Technical FAQ: Bifurcation and high-speed shimmy

In Technical FAQ, a math professor explains the phenomenon behind high-speed shimmy, and says it's not resonance

Photo: Watson

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This week we address one reader’s questions on countersteering a bike versus a car and an inquiry over the impact of bottom bracket drop on stability. A mathematics professor also checks in with a detailed explanation of high-speed shimmy sure to make your head hurt, but hopefully not as badly as a wobbly, high-speed crash.

Are you really countersteering?

Dear Lennard,
I have learned a lot of interesting technical facts from Zinn and the Art of Road Bike Maintenance and Zinn’s Cycling Primer books. The primer book has a lot of technical facts that are of interest even to the seasoned cyclist.

On page 96 of the primer, you show a photograph to demonstrate the principle of countersteering. Looking at the photograph, clearly the two leading racers are not countersteering. Their front wheels are definitely turned more into the turn that their back wheels. The front wheel projections into the photo are wider that the rear wheel projections.

Looking at this Wikipedia page, the motorbikes are clearly countersteering. Note also that the racers are moving their mass center toward the inside of the turn, keeping the bikes more upright, the opposite of what you describe on page 96.

One countersteers on a car or on a motorcycle when the rear wheels slide out of the turn due to the large motor torque or initiated by the torque. A FWD car typically does not oversteer and thus typically does not require countersteering.

On a bicycle, countersteering is the little turn to the right that one makes before making a left turn as you so well explain on page 95 and as described in the above Wikipedia article. On page 96 of the Zinn primer, I do not follow the logic. I also tried, while going slow and straight, to lean the bicycle and I kept going straight with a handlebar more or less straight. The tires made more noise but the bike still went straight. Maybe I was not going fast enough.

Dear Pierre,
On dry pavement, particularly at low speed, you won’t see it at the wheel, but you can see it in the rider’s arm position. Other than the initial reverse steer to initiate the lean, often the only time the wheel is actually turned the opposite direction from the turn is on dirt, snow, or ice, where traction is greatly reduced and the tires are sliding. On a road with good traction, despite the countersteering pressure from the hands, the front wheel is generally still turned into the turn or in line with the bike.

You still maintain the countersteering pressure on the handlebar, even if the wheel does not continue to be pointed in the opposite direction of the turn. Initiating the turn does involve the wheel turning the opposite direction from the way you want to go, though.

It definitely works, and I have taught innumerable people to do it, including lots of kids. Many times, it takes them a while to get it because they don’t commit the hand pressure. I imagine you’re having trouble getting it to work because you’re just bumping the handlebar in the opposite direction and then removing the pressure. Either that or you’re trying to lean your body, rather than allowing the steering action to drop you into a lean.

Try it riding across an empty parking lot. Ride in a straight line, and it doesn’t have to be very fast, but the faster you go, the better it works. Push the left end of the handlebar forward with the left hand to turn the wheel to the right, and maintain that pressure. The bike will immediately lean to the left and then turn left for as long as you have speed and maintain that pressure pushing the left end of the handlebar forward.

Of bottom bracket drop and stilts

Dear Lennard,
Your recent post about bottom bracket drop prompted me to send a question I have been thinking about for some time. Basically, how does bottom bracket drop affect bicycle handling? You compared two riders and bicycles, one short rider with short cranks, and one tall rider with longer cranks. Both bikes had the same amount of pedal clearance, but the bigger bike had less drop. Purely for the sake of this discussion, let’s assume that the two bikes offer identical positions to the riders — leg extension, reach to the handlebars, weight distribution over the wheels, and so on. If this were true, then the only significant difference between the two frames would be the bottom bracket drop.

Do you think this would affect handling? I have heard bottom bracket drop compared to stilts. Roughly, stilts with low footsteps are like bikes with more bottom bracket drop, and stilts with high footsteps are like bikes with less bottom bracket drop. This would lead me to believe that, everything else being equal, bikes with more bottom bracket drop (bottom bracket closer to the ground) might be more stable than bikes with small bottom bracket drop. Of course, the differences may be so small that we can’t feel them.

As a related question, how much higher are the hub axles on 29ers as compared to 26-inch mountain bikes? I imagine that the pedal clearance on 29ers is the same as 26-inch bikes, so the bottom bracket drop must be larger on the big-wheel bikes. Do you think this plays a part in the relative stability or comfort of 29ers?

I imagine that, because of your frame design philosophy, you probably have a great deal of experience building bikes with varying amount of bottom bracket drop, so I thought I would ask.

Dear Bruno,
The comparison with stilts doesn’t apply here, because the higher the step, the higher the center of gravity of the stilts and the person walking with them. On a bike, if you raise the bottom bracket the same amount that you lengthen the cranks, then the center of gravity of the bike and rider is the same either way, provided the seat height from pedal to top of saddle is the same on both bikes.

Consequently, when the rider is pedaling, the stability of the bicycles with similar geometry will be similar. That said, on a mountain bike, when descending standing on the pedals with the cranks horizontal, the rider’s center of gravity will be higher on the bike with the higher bottom bracket (less bottom bracket drop); this will decrease the stability of the bike.

With the same width tires on each, the axle is 31.5mm higher on a 29er wheel than on a 26-inch wheel. (The bead seat diameter of a 26-inch mountain bike wheel is 559mm, and on a 29er, it is 622mm; half of the difference between these numbers is 31.5mm.) The extra 31.5mm of bottom bracket drop on the 29er that maintains the same bottom bracket height as the 26er means that the bottom bracket is hanging lower between the wheels than on a 26er. I believe this adds to stability. Furthermore, the gyroscopic stabilization caused by the wheels is increased with the bigger, heavier 29er wheels.

A math prof explains high-speed shimmy

Dear Readers,
I received a letter from a math professor pointing out that bicycle high-speed shimmy is not a resonance phenomenon like I said it was in a recent column; it’s a nonlinear bifurcation phenomenon called “Hopf Bifurcation.” So, I asked him to expound on this, and I’m putting it in here, because there may be some of you who find this as fascinating as I do.

Dear Lennard,
A linear analysis leading to resonance is appropriate for any system where there is an oscillator that is being forced at a special frequency — the resonance frequency — and when this happens, the amplitude can simply build to infinity. This is not what happens in bicycle instability for two reasons: first, there is no periodic forcing that causes the high-speed wobble (in fact, it can happen on a smooth road); and second, there is not a phenomenon that shows a characteristic building of amplitude.

Instead, the high-speed wobble is a critical phenomena, which is typical of bifurcations and bifurcation theory in general. Below the critical parameter value, you see one thing, in this case a stable equilibrium characteristic of a smooth ride, and slightly above the critical parameter, the smooth ride is no longer stable (but it still exists as an equilibrium, but an unstable equilibrium, just as standing a stick upright is an equilibrium but unstable because if it tips even slightly away from the exact equilibrium, it quickly drifts away), but the now unstable equilibrium gives way to a stable periodic orbit, which is the wobble. And as the parameter increases, the amplitude of the wobble can increase to some larger but fixed amplitude.

Also, Hopf-born limit cycles are self-exciting if you like to see it that way, as opposed to resonance that requires an external forcing to excite the resonance frequency. (For resonance, think of a bridge with a special characteristic frequency and if soldiers march over it moving their feet just at that frequency, then you have resonance; this happened when I worked at the Naval Academy when some of the midshipmen forgot to break step and they actually did crack a walking bridge!)

There are many types of bifurcations, and they explain all sorts of critical phenomena on nature, and this type, the Hopf bifurcation, explains the onset of oscillations in all sorts of natural systems, from population dynamics, to chemical reactions, to airplane wing flutter, to stability of steerers in bicycles, trucks on trains (leading to derailment) to landing gear wobble on airplanes.

To many engineers not in the know, these oscillations and their onset are often mistaken for resonance, but it requires a nonlinearity in a certain way. This is often guessed by the critical onset of the phenomena as a parameter is adjusted.

For the bicycle scenario, the parameter in the engineering phase could be increasing the stiffness build of the bike — a stiffer bike will be more resistant. The several design parameters could all be “nondimensionalized” to a single, non-dimensional parameter as is necessary to explain with Hopf since it is a one-parameter phenomena (this concept is called co-dimension-1 bifurcation). Or once the bike is designed and built, then the parameter is speed. Each bike has a critical speed at which it will cross the Hopf bifurcation value, and then steady state becomes unstable and wobble becomes the stable state. This, by the way, is why it is very, very dangerous to try those 150 mph downhill bicycle attempts!

I am including some video of the flutter phenomenon which is an aeroelasticity phenomenon — airplane wings oscillate — you have probably seen it out of your window of your last plane ride (hopefully if an airplane hits a bump in the air, the wing oscillates back to its stable loaded state). But if the airplane goes faster, this state becomes progressively unstable. At a critical speed, the equilibrium becomes unstable, and there is born a limit cycle, which is a state where the airplane wings oscillates at a fixed amplitude. But the amplitude increases with increasing speed. Each airplane has a critical speed marked on its airspeed indicator (called VNE standing for “Velocity Not Exceed”). I have read that the FAA marks this fastest “safe” speed as 30 percent lower than the actual critical Hopf instability speed to build in a margin of safety. Often the wings fall off if you go Hopf. That’s bad.

Video 1 >>
Video 2 >>
Video 3 >>
Video 4 >>

The Tacoma-Narrows Bridge is also often cited as an example of resonance, but it broke also because of Hopf — almost for the same reason as the flutter in airplane wings.

By the way, in bicycle speak, Hopf bifurcation gives rise to what is called “weave” as the stable limit cycle.

And also, there is another characteristic bifurcation in the cycle mechanics called “capsize,” which is due to another bifurcation called the “pitchfork bifurcation”. (Pitchfork is also responsible for all sorts of exciting phenomena, including beams buckling.)

Funny timing, your waiting until today to ask about this, because it was today that I was working in great detail in my graduate differential equations class about the Hopf bifurcation, so I had collected some of these videos yesterday. …

I just stumbled across this webpage that seems to, at a glance, properly cite both Hopf and Pitchfork to explain weave and capsize.

Wow, here is a whole book that includes the proper discussion of Hopf, and trains and cars, etc. And here is an article focused on trains. Here is a decent write-up about the technicals of what is the Hopf bifurcation, including its normal form (a universal form hiding in the equations no matter what the application, if it is indeed Hopf.
—Erik M. Bollt
W. Jon Harrington Professor of Mathematics
Clarkson University

Dear Erik,
That’s cool! I was taught in college physics class that the collapse of the Tacoma Narrows bridge was an example of resonance. This is amazing to find out now that it was not. And bicycle high-speed shimmy I understood to also be resonance oscillation.

The good news is that the fix for the bicycle seems to be the same either way; making the frame torsionally stiffer and the wheels laterally stiffer deals with shimmy, whether it’s due to Hopf Bifurcation or resonance.

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