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So I was reading this article called Stability of a Bicycle by David E. H. Jones and in it he talked about how front projection affects the stability of a bicycle with a negative front projection making the bike more stable because when the bike rolls it might lean to the side and then the front wheel will turn into the direction of the lean, which would lower the center of gravity and that would stabilize the bike. Is that correct? I didn't really understand his explanation of how front projection affects stability. Can someone help me out?

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Jones refers to the horizontal distance between the front wheel axle and the steering axis as front projection. Front projection is the product of the fork offset (the displacement between the axle and steering axis, measured at angle perpendicular to the steering axis, not the ground) and the sine of the head angle (the angle between the ground and the steering axis). Fork offset is a measurement of the fork itself, front projection refers to the assembly of fork + frame.

The theory on negative front projection, as I understand it, is similar to the theory on trail.

Trail is the distance parallel to the ground between the steering axis and the center of the tire’s contact patch. With the fork pointed straight ahead, the contact area on most bikes will be behind the steering axis. This is because most bikes have a head angle between 0 and 90 degrees, and on most bikes, tan(head angle)*fork offset is less than the radius of the wheel. That is to say that for the same head angle, a larger front wheel and/or a smaller fork offset, will result in more trail. This pattern continues with negative fork offsets (I.e. the front axle being behind the steering axle, or mounting the fork backwards) as well.

Another measure that’s applied to steering geometries is mechanical trail. Mechanical trail is like ordinary trail except instead of measuring parallel to the ground, mechanical trail is the displacement perpendicular to the steering axis.

My understanding is that there are many theories on why a bicycle should be stable, but for me the easiest to visualize in terms of actively balancing, involves an inverted pendulum. If you’re tipping over to the left, you fix it by first pointing the handlebars towards the left, to bring the wheels under you. Likewise, if you were balancing a broomstick on your hand, and it was tipping to the left, you move your hand left to bring the fulcrum in line with its center of gravity.

The bike with negative front projection doesn’t get stable from turning into a lean simply because low center of gravity=stable; it gets stability because it has a greater tendency to turn in the direction of the lean, and turning in the direction of a lean is how you keep a bicycle upright. The backwards fork increases the tendency to turn towards a lean because the lever arm operating on the steering assembly (the mechanical trail) is greater.

Why does mechanical trail make a bike more stable? My understanding is that the friction force between the ground and the tire corrects the steering direction, and the more that force is offset from the steering axis, the more influence it has. When the bike is standing upright and rolling forward, mechanical trail tends to re-center the steering. Conversely, if you hold the bike upright and try to rotate the frame about the rear tire, because the front tire’s contact patch is behind the steering axis, it will rotate the steering assembly in the same direction. If you push the front of the frame to the left, you get a rightward force from the ground acting on the tire, and because it’s behind the steering axis, it creates a torque on the steering assembly turning the handlebars to the left. that translates into turning the front wheel into a lean when the bike is leaning, and helping it stay straight when the bike is upright.

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