Technical FAQ: A detailed battle against resonance in wheels

In Technical FAQ, a reader describes, in detail, his work to eliminate high-speed shimmy through attacking resonance via wheel tuning

Perhaps you remember the most recent discussion we had here on high-speed shimmy. A math professor and avid cyclist offered the “Hopf bifurcation” explanation for frightening phenomenon. I’ve received a lot of letters about that, but this one intrigued me because it gives a very clear recipe for alleviating shimmy problems based on a wheelbuilder’s many years of experience.

Dear Lennard,
As a custom wheelbuilder, I encounter many variations of wheels on many different types of bikes ridden by many different types of riders, from rank amateurs to pros. I get asked about shimmy all the time and how to solve it. The article on bifurcation phenomenon seems to address the core issue, however, I also suspect there is a harmonic resonance aspect as well.

As you know, it’s nearly impossible to build a perfectly round wheel with perfectly consistent spoke tension. The result of the imperfections will result in a device that has variance in the rotational energy. And, as the wheel rotates, the variance acts as an oscillator, thus generating a true harmonic amplitude response. Combine with the mass of the wheel, angle of dangle, etc., and yes, you do have harmonic resonance as a notable factor.

We use a Japan Robotics wheel analyzer in our building process to determine the rotational balance of the wheel. It has a 3-axis accelerometer and provides a simple, digital graph output similar to an oscilloscope. The period of rotation may be changed as well as the scale of amplitude. Furthermore, the wheel position is noted in 0.1-degree increments, so we can ‘see’ where the periodic energy is affected (nulled, or maxed).

In building some test wheels, I have purposely built them out of round and also with varying tension, then run the wheels on the analyzer to check results. As expected, the wheels all experience different flavors of oscillation pattern. And also, I noticed that ‘most’ wheels tend to go into a harmonic-induced oscillation with similar rotation envelope — roughly 6-8 fps. At the point of peak amplitude the wheel becomes a galloping gertie.

Now, my interest there is piqued, because I want to know how much energy is being transmitted non-tangentially — or out of round. Well, I made a very crude vertical-energy measuring device consisting of a scale and mini-rollers. This allows me to apply the rollers to the spinning wheel and measure the bounce, as it were. At speeds of peak resonance this results in a several kilograms of bounce force! I’m not sure exactly how much — I would need to build a much more sophisticated measuring tool, however, it is significant.

Now then, lets add in the bifurcation phenomena. I’m also a licensed pilot (Helicopter) and well accustomed to the issues that may result if one exceeds VNE (Never exceed speed). Mostly in aerodynamic structures, this phenomena is result of aerodynamic pressures that stress the structure (deflect it) in such a way to cause energy storage that cannot be damped: the wind moves the wing, it stresses and builds stored energy to the point the angle of attack changes and allows the wing to ‘spring’ back, whereupon it resumes an ideal angle of attack to pick up an even greater aerodynamic pressure, and so forth, until the wing eventually fractures from the excessive distortion.

On a bicycle, a similar situation may occur — only instead of aerodynamic input (though that can happen, too), you have the rider’s input via handlebars. Most riders cannot dynamically dampen themselves. The neuro-muscular response is too slow, which results in actually adding more energy to the bifurcation process eventually resulting in contact patch failure, then a crash.

So, let’s look at that contact patch and go back to the dynamic balance of the wheel. I’ll wrap this up very quickly. As you also know, friction between surfaces changes according to the weight or pressure applied to them. When a bicycle wheel goes into harmonic resonance, the pressure on the road will change substantially. Add in a small disruption, which the rider perceives as a ‘shimmy inducer’, and you will start the escalation of the bifurcation phenomena.

So, it is actually a combination of several systems that results in the dreaded wheel shimmy.

As a case study in correcting this, I rode the Haute Route two times, which provided many opportunities for high-speed descents over lengthy distances. The first time I rode I experienced wheel shimmy more than several times. I tried all the usual things — differing tire pressures, stem lengths, etc. I noticed that especially under conditions of duress (cold, tired, etc.) the shimmy was worse — that gave me my first insight into the biomechanical issues. Sometimes it got better with different wheels, sometimes not. The one thing in common was that none of my wheels were dynamically analyzed and balanced. The second time I rode, I used sets of wheels that I had specially balanced (laterally and radially) to at least 140 kph — hopefully 2x the max induction period. I experienced minimal shimmy under the same conditions (stages were same in some cases, duress, etc.) and my result was I could descend at much higher (90 kph+) speeds with confidence with nary a thought of impending shimmy.

I’m not saying it’s a magic bullet — but I do know in my case, and others I’ve corrected, it does help to consider the dynamic balance of the wheel along with the other usual suspects.

My personal ‘recipe’ for solving these issues is:
1. I rebuild the wheel so it’s finally round and equally tensioned. Probably less than five percent of the wheels I see from customers are built correctly (including all major brands and supposed hand builts);
2. I remount the tires then test on the JR Analyzer and apply corrective weights. If the weight correction is deemed excessive then I dig further — I’ve seen other issues causing this like tire construction, bent axle, bad bearings, etc. So, it’s not just adding weight that corrects the wheel;
3. I check all headset bearings and ensure 100-percent smoothness and alignment;
4. I check rider position and especially determine if they are riding at a neutral position, thus reducing the feedback potential into the stem; and
5. I check the alignment of the bike itsel; if it does not track straight, the rider will try to correct. This results in the neuro-muscular effect becoming significant to trigger bifurcation.

Based on all of that, I am generally able to reduce or eliminate most shimmy issues.

I know the above is hardly qualitative; however, it is factual based on the hundreds of cases that I deal with yearly. I guess I could take the time and put some math to it, but I’m rusty and there are other engineers with far better tensor skills than I!

Thanks again for your insight into the world of cycling!
— Tim Smith, Founder, GS Astuto

Dear Tim,
That is very cool. Thanks for taking the time to write it. I love that you have come up with a recipe for fixing shimmy. Since I have worked out a similar recipe when it comes to designing frames and choosing the components to attach to them, I am excited to see another reasoned trial-and-error means of attacking the problem, no matter whether it is due to resonance or Hopf bifurcation. I always thought it was resonance, and, over three decades of building frames, I have managed to make many riders who were plagued by shimmy problems on their old bikes happy without ever having heard of Hopf bifurcation.

I ran your letter by Prof. Bollt, and he wrote this:

It is not impossible for a cycle system to have a resonance, but I was arguing that is not the source of shimmy in most cases you are likely to run across. The deciding factor being a resonance occurs at a specific frequency, and as you get below, or above that frequency you would expect the problem to go away. If that is what is observed, you are looking at resonance. If the problem just builds and does not go away with increasing speed, then it is the nonlinear phenomenon, the critical onset, the Hopf bifurcation.

Your friend is describing something as if it is half one way and half the other way. I do not think I can say specifically what he is saying, but let me say it would be either one or the other.

Part of the outcome of his discussion, and your own experience, is that a good experimental engineer can engineer away the problem by trial and error and good intuiting and experience without ever having encountered the word “Hopf”. True enough.

― Lennard