# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 5 Cable Motion-Induced Resonances
### Two-to-One Internal Resonance  
In this section, we investigate the phenomenon of the two-to-one internal resonance between two modes of the roller battery occurring when one mode of the system undergoes primary resonance due to the periodic excitation of the moving cable. Without loss of generalization, the two resonant modes are indicated as mode \(m\) and mode \(n\), respectively, and the corresponding frequencies have a ratio close to two (i.e., \(\omega_{n}\approx 2\omega_{m}\)). Two case-studies are investigated, namely, the case when the frequency \(\Omega\) of the periodic excitation is near the frequency of the mode \(m\) (i.e., primary resonance of mode \(m\), with \(\Omega\approx\omega_{m}\)), and the case when \(\Omega\approx\omega_{n}\) (i.e., primary resonance of mode \(n\)), respectively. As mentioned in Section 3, 2:1 internal resonance is possible in the mechanical system studied in this work since the frequencies of the third and first mode have a ratio close to two. The two-to-one internal resonance is caused by the quadratic nonlinearities of the system; therefore, the asymptotic treatment of Equation (7) starts by seeking its solution in the form:  
\[\mathbf{x}(t)=\epsilon\,\mathbf{x}_{1}(t_{0},t_{1})+\epsilon^{2}\,\mathbf{x}_ {2}(t_{0},t_{1}) \tag{35}\]  
where \(t_{0}\) is the nondimensional fast time scale and \(t_{1}\) is the slow time scale in which the nondimensional time derivative can be written as \(d/dt=\partial_{0}+\varepsilon\partial_{1}\), with \(\partial_{0}=\partial/\partial t_{0}\) and \(\partial_{1}=\partial/\partial t_{1}\), respectively, while \(d^{2}/dt^{2}=\partial_{0}^{2}+2\partial_{0}\partial_{1}\epsilon+\epsilon^{2} \partial_{1}^{2}\), with \(\partial_{0}^{2}=\partial^{2}/\partial t_{0}^{2}\), \(\partial_{1}^{2}=\partial^{2}/\partial t_{1}^{2}\), and \(\partial_{0}\partial_{1}=\partial^{2}/\partial t_{0}\partial t_{1}\), respectively. In this case, the resonant terms are generated at the quadratic order, therefore, the viscous damping force and excitation need to be rescaled so as to appear at the second perturbation order; hence, we set \(\mathbf{C}=\epsilon\,\mathbf{C}\) and \(\mathbf{f}_{0}=\epsilon^{2}\mathbf{f}_{0}\), respectively. By now substituting Equation (35) into Equation (7), and retaining only terms up to order \(\epsilon^{2}\), the following hierarchy of linear problems is obtained by equating to zero the coefficients of like powers of \(\epsilon\):  
\[\text{order }\epsilon \tag{36}\] \[\text{order }\epsilon^{2}\] \[\text{\bf M}\,\partial_{0}^{2}\mathbf{x}_{2}+\mathbf{K}\, \mathbf{x}_{2}= -\mathbf{C}\,\partial_{0}\mathbf{x}_{1}-\mathbf{g}_{2}(\mathbf{x} _{1},\mathbf{x}_{1})\] (37) \[-\mathbf{d}_{2}(\partial_{0}\mathbf{x}_{1},\partial_{0}\mathbf{ x}_{1})-\mathbf{i}_{2}(\mathbf{x}_{1},\partial_{0}^{2}\mathbf{x}_{1})-\mathbf{f}_{0} \cos\Omega t.\]  
Figure 7: (**a**) FRCs of the modal amplitude \(a_{m}\) (case \(m=1\)) and stability regions for two different values of the coefficient \(\xi\) which rescales the damping; the dotted line indicates the backbone of the mode \(m=1\). (**b**) Loci of the fold bifurcation points for \(\xi=(0,0.2,0.4,0.6,0.8,1)\). In the case of two interacting modes (i.e., mode \(m\) and mode \(n\)), the general solution of the \(\epsilon\)-order problem (36) can be written as:  
\[\mathbf{x}_{1}(t_{0},t_{1})=A_{m}(t_{1})\,\boldsymbol{\psi}_{m}\,e^{i\omega_{m }t_{0}}+A_{n}(t_{1})\,\boldsymbol{\psi}_{n}\,e^{i\omega_{m}t_{0}}+cc, \tag{38}\]  
where \(A_{m}(t_{1})\) and \(A_{n}(t_{1})\) are the complex amplitudes of the \(m\)th and the \(n\)th mode, respectively, while \(cc\) stands for complex conjugate. The first-order solution to Equation (38) is then substituted into the \(\epsilon^{2}\)-order problem. The singularity generated by the secular terms exhibited in the second-order problem can be removed by enforcing solvability conditions.