# A Fast Simulation Model of Pantograph-Stitched-Catenary Interaction in Long-Distance Travel
## 2 Reduced Pantograph-Stitched-Catenary Interaction Model
### Reduced Stitched Catenary Model  
Based on the reduced catenary model, the reduced stitched catenary model is formulated first. The small area around the moving pantograph is chosen, and the ALE method and MSM are used to model the catenary located in this small area [1]. The contact and messenger wire are considered as Euler-Bernoulli beams, and the stitch wire is considered as a part of the messenger wire with a couple of spring-damping elements supporting this part, which makes the structure of the present stitch catenary model familiar to the simple catenary system. Thus, the derivation process of the dynamic equation of the contact and messenger wire in the reduced stitched catenary model is just the same as that shown in [28], and it is not repeated here. The dynamic equations of the contact and messenger wire in the reduced stitched catenary model are  
\[\mathbf{M}_{m}\ddot{\mathbf{q}}_{m}+\mathbf{C}_{m}\dot{\mathbf{q}}_{m}+\mathbf{K}_ {m}\mathbf{q}_{m}=\mathbf{Q}_{flowm}+\mathbf{Q}_{Fm} \tag{1}\]  
\[\mathbf{M}_{c}\ddot{\mathbf{q}}_{c}+\mathbf{C}_{c}\dot{\mathbf{q}}_{c}+\mathbf{ K}_{m}\mathbf{q}_{c}=\mathbf{Q}_{flowm}+\mathbf{Q}_{Fc} \tag{2}\]  
where \(\mathbf{M}_{m}\), \(\mathbf{C}_{m}\) and \(\mathbf{K}_{m}\) are the mass, damping and stiffness matrices of the messenger wire, and \(\mathbf{M}_{c}\), \(\mathbf{C}_{c}\) and \(\mathbf{K}_{c}\) are the mass, damping and stiffness matrices of the contact wire, respectively. Rayleigh damping is considered as the numerical damping for convergence, and the damping coefficient is chosen as 0.02 [10]. \(\mathbf{Q}_{flow}\) is the additional generalized force vector generated by the relative motion between the reduced stitched catenary model and the material of both the contact and messenger wire, and \(\mathbf{Q}_{F}\) is the generalized force vector caused by the force act on the contact wire, including the pantograph-catenary interaction force and gravity. The expression of these matrices can be found in [28]. The stitch wire model is then expressed here. As mentioned above, the stitch wire is simplified to a part of the messenger wire, and the catenary suspension and the original messenger wire are modeled as a couple of spring-damping elements. The distance between these two spring-damper elements is \(d_{S}\), and the stiffness and damping of these two  
Figure 2: Schematic of the reduced pantograph–stitched-catenary interaction system: (**a**) stitched catenary system and (**b**) reduced pantograph–stitched-catenary interaction system. elements are \(K_{S}^{\prime}\) and \(C_{S}^{\prime}\), respectively. These values are chosen to ensure that the stiffness of the catenary at the stitch wire part is equal to the original stiffness. The dynamic force of the spring-damping elements can be calculated by  
\[F_{S}=K_{S}^{\prime}d_{S}+C_{S}^{\prime}\dot{d}_{S} \tag{3}\]  
where \(d_{S}\) is the displacement of the messenger wire at the location of the spring-damping element. Note that the location of these spring-damping elements also changes as the pantograph travels along the catenary based on the theory of the reduced catenary model. Let \(x_{SQi}\) be the initial position of the \(i\)th stitch wire model at \(o\)-\(xyz\), and the position of the \(i\)th dropper \(x_{Si}\) at any time can be expressed as  
\[x_{Si}=x_{SQi}-Vt \tag{4}\]  
where \(V\) is the operation velocity of the vehicle. The dropper and registration arm are then modeled. Based on the reduced catenary model, the dropper is modeled as bilinear spring elements, and the registration arm (attachment points) is modeled as a lumped mass \(m_{S}\) attached to the contact wire, which can present the shock to the pantograph and catenary. Details of these models are shown in [28]. Based on the dynamic equations of the messenger and contact wires, the stitch wire model, and the model of the dropper and registration arm, the reduced stitched catenary model is finally formulated as  
\[