Title: Improving the Quantification and Estimation of Damping for Bridges under Traffic Loading
Description: It is important for engineers and designers to be able to accurately estimate the damping within a structure; however, this is not a trivial task. Simplifications are often made in an effort to make damping estimation easier, but these simplifications rely on assumptions that may not be universally true. One important assumption is that the excitation input for a structure may be modeled as broad-band noise, but traffic loading on a bridge likely violates that assumption. Traffic loads are characterized by the velocities of the vehicles and trains crossing the bridge, which gives the input specific frequency content. This added complexity increases the difficulty in accurately estimating the damping. The problem of traffic crossing a bridge was studied by creating a finite element model of a bridge using a beam system that consisted of a series of stringers resting on top of a larger girder. Traffic loads were then simulated using moving point loads and moving masses to represent cars and trains crossing the bridge.

In addition to the traffic loading case, an ambient loading case was conducted using uniform broad-band noise as a means of comparison. The accelerations at several locations along the bridge span were recorded and used as input for a variety of operational modal analysis (OMA) methods. The OMA methods included both frequency domain techniques, such as Frequency Domain Decomposition (FDD), and time domain based identification, such as blind source separation (BSS). The results from the various OMA methods demonstrated how traffic loading creates distortion in the frequency response spectra of the bridge. This distortion had adverse effects for damping ratio estimation and in certain cases led to extreme errors. The mode shape estimates were not found to be affected by the distortion, but that meant that mode shape estimates could not be used to identify potentially erroneous damping estimates. The cause for the distortion was later identified as the driving frequencies produced by the vehicle-bridge interactions.

The term ``driving frequency'' refers to the frequency created by a car traveling over a bridge or, by analogy, by a moving load traveling over a beam. This frequency is directly correlated with the speed of the vehicle and the length of the bridge. By considering a single moving point load traveling across the bridge, the responses of the stringers and girder were studied and the effects of the driving frequencies were better quantified in both the time and frequency domains. It was found that peaks in frequency domain appear at the even multiples of each car's driving frequency, and as more cars travel across the bridge the peaks of closely spaced driving frequency multiples begin to merge. As the number of cars increases to a full hour-long simulation and the car velocities become uniformly distributed over a given interval, numerous peaks merge together to form sustained regions of elevated energy in the frequency domain. These regions distort the frequency response spectra of the bridge and obscure the modal information.

In order to deal with these distorted regions, a new approach to modal identification was proposed that focused on using partial information from the modal peaks. The peaks in the frequency domain were divided into left- and right-side spectra in order to take advantage of any undistorted portions of the modal peaks. These side spectra were analyzed using a curve-fitting approach based on combining optimization methods with clustering analysis. The presence of distortion presented certain challenges to traditional curve-fitting approaches, such as polynomial least squares, but the optimization algorithm was able to overcome these issues while also adding efficiency to the curve-fitting process. The clustering analysis was used to quickly find the optimal subsets within the optimization-based curve-fitting results. By performing curve-fitting to side spectra, different sets of modal parameters were produced that fit each side. It was found that the modal parameters for the intact or undistorted side compared favorably with the true modal parameters.

While this optimization and clustering methodology could not account for all types of distortion, it demonstrated large improvements as compared to traditional OMA approaches for the modes most severely impacted by the distortion. Another potential benefit of this method is that the distributions within the final clusters could be used to provide ranges of possible values for the damping ratios instead of only a single value.