# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 3 Continuous Wavelet Transform and Development of the Proposed Method
However, we will first take a step back and explain the physical principles of the methodology that will be utilized in the proposed framework. For this, we will discuss the components of the response of a bridge to a moving load (or vehicle). When the bridge is exposed to the dynamic loading applied by a moving load, there are three components to its response. The first one is the static response, which is the response created by the presence of the load at a certain location on the bridge. The other component is the dynamic response that is created by the movement of the load. These two components are present in the response of all bridges; damaged or undamaged. The third component is the damage component, that is visible only in the damaged bridges and close to the damage location. The damage component impacts both of the other two components of the bridge response; the static component and the dynamic component. However, the main impact of the damage component is on the static component while its effect on the dynamic component is much more subtle. For example, Figure 3a shows the displacement response of the bridge both in its undamaged and damaged states. The depicted displacement response is computed separately for each point on the span at the instant the back axle is exactly at that point. For comparison, Figure 3b depicts the first mode shape of the undamaged and damaged bridges, which have frequencies of 4.35 Hz and 4.20 Hz, respectively. Comparing Figure 3a,b, and considering that the change in the first mode frequency is relatively small, it is clear that the impact of the damage is mostly visible on the static response while its impact on the mode shape remains imperceptible. Thus, if we can estimate the static response of the bridge from the vibrations recorded on the vehicle, we can then, potentially, detect the presence of damage and estimate its location. Figure 3: (**a**) The displacement of the undamaged and damaged bridge under a moving vehicle (**b**) first mode shape of the undamaged and damaged bridges. Based on this fact, [PERSON] and [PERSON] [50; 51] state that, for the case of direct monitoring, a horizontal section of the wavelet coefficient maps at a scale significantly higher than the scale corresponding to the first mode frequency of the bridge will be free from the dynamic response and closer to the static response. As such, [PERSON] and [PERSON] [50; 51] showcases that the wavelet coefficients taken at such high scales can be used to detect and locate damage. In this article, we will build on this work by expanding it to drive-by monitoring methods while also eliminating the negative impacts of road roughness, which is arguably the parameter that most adversely affects the success of these methods. For this, let us first examine how damage effects the wavelet coefficient maps and how it can be used to detect damage. Figure 4 depicts the WCM of the accelerations recorded at the rear-axle of the vehicle as it travels with a constant speed of 2 m/s over the undamaged and damaged bridges with no road roughness. The Mexican hat wavelet was used in the continuous wavelet transform as it was previously shown to be very effective in damage detection [51]. Also depicted in Figure 4 is the scale that corresponds to the first bridge frequency, which is 4.35 Hz. The damage is introduced in the bridge by reducing the flexural stiffness of a one-meter segment of the bridge between 12 m and 13 m by 50%. Comparing Figure 4a with Figure 4b, we can observe that, damage clearly leads to disturbances in the wavelet coefficients at scales higher than 250 in the vicinity of damage, while the coefficients for an undamaged bridge remains constant throughout the entire length of the bridge.