The London Millennium Footbridge Revisited: Emergent Instability Without Synchronization

The pedestrian-induced instability of London’s Millennium Bridge is widely held up as the canonical example of synchronization in complex networks [9]. The popular explanation maintained that once the number of pedestrians reached a certain threshold, the pedestrians could supposedly synchronize their footsteps with each other at the bridge’s natural frequency. The result was the onset of dangerous sideways oscillations.

Multiple engineering analyses and publications have debated this original interpretation [4, 7, 8]. Nevertheless, the belief that a textbook example of coupled pedestrian synchronization caused the Millennium Bridge instability remains part of numerous presentations in print, film, and radio [6].

We propose an alternative theory by arguing that any synchronization in the timing of pedestrian footsteps is a consequence—not a cause—of the instability [2]; this result is consistent with observations on 30 bridges, including the Brooklyn Bridge and Golden Gate Bridge. We show that unsynchronized pedestrians produce negative damping — a positive feedback effect wherein energy is transferred from pedestrians who are trying not to fall due to perturbations that are caused by bridge motion. Prior studies found negative damping empirically from measurements on London’s Millennium Bridge itself [5], though the researchers believed that synchronization caused this effect. In contrast, we show that negative damping is more fundamental and can occur without synchronization.

To quantify the effective total negative damping, we use a mathematical model that assumes that possible coordination between pedestrians transpires solely because of sensory stimuli from the moving bridge. We parsimoniously assume that walking is fundamentally a process in which the stance leg acts as a rigid strut, thus causing the body’s center of mass to behave like an inverted pendulum in the frontal plane during each footstep. The step ends when the other leg meets the ground and—ignoring the brief double-stance phase that occurs in realistic gaits—the pedestrian switches to an inverted pendulum on that other leg, which prevents them from falling over. We consider a single lateral vibration mode for the bridge, which is forced by the motion of \(N\) pedestrians who walk perpendicularly to this vibration. We assume that the displacement of the lateral bridge mode \(x(t)\) is governed by a second-order equation of motion:

\[M \ddot{x} + C \dot{x} + K x = \sum_{i=1}^N\tilde{H}^{(i)}(x,y^{(i)}).\]

\(M\), \(C\), and \(K\) are respectively the mass, damping, and stiffness coefficients of the bridge mode, and \(y^{(i)}(t)\) is the lateral displacement of the \(i\)th pedestrian’s center of mass relative to the bridge (see Figure 1). The forcing term \(\tilde{H}^{(i)}\) is the lateral component of the \(i\)th pedestrian’s foot force on the bridge. According to Newton’s second law of motion, the lateral component of the center of mass for a pedestrian of mass \(m\) obeys the equation

\[m \ddot{y}^{(i)}+m \ddot{x} =-\tilde{H}^{(i)}(x,y^{(i)}), \quad i = 1,\ldots N.\] \(\tilde{H}^{(i)}\) is a piecewise smooth function with abrupt changes at foot transitions that are associated with the pedestrian’s gait. We utilized three variants of \(\tilde{H}^{(i)}\) that correspond to three different gait control strategies and yield three different models of pedestrian gait adaptation [2, 3, 8], one of which has a strong propensity for synchronization.

Using a multiple-scale asymptotic analysis, we derived a general expression that quantifies the average contribution to the bridge damping from the interaction force of a single pedestrian over one gait cycle. This quantity \(\sigma\) has three components: (i) A coefficient of lateral bridge velocity-dependent component of pedestrian foot force on the bridge \((\sigma_1)\), which ignores gait timing adjustment; (ii) a coefficient of lateral bridge velocity-dependent component of force due to the adjustment of pedestrian lateral gait timing \((\sigma_2)\); and (iii) a coefficient of lateral bridge velocity-dependent component of force due to adjustment of the forward gait \((\sigma_3)\). 

One should numerically calculate the expressions \(\sigma_{1,2,3}\) as time averages (integrals) of partial derivatives of \(\tilde{H}^{(i)}\), which are associated with the pedestrian’s lateral and forward gaits. These expressions are evaluated individually for each pedestrian \(i\) and depend on the pedestrian’s stride frequency \(\omega_i\), in addition to the bridge’s vibration frequency \(\Omega\). 

The terms \(\sigma_2\) and \(\sigma_3\) depend on the timing of pedestrian stepping behavior in response to bridge motion. Yet in all of our simulations, we found that \(\sigma_1\) plays the most important role in triggering large-amplitude vibrations. This effect is counterintuitive because in the absence of phase synchrony between the bridge and pedestrian, one may imagine that the lateral foot force on the bridge will average to zero. However, this is not the case; Figure 2 provides an explanation.

We calculate the total effective damping coefficient \(c_T\) as

\[c_T = c_0 + N\overline{\sigma}(\overline{\omega},\Omega) \equiv c_0+ \sum_{i=1}^N \left (\sigma^{(i)}_1(\omega_i,\Omega ) + \sigma^{(i)}_2(\omega_i,\Omega )+ \sigma^{(i)}_3(\omega_i,\Omega) \right ),\]

where \(c_0\) is the coefficient of the bridge’s inherent damping and \(\overline{\omega}\) represents the mean pedestrian stride frequency. We found that \(\overline{\sigma}\) is negative over a large range of pedestrian and bridge frequency ratios. The overall modal damping \(c_T\) hence becomes negative when the number of pedestrians exceeds a critical value:

\[N=N_{\rm crit} = - c_0/ \overline{\sigma}.\]

As a result, negative damping causes bridge vibrations to grow. In all of our simulations, the occurrence of negative damping and onset of bridge instability always precede the emergence of increased coherence among pedestrian footstep timings — even for the model that is highly prone to synchronization.

In a key scientific conclusion of our work, we argue that negative damping due to pedestrian attempts to maintain balance is likely the essential cause of most cases of lateral bridge instability. Indeed, our simulations revealed that increased coherence in the timing of pedestrian footsteps is part of a secondary nonlinear adjustment to the amplitude of vibration after initiation of the instability. This secondary effect typically produces saturation of the vibration amplitude, but it can further exacerbate instability in extreme cases.

We achieved these findings via asymptotic analysis, which is applicable to a wide class of foot force models. Moreover, we conducted a comprehensive review of the literature on real bridges that have experienced large-amplitude lateral pedestrian-induced vibrations. Based on this review, it is clear that any direct evidence of synchronization is scant at best. In contrast, our theory is fully consistent with all known observations.

Our findings indicate that large amplitude oscillations can occur for a wide range of bridge frequencies. Therefore, bridge engineering techniques that try to avoid the problem by ensuring that bridge frequency is not close to typical pedestrian stride frequencies (often referred to as frequency tuning) are potentially dangerous.

For a wide range of systems in nature and society, our work argues more generally that this macro-scale instability (in this case, the bridge motion) may emerge from micro-scale behavior (in this case, of many individual pedestrians) without any obvious causal synchrony. It also points to other examples in economic cycles and the tuning of remarkably sensitive hearing organs in mammals and insects.

This article is based on [2] and Igor Belykh’s minisymposium presentation at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually last year. It is dedicated to the memory of John Macdonald, who passed away unexpectedly as the proofs were being prepared. He was an inspiration and will be sorely missed.

Acknowledgments: This work was supported by the U.S. National Science Foundation under DMS-1909924 (to Igor Belykh, Kevin Daley, and Russell Jeter) and the Polish National Agency for Academic Exchange (NAWA) under PPN/PPO/2019/1/00036 (to Mateusz Bocian). 

References

[1] Barker, C. (2002). Some observations on the nature of the mechanism that drives the self-excited lateral response of footbridges. In Proceedings of the International Conference on the Design and Dynamic Behaviour of Footbridges. Paris, France.
[2] Belykh, I., Bocian, M., Champneys, A., Daley, K., Jeter, R., Macdonald, J.H.G., & McRobie, A. (2021). Emergence of the London Millennium Bridge instability without synchronization. Nature Comm., 12, 7223.
[3] Belykh, I., Jeter, R., & Belykh, V.N. (2017). Foot force models of crowd dynamics on a wobbly bridge. Sci. Advan., 3, e1701512.
[4] Bocian, M., Burn, J.F., Macdonald, J.H.G., & Brownjohn, J.M.W. (2017). From phase drift to synchronization — pedestrian stepping behaviour on laterally oscillating structures and consequences for dynamic stability. J. Sound Vib., 392, 382-399.
[5] Dallard, P., Fitzpatrick, A.J., Flint, A., Le Bourva, S., Low, A., Ridsdill Smith, R.M., & Willford, M. (2001). The London Millennium Footbridge. Struct. Eng., 79(22), 17-33.
[6] Fry, H. (2019). Secrets and lies. Lecture 3 — how can we all win? Royal Inst. Christmas Lect. Retrieved from https://www.rigb.org/explore-science/explore/video/secrets-and-lies-how-can-we-all-win-2019.
[7] Ingólfsson, E.T., Georgakis, C.T., & Jönsson, J. (2012). Pedestrian-induced lateral vibrations on footbridges: A literature review. Eng. Struct., 45, 21-53.
[8] Macdonald, J.H.G. (2009). Lateral excitation of bridges by balancing pedestrians. Proc. R. Soc. Lond. A, 465, 1055-1073.
[9] Strogatz, S.H., Abrams, D.M., McRobie, A., Eckhardt, B., & Ott, E. (2005). Crowd synchrony on the Millennium Bridge. Nature, 438, 43-44.