An Inverted Pendulum Model Describing the Lateral Pedestrian-Footbridge Interaction

Bin Zhen,Liang Chang,Zigen Song

doi:10.1155/2018/5730162

Abstract

In this paper, the lateral pedestrian-footbridge interaction is investigated by using the model of an inverted pendulum on a cart. The inverted pendulum and the cart separately represent the synchronous pedestrians and the footbridge. The pivot point of the inverted pendulum is considered to vibrate harmonically to model the walking motion of the pedestrians. The proposed inverted pendulum model avoids the difficulty of the determination of the lateral force induced by the pedestrians applying to the footbridge, which was usually treated based on a semiempirical approach in previous works. Moreover, the model can describe the whole process: how the lateral amplitude of the bridge increases from small to large. Measurement data showed that a normal pedestrian always keeps the ratio of 1/2 between the lateral and vertical step frequencies. The theoretical analysis for the inverted pendulum model indicates that such walking habit of pedestrians is the root of the frequency-locking phenomenon, which eventually results in excessive lateral vibrations of the bridge. Furthermore, such walking habit also is a key factor in the occurrence of the “jump phenomenon” in the London Millennium Bridge.

Highlights

The lateral pedestrian-footbridge interaction is investigated by using the model of an inverted pendulum on a cart. e inverted pendulum and the cart separately represent the synchronous pedestrians and the footbridge. e pivot point of the inverted pendulum is considered to vibrate harmonically to model the walking motion of the pedestrians. e proposed inverted pendulum model avoids the difficulty of the determination of the lateral force induced by the pedestrians applying to the footbridge, which was usually treated based on a semiempirical approach in previous works
Measurement data showed that a normal pedestrian always keeps the ratio of 1/2 between the lateral and vertical step frequencies. e theoretical analysis for the inverted pendulum model indicates that such walking habit of pedestrians is the root of the frequency-locking phenomenon, which eventually results in excessive lateral vibrations of the bridge
Observations showed that many footbridges with excessive lateral vibrations have a lateral natural frequency near 1 Hz, such as the London Millennium Bridge, the T-bridge, and M-bridge in Japan [3]

Summary

The Model Equation

Consider the inverted pendulum of length L and lump mass m on a cart of mass M, shown as in Figure 1. e inverted pendulum represents the synchronization pedestrians, and the cart represents a footbridge with a stiffness k. To analyze the lateral pedestrianfootbridge interaction, we assume that the pivot point of the inverted pendulum vibrates harmonically along the y-axis to model the synchronization pedestrians’ walking motion Under such excitation, the cart sways along the x-axis, which simulates the lateral vibrations of the footbridge. Since the ratio between the lateral and vertical step frequencies remains unchanged despite a small or large lateral amplitude of the bridge, frequency-locking phenomenon is easy to be explained To deeply understand this in theory, we discuss the stability of the inverted pendulum system near β3 1/2

Stability Analysis of the Inverted Pendulum System

Numerical Simulations

Findings

Conclusions