# Millenium Bridge,London,UK Millenium Bridge, 6/10/00 The opening day: Soon after the crowd streamed on to Millennium Bridge, the bridge started to sway.

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Millenium Bridge,London,UK

Millenium Bridge, 6/10/00 The opening day: Soon after the crowd streamed on to Millennium Bridge, the bridge started to sway from side to side. Many pedestrians fell spontaneously into step with the bridge’s vibrations, inadvertently amplifying them.

The Wobbly Bridge Attempts to limit the number of people crossing the bridge. Long queues, but dampened neither public enthusiasm for something of a white-knuckle ride, nor the vibrations themselves. Unexpected lateral vibration (resonant structural response) caused the bridge to be closed after two days.

Diagnosis Trouble was at 1 hertz (one complete cycle per second) Walking pace is approximately two steps per second (2 hertz) Although most force exerts down when walking, there is small sideways force every two steps (1 hertz)

Positive Feedback The bridge's movements lead to 'positive feedback‘: Synchronous Lateral Excitation. The natural sway motion of walking caused small sideways oscillations. In turn people on the bridge sway in step. Increasing amplitude of bridge oscillations Reinforcing the effect.

Probability of Coordination What is the probability that a thousand people walking at random end up walking exactly in step, and remain in lock step thereafter? If individual steps are independent, then probability is close to zero. But if there is a coordinating mechanism, the probability is close to 1 under the right conditions Bridge moves → Adjust stance ↑ ↓ Further adjust stance ← Push bridge

Millenium Bridge Analogy Bridge moves Pedestrians adjust stance

Millenium Bridge Analogy Prices and measured risks change Banks adjust balance sheet

Endogenous Risk Risk from shocks generated and amplified within the system (feedback effects) In contrast to exogenous risk, from shocks that originate from outside the system (storms, earthquakes)

Positive Feedback Preconditions for feedback: Marking to market — Sensitivity of equity cushion to price changes Risk management that depend on market variables — Sensitivity of equity cushion to risk measures Sensitivity of price and risk to portfolio changes — Uniformity of trading positions among participants

Leverage Cycle Banks increase leverage Balance sheets Balance sheets strengthen expand Asset prices rise Upwards

The Run on Repo How did breakdown in housing market lead to systemic crisis in financial sector? The Panic of 2007-2008 was run on the Repo market, which stopped functioning, leading to massive deleveraging of participants. The run occurred with failure of Lehman Brothers in 09/08. “Non rescue” forced lenders to reevaluate default probabilities of banks and freeze Repo markets. Uncertain about solvency of counterparties, worry about liquidity of collateral bonds; if all firms hold cash, collateral decline in price to find buyers.

Leverage Cycle Downward Cycle Banks reduce leverage Balance sheets Balance sheets weaken contract Asset prices decline

Postscript The problem was fixed by the retrofitting of 37 fluid-viscous dampers (energy dissipating) to control horizontal movement and 52 tuned mass dampers (inertial) to control vertical movement. This took from May 2001 to January 2002 and cost £5m. The bridge was successfully re-opened on 22 February 2002. The bridge has not been subject to significant vibration since.

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