Rolling Resistance
Phil Holman
Introduction
We are all familiar with the concept of rolling resistance and some of us
even know enough to assess which tires will roll better than others.
Others just take these things at face value but I wanted to investigate this
subject in more detail and understand fully the losses so when a question
comes up like "What does inserting a Mr. Tuffy do for rolling resistance" I
don’t just have to remember a simple fact that it changes a certain way but
I know enough about the subject to determine the answer. Testing is often
the best method, but trying to be one of the fifty expert opinions Damon
Rinard estimates a single test is worth is OK also. Some of this information
comes from reading articles by Jobst Brandt, Lennard Zinn and others but
most of it is my own. It may not be original and may not even be correct.
Comments, additions, opinions and critique are always welcome.
Example
I will use the example of a tubular tire of 1 inch diameter inflated to
100psi and supporting a weight of 100lb. The area of the contact patch will
therefore be: pressure divided by load = 100 / 100 = 1 square inch. I will
treat the innertube as an integral part of the tire
Deflections
1/ Flattening at the contact patch. The 1 inch tubular shape is flattened at
the contact patch with the road (assuming a flat road) and will be close to
elliptical in shape. For simplicity, I will assume this shape is a ½ x 2
inch rectangle (Easier to calculate the center of pressure).
2/ Smaller radius at side wall. The sidewalls will bulge out and take on a
smaller radius reduced from ½ inch by the amount of deflection of the
contact patch. A ½ in wide flat on the bottom of the tire will result in a
sidewall radius reduction to approximately .35 inch.
3/ Reduction in hoop stress. The upward reaction load at the contact patch
(100lb) will locally reduce the hoop stress in the tire. The hoop stress in
the undeflected tire wall is
50lb per running inch of tire (pressure x radius). With the reduced radius
of .35 inch the hoop stress will now be 35lb per running inch. This will
result in a total reduction in hoop tension over the 2 inch length of
contact patch from 100lb to 50 lb and will result in shrinkage of the
overall tire cross section periphery. I did measure a Conti Sprinter at two
different pressures to determine dimensional change for a common tubular
(+.040 inches on diameter from 50 to 100 psi).
4/ Tire to rim interface. There will be a sharp radius in the tire casing
produced as the tire enters the rim to accommodate the smaller sidewall
radius as it transitions into the remaining tire supported by the rim.
5/ At the contact patch there is also compression and expansion of the glue
adhering the tire to the rim.
Load Distribution
The question is how much load and energy do we contribute to each
deflection. We all know that rubber and glue is soft and pliable and that
the reinforcing threads in the tire are very strong and will not stretch as
much under pressure. However, each will contribute to the load and energy
required to deflect the tire in producing a flat contact patch against the
road. Each will also contribute to losses in energy as the contact patch
returns to its circular shape. Assigning percentages to each is beyond the
scope of this article.
Static
In a static condition the 100lb load is equally divided (50 lb each) into
two halves, one either side of a line through the center of the contact
patch directly below and parallel to the hub axle centerline.
Dynamic
When the wheel is rolling, the dynamic action of compressing the forward
half of the contact patch shape requires additional load above 50 lb.
Similarly, the trailing half of the patch as it returns to its normal shape
exerts less than 50lb due to the losses in the restitution properties of the
tire rubber, glue and threads. The total losses in work and energy of
compressing and returning a section of tire to round are converted to heat.
As the total load on the tire must equal 100 lb, the amount above and below
50 lb for each half of the patch must be equal and will be dependent on the
stiffness and restitution (rebound) properties of the tire.
Assumption of Losses
For the fictitious tire I choose, I will use values of 55 lb and 45 lb and
see what kind of rolling resistance force and coefficient this gives.
The center of pressure of the forward contact patch (1/2 x 2) will be ½ inch
in front of the center and will result in a retarding torque of 55 x ½ =
27.5 in lb
The center of pressure of the trailing contact patch is ½ inch behind the
center and will result in a driving torque of 45 x ½ = 22.5 in lb.
The resultant retarding torque will be 27.5 - 22.5 = 5 in lb
For a 700c wheel this will result in a rolling resistance force of: 5 in lb
divided by the wheel radius (13 in approx.) = .39 lb. The coefficient of
rolling resistance will be .39 divided by the load (100 lb) = .0039.
This is a fairly realistic value for a 700x 25 tire at 100 psi with a load
of 100lb.
Change in Tire Material
What happens when we change the tire material to one that is just as stiff
but has a higher restitution property. In the static condition the load is
still 100 lb on the 1 square inch contact patch but when we start moving,
the forward load of 55lb is now in combination with an increased trailing
load of say 46 lb. This adds up to 101lb and is not possible. What actually
happens is that the bike rises a tiny amount to reduce the size of the
contact patch to maintain a constant 100 lb. The distribution now becomes
54.5 and 45.5 lb with a proportional reduction in rolling resistance.
Smaller Wheels
For an identical contact patch on a smaller wheel the torque losses will be
identical but as rolling resistance force is a function of wheel radius, the
rolling resistance for a smaller wheel will increase. For all wheels with
identical contact patches the rolling resistance is inversely proportional
to the wheel radius. In reality for smaller wheels, due to the increased
curvature of the tire, the contact patch length would be slightly shorter
and wider tending to reduce the magnitude of the increase.
Mr. Tuffy
So what happens when we insert a Mr. Tuffy. For a start, this insert is only
a strip which covers maybe 1/3rd of the outer circumference of the tire and
therefore sees no hoop stress. It does get flattened against the road at the
contact patch and does see some increased curvature as it extends somewhat
up into the side wall of the tire. The tire is also considerably harder to
depress with ones thumb (not an accurate gauge).
I notice that after riding one of these for a while and changing tires, the
center of the strip is adhered to the inside of the tire but the outer edges
are not which suggests that the strip does slide over the inside of the tire
as it flexes. This will result in increased losses and is evidenced by Mr.
Tuffy shavings and dust inside the tire.
Mr. Tuffy is made from a material more plastic than elastic which will have
greater losses than tire material.
As the data supplied by Damon Rinard states that the rolling resistance with
a Mr. Tuffy doubles, the losses would also have to double. Our two static 50
lb loads at the contact patch now translate to a 60 lb forward and a 40 lb
trailing load when the wheel starts turning. This will result in a rolling
resistance of .0078.
Tire Pressure
It is also generally believed that increasing tire pressure reduces rolling
resistance.
This is generally true up to a point where the tire becomes so hard that it
loses its damping properties and results in lifting of the wheel and bike as
it moves over irregular surfaces. The net result is that the rolling
resistance, when adding in road surface effects, will start to increase.
Clinchers
There is very little difference in rolling resistance between clinchers and
tubulars. Both tires are subject to the same tire deflections, have similar
weight and construction with flexible side walls and are operated at
equivalent tire pressures.
Perfect Tire
A tire that has infinite tensile stiffness and zero bending resistance will
have a rolling resistance resulting from the losses associated with moving a
few air molecules around.
Don’t hold your breath waiting for this.
Wider Tires
Some wider tires yield lower rolling resistance than narrower ones at the
same pressures. The wider tires will have a wider and shorter contact patch
resulting in a shorter offset dimension when calculating the torque losses
even though the contact load and area are the same. This can be more than
enough to offset any increased losses due to a thicker tire.
Heavier Tread
Generally speaking on the same size tire with a heavier tread of identical
material, the rolling resistance will increase. The increase in bending
stress of the rubber is a function of the thickness squared and results in
larger losses even though the tire material properties are identical.
Tread Material
Rolling resistance will reduce with tire materials of higher restitution
properties and identical thickness. However these materials generally have
poorer traction and wear properties. Tire material selection is a compromise
of these properties.
Slicks v Treaded
Expansion of the rubber into the tread at the contact patch results in
higher losses and increased rolling resistance for treaded tires compared to
slicks
Phil Holman
Jobst Brandt
> ROLLING RESISTANCE OF BICYCLE TIRES
> Introduction
> We are all familiar with the concept of rolling resistance and some
> of us even know enough to assess which tires will roll better than
> others. Others just take these things at face value but I wanted to
> investigate this subject in more detail and understand fully the
> losses so when a question comes up like "What does inserting a Mr.
> Tuffy do for rolling resistance" I don't just have to remember a
> simple fact that it changes a certain way but I know enough about
> the subject to determine the answer. Testing is often the best
> method, but trying to be one of the fifty expert opinions Damon
> Rinard estimates a single test is worth is OK also. Some of this
> information comes from reading articles by Jobst Brandt, Lennard
> Zinn and others but most of it is my own. It may not be original and
> may not even be correct. Comments, additions, opinions and critique
> are always welcome.
> Example
> I will use the example of a tubular tire of 1 inch diameter inflated
> to 100psi and supporting a weight of 100lb. The area of the contact
> patch will therefore be: pressure divided by load = 100 / 100 = 1
> square inch. I will treat the inner tube as an integral part of the
> tire.
I think your analysis of tire losses begins with too complex a model,
by involving inflation pressure and contact patch, neither of which
directly define what is happening. For example, two tires, one of
which has half the load and half the inflation pressure will have the
same rolling resistance, deformations being exactly the same.
Rolling resistance is the flexing of casing, tread and inner tube
whose losses are visco-elastic, assuming there is no Firestone tread
separation, something that would further increase losses by sliding
friction. For this reason, thinner casings that can be achieved with
finer thread have less bending resistance and therefore, less rolling
losses. Because the finer thread is also weaker, for the same
material, this is only possible with a smaller cross section tire.
It is the stress dependence of casing thickness and rolling losses
that makes a difference in cross sectional size in tires. The stress
is directly proportional to the cross section of the tire, the stress
being proportional to the radius of curvature. This relationship is
why a tire is under less stress a the load point than elsewhere,
because the unsupported radius of curvature is smaller.
Losses of tubular tires in the rim and base tape interface is an
unfortunate side effect of pressure sensitive glue. It has the
additional disadvantage of wearing through the base tape and causing
an underside blowout with time. As I mentioned in previous posts,
aluminum rims for tubular tires show an imprint of the base tape from
friction, while rim glue turns grey with aluminum oxide.
The Mr Tuffy liner loss is best visualized by how much force it
requires to bend and how little snap back there is. Although tire
casing distortions are more complex than simple bending, the feel of
its resilience is diagnostic. That tire casings have a complex shape
change is demonstrated by the business card test between tire and
tube. A paper business card will be shredded to square confetti on
the diagonal by the cord angles.
Jobst Brandt <jbr...@hpl.hp.com>
Phil Holman
>change is demonstrated by the business card test between tire and
>tube. A paper business card will be shredded to square confetti on
>the diagonal by the cord angles.
>
>Jobst Brandt <jbr...@hpl.hp.com>
Is the orientation of the threads a refined consideration in tire design.
Hoop stress is double longitudinal stress in any unsupported pressurized
cyclinder. A tubular tire will pick up additional longitudinal stress by
being stretched around the rim. The orientation looks like 45 degrees on a
Conti Sprinter.
Phil Holman
Jobst Brandt
>> That tire casings have a complex shape change is demonstrated by
>> the business card test between tire and tube. A paper business
>> card will be shredded to square confetti on the diagonal by the
>> cord angles.
> Is the orientation of the threads a refined consideration in tire
> design. Hoop stress is double longitudinal stress in any
> unsupported pressurized cylinder. A tubular tire will pick up
> additional longitudinal stress by being stretched around the rim.
> The orientation looks like 45 degrees on a Conti Sprinter.
If the cloth were not on a bias, it would have to be manufactured in
the final toroidal shape that bias ply accommodates easily. You'll
notice that folding tires are flat belts in the package, yet they can
be inflated to be toroidal rings. Beyond that, the bias ply is what
keeps tubular tires pressed on the rim glue. Without constriction,
caused by a 45 degree bias, the tubular tire would not stay in place.
Whether rotating a cloth with fibers crossing at right angles
increases stress is probably not so in this case. I think the
ancients have been through this and arrived on a useful casing. After
all, cars use it too although the radial has taken over in many
applications. The radial was not obvious and not easy to make.
Michelin designed it for rubber tired railways and only after, found
it had benefits for cars as well.
The squirming of a bias ply tire on a car is large and causes tread
scrub that wears them out rapidly. That is mainly a result of the
donut shape with the ratio of major to minor diameters being less than
4:1, something that is not the case with a bicycle where a radial
would not change the contact deformation significantly, the tire being
basically an infinite radius hose (straight) being squashed at the
contact point.
If you want to see the difference, look at a loaded bias ply truck or
car tire and you'll notice that he narrowest point is at the load
point and bulges appear on either side, there where the cords that
pass though the load point bulge out. In contrast a radial has a
belly at the load point. That's how you can detect a radial tire, by
its conspicuous belly. When radials were first introduces, many folks
though their tires were under inflated because they could see the
belly. On a bias ply bicycle tire, the bulge is at the load point
because the supporting cords do not extend significantly beyond the
load point.
Jobst Brandt <jbr...@hpl.hp.com>