# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 4 Asymptotic Solution of the Equations of Motion
### Nonlinearity of the Lowest Normal Modes
\begin{split}\ddot{q}_{m}+c_{m}\dot{q}_{m}+\omega_{m}^{2}q_{m}& +c_{i2}q_{m}\ddot{q}_{m}+c_{d2}q_{m}^{2}+c_{k2}q_{m}^{2}\\ &+c_{i3}q_{m}^{2}\ddot{q}_{m}+c_{d3}q_{m}\ddot{q}_{m}^{2}+c_{k3} q_{m}^{3}+f_{m}\cos\Omega t=0,\end{split}  \tag{23}\]  
where \(c_{m}=\boldsymbol{\psi}_{m}^{\top}\mathbb{C}\,\boldsymbol{\psi}_{m}\) is the modal damping coefficient, \(f_{m}=\boldsymbol{\psi}_{m}^{\top}\mathbb{f}_{0}\) is the modal component of the excitation term, and:  
\[ \begin{split} c_{i2}&=\boldsymbol{\psi}_{m}^{\top} \mathbb{i}_{2}(\boldsymbol{\psi}_{m},\boldsymbol{\psi}_{m}),c_{d2}= \boldsymbol{\psi}_{m}^{\top}\mathbb{d}_{2}(\boldsymbol{\psi}_{m},\boldsymbol {\psi}_{m}),c_{k2}=\boldsymbol{\psi}_{m}^{\top}\mathbb{g}_{2}(\boldsymbol{ \psi}_{m},\boldsymbol{\psi}_{m}),\\ c_{i3}&=\boldsymbol{\psi}_{m}^{\top}\mathbb{i}_{3}( \boldsymbol{\psi}_{m},\boldsymbol{\psi}_{m},\boldsymbol{\psi}_{m}),c_{d3}= \boldsymbol{\psi}_{m}^{\top}\mathbb{d}_{3}(\boldsymbol{\psi}_{m}, \boldsymbol{\psi}_{m},\boldsymbol{\psi}_{m}),c_{k3}=\boldsymbol{\psi}_{m}^{ \top}\mathbb{g}_{3}(\boldsymbol{\psi}_{m},\boldsymbol{\psi}_{m},\boldsymbol{ \psi}_{m},\boldsymbol{\psi}_{m}),\end{split}  \tag{24}\]  
are the second- and third-order inertial, velocity-proportional, and stiffness modal coefficients, respectively. The same asymptotic procedure, based on the method of multiple scales, can be used to investigate the nonlinearity of the \(m\)th mode through the nonlinear equation of motion (23), whose solution is sought in the form \(q_{m}(t)=\epsilon\,q_{m,1}(t_{0},t_{2})+\epsilon^{2}q_{m,2}(t_{0},t_{2})+ \epsilon^{3}q_{m,3}(t_{0},t_{2})\). Therefore, after rescaling the excitation term as \(f_{m}=\epsilon^{3}f_{m}\) and the modal viscous damping force as \(c_{m}q_{m}=\epsilon^{2}c_{m}\dot{q}_{m}\), the lowest three perturbation problems can be written as:  
\[ \begin{split}\partial_{0}^{2}q_{m,1}+\omega_{m}^{2}q_{m,1}=& 0,\\ \partial_{0}^{2}q_{m,2}+\omega_{m}^{2}q_{m,2}=&-c_{k2 }q_{m,1}^{2}-c_{d2}(\partial_{0}q_{m,1})^{2}-c_{i2}q_{m,1}\partial_{0}^{2}q_{m,1 },\\ \partial_{0}^{2}q_{m,3}+\omega_{m}^{2}q_{m,3}=&-c_{m} \partial_{0}\partial_{m,1}-2\,\partial_{0}\partial_{2}q_{m,1}-2\,c_{k2}q_{m,1}q _{m,2}-2\,c_{d2}\partial_{0}q_{m,1}\partial_{0}q_{m,2}\\ &-c_{i2}\left(q_{m,2}\partial_{0}^{2}q_{m,1}+q_{m,1}\partial_{0}^{ 2}q_{m,2}\right)-c_{k3}q_{m,1}^{3}-c_{d3}q_{m,1}(\partial_{0}q_{m,1})^{2}\\ &-c_{i3}q_{m,1}^{2}\partial_{0}^{2}q_{m,1}-f_{m}\cos\Omega t.\end{split}  \tag{25}\]To study the nonlinearity of the \(m\)th mode the excitation term is set to zero, i.e., \(f_{m}=0\), and the modal viscous damping is neglected, i.e., \(c_{m}=0\). The solution of the first-order problem can be written as \(q_{m,1}=A_{m}(t_{2})e^{i\omega_{m}t_{0}}+\tilde{A}_{m}(t_{2})e^{-i\omega_{m}t_{ 0}}\), while the solution of the second-order problem, which does not show resonant terms, is given in the form:  
\[ \begin{split} q_{m,2}=&\frac{c_{k2}-(c_{d2}+c_{i2} )\omega_{m}^{2}}{3\omega_{m}^{2}}A_{m}(t_{2})^{2}e^{2\,i\omega_{m}t_{0}}-\frac{ 2[c_{k2}+(c_{d2}-c_{i2})\omega_{m}^{2}]}{\omega_{m}^{2}}A_{m}(t_{2})\tilde{A}_ {m}(t_{2})\\ &+\frac{c_{k2}-(c_{d2}+c_{i2})\omega_{m}^{2}}{3\,\omega_{m}^{2}} \tilde{A}_{m}(t_{2})^{2}e^{-2\,i\omega_{m}t_{0}}.\end{split}