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I think a wide range of tires is ok. Consider the surfaces you may be measuring on. If streets with lots of potholes, broken glass, nails and the like, you may want tires that will handle stuff like that, and inner tubes also. But if regular roads without too many hazards, you have lots of choices that will work. On gravel, bigger tires with a rugged tread can help you keep your balance.

One caveat from my experience, others may disagree but tires with too much tread relative to size have given me problems, behaving in unexpected and unpredictable ways. I've thought this may have to do with the negative coefficient of expansion of rubber but I haven't worked this out very far.
Bob offers good advice on tire selection. This is an area I'd like to see more practical research on. I've been meaning to do some, and I suspect Pete Riegel and others must have visited the topic. I don't have the answers today - but I have some clues...

I think there may be a trade-off that you, the rider, need to balance based on your knowledge of the features of the course to be measured and the weather expected during the rides. You may want to have a selection of tires available for different situations.

One of the biggest challenges to good calibrated bike measurement is change in ambient temperature during the rides or between calibration and rides. A result of temperature change is a calibration finish-constant that varies from our working-constant(s). Big temperature changes may yield big variances in your tire diameter, and therefore differences in calculated measured distance over the course.

Our procedures suggest strongly how to handle it. We will usually err on the side that produces the longer course. In Canada they take the average constant. In the USA we take the larger constant, but are given discretion to take the average - if we can justify it. But, I think I'd rather not have to "err" at all!

Here's a pertinent observation from a point-to-point marathon team measure this summer: I had cross-bike tires. These tires are larger than racing bike tires and have some tread (i.e., more air volume, more rubber). The other riders essentially had the same diameter rims, but used the skinny little racing style tires. We pumped up to spec, calibrated, rode the course, then re-calibrated as usual.

I felt I got a slightly better ride than the other two riders. I got better traction, rode up all hills in a straight line, never had to walk, held the SPR well over both solid and loose terrain. That part is good. Note, there was a significant temperature change during the ride.

But upon calculating the finish constant It looked to me that I suffered the biggest variance in tire diameter due to the ambient temperate change! That is less good.

We theorized that the physically lower-profile tire may experience less volumetric (air) expansion and that would be a real advantage toward attaining an accurate measure, as opposed to just a measure that is not short.

So, which tire type would generally be the best?

I wish I knew.
I have experimented with several setups in the past couple of years. I currently use a skinny 120 lb racing tire on a 700 C bike for longer, steeper rides and an insert on a 26" bike which is on either a 559 or 590 rim for flat, in town 5Ks. Both do quite well. Any given 26" rim will NOT fit any 26" bike. Finding a rim and tire to match a given bike can feel like hitting the trifecta. My 26" airless insert will add the least unnecessary distance to a course. When using 2 riders, I use whatever set up will best match my partner's for the sake of agreement.

What does not work well? I got a mountain bike with 65 lb tires after seeing Pete Riegel's meter mounted on one somewhere. The first time I measured a 5K on rapidly rising temperatures, I had to go back and redo a mile. That lead me to a 75 - 95 lb tire for that bike. I had to redo a mile for agreement on an out and back 10 miler pumped to about 85 lb. At 95 lb, I had no problems with agreement. But I was making courses longer than buddies with 120 lb tires.

My first airless tire was the insert commonly available at Wal-Mart and it works well. Only problem is that it unpredictable regarding recalibration being shorter than calibration: the counts are almost exactly the same but .5 counts longer on one of the 4 recal rides and you theoretically have to go add distance to the course.

65 lb class airless tires (not necessarily inserts) are VERY hard to pedal. You sometimes have to (had to) pedal down hill. Furthermore, the best whole airless tire I could find for the old mountain bike would sometimes show a radical difference at recalibration like the tire settled or repositioned itself. Duane Russell has airless tires on a 700 C bike that work fine. I figured that was worth a try so I got a 700 C bike this spring. Then found out my bike shop could not find rims for the ones Duane uses. Also, airless tire sources have been tough to communicate with when I have a problem. The Wal-Mart mountain bike class airless insert I had on a rear tire was also a contributor to pedaling difficulties.

Mountain bike treads and be such that they will pick up small sticks or rocks after calibration and artificially lengthen a course. Finer treads and treads "over toward" bald are better about this.

All of this is based on my admittedly limited experience.

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