## 3 Numerical Model
### Implementation of Acceleration-Based Moving Force Identification  
In this paper, the model of Figure 1 is used, in the forward problem, to calculate the'simulated measured' accelerations on the bridge. A-MFI is then applied, in an inverse  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Symbol/Units** & **Mean Value/Value** & **Standard Deviation** \\ \hline Body mass & \(m_{s}\): kg & From database & \\  & \(M_{u,1}\): kg & 1100 & 300 \\ Unsprung masses & \(M_{u,2}\): kg & 700 & 250 \\  & \(K_{s,1}\): N/m & 0.4 \(\times\) 10\({}^{6}\) & 0.1 \(\times\) 10\({}^{6}\) \\ Suspension stiffnesses & \(K_{s,2}\): N/m & 1 \(\times\) 10\({}^{6}\) & 0.3 \(\times\) 10\({}^{6}\) \\  & \(C_{s,1}\): N/m & 10 \(\times\) 10\({}^{3}\) & 2.5 \(\times\) 10\({}^{3}\) \\ Suspension damping & \(C_{s,2}\): N/m & 20 \(\times\) 10\({}^{3}\) & 5 \(\times\) 10\({}^{3}\) \\ Tire stiffnesses & \(K_{s,1}\): N/m & 1.75 \(\times\) 10\({}^{6}\) & 0.5 \(\times\) 10\({}^{6}\) \\ Tire stiffnesses & \(K_{s,2}\): N/m & 1.75 \(\times\) 10\({}^{6}\) & 0.5 \(\times\) 10\({}^{6}\) \\ Axle spacing & \(D_{1}\) + \(D_{2}\): m & From database & \\ Speed & Vel: m/s & From database & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Vehicle population properties.  
Figure 1: Coupled vehicle-bridge system with half-car model (based on [37]).  
calculation, to determine the applied force history. It is acknowledged that using the same model for the forward and inverse problems suggests levels of accuracy unlikely to be replicated in the field. Nevertheless, this process is important as it demonstrates the process and, as the applied force histories are known, it is used to identify the sensitivity of the system to various sources of inaccuracy.  
An illustrative baseline example is considered of a single 18 t vehicle travelling across the bridge on a Class A road profile at a speed of 25 m/s. Tikhonov Regularization requires a plot of the residual norm against the smoothing norm (Figure 2a), from which the point of maximum curvature is extracted (Figure 2b). In this case, that point corresponds to a value, \(\lambda\) = 7.94 \(\times\) 10\({}^{-9}\) (see Equation (21)), which is deemed to be the optimal trade-off between an ill-conditioned solution to the original problem and a well-conditioned solution to a slightly different problem. In Figure 2a, the \(x\)-axis represents the residual norm of the error and the \(y\)-axis denotes the smoothing norm of the regularised solution. In Figure 2b, the \(y\) coordinate represents the curvature at a specific value of \(\lambda\) in the L curve.  
Using the optimum \(\lambda\) value, the force history is calculated for each axle and compared to the actual force histories, available from the forward problem calculation, in Figure 3. The bridge is present from \(x\) = 0 to 20 m and the distance between the two axles is 5 m. It can be seen in Figure 3 that, while the first axle enters the bridge at \(x\) = 0 m, the second does not arrive until \(x\) = 5 m. Similarly, the first axle leaves the bridge at \(x\) = 20 m and the second at \(x\) = 25 m. It can be seen in Figure 3 that, for this idealized case, the calculated axle forces match the real ones reasonably well. This is especially true in the context of bridge Weigh-in-Motion theory, where many systems are at, or less than, Class B accuracy, i.e., 95% of axle forces within \(\pm\)15% of the exact values [45]. It can be seen in Figure 3 that the difference between the real and calculated axle forces increases when the second axle enters the bridge span. The sum of the averages of the two calculated axle force histories gives a calculated GVW of 17.84 t which, when compared to the actual GVW of 18 t, is an error of less than \(-\)1%. It should be noted that the error is calculated by (inferred-real)/real.  
Figure 2: (a) L curve for 18 t vehicle passing at 25 m/s and (b) curvature of L curve vs. \(\lambda\).  
The results of this simulation demonstrate that this new A-MFI algorithm has the potential to estimate GVW with good accuracy.