# Pantograph-Catenary Interaction Prediction Model Based on SCSA-RBF Network
## 2 Pantograph-catenary Finite Element Model [7]  
During the dynamic analysis of the pantograph-catenary system, the pantograph is usually equivalent to a three-mass model with three vertical degrees of freedom (DOFs). The pantograph dynamic equation is as follows:  
\[\begin{bmatrix}m_{3}&0&0\\ 0&m_{2}&0\\ 0&0&m_{1}\end{bmatrix}\left[\begin{array}{c}\tilde{z}_{3}(t)\\ \tilde{z}_{2}(t)\\ \tilde{z}_{1}(t)\end{array}\right]+\left[\begin{array}{ccc}c_{3}&-c_{3}&0 \\ -c_{3}&c_{3}+c_{2}&-c_{2}\\ 0&-c_{2}&c_{2}+c_{1}\end{array}\right]\left[\begin{array}{c}\dot{z}_{3}(t)\\ \dot{z}_{2}(t)\\ 0&-k_{2}&k_{2}+k_{1}\end{array}\right]\left[\begin{array}{c}z_{3}(t)\\ z_{2}(t)\\ z_{1}(t)\end{array}\right]=\left[\begin{array}{c}-F_{\rm c}(t)\\ F_{2}\\ F_{1}\end{array}\right] \tag{1}\]  
where \(m_{n},k_{n}\), and \(c_{n}\) are the equivalent mass, stiffness, and damping, respectively. \(F_{n}\) is the lifting forces, \(z_{n}(t)\), \(\dot{z}_{n}(t)\) and \(\ddot{z}_{n}(t)\) are the vertical displacement, velocity, and acceleration, respectively. Subscripts \(n=3\), 2, and 1 represent the three lumped masses equivalent to the vertical vibration of the panhead, vertical bending vibration of the upper frame, and rotation of the frame, respectively. \(F_{\rm c}(t)\) is the contact force.  
A finite element model of the 30 spans two-dimensional stitched catenary system was established, referring to the geometric and material parameters of the actual catenary. The model included contact wires, messenger wires, stitch wires, droppers, steady arms, and various clamps. The contact wires, messenger wires, and stitch wires were treated as slender rod models with prestress due to their large length in comparison to their cross-sectional sizes and significant pretension. The element length for these wires was set at 0.2 m. The droppers were equivalent to nonlinear spring elements that only bear tension, and each dropper was only divided into one element. The steady arms were equivalent to a spring-lumped mass system, and each steady arm was only divided into one spring-lumped mass system. Each clamp was equivalent to a lumped mass. The proportional damping coefficients of the catenary dynamic system, \(\alpha\) and \(\beta\), were set to 1.25 \(\times\) 10\({}^{-2}\) s\({}^{-1}\) and 1.00 \(\times\) 10\({}^{-4}\) s, respectively. Based on the above assumptions, the dynamic equations of the catenary could be obtained. The penalty function method, with a contact stiffness of \(k_{\rm c}\) = 50,000 N/m, was used to simulate the coupling behavior between the panhead strips and contact wire, and the dynamic equations of the interaction of the pantograph-catenary system could be accordingly obtained. By combining the above equations, the dynamic equation of the pantograph-catenary system could be obtained. The maximum integration step of time was set to 0.00167 s, and the interval time for output was set to 0.005 s; that is, the sampling frequency was set to 200 Hz. At a given operating speed, pantograph-catenary parameters, and contact stiffness, the time-domain signals of the contact force and panhead acceleration, namely \(F_{\rm c}(t)\) and \(\ddot{z}_{3}(t)\), were calculated using Ansys 19.2\({}^{\rm\textregistered}\) commercial software.  
The FEM calculated values of the above pantograph-catenary finite element model (Figure 1) have been verified using the measured data from line tests on the Datong-Xi'an High-speed Railway Line [7] and the trade standard EN50318:2018 [22,23]. The computational cost of the pantograph-catenary finite element model was about 10 core hours when calculating a single pantograph-catenary interaction case.