Foot force models of crowd dynamics on a wobbly bridge

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Science Advances  10 Nov 2017:
Vol. 3, no. 11, e1701512
DOI: 10.1126/sciadv.1701512

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  • RE: Authors' response to an eLetter by At L. Hof
    • Igor Belykh, Professor of Applied Mathematics, Georgia State University
    • Other Contributors:
      • Russell Jeter, Ph.D. student, Georgia State University
      • Vladimir Belykh, Professor of Mathematics, Volga State University of Transport

    Thank you for your comments on our work.

    Our work represents an effort to use methods from applied dynamical systems to predict the critical crowd size for the onset of large-scale oscillations, much in the vein of Strogatz et al. in [1], where the phase oscillators are replaced with Van der Pol and biomechanically-inspired inverted pendulum models. Our analysis does not claim that synchrony is the cause of lateral wobbling in pedestrian bridges. On the contrary, in the abstract we state: “Pedestrian phase-locking due to footstep phase adjustment, is suspected to be the main cause of its large lateral vibrations; however, its role in the initiation of wobbling was debated.” The notion of synchrony being the cause of wobbling in the London Millennium Bridge has been debated as soon as immediately after the incident, see Josephson’s letter “Out of step on the bridge” to the Guardian Newspaper in [2] and later papers by Barker [3] and Macdonald [4].

    Furthermore, in the Conclusions we state: “The initiation of wobbling without crowd phase locking was previously observed during periods of instability of the Singapore Airport’s Changi Mezzanine Bridge [5] and the Clifton Suspension Bridge [6]. Both bridges experienced crowd-induced vibrations at a bridge frequency different from the averaged frequency of the pedestrians, while the pedestrians continued to walk without visible phase-locking [4]. Our recent results [7] on the ability of a single pedestrian to in...

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    Competing Interests: None declared.
  • RE: “Foot force models of crowd dynamics on a wobbly bridge” By I. Belykh, R. Jeter and V. Belykh. Science Advances 2017; 3: e1701512
    • At L. Hof, associate professor, Center of Human Movement Sciences, University of Groningen, The Netherlands

    I have read this paper with great interest and admiration for the mathematical handling of the subject of pedestrian-bridge interaction. The results surprised me, however: Belykh et al. predict that synchrony among walkers is the cause of lateral wobbling of pedestrian bridges. Synchronization within pedestrians means that the stride frequency of each individual is changed by the lateral movement of the bridge. In my own experiments [1], I saw a shortening of stride time only at greater perturbations (over 30 mm), so a role in initiating bridge movement seems improbable.
    In the paper under discussion the pedestrians are modelled as Van der Poll-type oscillators, proposed as “a simplified, more analytically tractable model” for lateral movement in walking. It must be admitted that this model gives a good representation of unperturbed gait for a single walker [2]. On the other hand, Van der Poll oscillators are well-known to show phase-locking, and thus, when a number of them are coupled, synchronization [3].
    Our objection to this approach is that this phase-locking has no experimental basis. Foot position and stride width are indeed modulated by lateral velocity [4] but stride frequency hardly so. In a recent paper it was shown that frequency locking occurred only in 2 out of 137 cases [5]. Thus a Van der Poll-type oscillator as a model of pedestrian lateral movement has properties that are not supported by experiment.
    On the basis of the inverted pendulum...

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    Competing Interests: None declared.

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