## 5 Transient Analysis
### Healthy Bridge  
There are various window function types that can be utilized in the STFT for instantaneous frequency extraction; the Hamming, Gaussian and Hann are amongst the most popular options [51]. Figure 7 shows the difference between these windows compared to the eigenvalue analysis for the vehicle given in Section 3.2. The Hann window outperforms the other two alternatives when intending to match the eigenvalue curve. In order to optimize the trade-off between time and frequency resolution [52], the selected length for the Hann window is \(2^{10}\), the hop size is \(2^{5}\) and the total number of fast Fourier transform (FFT) points is \(2^{20}\). Figure 7 shows how the entrances and exits of the axles induce significant deviations between the eigenfrequencies and the instantaneous frequencies due to edge effects. The locations for capturing the frequencies \(F_{3}\) and \(F_{9}\) were selected to avoid these \"blind spots\". Figure 8 compares the effect of damping on the instantaneous forced frequency obtained from the bridge acceleration due to the same truck (Section 3.2) traveling at 1 m/s. Classical Rayleigh damping is employed here to model viscous damping on the bridge as a linear combination of the mass and stiffness matrices [53]. There are two constants related to the Rayleigh damping, pre-multiplying mass and stiffness matrices, defined as: \(\alpha_{R}=2\xi\omega_{1}\omega_{2}/(\omega_{1}+\omega_{2})\), and \(\beta_{R}=2\xi/(\omega_{1}+\omega_{2})\), respectively, where \(\omega_{1}\) and \(\omega_{2}\) are the first two angular frequencies of the bridge. The damped frequency, \(f_{d}\), is defined by \(f\sqrt{1-\xi^{2}}\)[54], where \(f\) is the undamped frequency and \(\xi\) is the damping ratio. As the  
Figure 7: Comparison of the eigenvalue curve and the STFT of the undamped bridge response due to the truck travelling at 1 m/s on a smooth profile using three different window functions. modal damping increases, the damped frequency decreases, but only slightly, given the low damping found in bridges. Values of \(\xi=1\%\), 2% and 3% in the figure do not show a significant deviation from the 0% damping adopted in this paper. Figure 9 shows the history of forced frequencies for four different vehicle speeds, namely 1 m/s, 2 m/s, 4 m/s and 8 m/s. The STFT at 1 m/s offers the most reliable results regardless of the axle location. Although a higher speed could be employed, there are limitations related to a worsening of the frequency resolution, and to the contamination of the STFT by wider and more pronounced edge effects. Therefore, extreme caution should be placed in the selection of the vehicle positions when employing high vehicle speeds. For the two locations selected in the paper, i.e., \(F_{3}\) and \(F_{9}\), the STFT of the midspan acceleration when the vehicle travels at 4 m/s is still able to provide accurate results. However, the STFT at 8 m/s resembles the eigenvalue curve only for a small portion between 10 and 11 m. It should be pointed out that a bridge span longer than the 15 m investigated here would potentially allow for higher speeds to be implemented. Firstly, a longer span captures a longer duration of the measurements, and, consequently, leads to a better frequency resolution. Secondly, the interference of axles entering and leaving the bridge that corrupts the forced vibration record will have a comparatively lesser impact. Figure 8: Comparison of the eigenvalue curve and the STFT of the bridge response due to the truck traveling at 1 m/s on a smooth profile for four bridge damping ratios. Figure 9: Comparison of the eigenvalue curve and the STFT of the undamped bridge response due to the truck travelling on a smooth profile at four different speeds. The simulations conducted in this paper adopt a smooth road profile. Figure 10 shows the impact of increasing the roughness coefficient of the road, i.e., the geometric spatial mean in m\({}^{3}\)/cycle.