# Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach
## 5 Cable Motion-Induced Resonances
### Primary Resonance
By solving the ensuing equations in terms of \(\cos\gamma_{m}\) and \(\sin\gamma_{m}\), and enforcing the trigonometric identity, the frequency-response function providing the nonlinear relationship between the real amplitude \(a_{m}\) and frequency detuning \(\sigma\) can be obtained in its implicit form as:  
\[\mathcal{G}(a_{m},\sigma)=\frac{\omega_{m}^{2}a_{m}^{2}}{\mathcal{F}_{0}^{2}} \bigg{[}\mathcal{C}_{m}^{2}+4\Big{(}\Gamma_{m}a_{m}^{2}-\sigma\Big{)}^{2} \bigg{]}-1. \tag{33}\]  
The equation that allows for describing the bifurcation behavior of the system is given by \(\mathcal{G}(a_{m},\sigma)=0\); moreover, from the stationarity of \(\mathcal{G}(a_{m},\sigma)\) with respect to \(a_{m}\) it is possible to calculate the loci of the fold bifurcation points as the four non-trivial solutions of the equation \(\partial_{a_{m}}\mathcal{G}(a_{m},\sigma)=0\), that is:  
\[a_{m_{(2,3,4,5)}}^{\Gamma}=\pm\sqrt{\frac{4\,\sigma\pm\sqrt{4\,\sigma^{2}-3 \,\mathcal{C}_{m}^{2}}}{6\,\Gamma_{m}}}\,, \tag{34}\]  
with \(a_{m_{(1)}}^{\Gamma}\) being the trivial solution (i.e., \(a_{m_{(1)}}^{\Gamma}=0\)). Finally, after removing the secular terms by means of Equation (31), the solution of the inhomogeneous third-order problem can be calculated as \(\mathbf{x}_{3}=\hat{\mathbf{x}}_{3,1}A_{m}^{3}e^{3i\omega_{m}t_{0}}+\hat{ \mathbf{x}}_{3,2}\hat{A}_{m}^{3}e^{-3i\omega_{m}t_{0}}\), where the vectors \(\hat{\mathbf{x}}_{3,r}\) (\(r=1,2\)) are expressed, again, as a linear combination of the \(N_{s}\) modes of the system, that is: \(\hat{\mathbf{x}}_{3,r}=\sum_{s=1}^{N_{s}}b_{r,s}\boldsymbol{\psi}_{s}\). The third-order solution is then substituted into the \(e^{3}\)-order problem and it is then projected into the modal space as: \(\boldsymbol{\Psi}^{\top}\mathbf{M}\,\partial_{0}^{2}\mathbf{x}_{3}+ \boldsymbol{\Psi}^{\top}\mathbf{K}\,\mathbf{x}_{3}=\boldsymbol{\Psi}^{\top} \mathbf{r}_{3}\). By then equating the coefficients of terms having the same frequency, it is possible to calculate the \(2N_{s}\) coefficients \(b_{r,s}\) and find the second-order solution \(\mathbf{x}_{3}\). Thereafter, the nonlinear equations of motion of the system expressed by Equation (6) are numerically integrated in time by using the built-in NDSolve algorithm present in the software Mathematica [37]. For each simulation, at a selected excitation frequency \(\Omega\), the maximum integration time was set to \(t_{max}=500(2\pi/\Omega)\), where \((2\pi/\Omega)\) is the excitation period, while the time step was \(\Delta t=(2\pi/\Omega)/40\). At steady-state, the state vector (\(\mathbf{x}\), \(\dot{\mathbf{x}}\)) was saved and assigned as the initial condition for the time integration run with the subsequent \(\Omega\). Finally, backward and forward frequency sweeps were performed so as to calculate the numerical FRCs. Figure 6 shows the frequency response curves in terms of the vertical component (i.e., along the direction \(\mathbf{e}_{2}\)) of the nondimensional displacement vectors \(\mathbf{u}\) of points \(P_{8}\), \(A_{2}\), \(B_{1}\), and \(C\), respectively, when \(\Omega\approx\omega_{1}\). The comparison between numerical time integration (gray dots) and the third-order asymptotic solution obtained via the method of multiple scales (solid and dashed lines) shows a very good agreement in most of the range of stable solutions. The asymptotic analysis allowed us to investigate the unstable solutions (dashed lines) and calculate the threshold values of amplitude and frequency, where _fold_ bifurcation occurs (i.e., at the boundary between solid and dashed lines). The analysis was performed for a value of the nondimensional maximum amplitude of the periodic force \(\lambda=8.27\times 10^{-3}\), so as to show the softening nonlinearity of the dynamic response. The value of the forcing amplitude is almost two orders of magnitude lower than the characteristic force of the system. This means that moderately high values of the forcing term can also induce a nonlinear resonance response of the system.