) and \(c_{d3}\)), and stiffness nonlinear terms (\(c_{k2}\) and \(c_{k3}\)), of the lowest four modes. \[ \begin{split}\mathbf{r}_{3}=&-\mathbf{C}\,\partial_{0} \mathbf{x}_{1}-\mathbf{\bar{D}}\,\partial_{0}\partial_{2}\mathbf{x}_{1}- \mathbf{\bar{g}}_{1,2}(\mathbf{x}_{1},\mathbf{x}_{2})-\mathbf{g}_{3}(\mathbf{ x}_{1},\mathbf{x}_{1},\mathbf{x}_{1})\\ &-\mathbf{\hat{d}}_{1,2}(\partial_{0}\mathbf{x}_{1},\partial_{0} \mathbf{x}_{2})-\mathbf{d}_{3}(\mathbf{x}_{1},\partial_{0}\mathbf{x}_{1}, \partial_{0}\mathbf{x}_{1})\\ &-\mathbf{\hat{i}}_{1,2}(\mathbf{x}_{1},\partial_{0}^{2}\mathbf{ x}_{2})-\mathbf{\hat{i}}_{2,1}(\mathbf{x}_{2},\partial_{0}^{2}\mathbf{x}_{1})- \mathbf{i}_{3}(\mathbf{x}_{1},\mathbf{x}_{1},\partial_{0}^{2}\mathbf{x}_{1})- \mathbf{f}_{0}\cos\Omega t.\end{split}  \tag{30}\]  
To investigate the primary resonance of the \(n\)th mode, the frequency of the excitation term is expressed as \(\Omega=\omega_{m}+\epsilon^{2}\sigma\), with \(\sigma\) as a detuning parameter. Moreover, to make use of complex algebra, \(\cos\Omega t=\frac{1}{2}e^{i\omega_{m}t_{0}+i\sigma\,t_{2}}+cc\); then, at the third order, the singularity generated by the resonant terms is removed when the following modulation equations hold:  
\[ \begin{split} 2\,i\,\omega_{m}\partial_{2}A_{m}+i\,\omega_{m} \mathcal{C}_{m}\,A_{m}+8\,\omega_{m}\Gamma_{m}A_{m}^{2}\,\bar{A}_{m}+\frac{1}{ 2}\mathcal{F}_{0}e^{i\sigma\,t_{2}}=0,\\ 2\,i\,\omega_{m}\partial_{2}\bar{A}_{m}+i\,\omega_{m}\mathcal{C} _{m}\,\bar{A}_{m}-8\,\omega_{m}\Gamma_{m}A_{m}\bar{A}_{m}^{2}-\frac{1}{2} \mathcal{F}_{0}e^{-i\sigma\,t_{2}}=0,\end{split}  \tag{31}\]  
where, \(\mathcal{C}_{m}=\mathcal{C}_{m}(\omega_{m})\) and \(\mathcal{F}_{0}=\mathcal{F}_{0}(\lambda,\omega_{m})\) are strongly nonlinear functions of the linear frequency of the \(m\)th mode. Note that, in case of the one-mode projection of the equations of motion, it turns out that \(\mathcal{C}_{m}\equiv c_{m}\), \(\mathcal{F}_{0}\equiv f_{m}\), and \(\Gamma_{m}\) is given by Equation (27). By expressing the complex amplitudes in polar form and separating the real and imaginary parts, the modulation equations in terms of the real amplitude and phase read:  
\[ \begin{split} a_{m}\partial_{2}\theta_{m}-\Gamma_{m}a_{m}^{3}- \frac{\mathcal{F}_{0}}{2\,\omega_{m}}\cos\gamma_{m}=0,\\ \partial_{2}a_{m}+\frac{1}{2}\mathcal{C}_{m}a_{m}+\frac{\mathcal{ F}_{0}}{2\,\omega_{m}}\sin\gamma_{m}=0.\end{split}  \tag{32}\]  
By solving Equation (32) for \(\partial_{2}a_{m}\) and \(\partial_{2}\theta_{m}\) and introducing the relative phase \(\gamma_{m}(t_{2})=\sigma\,t_{2}-\theta_{m}(t_{2})\), the fixed points can be sought to impose \(\partial_{2}a_{m}=0\) and \(\partial_{2}\gamma_{m}=0\), respectively.