The London Millennium Bridge's wobble was probably due to people adjusting their balance as they crossed. Credit: Punchstock

It's the way people balance themselves, rather than the timing of their steps that makes bridges wobble, researchers have found.

The Millennium Bridge, which straddles the River Thames in London, was opened in June 2000, but swiftly closed again because it started to wobble disconcertingly when pedestrians tried to cross it. The blame was initially placed on a human tendency to synchronize footsteps in time with the swaying bridge.

But John Macdonald, from the University of Bristol, UK, has now run models to show that the way people move their feet more, or less, to one side in order to keep their balance when walking, can add energy to a bridge's wobble, making it sway even more1. The effect doesn't rely on the synchronization of lots of people's steps in time with the bridge's sway and could explain what happened in London in 2000.

Empirical measurements by Arup, the company that was in charge of building, and fixing, the Millennium Bridge, seemed to show that people act as 'negative dampers', adding energy to the bridge's natural movement — but the firm had no real scientific explanation for their observations. Even so, their findings form the basis of a formula that Arup now uses to predict how many pedestrians should be allowed on a bridge.

Stepping out

For his work, Macdonald looked at the biomechanics of walking. He used a simple 'inverted pendulum' model, which shows that as you move forward your balance is determined by how far to the left or right you place your foot. If you're about to fall to the right, you'll move your left foot further out to compensate.

On a stable floor, the undulations from side to side cancel out in terms of the forces they exert on the ground. But when the ground becomes unstable, the side-to-side motions are no longer the same, and the result is that the pedestrian actually contributes some force in the same direction that the floor is moving in.

When Macdonald ran simulations for a moving bridge, he found that the extra force pedestrians provide as they walk across it is proportional to the bridge velocity, in agreement with the observations made by Arup: the pedestrian acts as a negative damper. The work is published in Proceedings of the Royal Society A1.

Macdonald had shown earlier this year in an experiment using the Clifton Suspension Bridge in Bristol, that pedestrians do not time their steps to match the wobble of a bridge2.

"In some bridges synchronization is not actually observed," says James Brownjohn, a civil engineer from the University of Sheffield, UK, who showed the same thing with the Changi Mezzanine Bridge in Singapore.

Marching lesson

Macdonald's theory is the best yet at explaining the mystery of wobbling footbridges, says Steven Strogatz, from Cornell University in Ithaca, New York. Strogatz had earlier proposed a mathematical model explaining how the crowd crossing the bridge might spontaneously synchronize their steps3. "By modelling how people balance themselves when they walk, [Macdonald] has gone a long way toward explaining what made the Millennium Bridge wobble on its opening day," he says.

The phenomenon of troops 'breaking step' is based on a different theory, and involves vertical, as well as the lateral movement being studied by Macdonald. Nevertheless, his investigations could be important for marching soldiers too. "A large enough group of soldiers could still get the bridge swaying laterally even if they break step," says Macdonald.

The finding will be useful in future bridge designs, says Macdonald. "Up to a critical number of people, there basically isn't a problem," he says, but once that number has been exceeded, each pedestrian acts as a small negative damper and the effects all add up. "If you have enough pedestrians, you're going to have a large amount of negative damping."

Credit: Punchstock