Low dimensional description of pedestrian-induced oscillation of the Millennium Bridge

When it opened to pedestrian traffic in the year 2000, London's Millennium Bridge exhibited an unwanted, large side-to-side oscillation, which was apparently due to a resonance between the stepping frequency of walkers and one of the bridge modes. Models for this event, and similar events on other bridges, have been proposed. The model most directly addressing the synchronization mechanism of individual walkers and the resulting global response of the bridge-pedestrian system is the one developed by Eckhardt [Phys. Rev. E 75, 021110 (2007)]. This model treats individual walkers with a phase oscillator description and is inherently high dimensional with system dimensionality (N+2), where N is the number of walkers. In the present work we use a method proposed by Ott and Antonsen [Chaos 18, 037113 (2008)] to reduce the model of Eckhardt to a low dimensional dynamical system, and we employ this reduced description to study the global dynamics of the bridge/pedestrian interaction. More generally, this treatment serves as an interesting example of the possibility of low dimensional macroscopic behavior in large systems of coupled oscillators.