From: calhpvz@nobelium.Berkeley.EDU (Eric Anderssen )
Date: Tue, 5 Jan 93 19:12:28 -0800
Subject: hat "drag" thing

Here goes.  I just read lotsa mail so this may ramble.  There seems to
be a lot of confusion about this rolling resistance thing.  First of
all--noone seems to be using any units--or at best haphazardly.
"DRAG" has units of FORCE.  The rolling DRAG does not increase with
velocity, because if it did, the power curve that someone wrote about
would be quadratic (go as velocity squared).  POWER is FORCE times
VELOCITY, so obviously the power consumption is linear in velocity if
the rolling drag is a constant (or is independant of velocity).
 
--A bit more on the velocity dependance--
 
I wrote a paper on rolling resistance for the solar car team here (UC Berkeley)
we were trying to write the power curves for different concept cars so as to
 
ascertain which would do us best with only a hairdryer's worth of power 1500W.
There are terribly few references upon which to draw--here are mine.
 
'The Mechanics of Pneumatic Tires'  Library# TL270  M4 1981
 
There is an earlier edition '71, but it contains the same
articles--these two are considered to be the bible of tire engineers.
Unfortunately, as you might expect, they are biased toward the
automotive industry, they do however talk about 'toroidal' tires as
they are commonly found on airplanes.  The latter discussions are
limited to performance models and alot of correlation is necessary to
get information on rolling resistance.  There are a few excellent
papers on the theory of the contact patch and how the various
curvatures affect the casing, adhesion, wear---etc.  Any good
engineering library should have these.  The above Lib# is library of
congress standard so in principle you culd just walk to it.  My other
two references were in periodicals, one of which I have the other, I
have only the transcript.
 
'Rolling Resistance of Bicycle Tires' by Rob Van der Plas.
 
in:  Bike Tech vol. 2, #2  April 1988
 
I had to call Rodale press for that one and get a copy of the article
in that Bike Tech has been out of print for a number of years.  Rodale
Press publishes "BICYCLING" magazine.
 
The last article is by Dr. Chet Kyle whom most of you should be fami
oops I on ly got a copy of it fom an obscure source and it looked like
copies of his notes.
 
The important things to note are the trends.  Larger wheel: less drag
 
wider tire: more drag (that's rolling--not air)  Higher pressure:  less drag
 
Suspension:  lessens drag--can make small wheel damn close to normal one.
 
Thin, supple casing:  less drag.  Lower "aspect ratio":  less drag
 
"aspect ratio" is the ratio of casing height to width of the tire.  Mounting
a narrow clincher on a wide rim, while not entirely advisable, will lower teh
aspect ratio and thus the drag.  For each of the above cases you can look at
where the energy goes to get an intuitive feel for each--I reccomend the first
book(s)--while somewhat laborious, they are good.
 
In my email to mr St Clair i closed with the fact, tho' I couldn't explain it
that the drag force is independant of velocity.  I suppose I'll try.
 
Hysteresis is the cause of the resistance to rolling.  It is cyclic in
nature and has units of energy/cycle--al else held constant.  To get
power multiply by cycles/second.  Now do a "force balance" using a
terminal velocity--this will yield a force.  Because the power curve
is linear the force is constant and is the slope of the power curve.
It is hard to show that the force is constant by only looking at the
forces in the contact patch.  The resultant force depends on spatial
derivatives of the hysteresis which is more or less knowing exactly
what the stress field in the casing is.  There is a "requires" missing
in the previous sentence.  As you will see if you read either book--
these models are both complex and quite inexact.
 
 
OH!  Dave Jones' article was in the April 1970 issue of Physics Today p34-40.
--I have the article in front of me--I'm not really that old.
 
Eric