The Millennium Bridge, London
The wobbly bridge
Suspension bridges always wobble a small amount, this is simply down to their design. Normally, this causes no problem as individual wobbles are not allowed to build upon one another and, on the whole, people tend not to be synchronised.
However, as the crowd began their maiden voyage across the bridge an unexpected effect began to emerge. The people were synchronising their foot steps with the movement of the bridge, causing it to wobble more and more. The reason behind this synchronisation is that, as the bridge began to wobble a small amount, the crowds unconsciously changed their pattern of movement to counteract it. However, instead of reducing the motion, they were causing a feedback loop, in which the more they tried to stop the motion, the more the bridge would wobble (you can see this in the youtube footage).
Resonance and natural frequencies
The reason this feedback loop was so problematic was that the natural resonant frequency for the sideways movement of the bridge was so closely matched to the average person's walking pace (which is about 1.7 Hz, or two steps per second).
Objects most readily vibrate at their natural frequencies. [Does anyone play a musical instrument?] Musical instruments are a clear example of the importance of natural frequencies and resonance. For example, when you pluck a string on a guitar it resonates most strongly at the fundamental frequency, where the string oscillates between the two fixed ends.
However, strings also resonate at other, higher, freqencies, called the harmonics, which are multiples of the fundamental frequency. [Does anyone know what the second harmonic is called?] The second harmonic, known as the octave in music, has twice the frequency of the fundamental. [Does anyone know what the third harmonic is called?] The third harmonic has three times the frequency of the fundamental and, musically, is a perfect fifth above the octave. (The opening two notes of the Star Wars theme and Twinkle Twinkle Little Star are examples of perfect fifths).
It only takes a little push...
If you've ever pushed someone on a swing you'll know that it doesn't take much effort to make them swing very high, as long as your pushes are in time with their swinging. This is exactly what happened when the Millennium Bridge first opened.
Like all suspension bridges the Millennium Bridge is flexible and swayed a little in response to wind and foot traffic. What was surprising, however, was that people reacted to this slight movement by gradually falling into step with the sideways motion of the bridge. This seems to have happened because the bridge naturally swayed at a frequency that was close to the average walking pace for people crossing the bridge, around 1.7Hz.
As people synchronised their steps in response to the bridge's movement, they each provided an additional push in time with the sway of the bridge. This in turn increased the distance the bridge swayed from side to side (in exactly the same way as a small push in time with a swing will make it go higher and higher) which then led to more people synchronsing their steps. This feedback loop continued until the movement became so large that people began to stop walking and hold onto the hand rails for support.
Very quickly the bridge was closed and stabilisers were installed to stop the motion from occurring. You can see these beneath bridge as you walk under it on the south bank. [Can anyone see the stabilisers?]
Dangerous yet beautiful
[Can anyone suggest another situation where resonance can be dangerous?] Unexpected resonance can be dangerous, whether for bridges exposed to wind and traffic or for buildings responding to the vibrations of earthquakes. However, resonance is also a thing of beauty. It is precisely the resonance of musical instruments that creates their beautiful sounds.
And these mathematical harmonies have also been important ideas in science and architecture. Palladio, the sixteenth century Venetian architect, is said to have created "frozen music" as his buildings had proportions that mirrored these harmonic ratios. There is a beautiful example of this frozen music just down the river in Greenwich – the Queen's House built in the seventeenth century by Inigo Jones was the first example of Palladian architecture in Britain.
Props: A long slinky toy or rope, 4 heavy metal washers, fishing line or fine string.
We have included some questions in italics in the text above to help engage the group and a crib sheet below to suggest how to interweave the demos into your explanation.
Natural frequencies demo
Use the rope/slinky to demonstrate standing waves that are created when you vibrate the rope at the fundamental or harmonic frequencies. Get one person to hold the rope still at one end and the other person to shake it up and down. (If you are using a slinky get them to hold onto the first few coils at either end). See if they can generate the fundamental frequency first, then, as they increase the speed of their vibration, the second and third harmonics should appear. See http://www.mindbites.com/lesson/4603-physics-in-action-standing-waves-on... for a good demonstration of this.
Feedback loop demo
Use the washers and line to make four pendulums, two identical long pendulums and two identical short pendulums, all suspended from another line. Hold the suspension line very taut and make all the pendulums hang still. Then start one of the longer pendulums swinging and you’ll see that the other long pendulum starts moving with a similar freqency, while the two shorter pendulums stay relatively still.
This is because even the small movements from the swinging pendulum, transmitted by the suspension line, will kick the other long pendulum into motion as these pushes are at the natural frequency shared by both pendulums. However as the natural frequency of a pendulum is determined by its length, these pushes are not at the resonant frequency of the shorter pendulums, hence they remain relatively still. You can see a video of this demonstration on YouTube.
The movement passes between each of the similar pendulums, illustrating the feedback loop that occurred when the Millennium Bridge first opened, where the movement of the bridge affected the movement of the people, which affected the bridge, and so on.
- Intro: problems with the bridge when it first opened (if using headsets you can start introducing this site as you walk over the last third of the bridge)
- Problem was that natural frequency of sideways movement of the bridge was very close to the average walking pace
- Natural frequency demo using the slinky
- If you've ever pushed someone on a swing...
- Feedback loop demo with pendulums
- Slight movements of bridge, caused people to walk in step, caused more movement in bridge...
- Unexpected resonance can be dangerous
- But resonance can also be beautiful, for example musical instruments and Palladio's frozen music
See this Site in a Tour
Written by Thomas Woolley, Marcus du Sautoy and Rachel Thomas.
The model of the Millenium bridge resonating is courtesy of Cambridge University Dept of Engineering.
Photo of the Millennium Bridge found at http://commons.wikimedia.org/wiki/File:London_millenium_wobbly_bridge.jpg / CC BY-SA 3.0
The animated GIF showing the harmonics of a vibrating string is from Plus magazine, used with permission.
Photo of the resonance frequency demonstration by David White / CC BY-SA 3.0