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 hadto 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 ofthe bridge were observed. Further investigations indicated that the oscillation was caused by lateral forces produced by the synchronization of steps taken by thepedestrians. 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 soresonance occurred. Consider an oscillating system of mass m and natural angularfrequency omega_n. When the system is subjected to aperiodic external (driving) force, whose maximum value is F_max and angular frequency is omega_d, theamplitude of the driven oscillations is
A= Fmax/sqr(k-m{omega d}^2)^2 + (bomegad)^2}
where k is the force constant of the system and b 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 rm N at a frequency of about 1 rm Hz. Given that the mass of the bridge is about 2000 rm kg per linear meter, how many people were walking along the 144-rm m-long central span of the bridge at one time, when an oscillationamplitude of 75 rm 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 1/e of its original value in a time t=6T, where T is the period of the undriven, undamped system.
Part B
What would the amplitude A of oscillation of the MillenniumBridge have been on the opening day if the damping effects had been three times more effective?