# Railway Bridge Condition Monitoring Using Numerically Calculated Responses from Batches of Trains
## 2 Model Description
### Vehicle Model  
The train is represented by the two-dimensional (2D) 4-axle railway carriage model shown in Figure 1. The vehicle model includes lumped masses, rigid bars, springs, and dampers. There are 10 degrees of freedom (DOFs) in this vehicle model: vertical translation and rotation about the centre of gravity for the car body and bogies and vertical translation for each wheelset. The wheels are fixed to the track with no separation being permitted. This reduces the system to six DOFs. In this model, each wheel is modelled with a mass (\(m_{w}\)). It is connected to the bogie by a primary suspension represented by a spring (\(k_{p}\)) and a viscous damper (\(c_{p}\)) in parallel. The bogies are represented as rigid bars with mass (\(m_{b}\)) and moment of inertia (\(I_{b}\)), whereas the main body of the vehicle is also represented as a rigid bar with mass (\(m_{v}\)) and moment of inertia (\(I_{v}\)). Springs (\(k_{s}\). ) and dampers (\(c_{s}\)) represent the secondary suspensions to connect the bogies to the main body. The distances between the car body centre of mass and the bogie pivots are \(L_{v1}\) and \(L_{v2}\). Similarly, \(L_{b1}\), \(L_{b2}\), \(L_{b3}\), and \(L_{b4}\) are the distances between the bogie centres of mass and the wheels. Vehicle models of this type have been used in many studies [35; 36].  
Assuming small rotations, a linearised system of equations of motion is adopted [37] and can be expressed in matrix form:  
\[M_{v}\ddot{u}+C_{v}\dot{u}+K_{v}u=F \tag{1}\]  
where \(M_{v}\), \(C_{v}\), and \(K_{v}\) are the mass, damping, and stiffness matrices of the vehicle, respectively. The vectors \(\ddot{u}\), \(\dot{u}\), and \(u\) represent the vehicle accelerations, velocities, and displacements, respectively and \(F\) contains the external forces applied to the vehicle. All matrices and vectors are introduced in the following equations:  
\[\ddot{u}=\left\{\begin{array}{c}\ddot{u}_{v}\\ \ddot{u}_{b1}\\ \ddot{u}_{b2}\\ \ddot{\theta}_{v}\\ \ddot{\theta}_{b1}\\ \ddot{\theta}_{b2}\end{array}\right\} \tag{2}\]  
\[\dot{u}=\left\{\begin{array}{c}\dot{u}_{v}\\ \dot{u}_{b1}\\ \dot{u}_{b2}\\ \dot{\theta}_{v}\\ \dot{\theta}_{b1}\\ \dot{\theta}_{b2}\end{array}\right\} \tag{3}\]  
\[u=\left\{\begin{array}{c}u_{v}\\ u_{b1}\\ u_{b2}\\ \theta_{v}\\ \theta_{b1}\\ \theta_{b2}\end{array}\right\} \tag{4}\]  
Figure 1: Two-dimensional (2D) 4-axle railway carriage model.  
\[F=\left\{\begin{array}{c}m_{v}g\\ m_{b1}g+k_{p1}r_{w1}+c_{p1}r^{\prime}_{w1}+k_{p2}r_{w2}+c_{p2}r^{\prime}_{w2}\\ m_{b2}g+k_{p3}r_{w3}+c_{p3}r^{\prime}_{w3}+k_{p4}r_{w4}+c_{p4}r^{\prime}_{w4} \\ 0\\ L_{b11}\big{(}k_{p1}r_{w1}+c_{p1}r^{\prime}_{w1}\big{)}-L_{b12}\big{(}k_{p2}r_ {w2}+c_{p2}r^{\prime}_{w2}\big{)}\\ L_{b21}\big{(}k_{p3}r_{w3}+c_{p3}r^{\prime}_{w3}\big{)}-L_{b22}\big{(}k_{p4}r_ {w4}+c_{p4}r^{\prime}_{w4}\big{)}\end{array}\right\} \tag{5}\]  
\[M_{v}=diag\big{[}\begin{array}{cccccccc}m_{v}&m_{b1}&m_{b2}&J_{v}&J_{b1}&J_{ b2}\end{array}\big{]} \tag{6}\]  
\[K_{v}=\left[\begin{array}{cccccccc}k_{v1}+k_{v2}&-k_{v1}&-k_{v2}&L_{v1}k_{v 1}-L_{v2}k_{v2}&0&0\\ -k_{v1}&k_{p1}+k_{p2}+k_{v1}&0&-L_{v1}k_{v1}&L_{b11}k_{p1}-L_{b12}k_{v2}&0\\ -k_{v2}&0&k_{p3}+k_{p4}+k_{v2}&L_{v2}k_{v2}&0&L_{b21}k_{v3}-L_{v22}k_{p4}\\ L_{v1}k_{v1}-L_{v2}k_{v2}&-L_{v1}k_{v1}&L_{v2}k_{v2}&L_{v1}^{2}k_{v1}+L_{v2}^{ 2}k_{v2}&0&0\\ 0&L_{b11}k_{p1}-L_{b12}k_{p2}&0&0&L_{b11}2k_{p1}+L_{b12}2k_{p2}&0\\ 0&0&L_{b21}k_{p3}-L_{b22}k_{p4}&0&0&L_{b21}2k_{p3}+L_{b22}2k_{p4}\end{array}\right] \tag{7}\]  
\[c_{v}=\left[\begin{array}{cccccccc}c_{v1}+c_{v2}&-c_{v1}&-c_{v2}&L_{v1}c_{ v1}-L_{v2}c_{v2}&0&0\\ -c_{v1}&c_{p1}+c_{p2}+c_{v1}&0&-L_{v1}c_{v1}&L_{b11}c_{p1}-L_{b12}c_{p2}&0\\ -c_{v2}&0&c_{p3}+c_{p4}+c_{v2}&L_{v2}c_{v2}&0&L_{b21}c_{p3}-L_{b22}c_{p4}\\ L_{v1}c_{v1}-L_{v2}c_{v2}&-L_{v1}c_{v1}&L_{v2}c_{v2}&L_{v1}^{2}c_{v1}+L_{v2}^{ 2}c_{v2}&0&0\\ 0&L_{b11}c_{p1}-L_{b12}c_{p2}&0&0&L_{b11}2c_{p1}+L_{b12}2c_{p2}&0\\ 0&0&L_{b21}c_{p3}-L_{b22}c_{p4}&0&0&L_{b21}2c_{p3}+L_{b22}2c_{p4}\end{array}\right] \tag{8}\]