This is a common misconception about why bikes stay balanced.
Generally the gyroscopic force caused by the spinning wheels is not enough to counteract other forces.
What actually keeps the bike upright is the steering. When you steer the bike slightly left (by pulling the left handlebar), the bike will lean to the right (because of the apparent force caused by inertia). Thus, if you are "falling" on the left side, steering left will bring you back upright.
You normally don't think of it in this terms, partially because once you learn riding a bike it all becomes automatic for you, and partially because the geometry of the bike which causes it to steer in the direction of a lean.
In any case, if you find a way to make it steer in the lean, then it will be self stable.
This is also why you can be stable on a bike at very low speed. The amount of steering becomes very apparent. You might be surprised how long you can stay balanced at extremely low speed if you just allow yourself to change direction.
Generally the gyroscopic force caused by the spinning wheels is not enough to counteract other forces.
What actually keeps the bike upright is the steering. When you steer the bike slightly left (by pulling the left handlebar), the bike will lean to the right (because of the apparent force caused by inertia). Thus, if you are "falling" on the left side, steering left will bring you back upright.
(This technique can be also used explicitly, see http://en.wikipedia.org/wiki/Countersteering)
You normally don't think of it in this terms, partially because once you learn riding a bike it all becomes automatic for you, and partially because the geometry of the bike which causes it to steer in the direction of a lean.
This is accomplished by having a "positive trail". See: http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_geometry and http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics.
Although there is also a demonstration of bike self-stability which doesn't depend on the positive trail: http://en.wikipedia.org/wiki/Two-mass-skate_bicycle
In any case, if you find a way to make it steer in the lean, then it will be self stable.
This is also why you can be stable on a bike at very low speed. The amount of steering becomes very apparent. You might be surprised how long you can stay balanced at extremely low speed if you just allow yourself to change direction.