Posts Notes Resume

Bicycle Dynamics I: The Setup, Terminology, and Notation

13 May 2014

This is the first in a series of posts on bikes and their dynamics. The topic often comes to mind while cruising around on my bike, so it’ll be fun to hash them out thoroughly on paper.

I know the topic has been studied to death, but even so beliefs about bikes based on misguided intuition are pervasive. Maybe adding one more source of facts will be the tipping point!

This first post will just setup terminology, notation, and other logistics. It will serve as a good accompanying browser tab for subsequent bike posts.

Your browser does not support SVG images!

Normally you start with back of the envelope calculations when solving a problem like this - start simple and move to more complex only if the first order model doesn’t answer your questions. I know I’m going to go farther regardless (for fun!), so I’ll start with a general bike model and simplify things down later.

The red-green-blue triads each indicate a right handed coordinate system with directions \(1\), \(2\), and \(3\)) corresponding to red, green, and blue, respectively. The \( \hat e_i \) coordinate system is inertial, while the rest are attached at different points on the bike. The other coordinate systems are:

The coordinate systems are chosen so that a simple rotation (rotation about a single of the coordinate axes by an angle) is enough to walk through the whole chain of coordinate systems. In other words:

The two additional angles, \(\theta_r\) and \(\theta_f\) indicate the rotation angle of each of the two tires in the \(\hat a_i\) and \(\hat c_i\) frames, respectively. We could attach another coordinate system to each of the tires that these angles rotate to, but it is unnecessary (other than if we wanted to preserve convention).

For now the human will just be represented as a point mass somewhere above the bike.

In the next post, the rest of the unknown dimensions in the bike diagram will be specified and I’ll write down the kinematic equations for the system.