Emergence of the London Millennium Bridge instability without synchronisation

Igor Belykh,Alan R Champneys,Mateusz Bocian,John H G Macdonald,Allan Mcrobie,Kevin Daley,Russell Jeter

doi:10.1038/s41467-021-27568-y

Abstract

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation. Yet, reviewing observational, experimental, and modelling evidence, we argue that increased coherence of pedestrians’ foot placement is a consequence of, not a cause of the instability. Instead, uncorrelated pedestrians produce positive feedback, through negative damping on average, that can initiate significant lateral bridge vibration over a wide range of natural frequencies. We present a simple general formula that quantifies this effect, and illustrate it through simulation of three mathematical models, including one with strong propensity for synchronisation. Despite subtle effects of gait strategies in determining precise instability thresholds, our results show that average negative damping is always the trigger. More broadly, we describe an alternative to Kuramoto theory for emergence of coherent oscillations in nature; collective contributions from incoherent agents need not cancel, but can provide positive feedback on average, leading to global limit-cycle motion.

Highlights

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation
The main features of this instability are: (1) bridges can exhibit large vibration amplitudes in more than one mode of vibration simultaneously, which need not be tuned to a particular walking frequency[13,21]; (2) a critical number of pedestrians is required in order to cause an instability[22,23]; (3) evidence of pedestrian footstep synchronisation[8,24] is scant, with the most definitive study estimating only 20% of the crowd walked in time with the bridge motion[25]; (4) engineering consultants Arup, who reengineered the London Millennium Bridge, found that each pedestrian added, on average, effective negative damping[22]; retrofitting additional dampers successfully cures the problem[26]
In conclusion, the question of what caused the instability of the London Millennium Bridge on its opening day can be referred to as a debate in the literature between the negative damping and synchronisation hypotheses

Summary

Introduction

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation. The main features of this instability are: (1) bridges can exhibit large vibration amplitudes in more than one mode of vibration simultaneously, which need not be tuned to a particular walking frequency[13,21]; (2) a critical number of pedestrians is required in order to cause an instability[22,23]; (3) evidence of pedestrian footstep synchronisation[8,24] is scant, with the most definitive study estimating only 20% of the crowd walked in time with the bridge motion[25]; (4) engineering consultants Arup, who reengineered the London Millennium Bridge, found that each pedestrian added, on average, effective negative damping[22]; retrofitting additional dampers successfully cures the problem[26]. One of the first to call into question the synchronisation explanation of the London Millennium Bridge instability was Nobel prize winner Brian Josephson, writing four days after the bridge’s opening:

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Conclusion