## 3 Numerical Model
### Sensitivity Study  
A sensitivity study was undertaken on the scan rate of the acceleration signal, inaccurate measurement of vehicle velocity, initial bridge conditions, and signal noise. The scan rate of an accelerometer dictates the frequency of measured accelerations. Linear interpolation can be used to artificially increase the sampling rate between successive scans, i.e., the number of points of calculation. While this does not provide any new information, it has significant numerical benefits in the A-MFI algorithm. For the same example from Figure 3, it can be seen in Figure 4a that the accuracy of the force history improves significantly with increasing sampling rate--with a very good match to the actual Axle 1 force history at a rate of 5000 Hz. However, there is an inaccuracy in the early parts of the Axle 2 force history in Figure 4b, which is not improved by increasing the sampling rate. Nevertheless, the average forces tend to be closer to the actual average forces and to the axle weights, as the sampling rate is increased. For this example, the calculated GVW reduces from a 21% error at a sampling rate of 500 Hz to 2.7% at a rate of 5000 Hz.  
When the input velocity is not accurately measured, the accuracy of the A-MFI results is affected. Again, simulations are carried out for the baseline 18 t half-car model travelling at a constant velocity of 25 m/s and giving an error in the calculated GVW of 2.7% when the correct velocity is used in the calculation. The errors that result from inaccurate speeds are found to be \(-8.9\%\), \(-4.4\%\), \(-13.0\%\), and \(-49.5\%\) for'measured' speeds of 23, 24, 26, and 27 m/s, respectively. Clearly, the accuracy is greatly influenced by the assumed speed, particularly when it is higher than the actual value.  
To investigate the repeatability of this effect, the analysis was run for a population of 500 vehicles (all 18 t half-cars with 25 m/s real velocity) on a Class A profile. The velocity used as the input was randomly selected within the range of 23 m/s to 27 m/s (+/- 8% error). The errors in inferred GVWs are illustrated in Figure 5 and confirm the general trend of the previous example.  
Figure 3: Actual and calculated force histories for baseline example with optimum \(\lambda\).  
One of the assumptions of A-MFI is that the bridge is stationary when the first axle arrives, a condition that is not satisfied if the test vehicle is preceded by another. To test the implications of pre-existing bridge vibration, another half-car vehicle is run over the bridge before the test half-car arrives. The test vehicle is assumed to enter the bridge at time, \(\delta t=0.2\), \(0.5\), \(1\), and \(2\) s after the first vehicle leaves-see Figure 6. When the vehicles are close to each other, the effect is significant. For this example, the error in the inferred GVW is \(-7.1\%\) and \(7.3\%\) for \(\delta t=0.2\) s and \(0.5\) s respectively. However, it should be noted that this implies a headway of \(1.2\) s which is small relative to most recorded gaps [46]. For the larger time intervals of \(1\) and \(2\) s, the GVW errors fall to \(2.3\%\) and \(-0.7\%\) respectively.  
Figure 4: Influence of sampling rate on the accuracy of A-MFI results: (**a**) first axle; and (**b**) second axle.  
Figure 5: Error in the inferred GVW due to inaccurate estimation of vehicle velocity.  
The influence of random signal noise on the accuracy of the inferred GVW is also investigated. For this example, noise of up to 5% results in GVW errors of up to about 5%, with no clear correlation between accuracy and level of noise.