# Detailed Explanation to Simple harmonic motion

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• wobbling bridge.doc

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. The combined forces exerted by the pedestrians as they were
walking in synchronization had a frequency very close to the natural
frequency of the bridge, and so resonance occurred. Â

, the amplitude of the driven oscillations is

,

is the damping constant.

We will use this simple model to study the oscillations of the
Millennium Bridge.

Part A:

is the period of the undriven, undamped system.

number of peopleÂ =

Hints: A Wobbling Bridge

is the period of the undriven, undamped system.

Hint 1.Â How to approach the problem

The flexible structure of a bridge behaves like an oscillating system.
If a periodic external force acts on the bridge at a frequency close to
its natural frequency, the structure of the bridge resonates and, unless
the structure has an adequate damping system, large-amplitude
oscillations may develop, just as observed in the Millennium Bridge.
Thus, model the bridge as a driven, damped oscillating system, as the
problem suggests, and impose the resonance condition; that is, make the
driving frequency equal to the natural frequency. From the formula of
the oscillation amplitude, given in the problem introduction, determine
the maximum value of the driving force that produced the resonance
amplitude observed in the bridge. From that result you can calculate the
number of pedestrians whose synchronized cadence contributed to the
total driving force.

Hint 2.Â Find the maximum value of the driving force when resonance
occurs

, the maximum value of the driving force?

.

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Hint 3.Â Find the damping constant

can be calculated as

,

of its initial value.

.

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Hint 4.Â Find the total number of synchronized pedestrians

as they walk in synchronization, that is, when the time between steps
is the same for every pedestrian?

.

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Hint 5.Â Find the mass of the bridge

-long section?

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Hint 6.Â Find the angular frequency

driving force?