# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 3 Modal Characterization  
Eigenvalue analysis is first performed to characterize the fundamental modes of the system. By linearizing Equation (7), i.e., setting \(\mathbf{i}_{o}=\mathbf{d}_{o}=\mathbf{g}_{o}=\mathbf{0}\) (\(o=2,3\)), neglecting the forcing term and the linear viscous damping force (i.e., setting \(\mathbf{f}_{0}=\mathbf{0}\) and \(\mathbf{C}=\mathbf{O}\), being \(\mathbf{O}\) the \(N_{s}\times N_{s}\) null matrix), the homogeneous system of linear equations governing the free undamped oscillations of the system assumes the classical form:  
\[\mathbf{M}\,\ddot{\mathbf{x}}+\mathbf{K}\,\mathbf{x}=\mathbf{0}\,. \tag{8}\]  
The \(N_{s}\) eigenvalues of Equation (8) provide the natural frequencies \(\omega_{s}\) (\(s=1,\ldots,N_{s}\)) of the system and the corresponding eigenvectors, conveniently normalized, deliver the \(N_{s}\) mode shapes of the roller battery. In particular, by referring to the \(N_{s}\times 1\) vectors and natural frequencies of the \(m\)th and \(n\)th modes of the system, i.e., (\(\mathbf{\psi}_{m}\), \(\omega_{m}\)) and (\(\mathbf{\psi}_{n}\), \(\omega_{n}\)), respectively, the following relationships are satisfied: \(\mathbf{\psi}_{m}^{\top}\mathbf{M}\,\mathbf{\psi}_{n}=\delta_{mn}\) and \(\mathbf{\psi}_{m}^{\top}\mathbf{K}\,\mathbf{\psi}_{n}=\omega_{m}^{2}\delta_{mn}\), being \(\delta_{mn}\) the Kronecker delta. The lowest eight nondimensional circular frequencies of the system are reported in Table 1 and the corresponding mode shapes of the roller battery are displayed in Figure 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Mode \(s\) & 1 & 2 & 3 & 4 \\ \hline \(\omega_{s}\) & 0.376 & 0.581 & 0.734 & 0.96 \\ \hline Mode \(s\) & 5 & 6 & 7 & 8 \\ \hline \(\omega_{s}\) & 1.02 & 1.109 & 1.163 & 1.757 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lowest eight nondimensional circular frequencies. Interesting considerations emerge from the analysis of the ratios between the lowest frequencies of the system. Figure 3 reports the ratios between the frequencies of modes \((2,3,4,5,6,7,8)\) with respect to the lowest frequency \(\omega_{1}\) (Figure 3a) and of the modes \((3,4,5,6,7,8)\) with respect to the second frequency \(\omega_{2}\) (Figure 3b), respectively. As shown in Figure 3a, the frequency of the third roller battery mode is close to twice the frequency \(\omega_{1}\) of the lowest mode while the sixth and seventh modes have frequency three times \(\omega_{1}\). The same integer ratios occur between the frequency of the seventh and eighth modes with respect to the frequency \(\omega_{2}\) of the second mode, i.e., 2:1 and 3:1 ratio, respectively, as shown in Figure 3b. When in a nonlinear dynamical system, the frequencies of two modes are in an integer ratio with each other, then the internal exchange of energy between the modes involved may occur, leading the system to dynamic interactions, i.e., the so-called internal resonances [34; 35; 36]. It is a matter of fact that the internal resonances may more easily be activated between modes having lower frequencies, such as in the case of the mechanical system investigated, shown by the first and third mode of the roller battery (see red bar and red line in Figure 3a). Therefore, when the frequency \(\Omega\) of the cable-induced harmonic excitation is close to the frequency of the system, primary resonance may occur; furthermore, if the latter is a multiple or an integer ratio of the frequency of a different mode then internal resonance may also take place. In the system investigated in this work, the cable-induced periodic excitation has a frequency dependent on the cable speed which never exceeds the nondimensional  
Figure 3: Ratios of the lowest eight frequencies of the system with respect to **(a)** the lowest frequency \(\omega_{1}\) and **(b)** the second frequency \(\omega_{2}\). Red and blue bars and dashed lines indicate 2:1 and 3:1 ratios, respectively. Figure 2: Shapes of the lowest eight linear normal modes of the roller battery system.