Supplementary information for Emergence of the London Millennium Bridge instability without synchronisation

Supplementary Note 1: details of the asymptotic calculation of σ1,2,3 We start from (22) (see the main paper). For ease of notation, we shall let ẋ = u, ẏ = v, ż = w and drop the superscript (i) in what follows, providing the meaning is clear. Also, let us define the vector ξ = (x,u,y,v,z,w) and let ξ0 = (0,0,y0,v0,χt + z0,c+w0) be the unperturbed limit-cycle motion. Now we can formally expand the functions H and G as power series

Note that the coefficients h ξ k , and g ξ k represent partial derivatives of H and G with respect to their subscripted arguments, evaluated along the unperturbed solution y = y 0 (t), z = χt + z 0 (t). Hence each of these coefficients is a T -periodic function of time.

First-order solution
Substitution of the zeroth-order solution into (20) yields, to leading order, where we have used the assumption (21) that the pedestrians are uncorrelated to assume

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The solution to (26) is the free vibration of the bridge, which can be written in the form where the amplitude X and phase φ are allowed to be functions of a slow time variable τ = εt.
Substitution of x 1 into (18) and (19) using (25) yields, to leading-order in ε, This is a linear system with periodic coefficients and periodic forcing. It can be solved as the sum of free and forced vibration terms. Under the assumption that ω i = Ω, and that the limit cycle in the absence of bridge motion is asymptotically stable, the free vibration part must decay to zero for large times. The only non-decaying part comes from the forced vibration. We find approximate expressions for this term by averaging the periodic functions h ξ and g ξ . Let an overbar represent the average of a quantity over each period T . That is Then the solution for the forced vibration problem can be written in the form where y c,s , z c,s are constant amplitudes of cosines and sines of period T , and y r and z r are remainder terms that contain all other harmonics. Expressions for y c,s , z c,s can be written in closed form as

Second-order solution
Substitution of the O(1) solution into bridge equation (20) at second order yields where means differentiation with respect to the slow time τ and We can now substitute the form of x 1 from (27) and of y seek the general solution to the forced vibration problem.
In order to find a consistent asymptotic solution, under the formalism of the method of multiple scales 1 , we must avoid secular terms on the right-hand side of (31). That is, the components of cos(Ωt + φ ) and sin(Ωt + φ ) in the forcing term must vanish. Let us consider the three terms on the right-hand side (31) in turn. cos(Ωt + φ ) and sin(Ωt + φ ) to be

Consider first the term ∑
Note that each of these coefficients is a constant for each pedestrian, because we already averaged out the period-T i components in the definition (30). Therefore we can sum each of these N terms individually so where. means averaging over all pedestrianŝ Finally, the third term on the right-hand side of (31) iṡ Recalling that Nε = ν, the vanishing of the secular terms on the right-hand side of (31) thus implies These equations can then be written in the form (23) and (24).

Supplementary Note 2: further details of numerical simulation algorithms 2.1 Procedure for adding pedestrians on the bridge
Initially the bridge-pedestrian system is simulated with a crowd size of two pedestrians. Every T add of simulation time, we insert another pedestrian with a uniform random phase initialised to the limit cycle solution of the model in the absence of bridge motion. Subsequently, we advance the simulation T add

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seconds, compute the maximum amplitude of the bridge and mean order parameter of the pedestrians over the current interval, and save the entire state.

Implementation of negative damping criterion
To compute σ 1 : According to the formula (9), for each pedestrian, we need to compute where u =ẋ. We consider specifically the case of Models 1 and 3 in which there is a jump in the force H.
where Θ is the Heaviside step function, and p(t s ) is given by (13). The principle is easily generalised to any function H(y) with jumps at t = t s .
That is H during step s is given by H s and J = H s+1 − H s . For the specific case of Models 1 and 2 we can That is, we compute the integral as if the singularity were absent, provided we add an extra jump term every time t = t s for s = 1, 2, 3, . . .

It remains to show how to compute
from t s−1 to t s .

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Note that Hence, η(t) satisfies the variational differential equation To compute σ 1 for the particular case of Model 3, we proceed similarly; in this case, note that the jump term is due to the presence of the discontinuity in sgn(y) at each step and takes a similar form to that of the jump term of Model 1 in the case of a fixed step width p c .
Likewise, we may formulate a second-order variational equation, akin to (33), for the continuous part of Hẋ using the multivariate chain rulė Within the pedestrian step, Hẏ and H y have closed form and may be used to compute σ 2 (see next section).
To compute σ 2 . Here, according to (10), for each pedestrian we need to compute y c y s , h y and h v .
By definition, y c and y s would be the Fourier cosine and since coefficients, respectively, of the O(ε) components of y(t), assuming that the dominant bridge motion is cos(Ωt). Thus, given bridge motion x(t),

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we have where The integrals h y and h v can be computed similarly to h u . In particular, for Models 1 and 3 we get Hence, To compute σ 3 . The formula (11) requires evaluation of the forward dynamics, specifically a function z (i) (t) which is the fluctuating component of the pedestrian's progress Z (i) (t) where χ (i) is their average forward velocity. However, none of the models we consider have an equation for the forward dynamics z(t).

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Instead, Models 2 and 3 have an update rule that generates the time t s+1 as a function of t s . Because everything is averaged over a cycle, we can assume that z(t) is a continuous variable that at t = t s gives the perturbation to the position along the bridge span of the pedestrian's centre of mass from its position if it were walking at a constant velocity.
That is, we can assume that for t ∈ [t s ,t s+1 ) so that z becomes a piecewise-linear function of time Then the functions z c and z s can be defined similarly to y c and y s above. Also Hence, we need only consider the first term in (11), which necessitates computation of z s . Analogously, with (34), we have Fig. 4.

Calculations of the scatter plots and analytical curves in
To generate the scatter plots Fig. 4 (top row) and estimate average damping coefficient per pedestrian σ , we simulate the pedestrian system (5) with imposed bridge motion, wherein the bridge is taken to be sinusoidal at its natural frequency, i.e.

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where X b = 0.006 m is the constant amplitude of the imposed bridge motion. Only a single pedestrian is placed on the bridge and the bridge's acceleration is fixed to the second derivative of the sinusoid: x = −X b Ω 2 sin Ωt, regardless of the pedestrian's foot force. This scenario accurately describes the case for a low mass ratio, such that the force from an individual pedestrian has negligible effect on the structure.
We perform this simulation over a uniform 1200-point grid of the pedestrian and bridge natural frequencies For each frequency ratio [Ω/ω], we numerically compute the force H (1) by simulating equation (5) for y 1 over 50 foot steps for Model 2 and over 10,000 steps for Model 3. To calculate the component of H (1) in phase with the bridge velocity, we modify the formula (9) from Ref. 2

so that
We then compute a histogram ofĤ u , parametrised by a discrete set of K numerically computed phase offsets ϕ of the pedestrian step time. To calculate the damping coefficient of the pedestrian, we use the histogram to calculate the expected value H u ofĤ u for each ϕ. We then apply the scaling in equation (11) from Ref. 2 to the resulting average. Thus, we obtain the following formula where the summation is calculated over the discrete bins of the histogram. This formula is used to generate the blue points in the top panels of Fig. 4 for different frequency ratios [Ω/ω].

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The analytical curve for σ depicted by the green solid line in the top plot of Fig. 4 (a) is calculated via √ g L cos πΩ ω , and a 2 = e π ω √ g L sin πΩ ω .
Formula (36), derived in Ref. 3 , estimates the negative damping contribution of the pedestrian described by Model 1 via σ 1 . As a result, it does not account for the effect of step timing adaptation and, therefore, it yields estimates that differ from the numerical results for Model 2 that does allow for pedestrian step timing adaptation.
We also use formula (36) to estimate the critical crowd N crit as a function of the frequency ratio Ω/ω (the green solid line in the bottom plot of Fig. 4 (a)). This is done by setting the total bridge damping c T = 0 in (8). Therefore, we obtain the condition 2εζ Ω + N crit σ /M = 0 which yields the following estimate for the critical crowd size where σ is estimated via (36). The analytical expression (37) estimates the critical number of pedestrians N crit described by Model 1 precisely. However, it becomes less accurate for pedestrians described by 50π/ω s. These simulations were performed for the same bridge parameters as in Fig. 3 but for identical pedestrians with the same angular stride frequency ω, varied across the range of frequency ratio [Ω/ω], and the same default values of b min , L, and m given in Table 3. As in the simulations of Fig. 3, the initial conditions for the pedestrian phases were chosen randomly from a uniform distribution. Zero bridge amplitude and velocity were chosen as the initial conditions for the bridge.

Faster addition of pedestrians to the bridge
To better understand the role of time interval T add at which pedestrians are added sequentially to the bridge, we perform numerical simulations similar to those reported in Fig. 3  To elucidate the contributions of the damping per pedestrian terms σ 1 , σ 2 , σ 3 , we also include the additional plots (third row) in Supplementary Fig. 1. These plots indicate that while σ 1 is the only factor that matters for the onset of instability for non-adaptive Model 1, σ 2 , that accounts for the adjustment of pedestrian lateral gait timing, contributes to the overall negative damping to a lesser (Model 2) and greater (Model 3) degree. In all cases we find that σ 3 makes little contribution.

Extreme worst-case, complete resonance
To provide further evidence that pedestrian synchronisation is not necessary for bridge instability, we consider the worst-case scenario in which the pedestrians have identical natural stride frequency ω (but  Fig. 4. Other parameters are as in Fig. 3 .

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initially random phase) which also coincides with the bridge natural frequency Ω. It seems to be the ideal resonance scenario for the emergence of synchronisation among the pedestrians and with the bridge, and therefore, we could expect synchronisation to emerge at smaller crowd sizes and coincide with the onset of bridge instability. We also note that this case violates the central assumption (21) that underlies the above asymptotic derivation of the coefficients σ 1,2,3 .
The results are shown in Supplementary Fig. 2. Despite violating the above assumption, observe that the negative damping criterion still predicts the onset of bridge instability for Models 1 and 2. In particular, note how significant synchronisation now occurs, after the onset of large vibrations, for Model 2 with an order parameter greater than 0.5. For Model 3, the negative damping criterion no longer provides accurate information, but note that the large increase in negative damping at around 50-100 pedestrians precedes the onset of significant instability, which in this case leads to complete synchrony (order parameter r = 1).