Lateral periodic vibrations of footbridges under crowd excitation

In this paper, to investigate the lateral vibrations of a footbridge under crowd excitation Nakamura's model and its modified version are studied by using the energy method. The periodic solutions of Nakamura's model and its modified version are obtained analytically, the efficiency of which is demonstrated by comparing with numerical results. Our analytical solutions are very convenient to apply to the structural calculation and design of footbridges. Our analysis shows that the introduction of the time delay of interaction between pedestrians and the bridge in Nakamura's model makes the model accord better with the measurement data. From the modified Nakamura's model, the increase in the time delay decreases the lateral amplitude of a footbridge under crowd excitation, which suggests us that increasing the time delay of the interaction between pedestrians and the bridge maybe a new approach to reduce the lateral vibrations of a footbridge. In addition, our calculations indicate that delay differential equations can be solved more simply by using the energy method than the Galerkin method.