Forget finding a theory of everything – we can't even explain how bicycles work
IT IS almost dark at the Technical University of Delft in the Netherlands, but that hasn't stopped the world's foremost authorities on the bicycle from careering around a car park on two wheels. It's clear from the carnage that most are theorists. They might work with the maths behind bicycling on a daily basis, but it looks like some of them don't ride bikes much.
Or perhaps that's being a little harsh. Some of these bikes are ridiculous. They are meant to be improvements on the standard bicycle, but it is clear that a near-perfect design evolved decades ago. The 1970s bike made of flexible plastic might have won design awards, but it is terrifyingly shaky. Then there is the bike with two wheels on its rear axle which is hair-raisingly unstable when cornering. And the electric bikes, though fun, have a tendency to zoom away as soon as the rider starts to pedal, leaving them grabbing for the brakes.
Proving a particular problem are the low-slung recumbent bikes, in which the rider sits close to the floor. Spectacular crashes of these machines result in a mess of corduroy wrapped around metal. Trousers are extricated slowly and painfully. Then - with all the stubbornness that comes with years of wrestling seemingly intractable mathematical problems - the rider gets back in the saddle.
Such tenacity is vital, for bicycles have long been a headache for theorists. While 150 years of evolution have turned the standard, two-wheeled velocipede into a thing of beauty, we still don't understand exactly how it works. "We have the equations," says Andy Ruina of Cornell University in Ithaca, New York. "It's just that we don't know what they mean."
Ruina is among the mathematicians, physicists and engineers who gathered in Delft late last year to consider the physics of the bicycle - and to test the theories out in the car park. He is also among those who have just brought the situation to a head by publishing a paper on the humble bicycle in science
There's a new big hole in physics, the paper claims. Forget finding a theory of everything or working out what most of the universe is made of; we can't even explain how bicycles keep themselves upright while rolling along. "Why does this bicycle steer the proper amounts at the proper times to assure self-stability?" the paper asks. "We have found no simple physical explanation."
The good news is that we now have some important clues. Though we still don't know exactly what keeps a bike upright, those behind the Science paper have narrowed it down to a few aspects of the design. That means there is a real chance that a radical new take on the bicycle can be developed. No more wobbly, crash-prone plastic experimental machines: we are about to enter the age of the mathematically designed bike. "We believe manufacturers could, should and eventually will take a look at what we're doing," says coauthor Jim Papadopoulos of the University of Wisconsin-Stout at Menomonie.
Yet this isn't the first time researchers have claimed to have solved the riddle of the bicycle. Back in 2007, Ruina, Papadopoulos and others published a paper on bicycle stability in Proceedings of the Royal Society A (vol 463, p 1955). Newspaper headlines gushed about "The mathematical way to ride a bike", saying the team had "established conclusive equations that describe what gives a pushed bike its stability". But it wasn't as big a leap as everyone hoped. "The equations started being written in 1898," Ruina says. "We just finally wrote them down clearly and definitively."
The next step was to examine those equations for clues to how bicycles become "self-stable". Push a bike along at the right speed, then let it go and it will roll along, even if sideswiped by a considerable force. It will wobble, but as it starts to lean, the spinning wheels, wobbling handlebars and its distribution of mass act together to steer into the fall, keeping it upright.
The authors of the 2007 paper found this behaviour was influenced by 25 variables in the equations. But just two of those were thought to be the vital ones. First is the "trail", which is defined as the horizontal distance between where the front wheel makes contact with the ground and where the front forks that hold the wheel in place would hit the ground if they were extended (see diagram). Trail determines how easily a bike in motion self-corrects its steering when turning.
The second variable deemed essential is the gyroscopic effect of the front wheel. Forces on a spinning wheel act to keep its axle in a constant orientation because the wheel's angular momentum is always conserved. This effect is exploited by navigational gyroscopes to monitor an aircraft's pitch and roll, and was first pronounced crucial for bicycle self-stability by two German giants of mathematics, Arnold Sommerfeld and Felix Klein, in 1910.
Yet it turns out that neither trail nor gyroscopic effects are necessary for self-stability. Ruina credits Papadopoulos with the design that busted the myth: staring at the equations, he says, Papadopoulos imagined a bike that would trash all the cherished ideas about where stability comes from. Then, last year, Arend Schwab and his students at Delft turned Papadopoulos's vision into reality.
The resulting bike has an extra wheel above the front wheel that spins in the opposite direction - countering the gyroscope effect. Yet when this bike is pushed off course it still self-corrects. The team then rebuilt the bike with no trail, and found that it was still stable. And when they rebuilt the bike with no gyroscopic effect and negative trail, where the front forks are angled to touch the ground behind the contact point of the wheel, it, too, was stable. "No one knows why," Ruina says.
Or, more precisely, no one knows exactly why. But Ruina and the others have now assembled everything they need to find out. Though none of the parameters such as trail or gyroscopic effects are vital on their own, there is a subtle interplay between them that gives a bike its stability. There are half a dozen other factors involved too, such as the geometry of the steering axis, the distance between the wheels, or more important perhaps a particular mass distribution. Untangling all the effects still seems to be a job for a computer. "We are trying to find the simple conditions, but it's quite tricky to put your finger on what's going on," Papadopoulos says.
A bike less ordinary
Yet exploring the way these factors work together should provide a great leaping-off point for a new generation of bicycle designs. "The more we understand about the dynamics and control of these machines, the more likely it is we will be able to design something out of the ordinary," Schwab says.
The first mathematically enhanced bike is already on the road. Dutch cycle maker Raptobike has used the mathematics described in the 2007 paper to improve its recumbent bikes. The length of recumbent bikes has to be adjusted for each rider, but this can create problems. When the frame is made longer the centre of gravity becomes lower, and when the frame is set shorter the opposite occurs. This means that the bike's handling is different for taller and shorter riders, says Arnold Ligtvoet of Raptobike. Schwab's model allowed Ligtvoet to work out the perfect geometry for the bike's frame that would counter this effect. "The model was used to predict and calculate the change in handling and adapt the design to prevent the change," he says.
Another mathematical take on the traditional bicycle is in the pipeline, this time for older riders. Some older people find it harder to keep a bike stable, which is believed to be because they ride more slowly than the traditional bike is designed to go, and they don't always have the same reaction speed to messages from their eyes, vestibular system and other balance mechanisms.
Thanks to mathematics, though, these problems can be overcome. "We have used the information from Professor Schwab's laboratory to modify the geometry of the bike's frame so that it becomes more stable," says Rob van Regenmortel of Dutch bike manufacturer Batavus. "We had been using a trial and error system: this is the first chance we have had to do it in a scientific way."
The company expects the main changes to be in the angle of the seat and head tubes, which the handlebars attach to, as well as a longer distance between the two wheels. Changes to the geometry of the front fork may also be needed, van Regenmortel says.
Batavus hopes to include the bike in its 2013 range, but van Regenmortel doesn't see mathematics changing many other types of bike. "Applied to racing bikes it can give you information about performance, but those bikes have evolved quite well by trial and error," he says. "I don't think it will give us new ideas so much as confirmation that we've got things right."
Schwab is convinced there is room for improvement in other areas, though: folding bikes in particular. That's because their design has just been copied from old designs, he says, and that is not an ideal way to innovate. "The old process of trial and error could change the design eventually, but we want these machines now," Schwab says. "So we say we should put some more science into it."
For now, though, bicycle science is still in its infancy. "Down the road I hope someone will be able to say 'this is what makes a good bike - a bit of being nearly self-stable, this amount of effort to make it turn...'," says Papadopoulos. "But at the moment it's still a big mess. It's tricky to put your finger on what's going on."
Nonetheless, Ruina believes it is worth pursuing. "If you try to change the design of a bicycle any which way it gets worse, everybody knows that," he says. "But maybe if you change three or four things at one time, in the right way, there's another good bicycle out there."
Before mathematicians can really change the bike, however, there is a big problem to solve: how to model the lump of flesh sitting on it. The fact is, no one knows quite how we ride our bikes. "The human aspect of this is still a mystery for us," Papadopoulos says. Schwab agrees: "We need a model of the rider."
To ride a bike we seem to rely on a mix of feedback controls from various inputs, including the eyes, the vestibular system - even the nerves that sense what's happening at our elbows and knees. But, having shown it is not simply the gyroscopic effect nor the trail that keeps a bike upright, Schwab has now shown that the received wisdom about the rider has to go too. "We did a series of experiments and found that, contrary to what people think, you don't lean out your upper body to steer," he says. "Your upper body stays in the plane of the bicycle, and most of the control is done by turning the handlebars."
Even the obvious inputs aren't necessary. We know, for instance, that you can ride a bike blindfold; that was discovered by Tony Doyle in 1987 (New Scientist, 30 April 1987, p 36). "We were expecting to find it difficult, but it was easy," he says. "Even my grandchild rode blindfold."
At low speed, however, the rider's body does come into play, but not in a way anyone expected. To keep the bike stable, people waggle their knees from side to side. "It was incredible to see it," Schwab says. "We observed it in every circumstance: on the open road and on the treadmill."
All in all, "as easy as riding a bike" is turning out to be a rather misleading saying. "I have a colleague who studies how pilots control aircraft, and he says riding a bike is much more complex," says Mont Hubbard of the University of California, Davis.
Hubbard is one of the few people in the world to be specifically funded to study the science of bike riding. When he was awarded a grant by the US National Science Foundation, The Washington Times called it an outrage. Hubbard is unrepentant: "It's about studying human operators and the limits of control." His group also studies the human factors behind air crashes, and understanding the way we ride a bike - how we deal with multiple sensory inputs, and so on - feeds directly into that. "We're trying to answer an important question: how complicated a system can a human deal with?" Hubbard says.
The answer to that question may come from a rather unexpected quarter: the Belgian seaside. Engineers at the University of Mons in Belgium have constructed a beast of a bike they call the Anaconda. It consists of a two-wheeled bike connected to a line of steerable monocycles. The joints between the constituent cycles are hinged so that the bike will twist and turn chaotically - unless, that is, the people riding it are all experts at control.
The behaviour of the Anaconda is being mathematically modelled for a PhD project, but it is also planned to become a seafront attraction. On a sunny day, you and your friends could spend hours getting to grips with the way it snakes along.
It is certainly not an easily acquired skill; last year, a team of European Space Agency astronauts took on the challenge and failed. Olivier Verlinden, one of the Anaconda's creators, wasn't surprised. "They only had a few minutes to learn," he says. Given longer, however, teams of schoolchildren have managed to tame the Anaconda. "We think that it is especially suited to young people who want to have fun - and are not afraid of falling," Verlinden says.
So while our standard road bikes are resistant to change, perhaps there really is room for some radical new cycles to hit the road. And who knows, maybe the surprise hit of the 2016 Olympic games will be the 100-metre Anaconda sprint?
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