# 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  
By setting \(\mathbf{f}_{0}=\mathbf{0}\) and \(\mathbf{C}=\mathbf{O}\) in Equation (12), it is possible to study the nonlinearity of the roller battery nonlinear normal modes by solving, in cascade, the hierarchy of per turbation problems given by Equations (10)-(12). In particular, to study the nonlinearity of the \(m\)th mode, the general solution of the \(\epsilon\)-order problem, Equation (10), can be written as:  
\[\mathbf{x}_{1}(t_{0},t_{2})=A_{m}(t_{2})\,\boldsymbol{\psi}_{m}\,e^{i\,\omega_ {m}t_{0}}+\tilde{A}_{m}(t_{2})\,\boldsymbol{\psi}_{m}\,e^{-i\,\omega_{m}t_{0}}, \tag{13}\]  
where \(A_{m}(t_{2})\) and \(\tilde{A}_{m}(t_{2})\) are the complex and complex-conjugate amplitudes of the \(m\)th mode, respectively. The non-resonant terms are present in the second-order problem; therefore, after substituting the first-order solution (13) into Equation (11), the solution of the inhomogeneous \(\epsilon^{2}\)-order problem can be expressed in the form:  
\[\mathbf{x}_{2}(t_{0},t_{2})=\hat{\mathbf{x}}_{2,1}A_{m}^{2}e^{2i\omega_{m}t_ {0}}+\hat{\mathbf{x}}_{2,2}A_{m}\tilde{A}_{m}+\hat{\mathbf{x}}_{2,3}\tilde{A}_ {m}^{2}e^{-2i\omega_{m}t_{0}}, \tag{14}\]  
where the vector \(\hat{\mathbf{x}}_{2,r}\) (\(r=1,2,3\)) is the linear combination of the \(N_{s}\) modes of the system, that is: \(\hat{\mathbf{x}}_{2,r}=\sum_{s=1}^{N_{s}}a_{r,s}\boldsymbol{\psi}_{s}\). The solution given by Equation (14) is substituted into the \(\epsilon^{2}\)-order problem and Equation (11) is then projected into the modal space through the \(N_{s}\)-by-\(N_{s}\) modal matrix \(\boldsymbol{\
abla}\), whose \(s\)th column is the vector \(\boldsymbol{\psi}_{s}\) (i.e., the \(s\)th mode), as:  
\[\boldsymbol{\
abla}^{\top}\mathbf{M}\,\partial_{0}^{2}\mathbf{x}_{2}+ \boldsymbol{\
abla}^{\top}\mathbf{K}\,\mathbf{x}_{2}=\boldsymbol{\
abla}^{ \top}\mathbf{r}_{2}\,, \tag{15}\]  
where \(\mathbf{r}_{2}=-\mathbf{g}_{2}(\mathbf{x}_{1},\mathbf{x}_{1})-\mathbf{d}_{2}( \partial_{0}\mathbf{x}_{1},\partial_{0}\mathbf{x}_{1})-\mathbf{i}_{2}( \mathbf{x}_{1},\partial_{0}^{2}\mathbf{x}_{1})\) is the vector collecting the second-order inhomogeneous terms. By then equating the coefficients of terms having the same frequency in Equation (15), it is possible to calculate the \(3N_{s}\) coefficients \(a_{r,s}\) and find the second-order solution \(\mathbf{x}_{2}\). First- and second-order solutions given by Equations (13) and (14), respectively, are then substituted into the \(\epsilon^{3}\)-order problem. The singularity generated by the secular terms exhibited in the third-order problem can be removed by enforcing solvability conditions [35]. To this end, by now introducing the general solution of the adjoint homogeneous problem of Equation (12):  
\[\mathbf{x}_{0}^{*}=i\,\omega_{m}\boldsymbol{\psi}_{m}\,e^{-i\,\omega_{m}t_{0 }}+cc, \tag{16}\]  
where \(cc\) stands for complex-conjugate, the solvability condition is enforced by requiring the orthogonality between (16) and the inhomogeneous term (right-hand side) in Equation (12) as:  
\[\tfrac{\omega_{m}}{2\pi}\int_{0}^{\tfrac{2\pi}{\omega_{m}}}\mathbf{x}_{0}^{* \top}\mathbf{r}_{3}\,dt_{0}=0, \tag{17}\]  
where:  
\[ \begin{split}\mathbf{r}_{3}=&-\widehat{\mathbf{D}} \,\partial_{0}\partial_{2}\mathbf{x}_{1}-\widehat{\mathbf{g}}_{1,2}(\mathbf{ x}_{1},\mathbf{x}_{2})-\mathbf{g}_{3}(\mathbf{x}_{1},\mathbf{x}_{1},\mathbf{x}_{1}) \\ &-\widehat{\mathbf{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})\\ &-\widehat{\mathbf{i}}_{1,2}(\mathbf{x}_{1},\partial_{0}^{2} \mathbf{x}_{2})-\widehat{\mathbf{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}).\end{split}