Motorcycle steering geometry

I find the physics of steering a motorcycle to be enlightening and thought provoking. In abstract, motorcycle steering is not so different than bicycle steering. But in practice and when applied correctly, the knowledge of these steering principles can be powerful to enhance an operator's abilities and control.

A motorcycle and its connection to the world

I begin by finding all the essential variables of the system. Here are some point and vector definitions which trace the bike's frame from the ground up.

wheel contact patch \( P \)
Where the rubber meets the road.
wheel lean pivot point \( V \)
The point (in the tire) which has a fixed height at every lean angle from left to right. The point stays directly above the contact patch on a level surface.
center of wheel rotation \( C \)
In the front wheel's axis of spin and symmetrically at the middle of the wheel.
steering pivot point \( S \)
In the center of the steering column at the center of the circle traced by the wheel's center when pivoting the wheel's assembly on the steering column while holding the other wheel's assembly fixed.
minor wheel lean \( \vec{r} = V - P \)
Points upwards from \( P \) to \( V \). Normal to the contact patch.
major wheel lean \( \vec{w} = C - V \)
Points upwards from \( V \) to \( C \).
wheel turn arm \( \vec{t} = S - C \)
Points inwards from \( C \) to \( S \).

The front and back wheels have similar points defined called \( P_f \), \( V_f \), \( C_f \) and \( S_f \), and \( P_b \), \( V_b \), \( C_b \) and \( S_b \), respectively. Also, the front and back wheels have similar vectors defined called \( \vec{r}_{f} \), \( \vec{w}_{f} \) and \( \vec{t}_{f} \), and \( \vec{r}_{b} \), \( \vec{w}_{b} \) and \( \vec{t}_{b} \), respectively.

Some more vector definitions:

contact patch vector \( \vec{p}_f \) and \( \vec{p}_b \)
Points forward tangent to the rolling path of the wheel.
patch-to-patch vector \( \vec{p}_s \)
Points backward from the front to the back wheel patches.
axle axis \( \vec{x}_f \) and \( \vec{x}_b \)
Points left (of the driver) parallel to the wheel's axis of rotation.
steering column axis \( \vec{x}_s = S_b - S_f \)
Points upwards from \( S_f \) to \( S_b \) to finish the frame-tracing path.


Note that any vector parallel to the wheel's axle is also perpendicular to the plane of rotation of any point on the wheel. The wheel's contact patch vector, the wheel's major lean vector, and the wheel's axle axis are all mutually perpendicular. For example, when I consider the back tire, I observe any one of the following relationships, equivalently:

\[ \hat{p}_{b} = \hat{x}_{b} \times \hat{w}_{b} \]
\[ \hat{x}_{b} = \hat{w}_{b} \times \hat{p}_{b} \]
\[ \hat{w}_{b} = \hat{p}_{b} \times \hat{x}_{b} \]

In what appears to be an improvement, I can avoid the length calculations, yet still ignore any constant multipliers such as variability in length.

\[ \vec{p}_{f} \times \vec{x}_{f} \times \vec{w}_{f} = \vec{0} \]
\[ \vec{p}_{b} \times \vec{x}_{b} \times \vec{w}_{b} = \vec{0} \]

Next, I can close the frame-tracing path with the patch-to-patch vector to find another relationship.

\[ \vec{r}_{f} + \vec{w}_{f} + \vec{t}_{f} + \vec{x}_{s} = \vec{t}_{b} + \vec{w}_{b} + \vec{r}_{b} + \vec{p}_{s} \]


I want to illustrate what I'm talking about, so I can bring some life to this discussion.