Fig. 2: Outline of the mathematical model of pedestrian-induced lateral instability.
From: Emergence of the London Millennium Bridge instability without synchronisation
a Simulations are run for a coupled bridge-pedestrians system with pedestrians added sequentially at fixed time increments Tadd apart. The addition of the nth pedestrian (n = Ncrit) causes the overall damping coefficient to become negative hence the amplitude of motion to increase rather than diminish. b Inverted pendulum model of bridge mode and pedestrian lateral motion. Here, y is the lateral position of the pedestrian’s centre of mass (CoM), while p defines the lateral position of the centre of pressure (CoP) of the foot, both relative to the bridge. L is the equivalent inverted pendulum length and m is the pedestrian mass. The displacement x of the bridge in a lateral vibration mode is represented by an equivalent platform with mass M, spring constant K and damping coefficient C. \(\tilde{H}\) is the lateral component of the pedestrian’s foot force on the bridge deck. In return, the bridge motion causes an inertia force \(-m\ddot{x}\) on the pedestrian’s centre of mass. The pedestrians are depicted as “crash test” dummies with flexible hips; however, the actual inverted pendulum model is simpler, with pendulum-like legs connecting to the CoM.
