The rollout of a Coker wheel

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A Coker is a famous and fairly commonly used unicycle with a large wheel. The wheel is nominally 36" in diameter but exactly how large it is, is the very subject of this page. The exact wheel size is of particular interest for riders who use a cycle computer on their Coker. Over the years, in the newsgroup rec.sport.unicycling many rollout values have been given and in 2003 I offered to integrate all available data into a model. For the Coker only one tyre type is available (manufactured by Coker), which simplifies things as far as 'the' wheelsize is concerned. On the other hand, two types of rims are commercially available (standard and Airfoil) but hardly anyone specified which rim they use and I have ignored a possible size difference between the two. 

Before going into 'how I did it', here is the result. Download Rollout_predictor.xls (by rightclicking and then Save As), open it in Excel 4 or higher, and enter your rider weight, tyre pressure and tread wear. See screenshot if you want, use your Back button to return here. You will get three numbers:

  • unloaded straight rollout, this is the distance that the wheel rolls during one full revolution on a flat surface if you push it in a straight line, under the weight of the unicycle alone. 
  • loaded straight rollout (with rider); this is the distance that the wheel rolls during one full revolution on a flat surface in a straight line, under the weight of the unicycle with a rider on it. 
  • dynamic road rollout (incl wobble); this is the distance covered during one wheel revolution under realistic road riding conditions. 

Enter the number of your choice (see below) in your cycle computer, and you're off to go!

Why these different numbers? The unloaded straight rollout is used by many Cokeurs, partly because it is the easiest and most straightforward number to measure. Also, since it is rider-independent, it is the best basis for comparison. However, because the tyre will always compress when the unicycle is ridden, the unloaded straight rollout gives too high speed and distance values, which can be a reason to prefer a loaded rollout value (I phrased that carefully). There are two loaded rollouts because there are (at least) two schools of thought as to what 'speed' and 'distance' mean in relation to unicycling. If one wants to know the speed and distance that the tyre contact patch has travelled along its path, including all the wobbling and turning that one may do, then one should base oneself on the loaded straight rollout. However, if one is interested in road speed, or road distance (i.e. the distance from A to B), one should use the dynamic road rollout. "Dynamic" means to include all deviations from riding a straight line, such as single-cycle and multi-cycle wobbling (both horizontal and vertical), the way one torques up a hill, etc. U-Turn, fanatic Cokeur and builder of the Strongest Coker Wheel in the World, considered these factors in more detail in a post in rec.sport.unicycling

Some issues and conclusions worth mentioning:

  • For starters: I had to work with a very limited dataset. While the model predicts the input rollout values on average within 10 millimeters, the accuracy would certainly increase with more data. Oh well.
  • For every kg increase in rider weight, the (loaded) rollout decreases by about 0.32 millimeter.
  • For every bar increase in tyre pressure, the (loaded) rollout increases by almost 15 millimeters.
  • For every additional % in tyre wear, the rollout decreases by almost 0.3 mm. From that number, I can derive that the tread height (or more precisely, the difference in radius between a new tyre and a completely worn-down one) on the Coker tyre is 4.5 mm. I've never seen a Coker tyre in person, but it seems realistic. 
  • A modeling result is that the effects of dynamic riding (as opposed to riding in a perfectly straight line) decrease the rollout by about 5.5 millimeters. In fact I think this is too low, and more data might give a more realistic value. Ken Fuchs was the only individual who measured both straight rollout loaded with rider and dynamic road rollout under comparable conditions, and assessed a 20 mm difference. That looks more realistic to me. However, dynamic riding effects will be very much rider-dependent anyway.
  • Lastly, a word of warning. Some people measure the distance between wheel axle and riding surface and take that as a "loaded wheel radius" to calculate the loaded rollout. This is based on a wrong assumption. The effective rolling radius of a loaded wheel is in fact somewhere between said distance and the unloaded radius. The tyre contact patch does not have a single radius. At the ends of the centre line the 'radius' is the unloaded tyre radius while in the middle the 'radius' is the distance from the wheel axis to the ground. At the sides of the tyre contact patch the situation is even more complicated (3D).

* * * * *

For those who are interested in how I arrived at my model, I will now give some detail. Warning: this is not hard science, but the best I could do based on the sparse and inconsistent data at hand. 

Firstly, I collected all data that I could lay my hands upon. This was all from the newsgroup rec.sport.unicycling over the last couple of years (since the Coker unicycle came into existence). Some of the data were not given in a very 'hard' way, and I have firmed them up before the analysis. The available data are presented in this table:

name

rider weight lbs

tyre press psi

tread wear %

unloaded rollout mm

loaded rollout mm

dynamic rollout mm

undefined rollout mm

cycle comp value*) mm

remarks

John Childs

 

 

 

 

 

 

2775

 

Not used as the type of measurement wasn't specified

U-Turn

 

60

 

 

 2800

 

 

 

 U-Turn contributed few numbers but made some very useful comments

Ken Fuchs

 

32 

15

2839

2789

2769

 

 

 Deluxe Coker

Nathan Hoover

165

45

50

 

 

2776

 

 

 pre-Deluxe

Tom Blackwood

 

 

 

 

2756

 

 

 

 

Mark (cokerhead)

237.5

47.5

 

 

 

 

 

2810

 Not used. Based on road markings, bicyclists and car odometers, erring on the long/fast side

duaner

150

32

50

 

 

2782

 

 

 

Chuck Webb

 

 

 

 

 

2800

 

 

 Based on making small adjustments based on road markings etc. 

Mikefule

 

 

 

2873

 

 

 

 

 Not used, was based on (measured? nominal?) diameter of 36"

David Stone

192.5

52.5

 

2852.7

 

 2789.2

 

2810

 

Michael Grant

 

 

 

2819

 

 

 

 

 

Danny Colyer

 

 

 

2830

 

 

 

 

 Number allegedly based on past Coker threads (which I (Klaas Bil) wasn't able to find)

Ken Fuchs (again)

 

35

75

2858

 

 

 

 

 Not used, was measured with a tape around the wheel which will yield a too large value

Kevin Gilbertson (Gilby)

 

 

 

 

 

 

2820

 

 Not used; seemed unreliable. Quote: "I've heard measurements range from 277 cm to 285 cm for the circumference of the Coker wheel. I've had mine set to 282 cm before."

Joe Marshall

 

 

 

2870

 

 

 

 

 Not used, seems based on the nominal diameter of 36"

*) A value for the cycle computer setting is only included in the table if it is remarkable or if there are no other rollouts given by this individual.

The italic lines in above table have not been used in the analysis, for the reasons specified in the "remarks" column. 

As you can see, no-one reported a 'full' dataset. To increase the number of data points available for the analysis: 

  • I assigned the average rider weight (84 kg = 186 lbs) to the riders who did not enter theirs, 

  • I assigned the average tyre pressure (2.9 bar = 43 psi) to the tyres of the riders who did not enter theirs, and 

  • I assumed that tyres with an unstated wear status were worn halfway down.

I have built a numerical model of tyre compression, from which I have concluded that there is a near-linear relationship between compression on the one hand, and both load and tyre pressure on the other hand. (If anyone is interested in the details let me know.) Therefore I have assumed linear relationships between the three input parameters rider weight, tyre pressure and tread wear on the one hand, and the three 'output' parameters (i.e., predicted values for the three types of rollouts) on the other hand. Furthermore I assumed that the unloaded rollout was not affected by rider weight (duh), and that the effect of tyre pressure on unloaded rollout was only 10% of the tyre pressure effect on loaded rollout (because a Cokeur is typically 9 times as heavy as a Coker unicycle). Then I performed a multivariate least-squares linear regression for the whole dataset at once, i.e. solving for the minimum sum of the squares of the differences between the given and the predicted rollout values. This processing was done iteratively, using the Solver Add-In of Excel with the default settings. The parameters that were varied in doing the minimisation of the sum of squares were:

  • a start value for the rollout (in fact this is the unloaded straight rollout at the average rider weight, average tyre pressure and 50% tyre wear, a value that can be calculated directly from the input data)

  • a coefficient to account for rider weight

  • a coefficient to account for tyre pressure

  • a coefficient to account for tyre wear

  • a (negative) amount of rollout to add for a loaded versus an unloaded rollout

  • a (negative) amount of rollout to add for a dynamic versus a (straight) unloaded rollout.

 The optimised values for these parameters are included in rows 3 and 6 of Rollout_predictor.xls (just increase row height to view them). You will note that, contrary to my usual scientific religion, I did the actual calculations in the imperial unit system. 

 

Klaas Bil, July 2004

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