# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 2 Nonlinear Dynamic Characterization
### Nondimensional Vector Form of the Equations of Motion  
The equations of motion are then nondimensionalized by adopting the characteristic time \(t_{c}=\sqrt{\rho A_{hb}L_{hb}^{4}/E\,I_{hb}}\) and the equivalent beam span \(L_{hb}\) as the characteristic length, respectively. The later can be used for the nondimensionalization of the systems geometric parameters, and for the unknown generalized coordinates corresponding to \(p_{2},\ldots,p_{ne}\) and \(q_{1},\ldots,q_{N_{b}}\), respectively. Furthermore, it turns out that \(f_{c}=\rho A_{hb}L_{hb}^{2}/t_{c}^{2}\) can be used to nondimensionalize the forces acting in the system: in this case, the nondimensional maximum force acting on each roller can be calculated as \(\lambda=F/f_{c}\). The numerical simulations are then carried out considering \(ne=20\) finite elements for the cable discretization and \(N_{b}=4\) trial functions for the hoisting beam discretization; therefore, the system dynamics are described by \(N_{s}=30\) DOFs. The geometric and mechanical properties of the roller battery considered in this work are taken in consonance with the data reported in [13].  
The equations of motion reported in Equation (6) involve strongly nonlinear functions of the system DOFs (i.e., nonlinear combinations of trigonometric functions); nevertheless, their time integration can be quite easily performed by using well-established numerical integration schemes. On the other hand, their analytical treatment requires an asymptotic expansion of the nonlinear functions present in Equation (6). Since the system investigated includes only geometric nonlinearities, the most relevant effects of these nonlinearities are given by the approximating second- and third-order polynomial terms. Therefore, a third-order Taylor expansion of Equation (6) was performed, and the obtained system of nondimensional nonlinear equations of motion can be conveniently recast in vector form as:  
\[\begin{split}\mathbf{M}\,\ddot{\mathbf{x}}+\mathbf{C}\,\dot{ \mathbf{x}}+\mathbf{K}\,\mathbf{x}&+\mathbf{i}_{2}(\mathbf{x}, \ddot{\mathbf{x}})+\mathbf{d}_{2}(\dot{\mathbf{x}},\dot{\mathbf{x}})+\mathbf{ g}_{2}(\mathbf{x},\mathbf{x})\\ &+\mathbf{i}_{3}(\mathbf{x},\mathbf{x},\ddot{\mathbf{x}})+ \mathbf{d}_{3}(\mathbf{x},\dot{\mathbf{x}},\dot{\mathbf{x}})+\mathbf{g}_{3}( \mathbf{x},\mathbf{x},\mathbf{x})+\mathbf{f}_{0}\cos\Omega t=\mathbf{0},\end{split} \tag{7}\]  
where \(\mathbf{x}\) is the \(N_{s}\times 1\) vector collecting the \(N_{s}\) generalized coordinates, while \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the \(N_{s}\times N_{s}\) mass, viscous damping, and stiffness matrices of the system, respectively. Moreover, \(\mathbf{f}_{0}\) is the \(N_{s}\times 1\) vector of the nondimensional amplitudes of the external excitation, while \(\mathbf{i}_{0}\), \(\mathbf{d}_{0}\), and \(\mathbf{g}_{0}\) (\(o=2,3\)), are the inertial, velocity-proportional, and stiffness \(N_{s}\times 1\) nonlinear operators of \(o\)th order, respectively, arising from the system geometric nonlinearities. Finally, \(\mathbf{0}\) is the \(N_{s}\times 1\) null vector and the natural initial conditions are considered (i.e., \(\mathbf{x}(0)=\mathbf{0}\), and \(\dot{\mathbf{x}}(0)=\mathbf{0}\), respectively).