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
  • Fig. 1 A Huygens-type setup as a mechanical model for lateral vibrations of a bridge.

    (Left). The platform with mass M, a spring, and a damper represents lateral vibrations of the bridge y. Pedestrians are modeled by self-sustained oscillators, representing walker lateral balance that are capable of adjusting the position of their centers of mass and are subjected to lateral bridge motion. (Middle) Inverted pendulum model of pedestrian lateral movement. Variable x is the lateral position of pedestrian’s center of mass. The constant p defines the lateral position of the center of pressure of the foot. L is the inverted pendulum length. (Right) The corresponding limit cycle in the inverted pendulum model (Eq. 3) along with its acceleration time series. Parameters are as in Fig. 8.

  • Fig. 2 Illustration of the amplitude balance conditions (Eqs. 10 and 12).

    The light gray area indicates the absence of wobbling, guaranteed by Eq. 15. Intersections between the blue (solid) curve (Eq. 10) and a dashed line (Eq. 12) generate phase-locked solutions for the number of pedestrians exceeding a critical number nc = 165. Parameters are as in Fig. 3.

  • Fig. 3 Numerical verification of the analytical prediction (Eqs. 15 to 17): Phase locking with frequency equal to 1.

    (Top) Abrupt onset of bridge wobbling from random initial conditions as a function of the number of walkers in the van der Pol–type pedestrian-bridge model (Eq. 5). (Middle) Time series of phase locking among the pedestrians (blue solid line) and with the bridge (red dashed line) at the common frequency equal to 1 and n = 165. (Bottom) Adjustment of the averaged pedestrian and bridge frequencies. Note the bridge frequency jitters due to bridge beating before the onset of phase locking. The emergence of phase locking at frequency equal to 1 at n = 165 is in perfect agreement with the analytical prediction. Parameters are Ω = 1.2, λ = 0.5, h = 0.05, m = 70, M = 113,000, and a = 1.0. For n ∈ [1,165], parameter ω = 1.097 is fixed and chosen via Eq. 17 to ensure that the phase-locked solution appears at nc = 165. For n ∈ [166,300], ω is varied via Eq. 17 to preserve the amplitude-phase balance.

  • Fig. 4 Analytically intractable case of Ω = 1.

    Identical pedestrians with ω = 0.73, λ = 0.23, and a = 1. Other parameters are chosen to fit the data for the London Millennium Bridge: h = 0.05, M = 113,000, and m = 70. (Top) A hysteretic transition between nonwobbling and wobbling states as a function of crowd size n. Increasing (decreasing) n leads to the onset (termination) of bridge wobbling at n = 165 (n = 135). Notice the appearance and disappearance of the small bump in bridge wobbling n ∈ [10,40] due to a chimera state where a subgroup of pedestrians becomes phase-locked. (Middle) Corresponding average phase difference ΔΦ among pedestrians’ movement. (Bottom) The largest transversal Lyapunov exponent λ for the stability of complete phase locking between pedestrians. Its negative values (occurring around n ≈ 165) indicate the stability of the phase-locked solution.

  • Fig. 5 Pedestrian motions represented by an amplitude color plot for a crowd size before (n = 160) (left) and after n = 166 (right) the onset of phase locking at the critical number of walkers (n ≈ 165).

    The full video demonstrating the onset of phase locking as a function of the addition of pedestrians on the bridge for identical models (Eq. 5) is given in the Supplementary Materials.

  • Fig. 6 Effect of bridge feedback on the period of walker oscillations.

    As the total mass of the pedestrian-bridge system increases with n, the period of phase-locked oscillations also increases.

  • Fig. 7 Nonidentical oscillators with ωi ∈ [ω, ω+].

    Schematic diagram similar to the identical case of Fig. 2, where the cubic balance curve and inclined lines become a cubic strip and horizontal lines, respectively. The dark gray area denotes the range of phase locking, expressed through permissible amplitudes Bi and phases via Eqs. 22 and 23.

  • Fig. 8 Inverted pendulum model (Eqs. 1 to 3) of nonidentical pedestrians with randomly chosen parameters ωi ∈ [0.6935, 0.7665] (10% mismatch).

    Diagrams similar to Fig. 4. The onset of bridge oscillations is accompanied by a drop in the average phase difference between the pedestrian’s foot adjustment. The initial drop corresponding to the initiation of the bridge wobbling and partial phase locking is less significant, compared to the well-established phase locking at larger crowd sizes over 200 pedestrians. The individual pedestrian parameters are λ = 2.8, p = 1, and a = 1. Other parameters are as in Fig. 4. Models 1 to 3 with identical walkers produce similar curves with nearly the same critical crowd size; however, the phase difference drops to 0 (because complete phase locking is possible for identical oscillators).

  • Fig. 9 Diagrams for the mismatched inverted pendulum pedestrian-bridge model (Eqs. 1 to 3), similar to Fig. 5.

    The full videos for both identical and mismatched inverted pendulum models are given in the Supplementary Materials.

Supplementary Materials

  • Supplementary Materials

    This PDF file includes:

    • Legends for movies S1 to S3

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    Other Supplementary Material for this manuscript includes the following:

    • movie S1(.avi format). Identical van der Pol.avi is related to Fig. 5.
    • movies S2 and S3 (.avi format). Inverted pendula identical.avi and inverted pendula 10% mismatch.avi are related to Fig. 9.

    Files in this Data Supplement: