Technical Q&A with Lennard Zinn: The great rotating-weight debate
Back at Christmas, I threw in a letter from a reader claiming that rotating weight makes almost no difference on a wheel – that it takes negligible energy to bring it up to speed, and that the only thing that really matters when climbing is the overall weight of the bike, not how it’s distributed. Since then, I have gotten a lot of mail about this, and a trip to France last week piqued my interest in this subject again. Perhaps some of you remember when I did a test in VeloNews seven years ago (in the 6/28/99 issue) of wheel inertia by building a rotational pendulum in my garage. From twisting the rod running down through the hub with the wheel fixed from rotating relative to the rod, I measured the period of the back-and-forth twisting oscillations and from it deduced relative wheel rotational inertia. I was just at a Mavic tech seminar in Annecy, France, last week and saw a test machine set up virtually the same in the Mavic test lab (although it had a digital counter, rather than a guy with a stopwatch!). Having seen first hand from both of these setups that the period changes significantly with added weight out at the rim, particularly with adding a thicker innertube or a larger-diameter and/or heavier tire, I know that the extra energy required to accelerate additional rotating weight, whether it is in the shoes and pedals or in the rim and tire is not zero. I also know that very few people (probably none, actually) climb smoothly, without accelerating the wheel (and pedals and bike) each time a foot pushes down and decelerating it each time it goes across the top or bottom, particularly when climbing out of the saddle. This of course is more extreme in the case of accelerating out of corners in a criterium, but then you do not have the additional energy required of carrying the entire weight up a climb. And you probably know that Ondrej Sosenka believes that higher rotational inertia to maintain momentum through the dead spots is an advantage on steady efforts on the track, since he used super heavy rims when he broke the hour record last year.Tom Compton’s site also has some interesting calculator pages on subjects like this. Be sure to click on the TOOL KIT button to change to different scenarios, like criterium cornering, sprinting, etc. (More than just an übergeek, Tom is also the father of cyclocross champion Katie Compton)
Finally, here is a small sampling of the letters I received on the subject:
In your 12/27 column, Peter’s “you can accelerate a wheel to climbing speed with your finger, therefore it’s insignificant” statement was enough topush me off the dime.It’s obvious if two equal masses are both traveling (translating) at the same velocity, but only one is rotating, then the one that is rotating holds more kinetic energy than the one that isn’t. That energy comes from only one place, the rider, and here are the numbers. I mathematically “built” a wheel from Dura Ace 10 speed hubs, 32 DT Competition (2/1.8/2) spokes, Mavic Open Pro rims, and Michelin tires and tubes from weights taken from weightweenies.com. I did it both with brass spoke nipples, and with alloy nipples, for a savings in weight at the rim of about 20 grams per wheel. Next I made some simplifying geometric assumptions, and calculated polar moment of inertia for each component including rims, tires, tubes, spokes, nipples, hubs, and cassette. With both linear inertia (mass) and polar moment of inertia in hand, I looked at accelerating from 20 mph to 30 mph over a period of 10 seconds. Power for this acceleration falls out as follows:
Using Brass-nippled wheels Linear term (the same whether the mass is rotation or not, i.e. frame or wheels): 210 w
Rotating term (relevant to wheels only):115.5 w
Total power: 325.5
Note that ~35% of the kinetic energy stored by this acceleration went into rotation.
Using Alloy-nippled wheels:Linear term: 206.6 w
Rotating term: 112.6 w
Total power: 319.2 w
Switching to alloy nipples (20 grams per wheel saved) saved me 6.3 watts total to perform this acceleration. If I shaved the same amount of mass off my frame instead, I would enjoy only the savings from the linear term, or 3.4 watts. Everybody gets to decide for themselves whether 3 watts is “significant” or not, but I wouldn’t at all be surprised if the human engine is sensitive enough to feel the difference.
Your analysis of rotational inertia neglects the simple fact that bikesare human-powered. Biomechanical forces being what they are,your conclusion errs by a factor of 10,000 or so.Try pedaling with one leg sometime. You will find yourself lurching forward on the down stroke then drastically slowing as you spastically struggle to get the crank back in position for round two. Of course using two legs and good cycling form helps matters, but only to a certain extent.The lesson is that all cyclists apply effective power to the drive train in pulses – the emphasis being n “effective”. Only the component of force that is perpendicular to the crank at any instant contributes to forward motion. Therefore the right pedal mostly applies effective force between 90 and 180 degrees whereas the left pedal is doing the same mainly between 270 and 360 degrees. The blue line below illustrates the general shape of effective force vs. crank angle for a single pedal. To get the combined force of both pedals in general you can add anothercopy of the curve shifted by 180 degrees. The end result is that you put TWO discrete pulses of useful force into the drive train every revolution.
For a nice explanation of pedal forces see this document.
By the way, the well understood dead spots in pedal stroke account for all manner of crank contraptions such as BioPace, Rotor, O’Symetric, etc.,which aim to diminish the effect or alternately exaggerate it (but that’sanother topic entirely).
As you may have extra time to recall while propelling your heavy wheels uphill, F=M x A. Assuming rotating mass to be constant we can deducethat the acceleration curve is identical to the effective force curve (only with different units). More interesting perhaps is the plot of instantaneousvelocity vs. time, which we obtain by integrating the acceleration curve. Integrating reveals the red curve above, which is roughly sinusoidal with one positive pulse per cycle (two pulses when accounting for both pedals). In this case it’s easy to see that the constant term or offset simply represents the average angular velocity. We now have a periodic function that describes a crank that is rotating at a variable speed. In more simplistic terms – when you stomp the pedal the bike goes faster, when you don’t it slows down. As an example, you may be moving up the hill at an average rate of 10 MPH but in reality you could be varying from 9-11 MPH through each revolution.”So what?” you might ask. Well Peter, the bottom line is that you are accelerating your wheels twice per revolution and that can add up to some serious effort. Not just one measly finger flick as you commence your ride as you once opined. So how far has your analysis strayed? Well for starters it’s really two wheels you have to accelerate (2X). Then twice per revolution (2X). Let’s see – 80 RPM if you’re doing good or suffer an unsightly triple chainring (80X). And there’s still 60 minutes/hour even at such a high elevations (60X). Then I know I personally optimized my rig to get me through a 3-hour climb in the Appalachians – that’s another factor of three (3X). So that’s 2×2 x80 x60 x3 = 57,600 flicks of the finger per 3-hour ride. But then we must consider that instead of accelerating the wheel to 10 MPH we’re really accelerating it by, say, 2 MPH (from 9– 11 MPH per our example). That’s a factor of 5 or back in your favor. So then we have 57,600 divided by 5… amongst friends let’s just call it a nice round factor of 10,000. So how could we interpret this whopping discrepancy? Well, instead of flicking your deliberately dense 3-pound wheel/tire combo, picture yourself spinning a 15-ton wheel up to speed! Still can’t picture it? Thirty thousand pounds is roughly the weight of three and a half Hummer H2’s (8500 lbs GVW). Allowing some margin for rolling resistance. I surmise that the effort spent accelerating your heavy wheels over thecourse of a 3-hour mountain ride is akin to the effort required to towthree Hummers up to 10 MPH! But don’t worry though, from what I read somewhere, after you get them up to speed, it doesn’t take much effort. Still don’t believe me? Try another experiment. Ride up your favorite mountain pass with a pound of fishing weights in your jersey pocket. Then do it again with the weights mounted out on the spoke nipples. You won’t need a stop watch to figure out how much the second configuration sucks. Now do you see why folks plunk down $5000 to ride easily destroyed carbon wheels? My own crusade against rotational inertia motivated me to re-appropriate a discontinued 650c tri-bike for climbing purposes.I find the combined effect of wheel/tire weight savings and reduced moment of inertia (related to radius ^2) save an entire Hummer’s worth of effort over my mountain course of choice.
Technical writer Lennard Zinn is a frame builder (www.zinncycles.com), a former U.S. national team rider and author of several books on bikes and bike maintenance including the pair of successful maintenance guides “ Zinn & the Art of Mountain Bike Maintenance” and “Zinn & the Art of Road Bike Maintenance.”Zinn’s regular column is devoted to addressing readers’ technical questions about bikes, their care and feeding and how we as riders can use them as comfortably and efficiently as possible. Readers can send brief technical questions directly to Zinn. Zinn’s column appears here each Tuesday.