Ooo, another physics discussion
Gyroscopic forces work a little differently than most people think. When a bike/motorcycle begins to tip to the side (roll, ignoring steering here), the gyro torque actually causes a yawing torque on the bike which twists it into the corner, much like steering on a car. At the same time, if you were to yaw the bike through steering, the gyro effect exerts a roll torque that tips you into the corner. This is probably why when riding straight, you don't have to steer left to tip the bike to the right (at least not on my friend's Harley
), then steer again to the right to make a right turn. Just steer right and it rolls into the turn all by itself, no countersteering required at all. I'm very far from being anything remotely close to an experienced rider, but this is what my experience was on that one bike. And when it came time to straighten up the bike again I could just steer it straight without giving a quick input to the right in an attempt to kick the bike upright or do some fancy weight shifting with my behind. Without the gyro effect I suspect this wouldn't work so well.
In a car the effect is similar. When you steer the wheels or the car is rotating to the right in a turn, the gyro effect causes the car to roll into the corner slightly, reducing weight transfer from what the regular steady state calculation would say it should be. The quicker you steer or the quicker the car is rotating, the greater the roll torque (it must be so to conserve angular momentum). This is generally an understeer tendency and therefore a stabilizing effect.
I do gyro in my sim and although I haven't tried it without it in the full sized cars in a very long time, the difference with and without gyro in Virtual RC Racing is very noticeable. Enough to where you'd go back to the setup screen and tweak the car a bit to get the balance back how you had it before. Maybe it's not so much on a 1:1 scale car, I don't know
A few years ago I did some calculations for a motorcycle to see if the effect would be significant for upright stability. The resulting torques were so large I thought I'd made a mistake and rechecked it several times, but it was quite significant even at highway speeds. On a bicycle it would be much lower of course since the moment of inertia of the tires is much lower, as are the rotational velocities.
There is more to the picture of stability of course, such as the other effects mentioned earlier by others (fork angle, etc.), but this is how the gyro effect by itself influences things.
Gregor Veble tried a motorcycle in his engine a few years back. If I recall correctly (this was a long time ago) he found he could make the bike work and be pretty stable without the gyro effect, but gyro helped it quite a bit. I'll have to ask him about it again when he's online next.