Fig. 4: Average damping coefficient per pedestrian \(\overline{\sigma }\) calculated via (35), given in the Supplement Information (top row) and the critical crowd size Ncrit (bottom row) as a function of numerically calculated bridge and pedestrian frequencies ratio \([{{\Omega }}/\overline{\omega }]\).
From: Emergence of the London Millennium Bridge instability without synchronisation
Simulations of Models 1 and 2 (a) and 3 (b) indicate the range of frequency ratio \([{{\Omega }}/\overline{\omega }]\) in which \(\overline{\sigma }\) is negative so that a single pedestrian, on average, contributes to bridge instability. Each ratio of \([{{\Omega }}/\overline{\omega }]\) corresponds to different combinations of Ω and \(\overline{\omega }\) (blue dots). Black dotted lines indicate the average of \(\overline{\sigma }\) and Ncrit for a given ratio. The red curve indicates the 5th percentile of the \(\overline{\sigma }\) distribution. The green curve is the analytical expression (36) for \(\overline{\sigma }\) (top plot) and analytical estimate (37) for Ncrit (bottom plot), given in the Supplementary Information and calculated for Model 1 with identical pedestrians with fixed ω = 5.655 rad/s and S.D. = 0. The magenta dot corresponds to the initial ratio \([{{\Omega }}/\overline{\omega }]\) used in Fig. 3, the yellow dot corresponds to \({{\Omega }}/\overline{\omega }=1\). See the Supplementary Information for the details of the calculations.
