Normalized formula for removing damping effect in recovering bridge mode shapes using a moving and a stationary vehicle

Recovering bridge mode shapes using the response of a moving test vehicle has become a technique of interest. However, the mode shapes recovered tend to be distorted by bridge damping. To remove the distortion effect, a normalized formula that utilizes the responses of a moving and a stationary vehicle is devised for recovering bridge mode shapes with no prior knowledge of bridge damping ratios. Specifically, the moving test vehicle is used to recover the global modal response of the bridge throughout its span length at different instants as the vehicle moves over, and the stationary vehicle to generate a reference response at a fixed location of the bridge for removing the damping effect. In this paper, closed-form solutions are derived for the dynamic responses of the bridge and two test vehicles. The contact response is used in place of the vehicle response to alleviate the vehicle frequency's masking effect on bridge frequencies. The formula for recovering the modal mode shape is derived by normalizing the instantaneous amplitude of the Hilbert transform (HT) for the bridge from the moving test vehicle with respect to that by the stationary vehicle. The following conclusions are drawn: (1) the damping distortion effect on bridge mode shapes can be removed by the normalized formula; (2) the normalized formula is robust with regard to bridge damping, stationary vehicle location, and vehicle speed for beam-type bridges; (3) the mode shapes recovered by the present method are acceptable, even in the presence of pavement roughness, when aided by an accompanying truck.