How can precession produce a torque in the direction of precession when it has no angular momentum?

Question

Updated question, also includes somewhat of an answer.

To clarify the point about precession having no angular momentum, if the torque that induces precession is suddenly removed, the gyro suddenly stops precessing (so no angular momentum about the axis of precession). A video example of this where the gyro support is allowed to fall away:

side view: https://www.youtube.com/watch?v=XHUrPrIYLjg&t=116s

top view: https://www.youtube.com/watch?v=XHUrPrIYLjg&t=182s

As to the title's question, Wikipedia's article on countersteering and gyroscopic effect mentions a roll moment (torque): "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."

https://en.wikipedia.org/wiki/Countersteering#Gyroscopic_effects

Assuming this applies to gyro's in general, then for a gyro supported at the end of it's axis, with a torque about the gyro's roll axis, the resulting torque about the yaw axis (precession torque) is proportional to the rate of roll, and not the amount of roll torque. This is why if a gyro is just released, it has to drop a bit before the resulting yaw torque starts the precession. With some type of dampening, the gyro achieves a steady state, where there is no rotation about the roll axis, just a precession rotation about the yaw axis and no net torque about the yaw axis (the rate of precession remains constant in an ideal situation).

As for electric unicycles (EUC), the principle method of turning an EUC is to tilt it, which causes the tire to steer due to camber effect. A rider tilts an electric unicycle to steer, and leans their body inwards for balance. A rider learns to coordinate tilt and lean depending on turning radius, speed, and factors like tire profile. Precession related effects are small because the rate of tilt is not that fast, and the resulting yaw torque is opposed by the total angular inertia of EUC + rider.

As for speed related issues: at moderate speeds an electric unicycle can be tilted with just pedal pressure. At higher speeds, although the EUC is not tilted much, the angular momentum of motor + wheel + tire resists any change in tilt angle, and requires the rider to exert an inward force on the outside upper pad and an outwards force on the outside pedal, to generate enough inwards torque to force the EUC to tilt. Spiked pedals are used to keep feet from sliding on pedals. Example video of turns at 30 to 50 mph:

https://www.youtube.com/watch?v=L3aNqosYgG0&t=1170s

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Electric unicycles (EUC) - general information

EUCs use forwards | backwards balancing algorithm similar to Segways, typically using 3 axis gyro, a 3 axis accelerometer, and an algorithm to determine which way is up (some EUCs need to be initially calibrated for this). This allows the EUC to sense tilt in any direction, regardless of acceleration, wind, uphill, downhill, ... .

Tilting the shell forwards | backwards results in acceleration | deceleration, and there is a balancing algorithm to keep the rider from falling.

If the EUC detects excessive left or right tilt, it shuts off the motor, assuming that the EUC has been dropped. This creates a shut off issue if cornering on a banked track, and a few EUCS allow the shut off angle to be increased to avoid this.

Like any uni-track vehicle, an EUC has to be counter-steered outwards from under the rider so that the rider leans inwards for balance. Once leaned, counter-steering is used to control lean angle and balance, tilt inwards more to lean less, tilt inwards less to lean more.

Answer 1

Just to be clear: when it comes to the motion of gyroscopes Eric Laithwaite is not an authority. When it comes to gyroscopes Eric Laithwaite managed to sidetrack himself. Any inferences or conclusions stated by Eric Laithwaite arise from him misunderstanding the mechanics of gyroscopic precession.

On the subject of gyroscopic precession my view is as follows:

There is for one thing the physics fact that gyroscopic precession is counter-intuitive.

But then comes the following: the standard way of accounting for gyroscopic precession is by application of the concept of angular momentum vector, using the angular momentum vector in combination with operations such as vector cross product. The concepts of angular momentum vector and vector cross product are themselves abstract concepts, and the act of combining the two is not accessible to intuitive understanding.

Now, we will agree on the following: the whole point of doing physics is to try and gain an understanding of things that are not immediately comprehensible.

Example: when water freezes the ice takes up more volume. All other substances display a monotonic relation of volume as a function of temperature: colder: less volume; hotter: more volume. So what's going on with water? Physics provides understanding: there is the shape of the water molecule, and how water molecules interact with neighbouring water molecules, all of that explains this unusual property of water. You can understand the structural integrity of ice in the way you can understand the structural integrity of, say, an icosahedron. I will refer to that type of understanding as transparent understanding.

Circling back to the concepts of angular momentum and vector cross product: while mathematically correct, they don't provide transparent understanding.

The exposition of gyroscopic precession that I posted in 2012 does not use the concept of angular momentum vector.

I assert that in the specific case of trying to gain a transparent understanding of gyroscopic precession the abstract concept of angular momentum vector is not helpful, but instead a hindrance.

I assert that when you gain transparent understanding of gyroscopic precession you will see the demonstrations by Eric Laithwaite in a new light. Everytime the gyro wheel goes in climbing motion there is an assist.

(This is the most vivid in the demonstration where Laithwaite lifts a very large, very rapidly spinning gyro wheel over his head. Laithwaite provides the assist with a swing in the horizontal plane.)

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Electric Unicycles

In response to the videos with electric unicycles:
The relevant case, of course, is when the high power electric unicycle is going at high speed.
(I have a normal unicycle myself. I'm not proficient, but on smooth surface I can get around.)

I gather the forward/rear balancing is the same technology as implemented in Segways.

In the case of cornering at high speed:
I expect the following steering authority is the main contributor.
As the rider approaches a turn: in preparation the rider allows a sideways lean to develop.
There is the center of mass, COM, of the rider plus EUC, and there is the contact patch of the wheel.
During cornering the contact patch has to cover a bit more distance than the COM, since the contact patch is further away from the center of cornering.
In order to provide the contact patch with that bit of extra velocity the rider must use his footplate input to coax the EUC to go a bit faster.
The COM and the contact patch are cornering together; the COM is taking the corner on the inside, the contact patch on the outside.

To get out of the cornering motion the rider must move the contact patch back under the COM. The rider must simultaneously coax the EUC to reduce the velocity a bit, so that it is once again going at exactly the same velocity as that of the COM.

I expect that any gyroscopic effect will be swamped by other larger effects.

I expect that the learning process falls in the realm of subconscious learning. The rider tries, the first attempts are wobbly and erratic. Over time the human subconscious balancing system will figure it out, and over the course of many tries, and nights of good sleep, the cornering becomes smoother and more confident.

I assume the rider is using footplate input all the time to keep forward/rearward balance, all subconsciously. It seems likely to me that the rider will not be consciously aware of the special footplate input that accompanies cornering.

About the video demonstrating high speed cornering, using three different tires. when in a tilted orientation: depending on the profile the tire will have a tendency of its own to move along a curve. By the looks of it: the rider must bring that tendency in alignment with the intended radius of cornering. So, in order to corner with the same radius of cornering with each tire: the rider has to set up the appropriate amount of tilt. As can be seen in the video, this appropriate amount of tilt of the wheel can be different from the amount of overal body lean that is necessary for cornering at the rider's velocity.