Bicycle physics is a broad and complex subject, perhaps more so than one can imagine. Although the number of components of a bicycle is small, the interaction between them and the dynamic principles involved, is complicated. This is especially true with regards to bicycle stability, which is the result of a complex dynamic interaction within the bike-rider system.

On this page I will explain some of the main aspects of the physics of bicycles, which should give the reader a greater appreciation of how bicycles work, from a physics perspective.

**Bicycle Physics – Stability**

Bicycles are inherently stable when riding. Even riderless bicycles are stable if given enough forward velocity. Much effort has gone into analyzing the factors which make a bicycle stable. It has been determined that “trail” (shown below) is often an important contributor to bicycle stability. For the traditional bicycle design, if trail is positive, meaning the projection of the steering axis with the ground is in front of the contact point of front wheel and ground, then the bicycle is more stable when riding (i.e. it’s less likely to fall over when riding it). If this projection is behind the contact point (negative trail) then the bicycle is less stable and the bicycle is more likely to fall down when riding it.

Based on the geometric parameters shown, the mathematical formula for trail is:

where *R _{w}* is the wheel radius,

*A*is the head angle as shown, and

_{h}*O*is the rake, as shown, also known as the fork offset.

_{f}When analyzing bicycle stability it is common to use two parameters; the lean angle and steering angle of the bike. The lean angle is the left and right angle the bike frame makes with a vertical plane, and the steering angle is the angle the front wheel makes with the plane of the bike (containing the bike frame). The figure below illustrates the lean and steering angle.

where *θ* is the lean angle and *α* is the steering angle. The sign convention for these angles is typically as follows, and is with respect to a rider sitting on the bike: Right lean is positive *θ* and left lean is negative *θ*. Right steer is positive *α* and left steer is negative *α*. For stability analysis both of these angles are the only independent variables needed to mathematically analyze bicycle stability. They completely describe the orientation of a bicycle as it travels in the forward direction. For a bicycle to be stable the lean and steering angle must have a tendency to “die out”, which means that these angles will fluctuate around zero with small positive and negative values. This in turn means that the bicycle tends to stay upright with little turning, while moving in the forward direction. It is interesting that locking the front steering will always result in a bicycle falling over. The physics of stability requires that the front wheel can freely steer.

As mentioned, analyzing bicycle stability