Generalized Inverted Pendulum Model for Pedestrian-Induced Lateral Vibration of Footbridge

Since the occurrence of pedestrian-induced large lateral vibration of the London Millennium Bridge in 2000, scholars have realized the complexity of pedestrian-induced lateral vibrations. Despite extensive research spanning over two decades, the underlying mechanisms between pedestrians and footbridges remain incompletely understood. Currently, there are two main popular models for explaining the pedestrian-induced lateral vibration of the footbridge: the synchronization lock-in model and the self-excited force model.

Among existing studies, the inverted pendulum model (IPM) essentially belongs to the self-excited force model, has gradually gained recognition. This model assumes that pedestrians maintain a constant step frequency and suggests that footbridge lateral vibration divergence can occur through pedestrian-bridge interaction without synchronization. Although the IPM theoretically elucidates the mechanism by which pedestrian-bridge interaction leads to self-excited forces, it still has its shortcomings: it overestimates the critical number of pedestrians required for triggering vibration divergence of the footbridge.

The underlying cause of this problem stems from the inverted pendulum model’s inherent limitation as a single-mechanism framework, which fails to consider the adjustments of pedestrian’s step frequency and instead solely relies on the adjustments of pedestrian’s step width. Consequently, this results in an underestimation of the virtual equivalent damping coefficient that is in phase with the vibration velocity of the footbridge. This study proposes a generalized-inverted pendulum model (G-IPM), in which the pedestrian walking phase evolution is effectively combined with the IPM, thereby the adjustments of pedestrian’s step frequency and step width can be considered simultaneously.

Compared to the single-mechanism-based IPM, the numerical simulation results indicate that the proposed dual-mechanism-based G-IPM requires fewer pedestrians to trigger the bridge vibration divergence, which is closer to the actual results. Additionally, parameter analysis reveals that varying pedestrian characteristic parameters exert different impacts on the vibration response of bridge. The proposed model pioneers the use of a dual-mechanism-based model, which is of significant theoretical importance in revealing the underlying mechanism of pedestrian-induced lateral vibrations of the footbridge.