# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 2 Nonlinear Dynamic Characterization
### Nonlinear Equations of Motion  
The equations governing the motion of the roller battery system, including the interaction between the hoisting beam and the cable are derived via the Euler-Lagrange approach. To this end, the potential and kinetic energies of the beam and cable are first calculated together with the potential energy of the fictitious springs and the kinetic energy of the rollers to calculate the Lagrangian of the system, from which the nonlinear equations of motion of the roller battery are obtained. In particular, the potential energy of the hoisting beam and the cable can be expressed as:  
\[ \begin{split}\mathcal{U}_{b}(t)&=\frac{1}{2}\int_{0 }^{L_{hb}}\partial_{ss}v(s,t)EI_{hb}\partial_{ss}v(s,t)ds\,,\\ \mathcal{U}_{c}(t)&=\frac{1}{2}\mathbf{X}(t)^{\top} \mathbf{K_{c}}\,\mathbf{X}(t)\,,\end{split}  \tag{1}\]  
respectively, where \(\partial_{s}\) represents the partial differentiation with respect to the beam arclength \(s\) (with origin at the beam clamped cross section), \(EI_{hb}\) is the hoisting beam flexural stiffness, while \(\mathbf{K_{c}}\), incorporating the cable pretension \(N_{0}\), is the global stiffness matrix of the cable. The energy stored by the fictitious springs connecting the cable and the rollers is computed as follows:  
\[\mathcal{U}_{fict}(t)=\frac{1}{2}\sum_{i=1}^{8}K_{i}\Delta L_{i}(t)^{2}\,, \tag{2}\]  
where \(\Delta L_{i}(i)=\mathbf{u}_{P_{i}}(t)\cdot\mathbf{e}_{2}-w_{e,i}(t)\) is the relative displacement along the roller local direction \(\mathbf{e}_{2}\) (corresponding to the transverse direction of the cable) and \(w_{e,i}(t)\) is the cable displacement at the node corresponding to the \(i\)th roller. On the other hand, the hoisting beam and the cable kinetic energy contributions to the dynamics of the system are given by:  
\[ \begin{split}\mathcal{T}_{b}(t)&=\frac{1}{2}\int_{0 }^{L_{hb}}\partial_{t}v(s,t)\rho A_{hb}\partial_{t}v(s,t)ds\,,\\ \mathcal{T}_{c}(t)&=\frac{1}{2}\dot{\mathbf{X}}(t)^{ \top}\mathbf{M_{c}}\,\dot{\mathbf{X}}(t)\,,\end{split}  \tag{3}\]respectively, where \(\partial_{t}\) and the overdot represent partial and total differentiation with respect to time \(t\), respectively. \(\rho A_{hb}\) is the hoisting beam mass per unit length, while \(\mathbf{M_{c}}\), includes the cable mass per unit length \(\rho A_{c}\), is the global mass matrix of the cable. Finally, the kinetic energy possessed by the rollers can be calculated as:  
\[\mathcal{T}_{R}(t)=\frac{1}{2}\sum_{i=1}^{8}m_{P_{i}}\dot{\mathbf{r}}_{P_{i}} \cdot\dot{\mathbf{r}}_{P_{i}}+\frac{1}{2}\sum_{j=1}^{4}m_{A_{j}}\dot{\mathbf{r }}_{A_{j}}\cdot\dot{\mathbf{r}}_{A_{j}}+\frac{1}{2}\sum_{k=1}^{2}m_{B_{k}} \dot{\mathbf{r}}_{B_{k}}\cdot\dot{\mathbf{r}}_{B_{k}}+\frac{1}{2}m_{tb}(\dot {v}(L_{hb},t))^{2}, \tag{4}\]  
where \(m_{P_{i}}\), \(m_{A_{j}}\), and \(m_{B_{k}}\) are the equivalent masses of the roller battery lumped at \(P_{i}\), \(A_{j}\), \(B_{k}\), respectively, and \(m_{tb}\) is part of the mass of the roller battery, acting at \(C\). Finally, the stored and the kinetic energies of the overall system are given by:  
\[ \begin{split}\mathcal{U}(t)&=\mathcal{U}_{b}(t)+ \mathcal{U}_{c}(t)+\mathcal{U}_{fict}(t),\\ \mathcal{T}(t)&=\mathcal{T}_{b}(t)+\mathcal{T}_{c}( t)+\mathcal{T}_{R}(t).\end{split}  \tag{5}\]  
The simplifications made in the model formulation proposed in [13] include the assumption of considering the cable as a straight, prestressed string. Nevertheless, when crossing the roller battery system, the cable is subject to a small curvature due to the slight change in slop of the cable passing from the first to the last roller. Although this simplification does not imply relevant errors in the description of the dynamic behavior of this system (as demonstrated by the comparison with experimental results in [13]), the actual cable curvature generates a force acting on each roller transversally to the cable direction.