The Riderless Bicycle: Why Bikes Stay Upright, and Why Scientists Got It Wrong for a Century

In Jacques Tati’s 1949 comedy Jour de Fête, a village postman loses his bicycle, and the bicycle keeps going without him. It rolls down the road on its own, upright and steady, past puzzled farmers, as if it had somewhere better to be. It is a sight gag, but it is also good physics. Give a modern bicycle a firm push and let go, and it will usually wobble a little, straighten up, and coast for a surprisingly long way before it finally tips over. A bicycle propped up and released while standing still falls in a second or two. A rolling one, with nobody on it, can stay up for something like 20 seconds.

That simple fact hid a question that took more than 140 years to answer properly. “Almost everyone can ride a bicycle,” the British chemist David Jones wrote in 1970, “yet apparently no one knows how they do it.” For most of the 20th century, the textbook answer was gyroscopes, or else the way the front wheel trails behind the steering axis like a shopping-cart caster. In 2011, a team of engineers built a bicycle with neither and showed that it balanced itself anyway.

If you read our recent story on Euler’s Disk, the heavy spinning coin that whirs faster and faster as it settles, or the one on the tippe top that flips itself onto its stem, you have already met the family. These are ordinary objects in motion whose behavior is easy to film and hard to explain. The bicycle is the most familiar member of that family, and for a long time it was the most misunderstood.

A running machine born in a hungry year

The bicycle began as a way to replace the horse. In 1816, the year after the eruption of Mount Tambora in Indonesia, Europe and North America lived through the Year Without a Summer. Crops failed, oats became scarce, and many horses starved or were slaughtered. Historians still debate how much this pushed Karl Drais, a forestry official in Mannheim, toward his invention, but the timing is suggestive.

On June 12, 1817, Drais rode his new machine from Mannheim to an inn on the road toward Schwetzingen, about 7 kilometers (roughly 4 miles) away, and back in a little over an hour. It had two wooden wheels in a line, a wooden frame, a padded saddle, and a steerable front wheel. It had no pedals. The rider sat astride it and pushed off the ground with his feet, then lifted them and coasted. Drais called it a Laufmaschine, a running machine; others called it a draisine, a velocipede or, in England, a dandy horse. Riders could reach about 10 miles per hour on good surfaces.

The important part, for our story, was the steering. Drais understood that a rider could keep two wheels upright only by steering, and his design let the front wheel pivot so the rider could correct the machine as it began to lean. That idea, balance by steering, is the core of everything that follows. Roads were rough and rutted, so riders took to the smoother sidewalks, and their run-ins with pedestrians led many towns to ban them. New York City barred them from footpaths and public places. The craze faded within a few years, but the principle of a two-wheeler that balances while it rolls had been shown to work.

High wheels, safety bicycles and a prize question

Pedals arrived in the 1860s, fixed directly to the front wheel. Since one turn of the pedals meant one turn of the wheel, the only way to go faster was a bigger wheel, and by the 1870s that logic had produced the high-wheeler, later nicknamed the penny-farthing. Its front wheel could be taller than a person’s waist. It was fast and thrilling, and a rider who hit a bump could be pitched forward over the handlebars headfirst.

The fix was the “safety bicycle.” John Kemp Starley’s Rover of 1885 used two wheels of similar size, a chain drive to the rear wheel, and a steerable front fork. Within a decade or so, its basic layout had become the shape of a bicycle, and it has barely changed since.

Scientists were interested from the start. In 1869, the Scottish engineer William Rankine published an article in The Engineer repeating Drais’s own claim that a rider keeps a bicycle up by steering toward the side it is falling. In 1876, Charles Spencer noted in his book The Modern Bicycle that a bicycle could be ridden by someone lying on the seat with their hands off the bars. In 1897, the French Academy of Sciences made bicycle dynamics the subject of its Prix Fourneyron. The mathematician Carlo Bourlet won it, and the French mathematician Emmanuel Carvallo produced a detailed analysis of his own.

Carvallo’s analysis was one of the first two predictions that a bicycle could balance itself. The other came from Francis Whipple, a Cambridge undergraduate. In 1899, apparently unaware of Carvallo, he wrote down the equations of motion for a simplified bicycle made of four rigid parts (the rear frame with its rider, the front fork and handlebars, and two wheels) joined by hinges. He worked out that an uncontrolled bicycle could be stable within a certain range of speeds, and he identified the two ways it can go wrong. One is a quick, growing side-to-side oscillation at low speed, now called “weave.” The other is a slow topple at high speed, now called “capsize.” Whipple went on to a career in meteorology. His equations were, as later work confirmed, essentially correct. But they were long, and few people checked them.

Two tidy explanations, both incomplete

Into that vacuum came two simple explanations that are still repeated today. The first is the gyroscope. A spinning wheel resists being tilted, and when it is tilted it tends to steer. Lean a spinning front wheel to the left, and gyroscopic precession swings it to the left. That is just what a rider needs, because steering toward a lean drives the wheels back under the rider. An influential 1910 book on gyroscopes by Felix Klein and Arnold Sommerfeld included a chapter on bicycles, written by Fritz Noether, the brother of the mathematician Emmy Noether, and it gave gyroscopic effects a starring role.

The second explanation is “trail,” a word from the geometry of the front fork. Draw a line down the steering axis to the ground. On almost every bicycle, the point where the front tire actually touches the ground sits a few centimeters behind where that line lands. That gap is the trail. Like the wheels on a shopping cart, which swivel to follow behind their pivots, a front wheel with trail tends to swing into line with the direction of travel. And when the bike leans, the trail geometry helps twist the front wheel toward the lean.

Both effects are real. The problem was the assumption that one or the other, or the two together, was the whole story.

The chemist who built unridable bicycles

David Jones was a chemist at the British firm ICI and a writer better known as “Daedalus,” the pen name for a long-running column of wildly inventive ideas in New Scientist and later Nature. In the late 1960s he decided to test the bicycle theories the direct way. He would build bicycles designed to break them.

His first “unridable bicycle,” URB I, carried a second front wheel spinning in the opposite direction, which canceled the gyroscopic effect of the real one. If gyroscopes kept bicycles up, it should have been impossible to ride. It was easy. Jones found it handled almost like a normal bike, though riding no-hands was close to impossible. But when he pushed it off with no rider, it collapsed quickly. Something about the gyroscope seemed to matter for the bike’s stability on its own, even though it barely mattered to a rider.

So he turned to trail. URB II had a tiny castor wheel that got so hot it nearly glowed, and the test was inconclusive. URB III had its front forks turned around backward, so the trail grew much larger. Pushed off riderless, it was amazingly stable. With Jones aboard, it was so stable that it was awkward to steer. Jones had noticed that if you wheel a bike along by the saddle and tilt it, the front wheel steers into the lean, even when the bike is standing still. His explanation was that turning the wheel into the lean lets the whole bike sink slightly lower, so gravity pulls the steering that way. To test the idea, he wrote a computer program to calculate how the height of the front end changes with lean and steering, and he plotted a range of designs, from penny-farthings to a Rudge of 1887. Ordinary bikes, he concluded, sat close to the edge of instability, which is what makes them responsive. Along the way, he tested a riderless bike weighed down with 30 pounds of concrete slabs, sending it hurtling around an empty parking lot.

Then he built URB IV, moving the front wheel about 4 inches ahead of its normal position. That removed its trail and, by his calculations, put it well into the unstable region. It was, he reported, “very dodgy to ride,” though not as impossible as he had hoped, and when released riderless at speed it crashed to the ground. He called it “a machine of absolutely no utility whatsoever,” adding that this set him “firmly in the mainstream of modern technology.” His April 1970 article in Physics Today, “The stability of the bicycle,” became a classic. His verdict was that trail and steering geometry mattered most, and that emphasis was widely repeated for decades. Later researchers would find his static, energy-based reasoning incomplete for a machine in motion, but his experiments had already shown that gyroscopes alone could not be the answer.

A century of equations, checked at last

The person who eventually reopened the case was Jim Papadopoulos, who began working on bicycle dynamics at Cornell University in 1985 with the engineer Andy Ruina and Scott Hand. Combing through the old literature, Papadopoulos found that papers over the previous century gave conflicting equations for the same simple bicycle. Some agreed with Whipple and some did not. A theory that cannot agree with itself cannot settle anything.

In 2007, Papadopoulos and Ruina, together with Jaap Meijaard and Arend Schwab in the Netherlands, published a careful benchmark in Proceedings of the Royal Society A. They derived the linearized equations of Whipple’s bicycle, verified them by two different methods, and published a reference bike with every number spelled out, 25 parameters in all, so anyone could check their own work. Their benchmark bike was self-stable between about 4.3 and 6.0 meters per second, roughly 10 to 13 miles per hour. Slower than that, it weaved. Faster, it slowly capsized. That high-speed fall takes seconds to develop, and a rider barely notices correcting it.

The bicycle with no gyro and no trail

With trustworthy equations in hand, the team could ask the old question directly. Is either gyroscopes or trail actually necessary? The equations said no. So they built a bike to prove it. Jodi Kooijman and Arend Schwab at Delft University of Technology, working with Ruina, Papadopoulos and Meijaard, built a small, odd-looking machine they called the two-mass-skate bicycle, or TMS. Its wheels were barely bigger than roller-skate wheels, too small for real gyroscopic effects. To erase even those, each wheel had a second disk spinning the opposite way. Its geometry gave it a slightly negative trail of about 4 millimeters, so the front wheel touched down a hair ahead of the steering axis, not behind it.

By the old explanations, it should have fallen over at once. Instead, when they launched it riderless at above about 2.3 meters per second (about 5 miles per hour), it rolled along upright. When they knocked it sideways, it steered itself out of the fall and recovered. It finally toppled only when it slowed below about 2 meters per second. “Although the experimental TMS bicycle looks like a folding scooter, it is still a bicycle,” they wrote in their paper, which appeared in Science on April 15, 2011: two wheels, two frames and three hinges.

The secret was where they put the weight. On the TMS, the front assembly (the fork, handlebar and front wheel) has its center of mass placed ahead of the steering axis and set low. When the bike starts to lean, that front mass tends to fall faster than the rest of the frame, and because it sits ahead of the steering axis, its fall swings the handlebars into the lean. The bike turns toward the side it is falling, its wheels run back under its center of mass, and it straightens up. That is exactly what Drais and Rankine said a rider does, only now the machine was doing it with no rider at all.

The researchers’ conclusion was more interesting than “gyroscopes are wrong” or “trail is wrong.” There is no single cause. Gyroscopic effects, trail and the placement of mass all feed into how the front end steers when the bike leans. A bike can be stable with all of them, or with only some of them, depending on how they are combined. They found one condition every self-stable bike seems to share: it must turn into a fall. And they noted that almost any self-stable bicycle can be made unstable by mis-adjusting only the trail, only the front-wheel gyro or only the front-assembly center-of-mass position. Their paper ended on a hopeful note: “The evolutionary process that has led to common present bicycle designs might not yet have explored potentially useful regions in design space.” In other words, there may be good bicycles nobody has tried building yet.

What you actually do when you ride

A rider on a bicycle is balancing an upside-down pendulum, a bit like balancing a broomstick on your palm. You keep a broomstick up by moving your hand under it. You keep a bicycle up by steering the wheels under you. If the bike starts to fall to the left, you turn slightly left, the contact points sweep left, and you are upright again. Most of this is unconscious, a constant stream of tiny steering corrections that, added up, trace the gentle snaking path visible in a bike’s tire tracks on wet pavement. Lock the handlebars so they cannot turn and a bicycle becomes almost impossible to balance, even at speed. That is why riding is easier fast than slow. At higher speeds, a small steering correction moves the wheels under you more quickly.

The bike’s own self-stability helps by doing some of that steering for you, which is why you can ride no-hands. Shift your hips and the bike leans, and the front end turns into the lean on its own.

Turning shows the same logic in a sneakier way. To turn right, a rider must first lean right, and the quickest way to start a lean to the right is to steer briefly to the left. This is called countersteering, and almost every rider does it without knowing. Motorcycle schools teach it explicitly, because at highway speed a deliberate push on the right handgrip, which steers the front wheel left for an instant, is the fastest way to make the bike swerve right. Once the bike is leaned, the steering swings back toward the turn, and the lean angle settles where gravity and the turn balance each other. Faster speeds and tighter curves mean steeper leans.

Why it still matters

It is easy to think that a question about bicycles was settled long ago. But the work since 2007 has real uses. Bicycle and motorcycle designers can now model how a new frame will handle before building it, instead of relying entirely on trial and error. Engineers use extended versions of these models, with tire behavior added, to study the dangerous high-speed wobbles that occasionally shake motorcycles and bicycles, which echo another kind of runaway oscillation we covered in the Galloping Gertie bridge collapse. Bikes for children, cargo bikes, folding bikes and recumbents all differ from the classic diamond frame, and each changes the balance of gyroscopes, trail and mass.

It has also changed how many children learn to ride. Training wheels keep a bike from leaning, which means they also stop it from steering into a lean, the exact skill a rider needs. The pedal-less balance bikes now common in parks are, in effect, Karl Drais’s running machine shrunk to toddler size. Kids push along with their feet, lift them, and discover that small steering corrections keep them up. When they move to pedals, the balance is already there.

Try it yourself

You can see the front end at work with any ordinary bicycle. Hold it upright by the saddle, then tilt it gently to one side. The handlebars will swing toward the lean on their own, just as Jones noticed. That is the front end steering into the fall, the same tendency that helps a rolling bike catch itself. Next time you ride, in an empty parking lot, at a steady speed, try a slight push forward on the left grip and notice which way the bike goes. It leans and turns left, the counterintuitive start of every curve you have ever ridden.

Like Euler’s Disk and the tippe top, the bicycle is a reminder that the most familiar machines can still hold surprises. Karl Drais built his running machine in 1817, and it took until 2011 for anyone to show clearly how a bicycle keeps itself up. Somewhere in that long gap, millions of people learned to ride without needing to know.

Further reading