Oh, reading further in that link, "One effect of turning the front wheel is a roll moment caused by gyroscopic precession. The magnitude of this moment is proportional to the moment of inertia of the front wheel, its spin rate (forward motion), the rate that the rider turns the front wheel by applying a torque to the handlebars, and the cosine of the angle between the steering axis and the vertical.".
Note the term "of turning the front wheel is a roll moment caused by gyroscopic precession."
As the link shows, you can countersteer at speeds slow enough so that gyroscopic effects are minimal. cjl is correct that the main effect is to move the front tire's contact patch out from under the COM, which causes the bike to tip in. There is a small gyroscopic effect as well at higher speeds, but only adds a bit in my experience.
Let's consider this intuitively. The classic presentation of the gyroscopic effect is a guy holding a spinning bicycle tire, sitting on a rotating chair. Compare the weigh of this bicycle tire and rotational velocity to the weight and rotational velocity of a motorcycle wheel at 40 mph. The bicycle pales by comparison. It can hardly be said that the precession of that motorcycle wheel is negligible. Clearly it is not.
And, the phenomena does not occur at low speed. There is a crossover speed at which the steering changes from as expected to counter steering. As speed increases, the effect becomes more pronounced. What has changed? What has changed is the rotational velocity of the wheel.
Do you ride? What do you ride?
The precession only exists while a torque is applied. Once the steering is completed, initially outwards to induce a lean, the precession stops since it's a reaction to torque, not angle.
There are two primary speed ranges, one below the speed of self-stability, one in the range of self-stability. For a mathematical model using razor thin tires, there is a third speed range (capsize speed), above which a bike would tend to fall inwards at an extremely slow rate, but with real tires, this speed range doesn't exist or only exists well beyond any speed that a bike could achieve.
Counter steering is in effect at all speeds. At slow speeds, it's much less noticeable as the steering input is very light. It's easier to see this effect in the case of unicycles, which don't have the self correction related to trail on a two wheeled bike.
"Counter steering is in effect at all speeds."
Yes. It is one of a number of effects that vary with things like wheel weight and radius, the bike's weight, angle of the front forks, distance from front to back wheels, and .... things I haven't considered. Some of these are in opposition to the counter steering affect, what we might call "normal steering.". It is apparent that the counter steering affect decreases as speed decreases, becoming negligible. My impression, from riding around a parking lot, is that at 20-25 mph, "normal steering" is predominant.
It begs the question if why is there no apparent speed at which counter and normal steering component simple cancel out and make the bike simply unsteerable?
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