# SchoolPhysics: Stability of a bicycle

SchoolPhysics Online, by Keith Gibbs, considers itself to be “an invaluable resource base for all 11 to 19 year old physics students and their teachers.” That may be true, unless those students and their teachers wish to learn about why a bicycle is easier to balance when it is moving forward than when it is stationary.

In his brief article on the stability of a bicycle,  copyrighted in 2007, Mr. Gibbs wastes no time in getting it wrong and opens his answer with:

This one is a case of the conservation of angular momentum.

Unless the bicycle is free falling in the vacuum of space, it will experience externally applied moments, and angular momentum will not be conserved.

Mr. Gibbs explains further with these two nuggets:

Now a stationary bike wheel has no angular momentum and so does not need a force to change the direction of the axle – in other words the bike can easily fall over.

However the rotating bike wheel has angular momentum and so requires a force, in some cases a considerable one, to make the axle of the wheel change direction, and so the wheel stays upright.

Of course, it is not a force that can cause the axle of a wheel to change direction, but a couple or moment, and the existence of angular momentum is not what necessitates a couple or moment in order to change the direction of an axle, just a non-zero mass moment of inertia.

Then, if the wheel is spinning, so that it also has angular momentum about its axle, the axle does not resist changing direction in response to an applied moment, it just changes direction in a different way, by precessing about an axis at right angles to both the axle and the applied moment. If it is not free to precess about this third axis, as is the case of the rear wheel if the tires are in contact with the ground and the front wheel if the steering is locked, then it will respond to the applied moment as though it were not spinning at all.

Instead, the feature that all ridable bicycles have in common, regardless of their geometry and mass distribution, is the ability to steer, and the current answer needs to go back to the drawing board before any more physics students or their teachers get a bum steer.

# A Motorcycle is a Gyroscope in the SPS Observer

This article by Dwight E. Neuenschwander, a professor in the Physics Department at Southern Nazarene University appears to have been published in the SPS Observer, the quarterly magazine of the Society of Physics Students (SPS) published by SPS and the American Institute of Physics (AIP). No publication date is given, but the article is currently hosted on the SPS Observer web site, which also hosts issues back to 2002, but no way to search them for a particular article or topic.

In any case, Prof. Neuenschwander makes the astounding claim that simply leaning a motorcycle will make the entire motorcycle precess in the direction of the lean:

If you lean to the left, the motorcycle turns left (likewise on bicycles). Why is this so?  … This induces a nonzero torque … Hence the angular momentum vector L rotates about a vertical axis, and the motorcycle precesses to my left.

Not just the front wheel precesses, but the whole motorcycle. No mention is made of steer angle, steer torque, or even friction between the tires and pavement. The whole bikes just yaws by the magic of gyroscopes!

Of course, no such thing happens. Instead, there are two possibilities. Either the front wheel is free to precess about the steering axis, does so to steer in the direction of the lean, and the friction between the tires and the pavement generate a yawing moment on the bike. Or the front wheel is not free to precess and the roll moment generated about the contact patches by the force of gravity acting on the center of mass simply causes the bike to roll until it strikes the pavement. Gyroscopes are not magic after all.

Near the end of the article, Prof. Neuenschwander writes

Hmmm… further research is needed….

We don’t need any further research like this.  What is really needed, instead, is reading up on the topic before writing about it. Sharp’s seminal paper on the Stability and Control of Motorcycles has been available since 1971, so the information is out there.

# Ask a scientist at Argonne National Laboratory

In the reply to Ask a Scientist, on Argonne National Laboratory‘s Newton, there are two answers to the question asking what is the relative contributions from angular momentum and counter steering.

The first response, by Unknown, is not too bad. It quickly dispenses with angular momentum as negligible, and then explains how counter steering can work just fine without angular momentum.

The second response, however, by Dr. Ken Mellendorf, is a muddled mess and should be deleted. He first perpetuates the misconception that the spinning wheels somehow resist leaning and steering, and he follows that immediately with the misconception that the frame itself resists steering simply by moving forward.

The wheels are spinning in a vertical orientation, aligned with the path of the bicycle. The faster they spin, the more difficult it is to change them. The bicycle is moving forward. The faster it moves, the harder it is to make the body of the bicycle change direction.

Spinning wheels have no resistance to roll moments if they are prevented from precessing about the yaw axis. Instead, a roll moment causes the front wheel to precess in the direction of the lean, and the rear wheel, which is prevented from precessing by the frame and friction in the two contact patches,  leans exactly as it would if it were not spinning.

Linear momentum is a vector quantity and so the linear momentum in one direction, such as forward, has no effect on linear momentum in an orthogonal direction, such as to the side. Thus the increased linear momentum from going faster is not responsible for the smaller steering inputs required to maintain balance. Instead, it is simply the fact that a give steering input works faster, that is causes a larger lateral acceleration of the contact patches, if the wheels are rolling forward faster.

Then, Dr. Mellendorf tries to tackle counter steering, and things really get crazy.

Once turned, the front wheel moves to the side. The body of the bike, however, tries to keep going forward. The “natural” thing for the bike to do is fall down as the front wheel pulls out from under it. The rider has to lean toward the inside of the turn to prevent this from happening. If the bike “tries” to flip to the right, The rider leans to the left to counter the effect.

If by “the rider has to lean” he means “the rider has to lean along with the bike”, why would the rider have to do anything other than stay with the bike as it does its “natural” thing? Does he also mean to say that the rider cannot lean relative to the bike? Better not tell these guys:

If by “the rider has to lean” he means “the rider has to lean relative to the bike,” better not tell these guys:

The fact is that a rider can stay perfectly in line with the frame of his bike, or lean relative to the bike either into the turn or away from the turn. All that matters is where the combined center of mass is located with respect to the tire contact patches, and the only time a rider must lean to the left if the bike “tries” to flip to the right is when the bike is not moving forward at all.

No, Dr. Mellendorf’s description is most definitely not how a bicycle works.

# What makes for bad bicycle science

As with most endeavors,  there are plenty of ways to make bicycle science bad. There are a few ways, however, that seem to be more popular than others. Here are some of the most common:

1. Ignoring or misinterpreting previous work

Most of the examples itemized in the posts on this site make this mistake. The UW-Madison Physics Department writes as though it were in a vacuum, while Physlink.com cites a useful work and then comes to the opposite conclusion of the author.

Despite flaws in its final analysis, Jones’ 1970 Physics Today article demonstrates the limited role of gyroscopic effect pretty clearly . Thus, anyone writing after 1970 that bike stability or ridability derives solely from gyroscopic effects or that bikes are almost impossible to ride without gyroscopic effects simply hasn’t done their homework.

2. Misinterpreting laws of mechanics

By far, the most popular law to flaunt is that of angular momentum, as demonstrated by Mental Floss.

Spinning wheels have no resistance to roll moments if they are prevented from precessing about the yaw axis. Instead, a roll moment causes the front wheel to precess in the direction of the lean, and the rear wheel, which is prevented from precessing by the frame and friction in the two contact patches,  leans exactly as it would if it were not spinning.

A related misconception is the assertion that angular momentum is somehow conserved when riding a bike and this conservation of angular momentum is why the bike stays upright. Instead, a roll moment from gravity or a steer torque on the handlebars from the rider easily modify the angular momentum.

The next most popular law to flaunt is that of linear momentum, as demonstrated by Rider Education of New Jersey.

Linear momentum is a vector quantity and so the linear momentum in one direction, such as forward, has no effect on linear momentum in an orthogonal direction, such as to the side. Thus the increased linear momentum from going faster is not responsible for the smaller steering inputs required to maintain balance. Instead, it is simply the fact that a give steering input works faster, that is causes a larger lateral acceleration of the contact patches, if the wheels are rolling forward faster.

3. Providing no equations or calculations, no instrumented physical experimentation, or any other sort of validation

This is more of a problem with articles in supposedly peer-review journals, such as in the European Journal of Physics.

Certainly, not every article is intended for a technical audience, but every assertion still needs to be based on reality. Claiming the gyroscopic effect is responsible for so and so without even doing a back-of-the-envelope calculation to show it is possible is just blowing smoke.

Direct human observation is notoriously unreliable, especially of small behaviors combined with large behaviors, such as the steer angle of a speeding motorcycle. Even the Wright brothers observed that most bicycle riders do not realize that they apply a steer torque to the left in order to turn right.

# Georgia State University Department of Physics and Astronomy: The Bicycle Wheel as a Gyroscope

The bicycle wheel article on the award-winning HyperPhysics website runs into trouble immediately, in the first sentence:

The angulur momentum of the turning bicycle wheels makes them act like gyroscopes to help stabilize the bicycle.

It later tries to temper that bold assertion with a disclaimer,

it should be pointed out that experiments indicate that the gyroscopic stability arising from the wheels is not a significant part of the stability of a bicycle,

but the damage is already done. We can allow that the gyroscopic pressesion of the front wheel does contribute to it steering in the right direction to correct for a lean, but that still leaves the rear wheel, which is the only possible interpretation of the plural “wheels”.

The rear wheel is prevented from precessing by the rear frame, which is in turn prevented from yawing by the contact patches of the two wheels, and when a spinning object is prevented from precessing, it reacts to an applied torque exactly as it would if it were not spinning at all. Thus the angular momentum of the turning rear wheel is no help at all in stabilizing a bicycle or a motorcycle.

# Mental Floss: Why is it so hard to balance on a bicycle that’s not moving, and easy on one that is?

After a great start, with correct statements such as

bikes are, however, dynamically stable, or stable when moving forward, because steering allows a rider to move the bike’s points of support around under the center of gravity and keep it balanced,

the wheels come of the cart in paragraph 3 with this mess:

Spinning wheels have angular momentum, and when you’re sitting on a bike, you and it and its wheels make up a system that obeys the principle of conservation of angular momentum. Unless torque, or twisting force, is applied from outside the system to change the wheels’ angular momentum, that momentum and the direction of the momentum remain constant. In a nutshell, once the wheels line up a certain way, they want to stay lined up like that. It’s easy for you to move them, but hard for an outside force to do the same, and so the bike is easy to keep balanced but doesn’t topple easily.

1. Unless you and your bike are in free fall in a vacuum, angular momentum will not be conserved.  The slightest lean will allow gravity and the ground contacts to form a couple that applies a large external moment to the system.

2. The wheels do not care which way they are lined up and will easily change orientation in response to external torques, such as the roll torque described in point 1 above or a steering torque.

The rear wheel is generally prevented from presessing in response to a rolling moment, and so will roll with the rear frame exactly as if it were not spinning at all.

The front wheel is generally free to rotate about the steering axis and so gyroscopic pressession will cause it to steer in the direction of an applied roll torque and lean opposite to the direction of the applied steer torque.

It pains me a little to point out this flaw because the article ends with a plug for the wikipedia article and even credits me for the picture I downloaded from elsewhere on wikipedia and cropped to show cyclist performing a track stand, but facts is facts, and this article gets a couple of key facts wrong.

# PhysLink.com: Do gyroscopic forces from the wheels make any significant contribution to the rideability of a bicycle?

PhysLink.com tries to answer the question, and the author of the reply does not beat around the bush:

Yes, the gyroscopic forces, better known as the angular momentum, of the wheels on a bike allow us to ride a bike. … The reason that you stay up on a bike is that angular momentum, like regular momentum, must be conserved if no external torques act on the object. … When you are riding a bike forward, the right hand rule gives the direction of angular momentum to be to the left, perpendicular to the wheel. This direction does not want to change, therefore the wheel wants to stay upright and it makes the bike very ridable.

Wow. Flat out wrong in many ways and no mention of steering at all.

1. The  David Jones Physics Today article from 1970,  demonstrates that a bike is quite ridable if the gyroscopic forces are canceled.

2. Gravity acting on the center of mass and the ground reaction forces create a couple that acts as a large external torque. If there were no external torques, bikes wouldn’t tip over at all and there would be not need for a way to balance them

3. The direction of the angular momentum vector can easily be changed by the application of an external torque, such as the one described in the previous point.

The author even goes so far as to suggest that he has seen Jones’ article but comes to the opposite conclusion that Jones does.

However, when the extra wheel was spun backwards, the bike became almost impossible to ride because the vectors for angular momentum cancelled each other out. It was like trying to balance a bike that was not moving.

Instead, a main point of Jones’ article is how hard it was for him to make an unridable bike.