Wheel Physics Question
 

It's been close to 15 years since I took physics in school, chances are good that somebody here can figure out this answer better than I can.

Practical observation: if you hold the axle of a spinning wheel, imagine spinning with the top moving away from you as though looking at a bike from behind, and tip the top of that wheel to the left, the reaction force pushes the axle forward on the right and back on the left: it turns to the left. Similarly, if you force the axle to rotate toward the left around a vertical axis (spinning on a horizontal axis) the reactive force pushes the top of the wheel toward the right.

The resulting theoretical question: how much work is required to overcome this angular momentum and gyroscopic precession, and accelerate the spinning wheel perpendicular to its primary axis of rotation?

I don't know. I assume you want a formula, because the rpm and tire/rim mass would obviously matter.

Obviously. The further from the center of the axle any given mass is, the further it moves and the more significant it is. I think that the marketing emphasis on overall wheel weight is misguided, and there should be more emphasis on moment of inertia, but I'm not sure.

Zero work is required, because the gyroscopic precession is helping you lean the bike into the turn. You are not trying to fight the forces of precession here. You are enlisting those forces to help you lean the bike over.

The force from the rear wheel is pushing the bike straight when it leans, and the force from the front is pushing the bike to tip away from the direction the wheel is being turned (generally back upright). The net effect of gyroscopic precession on the two wheels is to stabilize the bike, but that costs energy that doesn't come from nowhere.

This is all under the topic Conservation of Angular Momentum. The wheel is spinning in one orientation. You tilt it, and the reaction forces it experiences are from transferring that motion into another form so that everything zeros out. Just like conservation of energy. It's harder to observe the small changes in energy or momentum due to friction at the bearings or aerodynamic drag of the wheel spinning, but the big changes we can see and feel.

You're standing on a turntable on ball bearings. You are holding a spinning bike wheel with the axle horizontal, and the wheel is spinning away from you on top, like the front wheel on a bike. You tilt the wheel to the left; Now that spinning motion is partly clockwise from your position, so you and the wheel as a unit, start to rotate slowly counter-clockwise on the turntable, so the total angular momentum of the system stays constant.

Now same example, but start with the wheel held with the axle vertical above your head, the wheel spinning clockwise when viewed from above. You slowly lower the axle to horizontal at your waist, and you and the wheel rotate clockwise on the turntable. Now, you continue to lower/tilt the wheel until the axle is again vertical, but inverted from how it started, with the wheel rotating counter-clockwise from above; Now you and the wheel are rotating on the turntable clockwise, but twice as fast, in order to zero-out the wheel angular momentum, that the total momentum of the system stays constant. This is a common demonstration in physics 1 class (Newtonian physics).

The work done in rotating anything is:
work = net torque x angle turned (angle is measured in radians).
The above formula assumes that the total torque is constant (or the average) during the turn.

Nope. When the bike leans to the left, the precession force on the rear wheel tries to steer it to the left. So it's helping the bike turn.

Right, which is precisely how you steer a two wheeled vehicle. If you want to make a left turn, you first turn slightly to the right, in order to tilt the bike to the left. This is known as counter-steering. Gyroscopic precession helps you do that.

I still have trouble with that concept. Since it is obvious that turning to the left or right can be begun and completed with both hands off the handlebar, whatever counter-steering is taking place must be occurring only in some esoteric, theoretical sense.

A whole bunch, but that isn't really a thing when riding a bike (maybe momentarily in a crit pile up :)). The gyroscopic effect is your friend when riding a bike.

Counter-steering is an effective way to get the bike to lean, but it's not the only way. You can also get the bike to lean with your body weight.

Your intuition that MOI matters and should be included is correct; however, the empirical result is that the contribution of MOI is small, so the practical result is that we usually can get away with ignoring it unless you really really need to. When I did some stuff on Olympic Team Pursuit a while back we collected data on wheel MOI but in the end the contribution was small enough that, amongst the noise and resolution of the power and speed data, it didn't improve our analysis or predictions--so we were able to drop it. That said, I seem to recall a recent paper by Slawinski in Sports Engineering on velodrome power demands included some discussion of the wheel MOI.

That's the way I understand it, too. But in previous discussions on the subject here, people have argued that it's impossible to steer a bike without counter-steering -- that we're doing it whether we know it or not, even when we steer around a corner while riding no hands. Pretty sure those hardliners were all motorcyclists as well as bike riders.

A related intuition is that different types of biking result in different amounts of side-to-side rocking. MOI must be most important for BMX racing, and least important in the velodrome.

The practical answer is very little and it can be ignored in a bicycle physics model.

If the resistive force was higher then you would simply rock less for the same input. But the forces involved are small enough to simply ignore anyway.

If that was the case then you could deadlift 100 lbs with the same amount of work as 200 lbs, you just wouldn't lift the 200 lbs as high. On one level it's correct, but when you actually think about the mechanics it's so laughably wrong that it draws everything else you say into question.

I love a deep end of the pool measuring contest thread :roflmao2:

But unlike a deadlift you are not concerned about how far your bike rocks for a given load. The rocking is just a balancing reaction to the pedal force input. So if your bike rocks a fraction of a degree less because of gyro forces it doesn’t mean you are losing any pedal torque/power.

But as you pedal and rock your bike side-to-side your joints go through a certain range of motion. That range of motion does not change depending on how much force goes in to moving the bike side-to-side versus moving the bike forward.

But you are not putting any more force into rocking the bike just because there is fractionally more lateral resistance. It just rocks less for the same force input. It’s the same as when riding on your turbo trainer, which effectively simulates almost infinite gyro force. The bike doesn’t rock much on the trainer, but it doesn’t really have much effect on your power other than changing the dynamics of a full bore sprint, where a degree of rocking motion might be an advantage. But the difference in degree of rocking between different wheel sets on a bike will be vanishingly small

The reason that rocking in a sprint is an advantage is that what matters there is less efficiency and more raw power production and transfer. Rocking allows a rider to convert work done by the arms and core into forward speed. The trainer has a high threshold force needed to tip, but the force curve of tipping a trainer over is very different than the one to tip a rolling bike over.

I'm willing to grant that we're obviously talking about small numbers, but rolling resistance is also small numbers and people seem to think that's worth optimizing. Is gyroscopic force the reason that Dylan Johnson is wrong about 40mm tires on road bikes? That's my guess, but I don't know the math to prove or disprove it.

Rolling resistance is a fairly major component of any cycling power model. If you ignore it then your power model will be pretty inaccurate. However, you can simply ignore gyroscopic forces and MOI without significantly affecting model predictions.

Where can I get my hands on that model? I wish I had the expertise to create one myself, but I don't. Ignoring gyroscopic forces and MOI sounds like the kind of short cut I would be forced to do if I developed that model.

significant in terms of some percent of your power lost to it. But not very variable. Most every informed study I've read/seen shows rolling resistance varies so little in the realm of cycling and velocity.
What you bring to riding is mostly constant, whether you're going 10 mph, 20 0r more... so it is a factor of the equipment you bring and the road surface, but after that it's close to constant.
'Rolling Resistance as a factor of velocity'
there is a study read quite some years back where bicycle rolling resistance, as a variable of velocity showed an almost flat slope to RR as a factor of velocity...
it's one of those very marginal gains, if one is willing or needing to consider...
will a difference of 4-5 watts saved make much of a difference in my or anyone's ride?
Aerodynamics is the Big Variable in riding velocity.
Ride On
Yuri