ekatherine
New member
- Joined
- Nov 8, 2020
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When first opened, the Millennium Bridge in London wobbled from side to side as people crossed; you can see this on video at www.arup.com/MillenniumBridge. Footfalls created small side-to-side movements of the bridge, which were then enhanced by the tendency of people to adjust their steps to compensate for wobbling. With more than a critical number of pedestrians (around 160) the bridge began to wobble violently.
Without any pedestrians, the displacement x of a representative point on the bridge away from its normal position would satisfy
Mx¨ + kx˙ + λx = 0, where M ≈ 4 × 105kg, k ≈ 5 × 104kg/s, λ ≈ 107kg/s 2 .
1)Show that the level of damping here is only around 1% of the critical value.
The effective forcing from each pedestrian was found by experiment (which involved varying numbers of people walking across the bridge) to be proportional to ˙x, with F ≈ 300 ˙x. If there are N pedestrians, the displacement of the bridge satisfies Mx¨ + kx˙ + λx = 300Nx.˙
2)Find the critical number N0 of pedestrians, such that if there are more than N0 pedestrians the bridge is no longer damped.
3)Show that if there are 200 pedestrians then there will be oscillations with a frequency of approximately 0.8 hertz (oscillations per second) the amplitude (max size) of oscillation of which grows as e t/80 .
4)The problem was corrected by adding additional damping, in order to bring the damping up to 20% of the critical level. What would this do to the value of k, and how many people can now walk across the bridge without counteracting all the damping?
Without any pedestrians, the displacement x of a representative point on the bridge away from its normal position would satisfy
Mx¨ + kx˙ + λx = 0, where M ≈ 4 × 105kg, k ≈ 5 × 104kg/s, λ ≈ 107kg/s 2 .
1)Show that the level of damping here is only around 1% of the critical value.
The effective forcing from each pedestrian was found by experiment (which involved varying numbers of people walking across the bridge) to be proportional to ˙x, with F ≈ 300 ˙x. If there are N pedestrians, the displacement of the bridge satisfies Mx¨ + kx˙ + λx = 300Nx.˙
2)Find the critical number N0 of pedestrians, such that if there are more than N0 pedestrians the bridge is no longer damped.
3)Show that if there are 200 pedestrians then there will be oscillations with a frequency of approximately 0.8 hertz (oscillations per second) the amplitude (max size) of oscillation of which grows as e t/80 .
4)The problem was corrected by adding additional damping, in order to bring the damping up to 20% of the critical level. What would this do to the value of k, and how many people can now walk across the bridge without counteracting all the damping?