Application of nakamura's model to describe the delayed increase in lateral vibration of footbridges

Nakamura's model is proposed to describe lateral vibrations of a footbridge induced by pedestrians. The model is found to be a slow-fast system if the number of pedestrians on a footbridge increases slowly. Then, based on the delayed bifurcation theory, Nakamura's model can be used to describe the delayed increase in the lateral vibration of the footbridge. The critical number of pedestrians on the footbridge is calculated by using the delayed bifurcation theory to quantitatively demonstrate our analysis. By comparison with a previously obtained formula, the rationality of the authors' formula for predicting the critical number of pedestrians is illustrated. Based on the authors' formula, the experimental parameter k5300 used in the previously obtained formula does not seem to be always applicable to other bridges. Based on Nakamura's model, the value of the parameter k seems to be affected mainly by the properties of the pedestrians on the bridge rather than those of the bridge.