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.
