# Railway Bridge Condition Monitoring Using Numerically Calculated Responses from Batches of Trains
## 3 Inverse Newmark-\(\beta\) Method to Calculate AP  
As shown in previous work, for a road or a track on infinitely rigid supports, the profile can be determined by CE optimization using bogie vertical accelerations and angular velocities [10]. This approach is shown to be effective but computationally intensive and therefore impractical for kilometres of track data. [PERSON] et al. [40] propose a much more efficient Inverse Newmark-\(\beta\) integration process to calculate the AP. The Inverse Newmark-\(\beta\) concept is extended here from [PERSON]'s 2-axle 4-degree-of-freedom half car to a 4-axle railway carriage model. The standard Newmark-\(\beta\) integration scheme is used to calculate the accelerations, velocities, and displacements in a vehicle, \(\{\ddot{u}\}\), \(\{\dot{u}\}\), and \(\{u\}\) due to a specified vehicle excitation (e.g., surface profile), i.e., to find \(\{\dot{u}\}\) according to Equation (1). The inverse Newmark-\(\beta\) concept uses measured accelerations and rates of rotation to find the excitation forces or the surface profile that generates them. In the standard Newmark-\(\beta\) integration scheme, the acceleration, \(\{\dot{u}\}\), can be calculated from \(\{u\}\):  
\[\ddot{u}_{t+\Delta t}=(u_{t+\Delta t}-u_{t})a_{0}-a_{2}\dot{u}_{t}-a_{3}\ddot{ u}_{t} \tag{9}\]  
In the inverse problem, the displacement, \(\{u\}\) can be determined from measured acceleration:  
\[u_{t+\Delta t}=(\ddot{u}_{t+\Delta t}+a_{2}\dot{u}_{t}+a_{3}\ddot{u}_{t})/a_{0 }+u_{t} \tag{10}\]  
where \(a_{0},\ a_{2}\), and \(a_{3}\) are integration constants of the Newmark-\(\beta\) method. Figure 3: (**a**) Elevation of track-bridge model. (**b**) Detail of track-bridge model of (**a**). For the 4-axle car model, accelerations and rates of rotation on bogies, \(\tilde{u}_{b}\) and \(\dot{\theta}_{b}\), are known inputs. Other parameters, \(\dot{u}_{b},\ u_{b},\tilde{\theta}_{b}\), and \(\theta_{b}\) of the bogies can be calculated from the known inputs according to the inverse Newmark-\(\beta\) concept. Then, using the equations of motion of the main body, the displacement and rotation of the carriage (\(u_{v}\) and \(\theta_{v}\)) can be calculated. Accelerations and velocities of the body, \(\tilde{u}_{v},\ \dot{u}_{v},\ \dot{\theta}_{v},\ \dot{\theta}_{v},\ \dot{\theta}_{v},\ \dot{ \theta}_{v},\ \dot{\theta}_{v},\) and \(\ddot{\theta}_{v}\), can be calculated from \(u_{v}\) and \(\theta_{v}\). Therefore, all the terms on the left side of Equation (1) are known and on the right side of Equation (1), \(F\), can be found. Finally, the AP that excited the vehicle and caused the accelerations, \(r_{w}\), can be determined by solving Equation (5). Details of this processes are shown in Appendix A. Here, a 100 m railway track with a profile and a 4-axle railway carriage model vehicle of known properties are used to generate the bogie vertical accelerations and angular velocities. The vehicle properties and speed are shown in Table 1. These signals are then taken as inputs and used to back calculate the AP using the Inverse Newmark-\(\beta\) method. The calculated results are compared with the original 'true' AP in Figure 4. The true AP is the AP used to generate the bogie vertical accelerations and angular velocities. The calculated result is from the first axle of this train. The calculated AP has similar local features to the 'true' AP but is displaced downwards by a 'drift' effect that results from an accumulation of numerical inaccuracies. As the drift effect is very low frequency, it has a negligible effect on the train/track dynamic interaction.