## 3 Numerical Validations
### Truss Bridge Example
This is mainly due to the similarity of the forces, namely the value, frequency, and, especially, their close position. The bridge-force interactions generated by these forces interfere each other and make the forces difficult to be distinguished by the method. This influence is also enlarged by the vibration instability when the forces leave the bridge and then causes identification failure. Nevertheless, it is important to state that the error basically meet the characteristic of normal distribution, so it could be decreased by averaging. It is predictable that this method will have good performance in determining axle loads of vehicles during practical application. Influence of measurement noise and sensor location  
Looking into Figure 6d,e, with the growth of noise, only a slight increase of the peak values are found in the results; the frequency-phase characteristics remain unchanged. This shows good robustness of the proposed method to measure noise. Analyzing the results of Cases 2-4 (Figure 6a,b), which use different sensors, it is illustrated that forces are determined with better accuracy around the sensors. As the system matrix in the identification Equation (17), namely the TLSF response matrix \(\Lambda\), is constituted of the responses excited by the small TLSFs, the details in responses captured by the sensor reduce when the TLSFs get away and hence produces a larger error. Therefore, in real-world practice, it is recommended to utilize the sensors evenly on the bridge. Figure 6: Moving force determination results in the synchronized assessment. (**a**) Cases 1 and 2; (**b**) Case 3; (**c**) Case 4; (**d**) Case 5; (**e**) Case 6. Result discussion for SDI  
The identified virtual force \(p^{e}\) (equals to \(\lambda_{11}^{e}\)) and calculated nodal force \(p^{o}\) (equals to \(\lambda_{11}^{o}\)) are presented in Figure 8 according to Equation (18), and the fitted damage severity \(1-\zeta_{1}\) and R-squared values are listed in Table 3.  
Influence of damage severity and measurement noise  
Figure 8a,b illustrate the damage ratios determined in Cases 1 and 2. It is found that both 5% and 20% of damages are estimated. The errors are around 4%. However, if we look at the RPE, it reaches 94.6% in Case 1 while it is only 21.25% in Case 2. It deduces that the damage determination accuracy of this method is low correlated with the damage severity. It is because when the damage is transformed to external virtual force, its identification accuracy is dominated by the precision of the TLSF matrix, not the magnitude of the force itself. In this aspect, it is an advantage of this method, but it also makes the method insensitive to small damage. This drawback could be developed by introducing an appropriate objective function [31], which should be studied further. Comparing Cases 2 and 3, the influence of measurement noise mainly reflects the confidence of the fitting curves, while the effect on results is relatively low. * Influence of moving force  
Observing the fitted damage severity results of Cases 3 and 4, it is determined that the force-velocity has a minor effect on the SDI accuracy, which is similar to the MFI results. Conversely, the 2-axle force cases show that increasing the number of unknowns significantly impacts the precision of damage detection and the confidence of fitting. The response of a damaged structure is the superposition of an intact structure subjected to the moving force and virtual force, respectively. Therefore, the synchronized inversion of moving force and virtual force will interfere with each other. Hence the reason for the accuracy loss is not the number of moving forces, but the error produced during the method distinguishing the forces. This drawback may be improved by utilizing more sensors and optimizing the precision of the TLSF matrix. Figure 7: RPEs of moving force identification.