bike stability

The physics fraternity has for years tried to explain the stability of the bicycle. The process of peer review has not weeded out the grain from the chaff but instead left behind several citeable articles to act as proof for incorrect ideas. If you doubt this I suggest you Google the subject and look at the answers that are given. Gyroscopic force is one of the leading explanations and is a red herring. From this springs ideas such as countersteering and probably the study of other ideas which only bring us farther from understanding what is going on. David Jones experimented with the bicycle and demonstrated that gyroscopic forces were not needed. He then dismissed the actual mechanism as nasty variable friction forces and proceeded with a deus ex machina conclusion involving the change in the height of the center of mass. While one needs to understand a little physics to explain why a bicycle tends to remain upright in motion, it does not involve calculus. More recently others, have produced a pushable vehicle which remains upright for the most part and has no angular momentum. Of course, the same can be said of a ski bike which predates their gadget by at least fifty years. Truth is their gadget is simply better optimized for a riderless vehicle with no angular momentum. As pointed out by Lowell and Mckell, without angular momentum the trail of the front wheel need not be as large.
Wilson-Jones tackled the subject of motorcycle stability in 1951 and concluded that the machine remained upright by steering in the direction of lean. The question he tried to answer was then; “what is this force that steers the front wheel?” Wrong question. This is a case of misdirection which may be why so many wrong answers exist. The front wheel is free to pivot about the “steering axis” so we assume that the bicycle is steered by the front wheel. A car is steered in this manner. A pneumatic tire will generate a side force if its plane is not aligned with its direction of travel so when the steering wheel of a car is turned, a steering moment is generated that rotates the car about its vertical axis. This is an active steering that is different from the passive nature of a single track vehicle traveling in a straight line.
A better understanding of the powerful forces that keep a bike upright while moving can be gained by studying an inverted pendulum restrained to a plane. Visualize a mass being held up by a rigid column free to rotate about its base. As the pendulum tips off center, gravity creates a moment about the base that will send the mass falling down through an arc about the base. If, however, the base is on a frictionless surface, the motion will be the column sliding to one side while the mass free falls to the ground. If a bicycle is not moving, it will fall to one side with reaction forces at the ground preventing side motion. The side force is proprtional to the tilt angle and the weight of the bike. This camber force is also present when the bike is moving. The difference between the static condition and the dynamic one is the reason it is safe to ride a motorcycle at insane speeds. When a bike tips to one side while moving the camber forces generated push the bike in that direction. In fact the bikes tipping will become simply side motion. Leaning a bike directs the tire forces to push the bike in that direction bringing the base back under the center of mass. Of course, the bike is not restrained to motion in one plane so additional forces are needed to steer the bike.
The pneumatic tire has the property of generating a side force when not aligned with the direction of travel. A ski will also have this property. The tire force will be proportional to the angle of steer for small angles with the constant relating the two called the slip angle coefficient. Consider a bike traveling along a roadway and starting to tip. It will have a forward velocity aligned with the contact points of the front and rear tires. There will now also be a velocity to the side owing to the motion of tipping. The combination of these two motions is a new vector at some small angle to the old straight ahead position. The front wheel has a degree of freedom the rear does not have. It can pivot and the trail causes the wheel to align with the direction of motion. The rear wheel has much greater trail and so reacts much more slowly. While slip angle is minimized at the front wheel by its ability to rotate about the steering axis, the rear wheel only slowly adjusts so a much larger slip angle occurs at the back wheel. The rear wheel side force creates a steering moment that rotates the bike about its vertical axis and the center of mass quickly moves over the line defined by the two contact points of the tires. The bike then may overshoot and begin the process in the opposite direction although the action will be damped by a large visco-elastic mass perched on the saddle. In short, the bike tips to one side, side forces occur at the tires to create a positive moment to steer the bike over its centerline.
Copyright © 2012 Calvin Hulburt. All Rights Reserved.