# 1D CNN Based Detection and Localisation of Defective Droppers in Railway Catenary
## 2 Pantograph-Catenary System Modelling  
The contact and messenger wires in the catenary system are modelled by a 2D Euler-Bernoulli beam whose vertical dynamic behaviour follows:  
\[\rho A\frac{d^{2}w}{dt^{2}}=T\frac{d^{2}w}{dx^{2}}-EI\frac{d^{4}w}{dx^{4}}-c \frac{dw}{dt}+F(x,t) \tag{1}\]  
where \(\rho\), \(A\), \(w\), \(t\), \(T\), \(E\), \(I\), \(x\), \(c\) and \(F\) denote the density of the wire, the cross-sectional area of the wire, the vertical deflection of the wire, the elapsed time, the tensile force of the wire, the Young's modulus of the wire, the second moment of area of the wire, the horizontal location of the measured point, the damping coefficient, and the external force, respectively.  
To represent the kinetics by means of FEs, the system is discretised into small FEs, and the dynamic behaviour of each element node can be approximately a function of the states of itself and its adjacent ones (i.e., deflection and slope). Hence, the system dynamics can be assembled and obtained, as in Equation (2).  
\[\left\{\begin{matrix}\mathbf{M}*\ddot{y}(t)+\mathbf{C}*\dot{y}(t)+\mathbf{K}*y(t)=F(t)\\ \mathbf{C}=\alpha*\mathbf{M}+\beta*\mathbf{K}\end{matrix}\right\} \tag{2}\]  
where \(y(t)\) is the states of nodes; \(\mathbf{M}\), \(\mathbf{K}\) and \(\mathbf{C}\) are the assembled mass, stiffness, and damping, respectively (the damping coefficient is assumed to be proportional to mass and stiffness); and \(F(t)\) denotes the external force on the nodes including the pantograph-catenary contact force and those exerted by other catenary components (i.e., dropper, steady arm and messenger wire suspension).  
The dropper is represented by a spring model with slackening as in Equation (3), whose elasticity \(k_{d}\) changes with the spring state. In this study, as the defective dropper is of concern, the elasticity \(k\) varies, ranging from 100% to 0% of the nominal value to represent dropper damage conditions. It should be noted that the initial condition of the catenary system is affected by the defective dropper, which will be incorporated in the model.  
\[k_{d}=\left\{\begin{matrix}k&\Delta x>0\\ 0&\Delta x\leq 0\end{matrix}\right. \tag{3}\]  
The material properties and structural information on the catenary model are taken from [28] and an existing academic benchmark [26]. Besides that, a validated three-mass lumped-mass pantograph model [26] is used for the simulations. To generate the pantograph-catenary contact force, a virtual spring of 50,000 N/m is placed between the pantograph head and the contact wire [26; 28]. The above pantograph-catenary model is depicted in Figure 1 and the parameters are summarised in Table 1. The Newmark-beta method with the rule of average constant acceleration is applied for time integration in simulations. The model validation will be given in the next section.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Component** & **Parameter** & **Value** \\ \hline \multirow{6}{*}{Lumped-mass pantograph} & \multirow{2}{*}{Lower frame} & Mass (kg) & 6 \\  & & Spring coefficient (N/m) & 160 \\  & & Damper coefficient (Ns/m) & 100 \\  & & Mass (kg) & 9 \\  & & Spring coefficient (N/m) & \(1.55\times 10^{4}\) \\  & & Damper coefficient (Ns/m) & 0.1 \\  & & Mass (kg) & 7.5 \\  & & Spring coefficient (N/m) & \(7\times 10^{3}\) \\  & & Damper coefficient (Ns/m) & 45 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the pantograph–catenary model.