I see that modern bicycles have curved front wheel forks , how does that affect stability ?
You can adjust the trail this way.
Its (very!) complicated and I don't think there . Have a look at the Nature website, they currently have an interview on there with a guy who has done a lot of research into bicycle stability.
Curved forks reduce, (but not eliminate) trail (caster effect). This makes steering a bit easier, but increases the minimum speed for self stability. Motorcycles use a similar strategy by using a triple clamp that offsets the front forks in front of the steering pivot axis. In one case, the Honda CBR900RR of the 1990's had more offset than most riders liked, and it contributed to speed wobble. The offset was reduced in 1998:
Regarding gyroscopic effect, precession is a reaction to a torque. If a bike is leaning but in a coordinated turn, there's no torque about the roll axis, and no gyroscopic related steering (zero correction). In general, while the lean is increasing, gyroscopic effect tends to under-correct, and while the lean is decreasing during recovery, gyroscopic effect opposes the corrective steering, so gyroscopic effect mostly acts as a damper to the trail related corrective steering. In the video from post #2, there's a comment about "not too fast": this is a reference to "capsize" speed, a high speed where the gyroscopic effects dominate, resulting in an over damped system, resulting in a bike that tends to lean inwards at an extremely slow rate (almost zero rate of lean). To a motorcycle racer at high speed, the perception is that the bike tends to hold it's current lean angle, which may be a combination of the extremely slow lean rate and the small outwards roll torque related to the tires contact patch with the pavement being off center.
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