## 2 Research Methodology
\[STFT\big{\{}\ddot{u}_{g}(t)\big{\}}(\tau,\omega)=\int_{-\infty}^{\infty}\ddot {u}_{g}(t)w(t-\tau)e^{-i\omega t}dt\] (5) where \(\ddot{u}_{g}(t)\) represents the acceleration signal, \(w(\tau)\) is the window function and \(\omega\) corresponds to the frequency. Acceleration signals of short duration will lead to a poor frequency resolution. Therefore, the methodology will perform best with a longer duration of the signal. An ideal scenario would consist of a long vehicle crossing a long bridge at low speed, with speed having the greatest impact on accuracy. In practice, the latter limits the range of application to bridges where speed restrictions apply, i.e., located in an urban area, or close to a junction where a vehicle must slow down. Initially, the bridge can be assumed to be healthy, and the mean pattern of forced vibrations obtained here can be used to calibrate the VBI model proposed in step (4). 8. For every vehicle crossing, a set of frequency values is selected within the time-history of forced frequencies. These values of frequencies correspond to the same specific locations of the vehicle over the bridge. It is essential to choose locations where the frequency estimation is not hindered by edge effects derived from the application of the signal processing tool. In this paper, the methodology is tested using only two vehicle positions (\(i_{1}\), \(i_{2}\)) and two frequencies, but it could be potentially extended to include more positions over the bridge. 9. As a result of steps (1) to (5), a database of bivariate distributions of the eigenfrequency values corresponding to the healthy bridge and each damage scenario is generated based on the frequencies corresponding to the two selected vehicle positions. Alternatively, a multivariate distribution of higher order would be employed in case of selecting more than two values from the time-history of forced frequencies. Then, Equation (6) is applied to obtain the Mahalanobis distance [32; 33], \(MD_{r}^{d}\), between the points defined by the frequencies in the \(q=1\), \(2\), \(3\),..., \(p\) vehicle crossings and the statistical distributions obtained using eigenvalue analysis (steps (4) and (5)). \[MD_{r}^{d}=\sqrt{\{y^{d}-\overline{\mu}_{r}\}^{T}[S_{r}]^{-1}\{y^{d}-\overline {\mu}_{r}\}}\ \ r\ =\ 1,\ 2,\ 3,\ \ldots,\ R\] (6) where \(r\) represents a distribution from the eigenvalue database, \(R\) is the total number of the bivariate distributions, and \(\overline{\mu}\), and \([S_{r}]\) denote the sample mean and covariance of the reference bivariate distribution, respectively. \([S_{r}]\) is a square matrix with an order equal to the number of selected forced frequencies. For each vehicle crossing, there is an observation \(\{y^{d}\}\) (i.e., set of selected forced frequencies) which is used to measure the Mahalanobis distance between the observation and the reference bivariate distribution. The superscripts \({}^{T}\) and \({}^{-1}\) indicate the transpose of a vector and inverse of a matrix, respectively. 10. For vehicle crossing \(q=1\), it is possible to calculate \(MD_{1}^{1}\), \(MD_{2}^{1}\), \(MD_{3}^{1}\),..., \(MD_{R}^{1}\) according to Equation (6). If there are \(p\) vehicle crossings, a mean Mahalanobis distance is defined for each distribution \(r\) as follows: \(\overline{MD_{r}}=\sum_{q=1}^{p}\frac{MD_{r}^{d}}{p}\). The \(R\) distributions are ranked in ascending order according to the mean Mahalanobis distance \(\overline{MD_{r}}\). The location and severity associated with the statistical distribution yielding the lowest value of the mean Mahalanobis distance are used to estimate the damage. The Mahalanobis distance can be employed to assist in the selection of the frequencies in step (8). It is found that, generally, combinations of frequencies leading to high values of the Mahalanobis distance will yield more inaccurate results than combinations with low values of the distance.