By Joseph W. Connolly, Professor of Physics/EE at the University of Scranton

Well, he tried, and helping people “understand the physical behavior of the bicycle with minimum use of mathematics” is an admirable goal, but I am afraid this misses the mark.

After chapters on “Momentum-impulse”, “Work-energy-power”, and even “Temperature and heat”, oh my, the trouble starts in chapter 10 on “Rotational motion”, specifically section 10.7 on the “Role of angular momentum in a bicycle”. Prof Connolly starts with a warning that “angular momentum is a very important physical concept,” which “is commonly misunderstood and applied in a flawed manner,” and then he immediately proceeds to misunderstand it and apply it in a flawed manner.

He continues with the example of a “simple hoop,” which “remains upright” “as long as it is rolling,” but he completely misunderstands why. Then he incorrectly asserts that the reason a rolling hoop stays upright does not apply to bicycles. The reality is that a rolling hoop stays upright because gyroscopic precession causes it to steer in the direction of a lean, just as with the front wheel of bicycles. Of course, on bicycles, gyroscopic precession is neither sufficient nor necessary for self-stability because there are other factors at play, but that doesn’t mean it doesn’t play an important role. More on that later.

The trouble continues in chapter 11 on “Torque-applications to the bicycle”. Prof Connolly employs plenty of math when he analyzes wheelies and headers, but oddly obscures the importance of center of mass location relative to the rear or front wheel by immediately using lengths instead of variables so that the relationship cannot be seen in the result. Sure, the pedal force and the coefficient of friction are also important factors, but so is the center of mass location. I suppose this oversight can be chalked up as simply an editorial choice, but the elegant result relating center of mass location to coefficient of friction was so close, it seems a shame to leave it unstated.

The main event, however, is chapter 12 on “Centripetal acceleration-turning and bicycle stability”. He does correctly point out that bicycles stay upright by steering in the direction of a lean, that a bicycle that cannot steer will not stay upright, and that many bicycle can exhibit self-stability.

Then, however, in section 12.11.1 on “Weight distribution of the handlebar-fork-front wheel” he confuses the behavior of a stationary bicycle held in some orientation by a person with the behavior of a bicycle rolling forward and not being held in some orientation. When a person holds a stationary at some lean angle, of course the front assembly will rotate about the steering axis if its center of mass is not on the axis. As an unsupported rolling bicycle leans, on the other hand, the center of mass of the rear frame and the front assembly both roll about a common axis, the line between the two tire contact patches. A crucial point that Prof Connolly misses from the two-mass-skate bike article he even cites, is that

When the TMS bicycle falls, the lower steering-mass would, on its own, fall faster than the higher frame-mass for the same reason that a short pencil balanced on end (an inverted pendulum) falls faster than a tall broomstick (a slower inverted pendulum). Because the frames are hinged together, the tendency for the front steering-assembly mass to fall faster causes steering in the fall direction.

This is a dynamic phenomenon, which can be explained in a single sentence, but a static explanation just doesn’t cut it.

It gets worse. Prof Connolly continues though to the end of chapter 12, the last chapter of the book, without ever mentioning gyroscopic effects! Oof. Sure, the two-mass-skate bike article shows that they are neither sufficient nor necessary, in theory, but it never asserts that they don’t play an important role is common bicycles. In fact, the authors state the opposite in their Supporting Online Text Material:

The benchmark bicycle entirely loses its stability if it is only changed by the removal of the wheel gyro terms.

The benchmark bicycle has no stable region when the gyro is removed.

That seems pretty clear, and this glaring omission should be the nail in the coffin of “Understanding the Magic of the Bicycle”.

That may seem harsh, and Prof Connelly, who seems like a very nice man, probably didn’t go through all the effort of writing a book just to make money, but can we say the same about the publishers, IOP Concise Physics? They are part of IOPscience, which also published the misguided “On the stability of a bicycle on rollers” in their “European Journal of Physics” that I discussed back in October 2014. It seems that they really need to find an editor that understands bikes and/or just stop publishing books and articles about them.

The argument has stretched for weeks over on KQED, and I believe Calvin has finally revealed enough about his model for me to find a fatal flaw.

As previously reported, Calvin argues that the net ground reaction force acting on a bike as it leans must remain aligned with the midplane of the bike. This then means that the ground reaction force cannot create a moment about the steering axis through trail. With gyroscopic effect already minimized, and trail now neutralized, Calvin’s tire theory can come to the rescue.

The problem, however, is that the net ground reaction force does not remain aligned with the midplane of the bike unless the bike is modeled as a point mass on a massless rod. As soon as a finite mass distribution is considered, Calvin’s fundamental premise falls apart.

In an effort to induce Calvin to concede this flaw, I have derived expressions for the ground reaction force orientation for two different mass distributions: a point mass and a uniform rod, and asked Calvin to point out any flaws in my derivation.

So far, he has not managed to find any, and has simply reiterated his reasoning instead. Therefore, I have done my best to follow his reasoning and find where it goes awry. It turns out that Calvin appears to have made a common undergraduate mistake by forgetting to use the Parallel Axis Theorem when he inserts the mass moment of inertia into his derivation.

The details are in the images below. The outdented text is Calvin’s original explanation, and the indented text and equations are my detailed reply:

There we have it. Time will tell if it has any impact on Calvin’s thinking.

I suppose I should apologize for the long silence, since continued postings are what drives blog traffic, but things have been fairly quite in the world of bicycle science, and I didn’t have anything that needed highlighting. Or so I thought.

It turns our that our old friend Calvin Hulburt, first mentioned here back in 2014, has remained busy since then. He posted a long screed with pictures and everything in answer to questions on Quora here and here, and even gets a little feisty on KQED when he refers to the current active group of bicycle dynamics researchers as “Whipplites”.

(Yikes! They’re like mushrooms after a rain. Additional instances include ScienceLine, ScienceMag, DiscoverMagazine, MotorcyclistOnline, and MentalFloss.)

Sadly, he still has nothing new to offer. Despite our pleading with him on the  Single Track Vehicle Google Group to make a testable prediction, he declines to do so. Instead, he simply continues to push his theories in venues where they can go unchallenged.

I guess now the task includes creating user IDs on each of those sites, logging in, and posting a reply. Sigh.

Update: Well, Andy Ruina tackled Quora, and I took on KQED.

The discussion has been busy on KQED, and Calvin has finally made some testable assertions, but we seem to have come to an impasse. One of these assertions is that the net ground reaction force in a lean is always aligned with the midplane of the bike, and this means that it cannot create a steering torque via a finite trail.

I counter that this is only true if the bike is modeled as a point mass on a massless rod. If the mass is distributed, as on a real bike, the net ground reaction force no longer aligns with the midplane.

We need to look at equations to sort things out, the most efficient way to do that is with attached images, and KQED keeps deleting my comments that contain images, so I’m posting them here instead.

So the equations do support the notion that the net ground reaction force is perfectly aligned with the midplane of a bike, if the bike consists of a point mass on a massless rod.

Note that f = Ntanθ, where f is the horizontal ground reaction force, N is the vertical ground reaction force, and θ is the roll angle; not f = Wtanθ as Calvin indicates in his diagrams on Quora.  N only equals the weight, W in Calvin’s diagram, when there is no motion.

So the equations do NOT support the notion that the net ground reaction force is perfectly aligned with the midplane of a bike, if the mass of the bike is distributed evenly along its height, as in a uniform rod. The same would be the case if the bike were modeled as a uniform plate

The shape and mass distribution do not enter into the linear momentum balance, and since the weight and applied forces remain the same, so do those equations. The only difference is in the angular momentum balance, which produces the slower acceleration.

When the equations from linear momentum balance are combined with the equation from angular momentum balance, the single difference propagates into the horizontal and vertical components of the ground reaction force such that the resultant net ground reaction force no long aligns with the midplane of the bike.

At long last, this means that the ground reaction force, combined with a finite trail, can generate a steering torque and cause the front wheel to steer towards the direction of the lean.

I’d be tempted just to leave this one alone because it looks like a high school physics weekend project, but this kind of site, with its attractive pictures and simple text, is just the kind of place high school physics students might be tempted to get their information.

The section on Newton’s 1st law contains this gem

When the brakes are applied, the force of friction increases due to the increase in rolling coefficient of friction. This unbalanced force is what brings the bike to rest demonstrating Newton’s first law.

Yes, the brakes generate friction, which convert kinetic energy to heat energy to be dissipated, but this has nothing to do with the increase in rolling coefficient of friction, whatever that might be. And the deceleration due to braking is an example of Newton’s 2nd law,  not 1st.

Then, in the section on Newton’s 2nd law, we get this nonsense

As you can see from the equation, net force is proportional to its mass. If you double the mass, then the net force doubles as well.

Sure, if acceleration is kept constant, but you have to say that for the above to be true.

Finally, in the section on Newton’s 3rd law,

Additionally, the suspension of mountain bikes uses the principal of Newton’ third law. Riding over rough terrain of several bumps and depressions, suspension systems on mountain bike transform the reaction force more into the bike itself instead of into the rider.

If you can make sense out of this, let me know.

I guess this is what we get from anonymous content providers.