Engineers testing a new (hypothetical) pedestrian footbridge

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks

Problem

Engineers testing a new (hypothetical) pedestrian footbridge, distinct from the Millennium Bridge case discussed earlier, model its crowd as a network of phase oscillators: each pedestrian's stride phase is coupled to the neighbours whose footfall they can feel through the deck's sway, with a mean of c=10c=10 such neighbours (an Erdos-Renyi approximation to the sensing graph). Pedestrians' preferred stride frequencies scatter with a Lorentzian half-width Δ=0.35 rad s−1\Delta = 0.35\ \mathrm{rad\,s^{-1}}. As in the bridge-crowd account discussed for coupling that grows with crowd size, each additional pedestrian on the bridge adds a fixed increment to the effective coupling, α=3.0×10−4 rad s−1\alpha = 3.0\times10^{-4}\ \mathrm{rad\,s^{-1}} per person, so that a crowd of npn_{\rm p} pedestrians supplies coupling K=αnpK = \alpha n_{\rm p}. ((a)) Estimate the leading eigenvalue κ1\kappa_1 of the sensing network. ((b)) Estimate the network synchronisation threshold KcK_{\rm c}. ((c)) Estimate the crowd size npn_{\rm p} at which synchronised sway first sets in.

Browse the problem library