How to quantify precession related torque?

Question

Updated question

Wikipedia's article on countersteering and gyroscopic effect mentions a roll moment (torque) related to precession and steering torque, during the initial outward counter-steering torque used to lean a bike inwards before steering inwards to turn:

"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

According to the Wikipedia article, the calculation is based on the rate of counter-steering as opposed to the counter-steering torque, probably some terms canceling out.

In this clip of a video of a large gyro in precession due to a small torque from gravity (slight imbalance), it doesn't take much force to stop the precession, and the gyro just falls. Granted that there isn't a lot of torque applied to the gyro, but it is a large gyro spinning at high speed:

https://www.youtube.com/watch?v=6tFzzSG8SU0&t=476s

In a more general case, consider a 3 axis gimbal mounted gyro, but with the axis of precession locked. How to calculate the amount of torque on that locked axis depending on the angular momentum of the gyro and torque applied to the non-locked axis perpendicular to gyro's axis of rotation. The net outcome will be that the gyro responds to the non-locked axis torque as if the gyro was not spinning, but there would be a torque exerted onto the locked axis.

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I removed the part about electric unicycles, since the question is now more specific about precession related torque.

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.

Answer 2

Some comments on the updated question.

The particular gyro wheel demo starting at 7:02 into the gyro demo video that you linked to:

I will refer to the particular model of gyroscope used from 7:02 on as 'Pasco Demonstration Gyroscope'

While it is the case that the demonstrator uses an electric motor to (friction) spin up the gyro wheel, that wheel is not spinning fast. It's just that the surface of that wheel is so smooth: you can't assess the rate of spin.

What does give a clear indication of the rate of spin is the rate of nutation. Rate of nutation and rate of spin are correlated. (The moment of inertia of the counterweight relative to the swivel point causes the rate of nutation to be slower, but not much slower.)

Also: when the demonstrator manipulates the device very little force is sufficient to have a visible effect, again indicating that the rate of spin is quite slow.

In the updated version of your question you also refer to a setup that you describe as 'a 3 axis gimbal mounted gyro, but with the axis of precession locked'

Here's the thing: the key factor in gyroscopic precession is motion.

At 7:31 into the video the demonstrator adds a weight, so that there is a surplus of weight on the gyro wheel side. Once the gyro wheel is precessing: it is the precessing motion that gives rise to a tendency of the gyro wheel to pitch up. The gyro wheel settles on a rate of precesssion such that the tendency to pitch up matches the tendency from gravity to pull the surplus weight down.

When motion is prevented (locked axis) then the phenomenon of gyroscopic precession is prevented.

What that means is that preventing precessing motion is not in any way informative.

About nutation:

Nutation can occur both with and without an overall gyroscopic precession.

In the video, after spinning up the gyro wheel there is already a small nutation (in addition to the precessing motion) Then the demonstrator gives the precessing gyro setup a bit of a whack. The nutation caused by that whack has a larger amplitude than the initial nutation; it has the same frequency.

With a gyro setup that is balanced (hence no precessing motion): the same whack would give the same amplitude of nutation.

Incidentally:
We have that with any form of gyro wheel suspension the setup has more inertial mass than just the inertial mass of the gyro wheel.

An interesting case is the case where there is no suspension. For instance, a gyro wheel in an orbiting space station doesn't need suspension.

The dynamics of purely a gyro wheel, with no other inertial mass, is known under the name 'Feynman's wobbling plate'

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To your question in the comment section:
The gimbal mounted gyro that you are referring to:

That particular demonstration gyro has been in use for a long time, I've seen it in different videos. Back when I created a diagram I used the same color scheme for the two gimbal rings:

So:
Let's say the yellow frame is fitted with a torque sensing device, in such a way that when the yellow frame tends to swivel the torque sensor tells you the magnitude of the torque.

There is a relation between rate-of-precession and torque.

For the orientation of the gyro wheel as depicted in the diagram:
When a small weight is added such that there is a down-pitching torque there is a corresponding rate of precession (and direction of precession) such that the precessing motion gives a tendency to pitch up that offsets the downward torque from the added weight.

The larger the mass of the added weight the faster the precessing motion needs to be in order to offset the torque from the added weight.

That relation works the other way round too:
given a particular angular velocity of the gyro wheel there will be a specific added weight such that the corresponding precession rate is one revolution every ten seconds.

When you make the gyro wheel pitch then in response to that motion the gyro wheel tends to swivel. If you make the angular velocity of the pitching motion such that it corresponds to one revolution per ten seconds then the tendency to swivel will correspond to the same torque that corresponds to a precession rate of one revolution per ten seconds.

I define three axes:

• Roll axis - the gyroscope wheel spins around the roll axis.
• Pitch axis - motion of the red frame. As you can see, the gimbal mounting ensures the pitch axis is perpendicular to the roll axis.
• Swivel axis - motion of the yellow frame.

When the gyro wheel (in the orientation depicted in the diagram) is being turned around the swivel axis then in response the gyro wheel will go to pitching motion.

Likewise: When the gyro wheel is being turned around the pitch axis then in response the gyro wheel will go to swiveling motion.

About the initial torque: in order to start motion a torque must be applied. It is the motion that elicits the gyroscopic response.

Gyroscopic response is what sustains nutation:
Pitching down elicits swiviling motion, that swiveling motion elicits pitching up, pitching up elicits swiveling motion in the other direction, and that brings the cycle back to pitching down.