CODE SUPPLEMENT: A new paradigm for emergent instability: the real story of the London Millennium Bridge
Igor Belykh, Mateusz Bocian, Alan Champneys, Kevin Daley, Russell Jeter, John Macdonald, and Allan McRobie*
*Authors listed alphabetically.
The following is a source code supplement for "A new paradigm for emergent instability: the real story of the London Millennium Bridge" (under review, Nature Communication, 2021). negative_damping.py and the MATLAB file contain the code necessary to reproduce the results of this work; see the source for usage and other information. FBX files are used in Blender to generate the supplemental figures and videos. model2_scatter_plot and 4panel_plot are sample plot scripts for figures 4 and 3, respectively.
Synchronisation of coupled Kuramoto oscillators is a well-established paradigm for the emergence of order across a wide range of natural and engineered systems. The pedestrian-induced instability of the London Millennium Bridge is widely held up as a canonical example; the number of walkers reached a threshold necessary to synchronise their footsteps with each other at the bridge's natural vibration frequency. After a careful review of the observational, experimental, and modelling evidence we finally dispel the synchronisation myth and show 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 lateral damping on average, to initiate significant bridge vibration over a wide range of bridge natural frequencies. We present a simple formula that quantifies the effective total negative damping per pedestrian, for any foot force model, and the contributions towards it from three distinct components. We perform simulations on three different simple mathematical models, each designed to illustrate different effects, including one with a strong propensity for synchronisation. Our results also point to the subtlety of the problem that the precise degree of instability and triggering number of pedestrians can depend on pedestrian gait and balance strategies. Yet we argue that negative damping on average can be always be regarded as the trigger. More broadly, we have uncovered an alternative mechanism for emergence of coherent oscillatory behaviour in natural systems where, unlike the Kuramoto theory, large scale instability is not caused by coherence between individual oscillators. Counterintuitively, collective contributions from incoherent agents need not cancel each other out. Rather, each agent can provide positive feedback on average, leading to global limit-cycle motion.