Simon Brooke writes:

Take the wheel out, spin it in you hands and try to tilt it to
the left or right and note the force of the steering action.
This should convince you of its effect.

please explain. gyro reaction is _90 degrees to the applied
force_. i.e. my front wheel, spinning "forwards" tries to tilt top
rightwards when turned to the left. you seem to be implying that
gyro recation is responsible for banking the bike to the left when
steered left.

No, when you wheel the bike along holding onto the seat you steer
by banking the bike and the turn is a reaction to the bank.
Banking the bike to the left will tend to initiate a turn to the
left. But the gyroscopic force is quite small when the wheel is
spun at only walking speed and the geometry of the bike also
results in the wheel turning left in response to a left bank (even
when the wheel isn't rotating).

Well, just so. I agree that you can get strong gyroscopic effects
with a fast rotating wheel, but I'm completely unpersuaded that they
are significant at walking speed. So, again, has anyone done the
maths?

There are no "maths", math is an aggregate concept and has no plural...
except in GB maybe.

I guess you don't have a bicycle with a QR front wheel so you can't
perform the simple experiment of turning the wheel at "walking speed"
to fell the strong gyroscopic effect caused by tilting the wheel
manually to the left and right.

Jobst Brandt