Lateral Vibration Mechanism of Footbridges with Random Parameters Using Inverted Pendulum Model

The inverted pendulum model (IPM) is highly regarded for its effectiveness in simulating lateral pedestrian loads. However, most existing studies have focused on treadmill tests with constant amplitudes while neglecting pedestrians’ variable stepping behavior. Additionally, relatively limited research has been conducted in two key areas: the application of the IPM to footbridges with variable vibration amplitudes, and the probabilistic prediction of lateral instability in footbridges. To address these research gaps, this study focuses on the Millennium Bridge and adopts the lateral IPM to analyze human-induced lateral vibrations. First, the human-bridge coupling equation based on the IPM was formulated and solved using the Newmark-β method. Subsequently, the self-excitation mechanism of the IPM was examined from the perspectives of equivalent damping and equivalent mass. Key findings indicate that the self-exciting properties of the IPM can lead to divergence of the lateral response. The system’s equivalent damping is more affected by fluctuations in step frequencies and pedestrian numbers than the equivalent mass. Significant response divergence is more likely to occur when step frequencies are below the resonant frequency, particularly as equivalent damping approaches zero. At resonance, equivalent damping remains positive, effectively suppressing excessive vibration amplification, while divergence emerges when the maximum equivalent damping decreases over time. Finally, a detailed procedure was proposed for analyzing the lateral vibration response of the footbridge using the IPM under random parameters. Results show that lateral instability occurs with 95% probability when the critical pedestrian number falls within the range [136, 196]. These findings are reasonably consistent with experimental data and allow for satisfactory prediction of the critical pedestrian number associated with lateral instability.