# The Physics of Bicycle Suspension on dw-link.com

In their attempt to explain the physics of bicycle suspension, the folks at Dave Weagle’s dw-link site lay this egg:

Newton’s Third law of Motion states that “Every action has an equal and opposite reaction.” When a bicycle accelerates forward, the rider’s mass is transferred rearward. Without something to counteract this mass transfer, the rear suspension on most bicycles will compress under acceleration.

The most glaring error is the misinterpretation of Newton’s 3rd law. As any undergraduate Engineering Mechanics textbook will explain, the “mutual forces of action and reaction between two particles are equal, opposite, and collinear.” That quote is from Engineering Mechanics by Hibbeler. Engineering Mechanics by Schmidt, Engineering Mechanics by McGill and King, Engineering Mechanics by Bedford and Fowler, Dynamics by Tongue and Sheppard, Engineering Mechanics by Meriam and Kraige, and Vector Mechanics for Engineers by Beer and Johnston all agree.

Thus, the only possible reaction, by Newton’s 3rd law, to the friction force acting on the rear tire contact patch, which is responsible for a bicycle’s forward acceleration, is the friction force acting on the ground that accelerates the earth in the opposite direction.

Of course, the law they should invoke is Newton’s 2nd law, or more specifically, Euler’s 2nd law, which describes the acceleration of a body in response to the sum of external moments.

The next error is in suggesting that load transfer actually requires the movement of mass. Tony Foale, in his Motorcycle Handling and Chassis Design, on page 9-1 explains why he doesn’t even like the expression weight transfer, let alone mass transfer.

This is normally referred to as weight transfer, but that is really a misnomer. Weight is the gravitational attraction of all the particles in the bike towards the centre of the earth, and for convenience we usually consider the sum of these forces to act through the CoG. Neither acceleration nor braking can cause this weight to transfer elsewhere. As a result the use of the term ‘load transfer’ is preferable.

I tried explaining all this back in 2009 to a wikipedia user named Tremanaps, who I suspect was either Dave himself or a surrogate, but made no progress. In both cases, I was able to provide multiple reliable sources to support my interpretation, but Tremanaps could not. You can read the contemptuous tone in his replies on the Wikipedia bicycle suspension talk page.

One can only hope that the actual dw-link suspension design is better than their ability to explain it.

# From Stargazers to Starships at NASA.gov

In answer to a question about balancing a bicycle on NASA’s (yes, that NASA“From Stargazers to Starships” site by Dr. David P. Stern we are treated to this gem:

By slightly turning the handlebars right or left, you impart some of the rotation of the front wheel (“angular momentum”) to rotate the bike around its long axis, the direction in which it rolls.

That’s it, the whole story.

While it is true that the precession of the spinning front wheel in response to an input steer torque is about the roll axis, this effect is minuscule in comparison to the roll moments generated first by the laterally accelerating contact patches and then by gravity acting on the center of mass that is no longer directly above those contact patches. Dr. Stern’s explanation also implies that gyroscopic effect is necessary for balancing a  single track vehicle by steering it, which will come as quite a shock to the guy riding this thing:

As for the actual magnitude of the contribution to the overall roll moment on a bike from the precession of the front wheel, we can refer to the nice example provided by Professor Cossalter in his excellent Motorcycle Dynamics.

On page 304 of the second edition, he calculates the roll moment generated by gyroscopic effect for a motorcycle traveling at 22 m/s (79 km/h or 49 mph) to be 3.5 N-m (2.6 lb-ft) and compares it to the roll moment generated by the accelerating contact patches of 30 N-m (22 lb-ft), which is 8.6 times larger. He concludes with the note that the gyroscopic effect is present from the instant torque is applied at the handlebars, and the roll moment generated by the lateral force of the tires can take some time, ~0.1 seconds in this example, to build up. We should expect the gyroscopic effect on bicycles, with lighter wheels and lower speeds, to be proportionately smaller.

There is a little more nonsense at the end of Dr. Stern’s reply about how bikes with straight forks, such as motorcycles, have no fork offset and so more trail, more stability, and less agility.

The design of the front wheel “fork” is quite interesting, turning forward like the letter j. This is an “inverse caster” which makes the ride less stable–but allows a skilled rider nimble moves. If you ever look at a motorcycle, or at the bikes used in a circus on the high wire, they lack this “letter j” feature, the wheel axis is always in line with the handlebar shaft. That helps stability but reduces agility.

Can you imagine how much better these guys would be if there were some way to reduce trail and increase agility without curved forks.

Of course curved forks is only one of several ways to generate fork offset and thereby alter trail.

No, bicycle science is not rocket science, but it still requires that you do your homework.

# 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.

# Single Track Vehicles by Calvin Hulburt

The article by Calvin Hulburt, a self-described retired engineer, gets off to a seemingly good start, and even refers to some of the other bad bicycle science already published elsewhere. It does get into a little trouble early when it dismisses the two-mass skate bike, by Kooijman et al. as a mere gadget, but recovers by citing Lowell and McKell and then Wilson-Jones.

After that, however, Mr. Hulburt really heads into the weeds. His attribution of bike stability to tire forces may sound reasonable to the uninformed, as did Sokal’s paper, but his argument includes no calculations, no supporting evidence, and makes no testable predictions.

This is not Mr. Hulburt’s only attempt to get his “theory” accepted. Besides several youtube videos, and several insertions into the wikipedia article, he recently began a discussion on the Single Track Vehicle Google Group.  There, various members pleaded with him to make a testable prediction, but Mr. Hulburt declined, and replied instead with a parting dig about the two-mass skate bike paper.

That was met appropriately with a resounding chorus of silence by the rest of the group.

Probably the best comment before the end was from Andy Ruina, a professor in the Mechanical Engineering Department at Cornell University, who tried to prod Mr. Hurlburt into stating something useful with I don’t want to dismiss you, as you have great intuitions and energy. But do you know the line:  “Not even wrong“?

# 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.

# Everything2: Gyroscope

It is not exactly clear what Everything2 is about, but it is not Wikipedia, and so does not appear to have any requirement for assertions to be supported by reliable sources or consensus among contributors . The article about gyroscopes, is a good example of what that yields:

In the bicycle’s case, the gyroscope effect of the wheels will help keep the bicycle upright and resist falling over, because falling over would be a change in the rotational plane of the wheels. It also helps the bicycle maintain a course roughly straight ahead, because turning would also be a change in the rotational plane of the wheels, which allows a confident rider to take his hands off the handlebars.

Here we have more perpetuation of the misconception that spinning wheels somehow resist falling over and even turning. In reality, large external moments, from gravity and the rider, are applied that easily alter the angular momentum vector of each wheel.

# 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.

# Dave Moulton’s Blog: Head Angles and Steering

In an older post, in which he get’s into stability and handling, Dave helps perpetuate two old misconceptions:

Because you are riding straight the gyroscopic action of the spinning wheels, plus your own weight and momentum, is holding you vertical just as surely as if you were physically holding the front wheel.

1. Gyroscopic action simply does not work like this. Instead, either a spinning object precesses in response to an applied torque, as is approximately the case with the front wheel of a bike, or it is prevented from precessing and so reacts to an applied torque exactly as if it were not spinning at all, as is approximately the case with the rear wheel of a bike. Neither holds anything vertical. The front wheel steers to accelerate the contact patches laterally and create a roll moment to counter the roll moment of gravity on the leaning bike, and the rear wheel simply leans with the rear frame.

2. The forward moment of the bike and rider contribute zero resistance to tipping. Linear momentum is a vector quantity and so linear momentum in the forward direction is completely independent of linear momentum in the lateral direction.

3. The mass of the bike and rider resist acceleration due to the force of gravity exactly the same when moving forward as they would when stationary. Whatever velocity they may have makes no difference.

Dave is no longer accepting comments on this post, so I cannot point out these mistakes there. Perhaps he will see them here and make a correction.

# Dave Moulton’s Blog: The ideal handling bicycle

While Dave’s “Optimum Handling” line may or may not be helpful, his explanation of forces in a turn could be down right dangerous, if someone where to depend on it:

Because the bike and rider are leaning, the rider’s weight is being pushed outwards and downwards by centrifugal force, thereby actually pushing the tires down onto the road, which increases traction.

Yikes! If only that were true.

The sad fact, however, is that the centrifugal force is perfectly parallel to the pavement, and so can contribute nothing to the force of the bike onto the road, and the lean angle of the bike and rider has nothing to do with it.

Instead, the centrifugal force must be resisted by the available friction between the tires and the pavement, which remains the same whether the bike is leaning or not.

I’ve posted a comment for Dave on his blog, and perhaps he can correct this mistake.

# European Journal of Physics: On the stability of a bicycle on rollers

Unlike all the previous examples discussed, which are self-published and answerable to no one, the European Journal of Physics describes itself as a peer-review journal, and as such, it should be held to a higher standard. That makes the 2011 article “On the stability of a bicycle on rollers” by Cleary and Mohazzabi all the more troubling.

Jim Papadopoulos and I, with the help of Andy Ruina, responded with “Comment on ‘On the stability of a bicycle on rollers’“, but it was treated as though it were merely the other side of some controversial issue with no clearly correct answer. Instead of trying to paraphrase what we spent weeks polishing to perfection, let me just quote directly from our response:

The two key differences noted in the paper, however, are not correct. The first is the authors’ claim (page 1297) that ‘riding a bicycle on rollers explicitly tests the role of the centrifugal force’, repeated in other words as ‘if the bicycle and the rider both lean to one side, there is no centrifugal force to correct the lean on rollers . . . ’. This claim is most easily refuted for treadmill riding, to which the authors suggest their ideas also apply. Riding a bicycle on a constant-speed treadmill is actually (neglecting wind resistance) mechanically identical to riding on fixed, level ground at a constant speed. The most straightforward explanation for this is Galilean invariance, which says that the laws of mechanics are the same in all reference frames moving at a constant velocity relative to each other. To argue against this invariance, one would have to show that the mechanics have changed at some point in the following short series of experiments.

(1) Ride a bike on the deck of a long stationary ship straight toward the stern at 18 km/h, as indicated by a speedometer on the bike’s handlebars.

(2) Ride on the same ship that is now traveling forward at 18 km/h relative to the shore, straight toward the stern at 18 km/h, and thus remain stationary relative to an observer standing on shore.

(3) Bring the ship back to rest, and ride straight toward the stern at 18 km/h on a deckmounted treadmill whose belt is moving straight toward the bow at 18 km/h, and thus also remain stationary relative to an observer standing on shore.

In a video of just the bike and the surface on which it rides, the scene in experiment (3) cannot be distinguished from the scene in experiment (2) because the mechanics are in fact identical. And those, in turn, are identical to the situation in experiment (1). Thus, riding on a treadmill is mechanically no different from riding on fixed ground, and any differences between riding on rollers and riding on fixed ground are, in fact, differences between riding on rollers and riding on a constant-speed treadmill.

The second issue is the paper’s reinforcement of the common misconception that inertia or forward momentum somehow provides stability. This is indicated by the phrases ‘forward inertia . . . aids in bicycle handling for stability’ on page 1293 and ‘the loss of inertia limits the degrees of freedom in bicycle stability’ on page 1299. This point is again simply refuted by the Galilean invariance: in a frame of reference that is stationary with respect to the ground, a bicycle on a treadmill has no forward momentum, yet it has the same balance dynamics (as argued above).

We could not understand Cleary and Mohazzabi’s reply and so discontinued the discussion.