Change and calculus
10 June 2000
The swaying footbridge
Arup engineers; Dallard and colleagues
How can people walking normally make a steel bridge sway?
London’s Millennium Bridge opened on 10 June 2000 and swayed so much it closed two days later. As the deck leaned, walkers adjusted their steps to keep their balance, and those steps pushed it further. Engineers fitted dampers, which soak up motion like a car’s shock absorbers, and it reopened in 2002.
First, make a guess
As more walkers fall into step, what happens to the sway?
- Real frequency
- 0.50 Hz
- Fades to a third in
- 60.6 s
- Swings before it fades
- 30.3
- Damping ratio ζ
- 0.005
Walkers who fall into step push the pole right. When it crosses the line, the sway grows instead of fading: London’s Millennium Bridge, June 2000. Sound is shifted into hearing range: its pitch follows the pole’s height and its fade follows the pole’s distance from the line.
How they figured it out
On opening day, 10 June 2000, around 90,000 people crossed London’s Millennium Bridge, with up to 2,000 on the deck at once, and it swayed sideways by up to about 7 cm. Engineers at Arup ran crowd tests and found something surprising: the sway did not grow gradually. It started suddenly once enough people were walking.
People on a gently swaying deck widen their stance and time their steps with the sway, which pushes it further. The fix cost about £5 million: 37 fluid-viscous dampers and around 50 tuned mass dampers. The bridge reopened in February 2002 and has been steady since.
The equation
A guitar string’s equation plus one term: the crowd’s push. When it beats the damping, sway grows.
See it in real life
Soldiers break step
Pump a playground swing with small pushes in time and it goes higher and higher. Soldiers are told to break step when crossing bridges; in 1831, Broughton Suspension Bridge near Manchester collapsed under marching soldiers.
Where engineers use it
Structural design
Footbridges are now checked for crowd-induced lateral sway, a rule written because of this bridge.
Dampers
Tuned mass dampers and viscous dampers protect skyscrapers from wind and earthquakes.
Where it came from
- Laplace transform and poles 1812
Led to
In pole language: walkers in step dragged the bridge’s pole across the line into growth.