Bicycle and Motorcycle Dynamics - Longitudinal Dynamics - Stability

Stability

Though longitudinally stable when stationary, a bike may become longitudinally unstable under sufficient acceleration or deceleration, and Euler's second law can be used to analyze the ground reaction forces generated. For example, the normal (vertical) ground reaction forces at the wheels for a bike with a wheelbase and a center of mass at height and at a distance in front of the rear wheel hub, and for simplicity, with both wheels locked, can be expressed as:

for the rear wheel and for the front wheel.

The frictional (horizontal) forces are simply

for the rear wheel and for the front wheel,

where is the coefficient of friction, is the total mass of the bike and rider, and is the acceleration of gravity. Therefore, if

which occurs if the center of mass is anywhere above or in front of a line extending back from the front wheel contact patch and inclined at the angle

above the horizontal, then the normal force of the rear wheel will be zero (at which point the equation no longer applies) and the bike will begin to flip or loop forward over the front wheel.

On the other hand, if the center of mass height is behind or below the line, as is true, for example on most tandem bicycles or long-wheel-base recumbent bicycles, then, even if the coefficient of friction is 1.0, it is impossible for the front wheel to generate enough braking force to flip the bike. It will skid instead, unless it hits some fixed obstacle, such as a curb.

Similarly, powerful motorcycles can generate enough torque at the rear wheel to lift the front wheel off the ground in a maneuver called a wheelie. A line similar to the one described above to analyze braking performance can be drawn from the rear wheel contact patch to predict if a wheelie is possible given the available friction, the center of mass location, and sufficient power. This can also happen on bicycles, although there is much less power available, if the center of mass is back or up far enough or the rider lurches back when applying power to the pedals.

Of course, the angle of the terrain can influence all of the calculations above. All else remaining equal, the risk of pitching over the front end is reduced when riding up hill and increased when riding down hill. The possibility of performing a wheelie increases when riding up hill, and is a major factor in motorcycle hillclimbing competitions.