Walking crowds on a shaky surface: stable walkers discover Millennium Bridge oscillations with and without pedestrian synchrony

Abstract

Why did the London Millennium Bridge shake when there was a big enough crowd walking on it? What features of human walking dynamics when coupled to a shaky surface produce such shaking? Here, we use a simple biped model capable of walking stably in three dimensions to examine these questions. We simulate multiple such stable bipeds walking simultaneously on a bridge, showing that they naturally synchronize under certain conditions, but that synchronization is not required to shake the bridge. Under such shaking conditions, the simulated walkers increase their step widths and expend more metabolic energy than when the bridge does not shake. We also find that such bipeds can walk stably on externally shaken treadmills, synchronizing with the treadmill motion for a range of oscillation amplitudes and frequencies. Our simulations illustrate how interactions between (idealized) bipeds through the walking surface can produce emergent collective behaviour that may not be exhibited by just a single biped.

Keywords: emergent behaviour; feedback control; human–structure interaction; pedestrian dynamics; stability; walking.

Figure 1.

Biped model. (a) Point-mass biped with massless legs. The walking platform can only oscillate laterally, (b) inverted pendulum walking and (c) simulating a large crowd with fewer simulated bipeds.

Figure 2.

Walking on a shaky bridge. (a) Platform oscillation with P = 2, 4, 6 and 80 groups of pedestrians representing an equivalent number N = 80, 240 or 400 pedestrians. The steady state is independent of P, barring time-offsets owing to random initial phase. We see decaying oscillations for low N oscillations, with multi-step periodicity for intermediate N and two-step periodic oscillations for large N. (b) Platform steady-state oscillation amplitude (root-mean-squared (RMS) position of the steady-state motion) as a function of N, showing three qualitatively different regimes. (c) Bridge motion when the bipeds (P = 8) are identical (‘similar’) and non-identical (‘dissimilar’). (d) Order parameter variation showing that identical bipeds synchronize but non-identical bipeds do not. See electronic supplementary material, videos for walking animations. (e) The energy cost of walking goes up when the pedestrians shake the bridge, comparing the 400 pedestrian case (shaking) to the 80 pedestrian case (no shaking). (f,g) Walking on a shaken treadmill. The steady-state phase difference as a function of (f) platform oscillation amplitude and (g) platform oscillation frequency. Pedestrians entrain to platform oscillations for some frequencies and amplitudes. All quantities non-dimensional.