Dynamic Analysis for Lateral Vibrations of Footbridges under Crowd Based on a Two-Degrees-of-Freedom Model

Nakamura's model is widely used to describe lateral vibrations of a footbridge induced by crowd. The predicted responses of Nakamura's model were compared with measured data of T-bridge and M-bridge in Japan to demonstrate the validity. However, the predicted responses based on Nakamura's model almost always were stronger than measured data. Considering that both T-bridge and M-bridge are cable-stayed bridges, it seems to be not precise enough to simplify a cable-stayed bridge as a single degree of freedom system in Nakamura's model. In this paper, we establish a two-degrees-of-freedom model to describe lateral vibrations of a cable-stayed bridge. The cables have one degree, and the bridge deck has the other. Additionally, in this model we introduce a time delay in interaction between the bridge and pedestrians. By employing the center manifold theory, we find that a subcritical Hopf bifurcation occurs in the two-degrees-of-freedom model. We theoretically and numerically illustrate that the cables and time delay have significant influence on the lateral vibration amplitude of a footbridge under crowd. The appropriate increases of tension in the cables and time delay both can decrease the lateral vibration amplitude. The analysis for the proposed two-degrees-of-freedom model shows that the predicted responses of Nakamura's model can better agree with the measured date if we take the influence of cables and time delay into account.