Modeling walker synchronization on the Millennium Bridge

Abstract

On its opening day the London Millennium footbridge experienced unexpected large amplitude wobbling subsequent to the migration of pedestrians onto the bridge. Modeling the stepping of the pedestrians on the bridge as phase oscillators, we obtain a model for the combined dynamics of people and the bridge that is analytically tractable. It provides predictions for the phase dynamics of individual walkers and for the critical number of people for the onset of oscillations. Numerical simulations and analytical estimates reproduce the linear relation between pedestrian force and bridge velocity as observed in experiments. They allow prediction of the amplitude of bridge motion, the rate of relaxation to the synchronized state and the magnitude of the fluctuations due to a finite number of people.

Synchronization of clocks

“…Eqs. (41,42,59) [Nm] is significantly smaller than the values of other energies and is not considered in Figure 25(a,b) and Table 1.…”

“…On the opening day the Millennium Bridge, a pedestrian footbridge crossing the Thames River in London, was observed to exhibit a pronounced lateral wobbling as more and more people streamed onto the bridge [36,37,81,42,92,101,1]. This phenomenon apparently occurred due to a resonance between a low order bridge oscillation mode and the natural average stepping frequency of human walkers: a small initial oscillation of the bridge induces some of the walkers to synchronize the timing of their steps to that of the bridge oscillations, thus exerting a positive feedback force on the bridge that derives the bridge oscillation to higher amplitude, eventually resulting in a large steady-state oscillation.…”

“…(41,42) and (59) Eqs. (61,62) give relation between the pendula's amplitudes Φ 1 and Φ 2 and the phase angle β 2 .…”

“…This was one of the first observations of the phenomenon of the coupled harmonic oscillators, which have many applications in physics [89,16,[18][19][20]47,93,5,99], biology and chemistry [93,60,63,64,100,104]. In engineering very spectacular was synchronization of pedestrians on Millennium bridge [1,19,36,37,42,81,84,86]. Huygens originally believed the synchronization occurs due to air currents shared between two pendula, but later after performing several simple tests he dismissed this and attributed the sympathetic motion of pendula to imperceptible movement in the beam from which both pendula were suspended.…”

Collective motion of oscillatory walkers

“…We believe that by including the effects of phase, models for pedestrians and other animals can be improved. Moreover, the synchronization of pedestrians through a bridge [23,24] might be an interesting problem. It is also attractive to extend this model to two dimensions using the two-dimensional OV model [25], which generalization would require additional rules of phase synchronization.…”