# An Effective Tangent Stiffness of Train-Track-Bridge Systems Based on Artificial Neural Network
## 3 Application Examples
### A Light Rail Vehicle Running on the Seven-Span Bridge  
Using the same trained BPTS above (i.e., that obtained by a wheelset running with a constant speed), an ML95 trailer vehicle running on the bridge is studied to verify the BPTS derived herein, as shown in Figure 14. The detail of the model of the car body and bogie can be found in [22; 23; 24]. The vehicle consists of the car body, bogies, and wheelsets, respectively. The vehicle body and bogies are modeled using lumped mass points and rigid beams, while the wheelsets are simulated by the wheel nodes mentioned in Section 2.1 and the elastic beam (axis of the wheelsets). The primary suspension of the vehicle is simulated by spring-dampers I between the wheelsets and the bogies, which appear in black. The secondary suspension of the vehicle is simulated by spring-dampers II between the bogies and the car body, which appear in pink. The parameters of spring-dampers I and II can be found in Table 5, where \(k_{\rm I}\) denotes the stiffness of spring-dampers I, \(c_{\rm I}\) represents the damping of spring-dampers I, \(k_{\rm II}\) denotes the stiffness of spring-dampers II, and \(c_{\rm II}\) represents the damping of spring-dampers II. Both \(k\) and \(c\) are three-dimensional herein and correspond to \(x\), \(y\), \(z\) in Table 5. The rail profile used herein is the UIC50. The element of rail is simulated by an elastic beam, the parameters of which can be found in Table 6, where \(A_{r}\) is the cross-sectional area of the rail, \(I_{rx}\), \(I_{ry}\), \(I_{rz}\) represent the area moment of inertia about the \(x\), \(y\), and \(z\)-axes, respectively. During the static analysis, the lateral forces applied to each wheelset to make the lateral displacement of the vehicle is 1 mm, and then the forces are removed at the beginning of the dynamic analysis. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Time [s]** & **Model** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \multirow{3}{*}{0.500} & CTS & \(4.99\times 10^{3}\) & \(3.04\times 10^{3}\) & \(6.83\times 10^{1}\) & \(8.63\times 10^{-1}\) & \(2.38\times 10^{-4}\) & \\  & BPTS & \(8.37\times 10^{3}\) & \(6.58\times 10^{3}\) & \(4.65\times 10^{2}\) & \(0.36\times 10^{1}\) & \(9.07\times 10^{-2}\) & \\  & CTS & \(2.28\times 10^{4}\) & \(1.13\times 10^{4}\) & \(5.37\times 10^{1}\) & \(2.65\times 10^{-1}\) & \(7.54\times 10^{-5}\) & \\  & BPTS & \(1.25\times 10^{4}\) & \(8.45\times 10^{3}\) & \(8.51\times 10^{2}\) & \(1.88\times 10^{1}\) & \(0.83\times 10^{-1}\) & \\  & CTS & \(1.58\times 10^{5}\) & \(8.79\times 10^{4}\) & \(2.06\times 10^{2}\) & \(0.39\times 10^{1}\) & \(1.67\times 10^{-3}\) & \\  & BPTS & \(5.45\times 10^{5}\) & \(8.64\times 10^{3}\) & \(9.64\times 10^{2}\) & \(2.60\times 10^{1}\) & \(1.26\times 10^{-1}\) & \(4.31\times 10^{-2}\) \\ \hline \end{tabular}
\end{table} 
Table 4: The norm of the unbalance force in a few representative time steps (wheelset). Figure 14: The model of the train and the track. The parameters of the rail element. The BPTS trained previously using a wheelset is re-used in the TTB system; the lateral displacement of the left wheel of wheelset 1 is shown in Figure 15 for cases of running speed at 30 and 50 m/s, respectively. Additionally, the lateral displacement of the car body is shown in Figure 16. The responses obtained by the two tangent stiffness are identical. The convergence data are also recorded. The convergence rates are compared between the use of the two tangent stiffness, as shown in Tables 7 and 8. It can be seen that the convergence rate of BPTS is slightly less than that of CTS. However, the total calculation time is faster when using BPTS over CTS. The total calculation time using CTS is 11,508 s when the speed of the train is 30 m/s, while the calculation time using BPTS is 8264 s, which saves 28% of time. When the train speed is 50 m/s, the calculation times are 16,528 and 13,004 s for the use of CTS and BPTS, respectively, and the use of BPTS saves 21.3% of time.