# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 2 Nonlinear Dynamic Characterization
### Kinematic Descriptors  
The kinematics are described in the plane containing the direction collinear with the cable configuration as well as the vertical direction. In particular, two fixed frames are introduced, namely, the frame \(\left(\mathbf{e}_{1}^{0},\mathbf{e}_{2}^{0}\right)\) with origin in \(C\) and oriented as the horizontal and vertical directions, respectively. The frame \(\left(\mathbf{e}_{1},\mathbf{e}_{2}\right)\), centered in \(C\) and oriented along the roller battery in the inclined direction, thus rotated with respect to the fixed (global) frame by \(\varphi_{R}\) (see Figure 1). The geometric parameters characterizing the reference configuration of the roller battery, the cable, the cabin, and the grip are shown in Figure 1. In particular, along the direction \(\mathbf{e}_{1}\), \(d_{C}\) denotes the distance between \(C\) and \(B_{k}\) (\(k=1,2\)), \(d_{B}\) is the distance between \(B_{k}\) and \(A_{j}\), and \(d_{A}\) represents the distance between \(A_{j}\) and each of the two corresponding roller-cable contact points \(P_{i}\). Similarly, the distances \(h_{C}\), \(h_{B}\), and \(h_{A}\) represent the distances along \(\mathbf{e}_{2}\) between the mentioned hinges. The parameters characterizing the reference geometry of the cable are the side lengths \(L_{S1}\) and \(L_{S2}\) (i.e., the distances of the two supports from the first and last rollers \(P_{1}\) and \(P_{8}\), respectively). The roller battery degrees of freedom (DOFs) are the rotations \(\theta_{A_{j}}(t)\), \(\theta_{B_{k}}(t)\), and \(\theta_{C}(t)\), about the hinges \(A_{j}\), \(B_{k}\), and \(C\), respectively. Therefore, the current configuration of the roller battery can be described by the vectors \(\mathbf{r}_{P_{i}}(t)\), \(\mathbf{r}_{A_{j}}(t)\), \(\mathbf{r}_{B_{k}}(t)\), and \(\mathbf{r}_{C}(t)\) which provide, at time \(t\), the positions of the cable contact points \(P_{i}\) (\(i=1,\ldots,8\)) and of the hinges \(A_{j}\) (\(j=1,\ldots,4\)), \(B_{k}\) (\(k=1,2\)) and \(C\), respectively, or by the displacement vectors of the above mentioned points, that is, \(\mathbf{u}_{B_{k}}(t)\), \(\mathbf{u}_{A_{j}}(t)\), and \(\mathbf{u}_{P}(t)\), respectively. To discretize the time- and space-dependent function \(v(s,t)\) representing the vertical displacement of the centerline hoisting cantilever beam, having length \(L_{bb}\), the Galerkin  
Figure 1: (**Top**) Geometric characteristics of the roller battery system in compression towers. (**Bottom**) Degrees-of-freedom and FE discretization (i.e., diamonds symbols). method is adopted by considering a suitable number of trial functions to approximate the dynamics of the beam-like subsystem. In particular, the lowest \(N_{b}\) mode shapes the \(\phi_{b}(s)\) of the cantilever beam, which are employed to discretize the beam deflection in space, while \(q_{b}(t)\) (\(b=1,\ldots,N_{b}\)) represent the time-dependent generalized coordinates, hence, the beam deflection can be written as \(v(s,t)\approx\sum_{b=1}^{N_{b}}\phi_{b}(x)q_{b}(t)\), and the vertical displacement of point \(C\) turns out to be \(v_{C}(t)=v(L_{hb},t)\). On the other hand, due to the multiple contact points \(P_{i}\) (\(i=1,\ldots,8\)) between the cable and roller battery, the cable discretization is carried out according to the finite element (FE) technique. In particular, \(ne=20\) finite elements are used to discretize the cable length such that each contact point \(P_{i}\) coincides with a finite element node while the cable support points are the boundary nodes. First-order Lagrangian polynomials are employed to approximate the cable transversal displacement and to describe the kinematic function \(w_{e}(\zeta,t)\) in the \(e\)th element.