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Unit 12

In Sync

12.7 Mechanical Sync

METRONOMES

At the beginning of this unit, we caught a glimpse of the variety of situations in which synchronization can occur. Up until now, we have focused primarily on sync as it occurs with living things that are able to send, receive, and interpret signals. We hinted, however, at the fact that spontaneous synchronization is not limited to living beings. It seems to be a fundamental phenomenon in nature, occurring not only in the realm of biology, but also in chemistry and physics.

In fact, the first documented observations of a system coming into spontaneous order were solidly in the realm of physics. In the 1660s, the Dutch physicist Christiaan Huygens, known primarily for his contributions to probability, astronomy, and optics, found himself sick in bed, as the legend goes, observing two pendulum clocks. He noticed that no matter what configuration each started in, they would eventually begin swinging in sync with each other. Technically it was anti-phase sync:

pendulums

Huygens examined the situation and found that the two clocks were both resting on a loose, wobbly floorboard. He also noted that if the two clocks were placed at opposite ends of the room, no such synchronization occurred. He surmised that the motions of the two pendula transmitted tiny forces to each other via the loose plank, subtly slowing down or speeding up the frequency of each until they swung in anti-phase synchrony.

We can observe a similar phenomenon using a couple of metronomes. Imagine that we have two metronomes, both set to oscillate at the same frequency.

metronomes Item 1756/Oregon Public Broadcasting, created for Mathematics Illuminated, METRONOMES (2008). Courtesy of Oregon Public Broadcasting.

If we place these two metronomes on a solid, fixed surface, out of phase with each other, they will continue to oscillate out of phase with each other for as long as we care to watch. If we place the same two metronomes on a board that is allowed to move in a particular way, however, the situation is quite different.



metronomes Item 1757/Oregon Public Broadcasting, created for Mathematics Illuminated, METRONOMES ON A WOBBLY PLANK (2008). Courtesy of Oregon Public Broadcasting.


If the board connecting the metronomes sits atop two cans, so that it is free to move laterally, parallel to the motion of the metronome arms, it becomes a connection between the two metronomes that is capable of transmitting subtle shifts in momentum.

Imagine that the arm of metronome 1 is moving towards the left, while the arm of metronome 2 is moving towards the right. Let’s say that metronome 2 is closer to the right-most point in its cycle than metronome 1 is to the left-most point of its cycle.

metronomes Item 1781/Oregon Public Broadcasting, created for Mathematics Illuminated, METRONOME 2 IS ABOUT TO HIT ITS RIGHTMOST POSITION AND 1 ABOUT TO BE PULLED SLIGHTLY TO THE LEFT (2008). Courtesy of Oregon Public Broadcasting.


When metronome 2 reaches its right-most point, the motion of switching to start moving to the left imparts some small change in momentum that is equal and opposite to the change that drives the metronome arm to the left. In other words, it will shift the board ever so slightly to the right. This is a consequence of Newton’s third law of motion, which states that for every action, there is an equal and opposite reaction.

The effect of the board moving to the right is to accelerate, ever so slightly, the arm of the left metronome towards its left-most point.

This is similar to the forces involved when you try to pull the tablecloth out from under a setting of tall glasses. Unless you are extremely gifted and/or lucky, you are likely to cause at least a few glasses to fall. When they fall, they will fall in the direction opposite the movement of the tablecloth.

Tablecloth

This is how the board allows the two metronomes to influence each other. The net effect of the small changes transmitted from metronome 1 to metronome 2, and vice versa, will be that the metronomes eventually will come to oscillate in sync with each other.

metronomes Item 1804/Oregon Public Broadcasting, created for Mathematics Illuminated, TWO METRONOMES ON A WOBBLY PLANK (2008). Courtesy of Oregon Public Broadcasting.


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THE MILLENNIUM BRIDGE

This concept of oscillators, connected by some medium that can transmit signals between them, seems to be at the heart of the phenomenon of synchronization. We’ve seen how sync arises in a variety of contexts, both biological and mechanical. In our final example, we will see how sync occurred in a system comprised of both biological and mechanical elements.

The Millennium Bridge was constructed across the River Thames in London in the late 1990s to commemorate the beginning of a new millennium in the year 2000.

Millennium Bridge Item 3228/Alexander Hafemann, LONDON THAMES MILLENNIUM BRIDGE (2007). Courtesy of iStockphoto.com/ Alexander Hafemann.


Millennium Bridge Item 3229/S. Greg Panosian, LONDON ARCHITECTURE AT NIGHT (2007). Courtesy of iStockphoto.com/ S. Greg Panosian.


On the day the bridge was opened to the public, crowds of people assembled to walk across the newest landmark in the city. As the bridge filled with people, something remarkable, and somewhat frightening, began to take place. The bridge started swaying, with no observable cause. The winds were calm, and yet the bridge began to sway with more and more severity.

bridge

Video from that day shows that, as the bridge swayed, the pedestrians began to compensate by adopting a staggering, side-to-side gait. Moreover, groups of them began to stagger in sync with one another, completely unintentionally.

bridge

The synchronized staggering of the people, begun as a response to the initially slight swaying movements of the bridge, served to amplify the oscillations until the bridge swayed quite violently. In this case, the walking surface of the bridge served the same function as the plank in the metronome example that we just examined; it transmitted small changes in lateral momentum between people to the bridge structure, reinforcing the oscillations that had already begun. The more the bridge shook, the more people compensated in their walking motion, and as more people began to stagger in sync with each other, the bridge shook more violently, creating a sort of feedback loop.

After a few days, the bridge was closed due to safety concerns and construction crews reinforced it to prevent so much lateral flexibility. No one was injured in the event, and it might have been written off as just an odd coincidence were it not for mathematicians taking an interest in the phenomenon and seeing it as a startling example of the mathematics of synchronization.

The following is an interview with Roger Ridsdill Smith, Director, Ove Arup and Partners Ltd. and Project Director for the London Millennium Footbridge

What was Arup's role in the design and construction of the Millennium Bridge?

Arup have been the Engineer for the bridge, from its inception to completion of the modification works.

Arup won the international competition (over 200 entrants) in 1996 as the Engineer in a team with Foster and Partners (Architect) and Sir Anthony Caro (Artist).

Describe what happened to the bridge on 10 June 2000

It is estimated that between 80 000 and 100 000 people crossed the bridge during the first day. Analysis of video footage showed a maximum of 2000 people on the deck at any one time, resulting in a maximum density of between 1.3 and 1.5 people per square metre.

Unexpected excessive lateral vibrations of the bridge occurred. The movements took place mainly on the south span, at a frequency of around 0.8 Hz ( the first south lateral mode), and on the central span, at frequencies of just under 0.5Hz and 1.0 Hz (the first and second lateral modes respectively). More rarely movement occurred on the north span at a frequency of just over 1.0 Hz, (the first north lateral mode).

Excessive vibration did not occur continuously, but built up when a large number of pedestrians were on the affected spans of bridge and died down if the number of people on the bridge reduced, or if the people stopped walking. From visual estimation of the amplitude of the movements on the south and central span, the maximum lateral acceleration experienced on the bridge was between 200 and 250 milli-g. At this level of acceleration a significant number of pedestrians began to have difficulty in walking and held onto the balustrades for support.

No excessive vertical vibration was observed.

The number of pedestrians allowed onto the bridge was reduced on Sunday 11th June, and the movements occurred far more rarely. On the 12th June it was decided to close the bridge in order to fully investigate the cause of the movements.

What is Synchronous Lateral Excitation? Briefly, how did you model it mathematically?

The movement of the Millennium Bridge has been found to be due to the synchronisation of lateral footfall forces within a large crowd of pedestrians on the bridge. This arises because it is more comfortable for pedestrians to walk in synchronisation with the natural swaying of the bridge, even if the degree of swaying is initially very small. The pedestrians find this makes their interaction with the movement of the bridge more predictable and helps them maintain their lateral balance. This instinctive behaviour ensures that footfall forces are applied at the resonant frequency of the bridge, and with a phase such as to increase the motion of the bridge. As the amplitude of the motion increases, the lateral force imparted by individuals increases, as does the degree of correlation between individuals. It was subsequently determined, as described below, that for potentially susceptible spans there is a critical number of pedestrians that will cause the vibrations to increase to unacceptable levels.

How was the Millennium Bridge's swaying different from the swaying that brought down the Tacoma Narrows Bridge in 1940?

The movements that occurred on the Tacoma Narrows Bridge were a resonant response to forces exerted by wind rather than pedestrians. The pedestrian induced forces that cause Synchronous Lateral Excitation are self-limiting because above a certain level of movement, pedestrians stop walking.

How did ARUP fix the issue with the Millennium Bridge?

Although a few previous reports of this phenomenon were found in the literature, none of them gave any reliable quantification of the lateral force due to the pedestrians, or any relationship between the force exerted and the movement of the deck surface.

Arup therefore carried out tests in 3 universities, as well as crowd walking tests on the bridge itself, in order to quantify the force exerted on the structure. Arup then designed a system of passive dampers which are mobilized by the lateral movements of the bridge. These dampers are arranged beneath the deck over the full length of the bridge, as well as at the piers and at the south abutment.

In order to demonstrate that the solution performed satisfactorily, Arup carried out a crowd test with 2000 pedestrians — the most extreme dynamic test ever carried out on a bridge. The bridge movements were less than a sixth of the allowable movements.

The bridge reopened in February 2002.

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