# 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{26}\]  
At the third order, the singularity generated by the resonant terms is removed when the modulation equations, having the same form of Equation (19), hold; although, in this case, the nonlinearity coefficient of the \(m\)th mode has the simpler expression:  
\[\Gamma_{m}(\omega_{m})=-\frac{10c_{k2}^{2}+[(10c_{k2}-11c_{k2})c_{k2}-9c_{k2} ]\omega_{m}^{2}+(4c_{k2}^{2}-3c_{k3}-7c_{k2}c_{k2}+c_{k2}^{2}+9c_{k3})\omega_{m }^{4}}{24\,\omega_{m}^{4}}. \tag{27}\]  
After removing the secular terms by superimposing Equation (19), the solution of the inhomogeneous third-order problem can be written as:  
\[q_{m,3}=\hat{q}_{3,1}\,A_{m}^{3}e^{3\,i\omega_{m}t_{0}}+\hat{q}_{3,2}\,\tilde {A}_{m}^{3}e^{-3\,i\omega_{m}t_{0}}, \tag{28}\]  
and can be substituted into the inhomogeneous third-order problem; then, by equating the coefficients of terms having the same frequency, it is possible to calculate the unknown coefficients of Equation (28), which turn out to be \(\hat{q}_{3,1}=\hat{q}_{3,2}=\hat{q}_{3}(\omega_{m})\), where:  
\[\hat{q}_{3}(\omega_{m})=\frac{2c_{k2}^{2}-[(6c_{k2}+7c_{k2})c_{k2}-3c_{k3}] \omega_{m}^{2}+(4c_{k2}^{2}-3c_{k3}+9c_{k2}c_{k2}+5c_{k2}^{2}-3c_{k3})\omega_{ m}^{4}}{24\,\omega_{m}^{4}}. \tag{29}\]  
To study the nonlinearity of the \(m\)th normal mode and express the complex amplitudes in polar form, substitute their expressions into the modulation equations and separate the real and imaginary parts; this furnishes differential relationships in terms of the real amplitude and the phase of the \(m\)th mode in the same form of Equation (21). By introducing the relative phase \(\gamma_{m}(t_{2})\), the fixed points are obtained when \(\hat{\omega}_{2}a_{m}=0\) and \(\hat{\omega}_{2}\gamma_{m}=0\); hence, the frequency of the \(m\)th nonlinear normal mode can be expressed in terms of the real modal amplitude \(a_{m}\) and through the effective nonlinearity coefficient given by Equation (27) as: \(\omega_{m}^{\text{ml}}=\omega_{m}+\Gamma_{m}\,a_{m}^{2}\). The backbone curves of the lowest four nonlinear normal modes, that is, the nonlinear relationships between the modal amplitude \(a_{m}\) and the detuning \(\sigma\) (where \(\sigma=\omega_{m}^{\text{nl}}-\omega_{m}\)), are shown in Figure 4a. The dashed lines in Figure 4a indicate the backbones calculated by adopting the one-mode projection, while solid lines refer to the solution obtained by using the full modal basis discretization. As shown in the figure, the differences between the curves obtained with the two approaches are more evident for the lowest skewsymmetric mode (red lines) and the lowest symmetric mode (blue lines). Nevertheless, as shown in Figure 4b,c, the shapes of the corresponding modes are less influenced by the modal discretization. Figure 5 reports the values of the coefficients of the nonlinear terms of Equation (23) for the lowest four modes. It is shown that all coefficients of the quadratic terms are negative. On the other hand, the inertial and velocity-proportional coefficients of the cubic terms are positive for all modes, while the coefficient of the stiffness cubic term is always negative. Moreover, the coefficients of the stiffness terms are orders of magnitude higher than the inertial and velocity-proportional coefficients. This indicates that the nonlinearity of the roller battery is mainly governed by the stiffness terms while the nonlinear effects provided by the inertial and velocity-proportional terms are negligible. This is further confirmed by the values of the nonlinearity coefficients \(\Gamma_{m}\) reported in Table 2. In the latter