## 4 Eigenvalue Analysis  
### Healthy Bridge  
Figure 3 shows the first bridge eigenfrequency versus the location of the front axle of the vehicle for a healthy version of the bridge, obtained as described in steps (1) to (4) of the methodology presented in Section 2. The value of the eigenfrequency corresponding to a truck based on the mean values of Table 2 is represented by a black solid curve. The red dash-dotted and blue dashed curves correspond to the values of mean and mean \(\pm\) standard deviation, respectively, of the 1000 eigenfrequencies obtained for each position of the fleet (Section 3.3) on the bridge. The black solid and red dash-dotted curves become closer as the size of the sample increases, being a reasonable match for the 1000 trucks calibration sample. In Figure 3, there are four vertical dashed lines indicating the 2 nd and 3 rd axles entering the bridge (5.4 and 6.6 m) and the 1 st and 2 nd axles leaving the bridge (15 and 20.4 m). Vertical solid lines signal the frequency \(F_{3}\), associated with the 1 st axle located at  
Figure 3: Bridge eigenfrequencies vs. location of the 1 st axle on the bridge. Figure 2: Histogram of frequencies of vibration of the fleet for: (**a**) the 1 st mode, (**b**) the 2 nd mode, (**c**) the 3 rd mode, (**d**) the 4 th mode, and (**e**) the 5 th mode of vibration. 3 m and the frequency F\({}_{9}\) associated with the 1 st axle at 9 m. The selected locations at 3 and at 9 m seek a compromise between a low sensitivity to the variability of the fleet, and a relatively far distance from singularities associated with the entrance and exit of axles. The values of the eigenfrequencies at these two locations (\(F_{3}\) and \(F_{9}\)) will be used as references in subsequent analyses. It is noted that the standard deviation increases significantly after the 3 rd axle of the vehicle enters the bridge (\(x=6.6\) m), and it decreases when the 1 st axle is about to leave the bridge (\(x=15\) m). The eigenfrequencies for locations between 10 and 15 m are most sensitive to the variability of the truck fleet. The use of frequencies in this 10-15 m range would only be advisable in the case of having a large fleet available for testing, in order to get a good approximation of the mean value. It is worth mentioning that the variability displayed in Figure 3 belongs to a general population of 3-axle trucks, but could be further reduced by limiting the characteristics of the fleet. For example, the fleet was generated by sampling axle weights and axle spacings of a 3-axle truck randomly based on normal distributions from real WIM data as per Table 3. However, a WIM or BWIM system would allow measuring these static properties to narrow down on a subset of the general 3-axle truck fleet that could be used to derive a more specific Figure 3. Although some variability will remain as a result of the uncertainty on the dynamic properties of the subset, this uncertainty would be considerably smaller than when also covering for the static properties. Figure 4a,b show the number of vehicles versus the bridge eigenfrequencies (\(F_{3}\) and \(F_{9}\)) for locations \(x=3\) m and \(x=9\) m, respectively. The histogram of eigenfrequencies appears to follow a normal distribution with a mean value of 5.658 Hz and a standard deviation of 0.0006 Hz for \(F_{3}\), and with a mean value of 5.673 Hz and a standard deviation of 0.0035 Hz for \(F_{9}\). In agreement with the discussion above, the uncertainty around \(F_{9}\) is clearly larger than for \(F_{3}\). A bivariate distribution can be fitted to the 1000 \(F_{3}\) eigenfrequencies and 1000 \(F_{9}\) eigenfrequencies as per Figure 5. Here, the \(x\)-axis represents the individual distribution of \(F_{3}\) highlighted in red, the \(y\)-axis represents the individual distribution of \(F_{9}\) highlighted in blue, and the \(z\)-axis represents the probability. The ellipse highlighted in green is based on the individual mean and standard deviations of \(F_{3}\) and \(F_{9}\).