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.
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 ˆei coordinate system is inertial, while the rest are attached at different points on the bike. The other coordinate systems are:
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ˆli - Attached at the rear wheel’s contact point with the ground. ˆl1 is tangent to the rolling direction of the rear wheel, and ˆl2 is vertical.
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ˆai - Attached at the rear axle. Both the ˆa1 and ˆa2 directions lie in the plane of the bike frame, and the ˆa1 lies in the line connecting the bike’s rear and front axles.
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ˆmi - Attached at the rear axle. The ˆm1 direction lies in the plane of the bike frame and lies in the line connecting the bike’s rear and front axle (like ˆa1). The direction ˆm2 lies in the plane that includes the rider’s center of mass (denoted by the circle with white and black quadrants). This accounts for the rider’s lean.
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ˆbi - Attached at the frame to headset joint, both ˆb1 and ˆb2 lie in the plane of the bike frame but ˆb2 is inclined to align with the headset tube.
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ˆci - Attached at the front axle with ˆc1 and ˆc2 in the plane of the front wheel. ˆc3 is inclined so it aligns with the headset tube.
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:
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Rotating about ˆl1 by θl brings you to the ˆai coordinate system
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Rotating about ˆa1 by θm brings you to the ˆmi coordinate system.
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Rotating about ˆa3 by angle θh brings you to the ˆbi coordinate system
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Rotating about ˆb2 by angle θs brings you to the ˆci coordinate system
The two additional angles, θr and θf indicate the rotation angle of each of the two tires in the ˆai and ˆci 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.