# 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
 \tag{18}\]  
Equation (17), which annihilates the secular terms arising in \(\mathbf{r}_{3}\), governs the modulation in time \(t_{2}\) of the amplitudes \(A_{m}\) and \(\tilde{A}_{m}\) through to the following equations:  
\[2\,i\,\omega_{m}\partial_{2}A_{m}+8\,\omega_{m}\Gamma_{m}A_{m}^{2}\,\tilde{A }_{m}=0,\quad 2\,i\,\omega_{m}\partial_{2}\tilde{A}_{m}-8\,\omega_{m}\Gamma_{m}A_{ m}\tilde{A}_{m}^{2}=0, \tag{19}\]  
where \(\Gamma_{m}=\Gamma_{m}(\omega_{m})\) is a strongly nonlinear function of the frequency \(\omega_{m}\) of the \(m\)th mode and of the systems mechanical parameters. After removing the secular terms, the solution of the inhomogeneous third-order problem can be calculated as:  
\[\mathbf{x}_{3}(t_{0},t_{2})=\hat{\mathbf{x}}_{3,1}A_{m}^{3}e^{3i\omega_{m}t_ {0}}+\hat{\mathbf{x}}_{3,2}\tilde{A}_{m}^{3}e^{-3i\omega_{m}t_{0}}, \tag{20}\]  
where the vector \(\hat{\mathbf{x}}_{3,r}\) (\(r=1,2\)) is expressed, again, as a linear combination of the \(N_{s}\) modes of the system, that is: \(\hat{\mathbf{x}}_{3,r}=\sum_{s=1}^{N_{s}}b_{r,s}\boldsymbol{\psi}_{s}\). The solution given by Equation (20) is substituted into the \(\epsilon^{3}\)-order problem and Equation (12) is then projected into the modal space as: \(\boldsymbol{\
abla}^{\top}\mathbf{M}\,\partial_{0}^{2}\mathbf{x}_{3}+ \boldsymbol{\
abla}^{\top}\mathbf{K}\,\mathbf{x}_{3}=\boldsymbol{\
abla}^{ \top}\mathbf{r}_{3}\). By then equating the coefficients of terms having the same frequency, it is possible to calculate the \(2N_{s}\) coefficients \(b_{r,s}\) and find the second-order solution \(\mathbf{x}_{3}\). To study the nonlinearity of the \(m\)th normal mode, the complex amplitudes are expressed in polar form as \(A_{m}(t_{2})=\frac{1}{2}a_{m}(t_{2})e^{i\theta_{m}(t_{2})}\) and \(\tilde{A}_{m}(t_{2})=\frac{1}{2}a_{m}(t_{2})e^{-i\theta_{m}(t_{2})}\), respectively, where \(a_{m}(t_{2})\) is the real amplitude and \(\theta_{m}(t_{2})\) is the phase of the \(m\)th nonlinear normal mode. The polar form is then substituted into the modulation equations (19). Furthermore, the separation of the real and imaginary parts yields the following differential relationships in terms of the real amplitude and the phase of the \(m\)th mode:  
\[\partial_{2}a_{m}=0,\ \partial_{2}\theta_{m}=\Gamma_{m}\,a_{m}^{2}\,. \tag{21}\]  
By defining the relative phase \(\gamma_{m}(t_{2})=\sigma t_{2}-\theta_{m}(t_{2})\), with \(\sigma\) being a detuning parameter, it turns out that the fixed points of the dynamic system are obtained when \(\partial_{2}a_{m}=0\) and \(\partial_{2}\gamma_{m}=0\); therefore, the detuning can be calculated from Equation (21) as \(\sigma=\Gamma_{m}\,a_{m}^{2}\) and the frequency of the \(m\)th nonlinear normal mode can be expressed in terms of the real modal amplitude \(a_{m}\) as:  
\[\omega_{m}^{\text{nl}}=\omega_{m}+\Gamma_{m}\,a_{m}^{2}. \tag{22}\]  
The function \(\Gamma_{m}\) is also known as the effective nonlinearity coefficient of the \(m\)th mode: when \(\Gamma_{m}>0\), the nonlinearity is hardening, while, when \(\Gamma_{m}<0\), it is softening. One-Mode Projection of the Nonlinear Equations of Motion  
The nonlinearity of the \(m\)th normal mode can be qualitatively investigated by neglecting the contribution of the remaining \(N_{s}-1\) modes in the reconstruction of the second- and third-order solutions given by Equations (14) and (20). This approach, although simplified, creates a better understanding of the role of the system mechanical parameters into the effects of the geometric nonlinearity of the system. To this end, the solution of Equation (7) is expressed as \(\mathbf{x}=\boldsymbol{\psi}_{m}\,q_{m}(t)\), where \(q_{m}(t)\) is the \(m\)th generalized coordinate. Equation (7) is then projected into the \(m\)th mode by pre-multiplying it by \(\boldsymbol{\psi}_{m}^{\top}\) to obtain the \(m\)th nonlinear modal equation in the form:  
\[