The reprint of David E. H. Jones’s article made me wonder how one can understand a bicycle’s stability without considering the effect of the forward velocity. Any bicyclist knows that the smaller the velocity, the more unstable it becomes. The novice cyclist learns very quickly to keep the bicycle upright by steering in the direction of the lean to restore the balance. The result of this action is to convert the unbalanced force threatening to topple the bicycle into the centripetal force acting on a mass moving in a circular path. By equating torques, we find that R = v 2/g sin(A), where R is the radius of the path, v the velocity of the bicycle, g the acceleration due to gravity, and A the angle of lean. For a lean of 0.1 radians and a velocity of 4 m/s, a radius of 16 m will stabilize the lean. A smaller radius will reduce the lean. That this stratagem does not depend in any major way on the details of the bicycle’s design explains why many supposedly unrideable bicycles could be ridden after all. It also accounts for the greatly reduced stability at low velocity.
More interesting is the bicycle’s behavior sans rider or when the rider has no hands on the handle bars. Consider pushing a bicycle with one hand on the seat. Experience tells you that you can steer the bicycle by making it lean in the desired direction. The front wheel will pivot in that direction. A little thought will show that when the bicycle is made to lean, the force exerted by the ground on the front wheel no longer passes through the steering axis. That exerts a torque on the steering mechanism and turns the handle bars. Now the details of the design become significant. The torque is in the desired direction provided the front wheel contacts the ground behind the point of intersection of the extended steering axis and the ground, typically a distance of around 5 cm. For small angles of lean, the torque is proportional to the sine of that angle. A bicycle ridden “no-hands” is steered by shifting one’s weight to make it lean.
The riderless bicycle remains upright by the same sequence of events. If it starts to lean, the front wheel automatically steers in the direction of the lean. If the velocity is sufficiently large, the centrifugal force will reduce the lean, and the caster action will straighten out the steering. The resulting negative feedback keeps the bicycle upright.
