Emergence of the London Millennium Bridge instability without synchronisation

Abstract

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation. Yet, reviewing observational, experimental, and modelling evidence, we argue that increased coherence of pedestrians’ foot placement is a consequence of, not a cause of the instability. Instead, uncorrelated pedestrians produce positive feedback, through negative damping on average, that can initiate significant lateral bridge vibration over a wide range of natural frequencies. We present a simple general formula that quantifies this effect, and illustrate it through simulation of three mathematical models, including one with strong propensity for synchronisation. Despite subtle effects of gait strategies in determining precise instability thresholds, our results show that average negative damping is always the trigger. More broadly, we describe an alternative to Kuramoto theory for emergence of coherent oscillations in nature; collective contributions from incoherent agents need not cancel, but can provide positive feedback on average, leading to global limit-cycle motion.

Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning

“…)), and P 4 = (− π 2 − 1 2 arcsin( 2ω α ), 0, −α sin( 1 2 arcsin( 2ω α ))). The Jacobian matrix for the linearization of ( 6)- (7) at the equilibrium points is…”

“…Similarly to our calculation above, we observe that equation ( 7) includes the term k sin φ. Recall from equation ( 6) that dφ dt = γ, so we posit an approximation of γ(t) of the form γ(t) ≈ ζ + c cos 2φ + d sin 2φ for some constants c and d. Inserting the approximations of k(t) and γ(t) into equation (7) with m = 1 yields…”

“…The analysis of systems of coupled oscillators has been used extensively to study collective behavior in many systems [1], such as flashing in groups of fireflies [5], synchronization of pedestrians on the Millennium Bridge [6,7], pacemaker conduction [8], and singing in frogs [9]. In 1975, Yoshiki Kuramoto proposed a model of coupled biological oscillators as a system of a first-order differential equations in which each variable corresponds to the phase of an oscillator [10].…”

Dynamical System-Based Computational Models for Solving Combinatorial Optimization on Hypergraphs

“…The analog computing approach, focus of the present work, exploits the fact that combinatorial optimization problems entail the minimization of an objective function, and thus, exhibit a natural similarity to the minimization of energy in a dynamical system. Consequently, this has motivated the formulation of physics-based computational models [7], [17]- [22], inspired by dynamical systems, for solving such problems as well as others [23]- [26]. For instance, Wang et al, [7] showcased how the challenge of minimizing the Ising Hamiltonian (and the equivalent MaxCut problem) can be formulated in terms of the dynamics of coupled oscillators under second harmonic injection.…”

Dynamic Entrainment: A deep learning and data-driven process approach for synchronization in the Hodgkin-Huxley model

“…This event created resonance and most likely caused the bridge to collapse with the soldiers [7]. The instability of the London Millennium Bridge or so called "Wobbly Bridge", which was induced by pedestrians walking on it during the opening day, enforced the closure of the bridge for almost two years for further repairs and modifications to make the bridge more stable [8].…”