What would the amplitude of oscillation of the MillenniumBridge have been on the opening day if the damping effects had been three times more effective?
Introductory Text:
On June 10, 2000, the Millennium Bridge, a new footbridge over the River Thames in London, England, was opened to the public. However, after only two days, it had to be closed to traffic for safety reasons. On the opening day, in fact, so many people were crossing it at the same time that unexpected sideways oscillations of the bridge were observed. Further investigations indicated that the oscillation was caused by lateral forces produced by the synchronization of steps taken by the pedestrians. Although the origin of this cadence synchronization was new to the engineers, its effect on the structure of the bridge was very well known. Thecombined forces exerted by the pedestrians as they were walking in synchronization had a frequency very close to the natural frequency of the bridge, and resonance occurred.
Consider an oscillating system of mass 
and natural angular frequency 
. When the system is subjected to aperiodic external (driving) force, whose maximum value is 
and angular frequency is 
, the amplitude of the driven oscillations is
A=Fmax(k−mωd2)2+(bωd)2A=frac{F_{max }}{sqrt{left(k-m omega_{d}^{2}right)^{2}+left(b omega_{d}right)^{2}}}
,
where 
is the force constant of the system and 
is the damping constant.
We will use this simple model to study the oscillations of the Millennium Bridge.
—
Part A
Assume that, when we walk, in addition to a fluctuating vertical force, we exert a periodic lateral force of amplitude 25 
at a frequency of about 1 
. Given that the mass of the bridge is about 2000 
per linear meter, how many people were walking along the 144-
-long central span of the bridge at one time, when an oscillation amplitude of 75 mm was observed in that section of the bridge?Take the damping constant to be such that the amplitude of the undriven oscillations would decay to 
of its original value in a time 
, where 
is the period of the undriven, undamped system.
Answer ≈ 1810 people (to 3 significant figures).
—
Part B
What would the amplitude 
of oscillation of the MillenniumBridge have been on the opening day if the damping effects had been three times more effective?
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