Just remembered I wrote something about steering a while ago:
Motorcycle Dynamics:
There's been lots of controversy and dim-witted flame wars on rec.motorcycles concerning how a motorcycle steers. I myself have given this lots of thought, done many calculations, tried some experiments, and here in layman's terms is what I think.
When you initiate a turn by pressing the bars in the opposite direction (i.e. countersteering), the bike responds by leaning. This is due to the `outtracking' effect. Basically, you steer the bottom part of the bike out from under the centre of mass.
As the bike leans, gyroscopic forces cause the forks to turn into the proper direction. You don't notice this because the handlebar movement is very small. All you notice is that a constant
pressure to the left causes a lean to the right and subsequently a turn to the right. In effect, the handlebars end up turning in the direction opposite to that in which you are pushing them. But this depends upon the spinning of the front wheel. That's why it doesn't happen in the same way when driving very slowly. You can verify this by rigging up some kind of long pointer from the handlebar centre. Others have done this. The gyroscopic effect can be verified by playing with spinning bicycle wheels.
Rake and Trail:
To understand the importance of motorcycle/bicycle steering geometry, a good place to begin is with a simple caster wheel as found for example on grocery carts. This is a system that has trail but no rake. (Trail is the distance that the point of contact of the tire with the road trails the intersection of the steering axis with the road.) Basically, the wheel pivots to the necessary direction to follow the direction of motion. A similar system does not work on a single-track vehicle because such a vehicle leans when it corners. Try the following. Take such a wheel (say from an office chair), and lean it to one side, pushing it down to mimic the effect of the vehicle's weight. You will notice that it tries to flip outward, turning into the turn. In other words, if a motorcycle had such a geometry, you'd have to counter-steer very strongly just to keep it leaned over in a turn. Now orient the caster the 'wrong' way so that the trail is negative. Lean it and weight it. You'll notice that it tries to turn out of the turn. This suggests that steering would be neutral if trail were zero. This is correct, but zero trail is not an option since we want some of the self-stabilizing caster effect. The way around this is to tilt the steering axis back from vertical. The angle of tilt is called rake. Take a wheel on a steering axis with no trail. Now rake it backwards. Immediately you have trail equal to the tangent of the rake angle multiplied by the wheel's radius. It turns out that for neutral steering, this is too much trail. The optimum is about 2/3 this amount. On a motorcycle, this is achieved by moving the forks ahead of the steering axis. On a bicycle, the forks are curved forward.
Example: My motorcycle has a rake of 27.5 degrees, and a wheel radius of 13.5 inches. With no fork offset, the trail would be 13.5" x tan 27.5 = 7". A 2.5" offset reduces the trail to 4.5".