## 2 Formulation of the Pantograph-catenary Interaction System and Length of EVA
### Model of the Catenary System
In Equations (9) and (10), \(\mathbf{q}_{m}=\left[\mathbf{q}_{my}\), \(\mathbf{q}_{mz}\), \(\mathbf{q}_{omx}\right]^{T}\), and \(\mathbf{q}_{c}=\left[\mathbf{q}_{cy}\), \(\mathbf{q}_{cz}\), \(\mathbf{q}_{\theta cx}\right]^{T}\). The expressions of \(\mathbf{M}_{m}\) and \(\mathbf{K}_{m}\) are:  
\[\mathbf{M}_{m}= \begin{bmatrix}\rho_{m}A_{m}\int_{0}^{L}\left( \mathbf{Y}_{m}^{T}\mathbf{Y}_{m}\right)dx\\ \rho_{m}A_{m}\int_{0}^{L}\left(\mathbf{Z}_{m}^{T}\mathbf{Z}_{m}\right)dx\\ \rho_{m}\left(I_{ym}+I_{zm}\right)\int_{0}^{L}\left(\mathbf{\Phi}_{m}^{T} \mathbf{\Phi}_{m}\right)dx\end{bmatrix}  \tag{11}\]  
\[\mathbf{K}_{m}= \begin{bmatrix}E_{m}I_{mz}\int_{0}^{L}\left(\mathbf{Y}_{m}^{ \prime\prime}\mathbf{Y}_{m}^{\prime\prime}\right)dx\\ +T_{sm}\int_{0}^{L}\left(\mathbf{Y}_{m}^{\prime\prime}\mathbf{Y}_{m}^{\prime \prime}\right)dx\\ E_{m}I_{my}\int_{0}^{L}\left(\mathbf{Z}_{m}^{\prime\prime}\mathbf{Z}_{m}^{ \prime\prime}\right)dx\\ +T_{s}\int_{0}^{L}\left(\mathbf{Z}_{m}^{\prime\prime}\mathbf{Z}_{m}^{\prime \prime}\right)dx\\ G_{m}I_{pm}\int_{0}^{L}\left(\mathbf{\Phi}_{m}^{\prime\prime}\mathbf{\Phi}_{ m}^{\prime}\right)dx\end{bmatrix}  \tag{12}\]  
Based on the Rayleigh damping theory, \(\mathbf{C}_{m}=\alpha\mathbf{M}_{m}+\beta\mathbf{K}_{m}\), where \(\alpha=\beta=0.02\). Similarly, the expressions of \(\mathbf{M}_{c}\) and \(\mathbf{K}_{c}\) are:  
\[\mathbf{M}_{c}= \begin{bmatrix}\rho_{c}A_{c}\int_{0}^{l}\left( \mathbf{Y}_{c}^{T}\mathbf{Y}_{c}\right)dx\\ \rho_{c}A_{c}\int_{0}^{l}\left(\mathbf{Z}_{c}^{T}\mathbf{Z}_{c}\right)dx\\ \rho_{c}\left(I_{yc}+I_{zc}\right)\int_{0}^{l}\left(\mathbf{\Phi}_{c}^{T} \mathbf{\Phi}_{c}\right)dx\end{bmatrix}  \tag{13}\]  
\[\mathbf{K}_{c}= \begin{bmatrix}E_{c}I_{cz}\int_{0}^{L}\left( \mathbf{Y}_{c}^{\prime\prime}\mathbf{Y}_{c}^{\prime}\right)dx\\ +T_{sc}\int_{0}^{L}\left(\mathbf{Y}_{c}^{\prime T}\mathbf{Y}_{c}^{\prime} \right)dx\\ E_{c}I_{cy}\int_{0}^{L}\left(\mathbf{Z}_{c}^{\prime\prime}\mathbf{Z}_{c}^{ \prime\prime}\right)dx\\ +T_{sc}\int_{0}^{L}\left(\mathbf{Z}_{c}^{\prime\prime}\mathbf{Z}_{c}^{\prime }\right)dx\\ G_{c}I_{pc}\int_{0}^{L}\left(\mathbf{\Phi}_{c}^{\prime T}\mathbf{\Phi}_{c}^{ \prime}\right)dx\end{bmatrix}  \tag{14}\]  
and \(\mathbf{C}_{c}=\alpha\mathbf{M}_{c}+\beta\mathbf{K}_{c}\). The droppers, catenary suspension, stitch wire, and registration arm are then modeled. The droppers are modeled as a bilinear spring element with zero stiffness in compression, and the dropper forces are calculated as:  
\[F_{di}= \begin{cases}(d_{ci}-d_{mi})k_{id},\,d_{ci}-d_{mi}\geq 0\\ 0,\,d_{ci}-d_{mi}<0\end{cases} , \tag{15}\]  
where \(k_{id}\) is the \(i\)th dropper stiffness and \(d_{ci}\) and \(d_{mi}\) are the displacements of the contact wire and messenger wire at the dropper-connected locations based on the static shape of the catenary. The static force of every dropper should also be considered in the calculation process. The suspension of the catenary is modeled as spring-damper elements with stiffness \(K_{s}^{\prime}\) and damping \(C_{s}^{\prime}\). The registration arm is modeled as a lumped mass \(m_{\text{BT}}\) attached to the contact wire. As mentioned above, the stitch wire is simplified as two spring-damping elements attached to the messenger wire at the stitch wire location. The stiffness and damping of these two spring-damping elements should be calculated based on the stiffness and damping of the catenary suspension and dropper.