It is rare that most people appreciate the bicycle, but it is quite an extraordinary machine. Push a riderless bike, letting it roll freely at high enough speeds, and it can withstand pushes from the side – it will wobble a little, but quickly recover. In the conventional analysis, that is because the gyroscopic force of the front wheel, its mass and the spontaneous turn of the handlebars all act together to keep the bicycle rolling forwards. This has something to do with the gyroscopic effect, the force that keeps a spinning top upright. You can feel this by removing a wheel from your pushbike and spinning it while you hold the axle spindles. If you try to change the orientation of the wheel, you’ll feel it push back against you.

The first mathematical analysis of bicycles suggested that this is also what keeps a moving bike on its wheels. But although the equations were written down in 1910, physicists always had nagging doubts about whether this was the whole story.

The most definitive analysis came exactly a century later. It involved an experimental bicycle that had all its gyroscopic effects cancelled out by a system of counter-rotating wheels. The effort of building such a strange contraption was worth it: the resulting paper was published the prestigious journal

*Science*.

The publication plunged bicycle dynamics back into chaos. It turns out that taking into account the angles of the headset and the forks, the distribution of weight and the handlebar turn, the gyroscopic effects are not enough to keep a bike upright after all. What does? We simply don’t know. Forget mysterious dark matter and the inexplicable accelerating expansion of the universe; the bicycle represents a far more embarrassing hole in the accomplishments of physics.

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A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects

**A riderless bicycle can automatically steer itself so as to recover from falls. The common view is that this self-steering is caused by gyroscopic precession of the front wheel, or by the wheel contact trailing like a caster behind the steer axis. We show that neither effect is necessary for self-stability. Using linearized stability calculations as a guide, we built a bicycle with extra counter-rotating wheels (canceling the wheel spin angular momentum) and with its front-wheel ground-contact forward of the steer axis (making the trailing distance negative). When laterally disturbed from rolling straight, this bicycle automatically recovers to upright travel. Our results show that various design variables, like the front mass location and the steer axis tilt, contribute to stability in complex interacting ways.**

A bicycle and rider in forward motion balance by steering toward a fall, which brings the wheels back under the rider [supporting online material (SOM) text S1 and S2] (

__). Normally, riders turn the handlebars with their hands to steer for balance. With hands off the handlebars, body-leaning relative to the bicycle frame can also cause appropriate steering. Amazingly, many moving bicycles with no rider can steer themselves so as to balance—likewise with a rigid rider whose hands are off the handlebars. For example, in 1876, Spencer (____1____,____2____) noted that one could ride a bicycle while lying on the seat with hands off, and the film____3__*Jour de Fête*by Jacques Tati, 1949, features a riderless bicycle self-balancing for long distances. Suspecting that bicycle rideability, with rider control, is correlated with self-stability of the passive bicycle, much theoretical research has focused on this bicycle self-stability.The first analytic predictions of bicycle self-stability were presented independently by French mathematician Emmanuel Carvallo (1897) (
, where

__) and Cambridge undergraduate Francis Whipple (1899) (____4____,____3____). In their models and in this paper, a bicycle is defined as a three-dimensional mechanism (____5____Fig. 1A__) made up of four rigid objects (the rear frame with rider body B, the handlebar assembly H, and two rolling wheels R and F) connected by three hinges. The more complete Whipple version has 25 geometry and mass parameters. Assuming small lean and steer angles, linear and angular momentum balance—as constrained by the hinges and rolling contact—lead to a pair of coupled second-order linear differential equations for leaning and steering (SOM text S3) (__). Solutions of these equations show that after small perturbations, the motions of a bicycle may exponentially decay in time to upright straight-ahead motion (asymptotic stability). This stability typically can occur at forward speeds____6__*v*near to*g*is gravity and*L*is a characteristic length (about 1 m for a modern bicycle). Limitations in the model include assumed linearity and the neglect of motions associated with tire and frame deformation, tire slip, and play and friction in the hinges. Nonetheless, modern experiments have demonstrated the accuracy of the Whipple model for a real bicycle without a rider (__).____7__