by Matt Parker
Resonators gonna resonate
If something resonates with you, it means you have really connected with it; it’s struck a chord with you. This figurative use of ‘resonate’ took off in the late 1970s and has remained surprisingly true to the literal use of ‘resonate’ from about a century earlier. From the Latin word resonare, which roughly means ‘echo’ or ‘resound’, in the nineteenth century ‘resonance’ became a scientific term to describe infectious vibrations.
A crude analogy for resonance is that of a pendulum, often modelled as a child in a swing. If you are charged with pushing the child and you just thrust your arms out at random intervals, you will not do very well: you’d hit the child coming towards you and slow them down as often as you’d give the swing a push as it’s going away and speed it up. Even a regular pushing rate that did not match the movement of the swing would leave you pushing empty air most of the time.
Only if you push exactly at the rate which matches when the child is directly in front of you and starting their descent will you achieve success. When the timing of your effort matches the frequency the swing is moving at, each push adds a little more energy into the system. This will build up with each push until the child is moving too fast to easily inhale and their screaming will finally cease.
Resonance in a musical instrument is this on a much smaller scale: utilizing the way in which a guitar string, a piece of wood or even contained air will vibrate thousands of times per second. Playing the trumpet involves tightening your lips then throwing a cacophony of messy frequencies at it. But only those that match the resonant frequencies of the cavity inside the trumpet build up to audible levels. Changing the shape of the trumpet (via convenient levers and valves) changes the cavity’s resonant frequency and a different note is amplified.
The same thing works inside any radio receiver (including contactless bank cards). The antenna is receiving a mess of different electromagnetic frequencies from TV signals, wifi networks and even someone nearby microwaving their leftovers. The antenna is then plugged into an electronic resonator made of capacitors and coils of wire that perfectly matches the specific frequency it wants to pay attention to.
While resonance is great in some situations, engineers often have to go to a lot of effort to avoid it in machines and buildings. A washing machine is incredibly annoying in that brief moment when the spin frequency matches the resonance of the rest of the machine: it takes on a life of its own and decides to go for a walk.
Resonance can affect buildings as well. In July 2011 a thirty-nine-storey shopping centre in South Korea had to be evacuated because resonance was vibrating the building. People at the top of the building felt it start to shake, as if someone had banged the bass and turned up the treble. Which was exactly the problem. After the official investigation had ruled out an earthquake, they found the culprit was an exercise class on the twelfth floor.
On 5 July 2011 they had decided to work out to Snap’s ‘The Power’, and everyone jumped around harder than they usually did. Could the rhythm of ‘The Power’ match a resonant frequency of the building? During the investigation, about twenty people were crammed back into that room to recreate the exercise class and, sure enough, they did have the power. When the exercise class on the twelfth floor had ‘The Power’, the thirty-eighth floor started shaking around ten times more than it normally did.
Shakes on a plane
The Millennium Bridge’s 1-Hertz resonant frequency was only for oscillations in a specific direction: side to side. People stepping up and down should not be a problem; and even the 1-Hertz sideways back-and-forth movement of humans walking should not have been a problem, as everyone is likely to be stepping at different times. For anyone pushing with their right foot, another person would be pushing with their left and all the forces would pretty much cancel each other out. This sideways resonance would only be a problem if enough people walked perfectly in step.
This is the ‘synchronous’ in ‘synchronous lateral excitation’ from pedestrians. On the Millennium Bridge, people did start to walk in step, because the movement of the bridge affected the rhythm at which they were walking. This formed a feedback loop: people stepping in synch caused the bridge to move more, and the bridge moving caused more people to step in synch. Video footage from June 2000 seems to show over 20 per cent of pedestrians walking in step – more than enough to get the resonant frequency ringing and the middle of the bridge swaying about 7.5 centimetres in each direction.
Fixing it was a costly two-year retrofit, during which the bridge was completely closed. Removing the wobble cost £5 million, on top of the original £18 million build. Part of the difficulty was breaking the pedestrian-bridge feedback loop without changing the aesthetics of the bridge. Hidden beneath the footpath and around the structure are thirty-seven ‘linear viscous dampers’ (tanks with a viscous liquid that a piston moves through) and around fifty ‘tuned mass vibration absorbers’ (pendulums in a box). These are designed to remove energy from the movement of the bridge and damp the resonance feedback loop.
It works. Originally, the bridge’s sideways movement had a damping ratio of below 1 per cent for resonant frequencies below 1.5 Hertz. They are now all damped by 15 to 20 per cent. This means enough energy is removed from the system to nip a feedback loop in the bud. Even frequencies up to 3 Hertz are damped by 5 to 10 per cent – I guess in case a bunch of people decide to all run across simultaneously and in step. When it was reopened, the Millennium Bridge was described as ‘probably the most complex passively-damped structure in the world’. Not an epithet most of us would aspire to.
This is how engineering progresses. Before the Millennium Bridge, the maths of ‘synchronous lateral excitation’ from pedestrians was not at all well understood. Once the bridge had been fixed, it was a well-investigated area. As well as studying the footage from when it was open, tests were run with an automatic shaking device placed on the bridge. And groups of volunteers walked backwards and forwards on it.
In one test, progressively more people were made to walk over the bridge and any wobbling was closely measured. Opposite, you can see the plot of increasing numbers of pedestrians and the sideways acceleration of the bridge. A critical mass of pedestrians is reached at 166, well below the 700 or so on the bridge when it opened. Not the most scientific plot ever: it does make me wonder what the unit of ‘bridge deck acceleration’ is. And my favourite part of this graph is that, because it shows pedestrians as well as acceleration at the same time, the axis allows for there to be a negative number of pedestrians on the bridge. Or, technically, normal pedestrians moving backwards in time. Which, if you’ve ever been stuck behind tourists ambling through London, you know is actually possible.
You do not want to be the 167th person on this bridge.
Prior to the Millennium Bridge incident there had been a few hints that synchronized pedestrians could set a bridge shaking sideways. In 1993 an investigation was carried out on a footbridge which wobbled sideways when two thousand people crossed it at the same time. Before that, there was a 1972 investigation into a bridge in Germany with similar problems when three hundred to four hundred people walked on it simultaneously. But none of this had seemingly made it into the building regulations for bridges. Everyone remained obsessed with vertical vibrations.
Ups and downs
The vertical up-and-down impact from a human walking is around ten times greater than the side-to-side force, which is why the lateral movements had been ignored for so long. The vertical vibrations of bridges had been noticed much sooner. Solid stone or wood bridges do not have resonant frequencies which can be easily matched by human footsteps. But after the Industrial Revolution of the eighteenth and nineteenth centuries, engineers started experimenting with novel bridge designs involving trusses, cantilevers and suspension cables. Eventually, a modern suspension bridge was built within the resonance reach of humans.
One of the first bridges to be destroyed by synchronized pedestrians was a
suspension bridge just outside Manchester (in what is now the city of Salford). I believe that this Broughton Suspension Bridge was the earliest bridge destroyed when people walked over it at the resonant frequency. Unlike the Millennium Bridge, which had a feedback loop to synchronize the pedestrians, on Broughton Bridge the people crossing it had to do all the work themselves.
The bridge was built in 1826, and people crossed it with no problem at all until 1831. It took a troop of soldiers all marching perfectly in synch to hit the resonant frequency. The 60th Rifle Corps of seventy-four soldiers were heading back to their barracks at about midday on 12 April 1831. They started to cross in rows of four and pretty quickly noticed that the bridge was bouncing in rhythm with their steps. This was apparently quite a fun experience and they started to whistle a tune to go with the bouncing. Until about sixty soldiers were bouncing on the bridge at once and it collapsed.
Around twenty people sustained injuries from the sixteen-foot fall into the river; luckily, nobody died. The discussion during the aftermath identified the vibrations as having put the bridge under a greater load than the same number of people standing still would have. Similar bridges were scrutinized; the knowledge of this type of failure was now out there. Thankfully, it did not take loss of life for humans to learn about the resonance in suspension bridges. To this day, there is a sign on the Albert Bridge in London warning troops not to march in time across it.
But they must not break dance.
In a twist
Not all such knowledge is so easily discovered or even remembered. During the mid-1800s the rail network was exploding across England, which required a slew of new railway bridges able to support a fully loaded train. A bridge to carry a train is harder to design than a foot or traffic bridge. Humans and carriages have some level of built-in suspension; they can deal with a road surface which is moving around a bit. A train has no such tolerance. The track needs to remain absolutely stationary, which makes for some very stiff railway bridges.
In late 1846 a railway bridge designed by engineer Robert Stephenson was opened over the Dee River in Chester. The bridge was longer than previous bridges that Stephenson had designed but he tightened and reinforced it to help it cope with heavy loads without it moving too much. It was a classic step forward in engineering: take previous successful designs and make them do slightly more while using slightly less building materials. The Dee Bridge fulfilled both these criteria.
It opened, and it worked fantastically. The British Empire was all about trains and British engineers prided themselves on their stiff upper bridges. In May 1847 the bridge was modified slightly: extra rock and gravel were added to keep the tracks from vibrating and to protect the bridge’s wooden beams from burning embers produced by the steam engines. Stephenson inspected the work and was satisfied that it had been done correctly. The extra weight this put on the bridge was within the expected safety tolerances. However, the first train to cross after the work did not make it to the other side.
It was not that the bridge could not support the extra weight but rather that the combination of length and mass opened up a whole new way for bridges to go wrong. It turns out that as well as vibrating up and down and side to side, bridges can also twist in the middle. Six trains had passed over the bridge perfectly safely on the morning of 24 May 1847, before the extra mass of broken rocks were added that afternoon.
As the next train was crossing the reopened bridge, the driver felt the bridge moving beneath him. He tried to get across as fast as he could (steam trains are not known for their acceleration) and only just made it. That is to say, the driver in the engine made it. The five carriages he was pulling did not. The bridge twisted to the side and the carriages were dumped into the river below. Eighteen people were injured, and five died.
In some senses, a disaster like this is understandable. Obviously, we should do whatever we can to avoid engineering mistakes, but when engineers are pushing the boundaries of what is possible, occasionally a new aspect of mathematical behaviour will unexpectedly emerge. Sometimes the addition of a little bit more mass is all it takes to change the mathematics of how a structure behaves.
This is a common theme in human progress. We make things beyond what we understand, and we always have done. Steam engines worked before we had a theory of thermodynamics; vaccines were developed before we knew how the immune system works; aircraft continue to fly to this day, despite the many gaps in our understanding of aerodynamics. When theory lags behind application, there will always be mathematical surprises lying in wait. The important thing is that we learn from these inevitable mistakes and don’t repeat them.
The twisting action of the bridge has since become known to engineers as ‘torsional instability’, which means that a structure has the capability to twist freely in the middle. I think of torsional instability as the movement no one expects. Most structures don’t have the right combination of size and length to twist noticeably, so torsional instability is forgotten about until a new construction dips just below the threshold where it manifests and then, suddenly, it’s back!
After Dee Bridge (and similar accidents), engineers took a long, hard look at the cast-iron girders it had been built from and decided to use stronger wrought iron from then on. The official report blamed the disaster on a weakness in the cast iron. Stephenson went with the creative suggestion that the train had derailed on its own – basically arguing that the train broke the bridge, not the other way around. Nobody bought it. But he did raise the very good point that, in all the previous bridges he had built, the cast-iron girders were fine. None of their theories had hit upon the true cause.
They almost unmasked the true culprit of torsional instability at the end of the report. The civil engineer James Walker and Inspector of Railways J. L. A. Simmons closed their accident report by admitting that Stephenson’s other bridges had not fallen down, but they were all ‘of less span than the Chester [Dee] Bridge’ and ‘the dimensions of the parts [were] proportionally less’. For a brief moment, they admit that there could be something else going on with the scale of the bridge, but they still ended up blaming the weakness of the girders. They didn’t make that final step. And the increased reinforcement of future bridges was enough to drive torsional instability back into hiding. For a while.
Torsional instability came back with a vengeance in the Tacoma Narrows Bridge (Washington State, US). Designed in the 1930s, it was part of the new art deco visual aesthetic; the main designer, Leon Moisseiff, said that bridge engineers should ‘search for the graceful and elegant’. And that it was. A thin, ribbon-like, streamlined bridge, it looked incredibly graceful. As well as looking good, it was cheap. By using substantially less steel, Moisseiff’s design was about half the cost of the bridge proposed by his competitor.
Opened in July 1940, the bridge quickly proved that being cheap to build had come at a cost. The thin road surface would move up and down in the wind. This was not yet torsional instability but the classic up-and-down bounce that had troubled many a bridge. But it seems, in this case, there was not enough bounce for it to be dangerous. People were told that it was perfectly safe to drive across Galloping Gertie, as it had been nicknamed by the locals. (It seems Americans are even more creative at naming structures than Londoners, who probably would have gone with The Wavy Bridge.)
Having been reassured by experts that it was safe, people viewed it as a kind of fun ride, while engineers scrabbled to work out how that movement could be damped. Then, in November 1940, the bridge collapsed spectacularly. This has become an iconic example of engineering failure, because it happened to be near a local camera shop, where the owner, Barney Elliott, had new-fangled 16-millimetre Kodachrome colour film. Elliott and his colleague managed to capture the bridge’s demise.
Tacoma Narrows Bridge, 1940. Moments later, a guy jumped out of that car and ran for his life.
But the notoriety of this bridge’s collapse has come with a down side: the wrong explanation. To this day,
the Tacoma Narrows Bridge disaster is held up as an example of the dangers of resonant frequencies. Like the Millennium Bridge, it is argued that the wind travelling down the Tacoma Narrows matched a resonant frequency of the bridge and tore it apart. But, unlike the Millennium Bridge, that is not true. It was not resonance that brought down this bridge.
It was the other villain of the Millennium Bridge: a feedback loop. A feedback loop which had teamed up not with resonance but with torsional instability. The sleekness of the design made it very aerodynamic. As in, the air made it dynamic. Whereas other proposed designs for the Tacoma Narrows Bridge had a metal mesh which wind could blow through, the bridge that was built had flat metal sides, perfect for catching the wind.
The actual feedback loop was ‘flutter’. Under normal circumstances, the bridge would twist in the middle a bit but quickly spring back to normal. But with enough wind the flutter feedback loop would drive torsional instability to very noticeable levels. If the side of the bridge which was upwind were to lift slightly via some classic torsional twisting, then it would act like an aeroplane wing and be pushed higher by the wind. When it rebounded and dipped down, the wing effect would go the other way, pushing it further down. So each time the twist went up or down, it would be helped along by the wind and the size of the oscillations would increase. If you blow hard enough over a taut ribbon, you can see this effect for yourself.
In the wake of the Tacoma Narrows Bridge disaster, similar bridges were reinforced. Aerodynamic flutter was added to the long list of things an engineer had to worry about when designing a bridge. Engineers are now generally aware of torsional instability and design bridges accordingly. Which should mean we’ve seen the last of it. But sometimes, lessons learned in one area of engineering don’t get passed on to another. It turns out that torsional instability can also affect buildings.