Rolling Tire Characteristics

Thank you Pete.

Here is my summary of the argument, and my new experimental result which agrees closely with my thin membrane model of the bicycle tyre and differs from the motorbike tyre theory. I see no reason to expect that the coefficient of friction in normal conditions is small enough to allow much slipping to take place, which might affect the calibration constant by a large amount. I do not rule out very small effects due to slipping right at the edge of the contact patch.

Several times I have noticed that measurers have claimed that the coefficient of friction between the bicycle tyre and the road surafce affects the calibration constant, and I agree that the could be effects especially on icy roads. However, I have always been of the opinion that for ordinary road surfaces and for bicycle tyres like a racing or touring bike, there was not much slip in the contact patch between the front wheel tyre and the road. I was also of the opinion that of far greater importance in its effect on the calibration constant is the surface texture of the road. Ten years ago, I wrote a series of articles on this which appeared in Measurment News. Copies of those articles can now also be down loaded from http://coursemeasurement.org.uk/reports/

3 days ago Jim Gerwick made a posting on this board https://measure.infopop.cc/eve/forums/a/tpc/f/69510622/m/8761022113 in which he said

quote:

Also, I think the different friction coefficient for synthetic rubber vs. asphalt may be a factor.

I replied saying

quote:

Incidentally, I don't think anyone has proved that the coefficient of friction makes a difference, although I do suspect it may possibly make a contribution to the variation in calibration constant on different surfaces. What we do know for sure is that the roughness of the surface has an effect on the calibration constant. I took data on this including on a rubber track over ten years ago.

Mark Neal then entered the discussion and suggested that the tyre might be slipping over the road surface at some points in the contact patch, since the ground has to provide forces to which compress the tyre wall in the area of the contact patch, in order that the tyre can "squeeze" through the contact area (this is my short interpretation of Mark's theory). Mark bolstered his theory by quoting from a book about motobike tyres. This book quotes an approximate equation:

Ro=R-(R-H)/n where R is is the unloaded radius of the tyre, Ro is the effective rolling radius (ie the radius which gives the effective circumference and hence the calibration constant of the tyre), H is the height of the axle above the ground, and n is a constant which has the value of approximately 3 for a motor cycle tyre.

Models of car and presumably motor bike tyres are quite different to my simple model of a pneumatic bicycle tyre. I assume the bicycle tyre inner tube and tyre casing behave like a perfect thin membrane which can only transmit forces from one part of the tyre to another part in the plane tangential to the plane of the membrane at any point. In the theory of car tyres the flexural stiffness of the tyre wall plays a more signicant role, and when it is bent the tyre wall can transmit bending forces from one part of the tyre to another. It is like the difference between a thin walled toy balloon and a stiff walled ball - the former being my concept for the bicycle tyre, the latter more like a car tyre.

If my model of the bicycle pneumatic tyre rolling along the road is correct, then I predict a value of n=1. If the theory Mark suggests applies, then the value will be larger than 1, perhaps as large as 3 for a situation like a motorbike tyre.

This morning I carried out an experiment to determine n for my tyre.

Some time ago I measured the difference between my riding calibration: 10901.8 cts/km, and my walking calibration:10802.7 cts/km. The riding calibration is 0.92% larger.

The J-O and J-R counters produce 260/11 counts per revolution of the front wheel. So we can calculate the effective rolling radius for each case, obtaining the following values:

Riding Ro = 345.1 mm Walking Ro = 348.2 mm

Now the value of the walking Ro, will be very close to the value R, the unloaded radius of the tyre, because when walking the load on the front wheel is only a few Kgs of the bike, whereas when I ride about 25kg of my weight is loaded on the front wheel. So for this approximate calculation I am going to assume that the walking Ro = R.

This morning I measured R-H by photographing how much the tyre compressed when I added my weight in the riding postion. I can give full details of the experimental set up separately, but basically my son photographed the movement of a pointer attached to the steel tyre rim above the contact patch relative to a vertical ruler which was attached to the ground. We took 3 pairs of readings - we can easily read the position of the pointer in the photographs with a precision of 0.2mm.

The results were :
Loaded: 15 mm 14.8 mm 14.5 mm Unloaded: 18.3 mm 18.4mm 18.3mm (N.B. the zero of the scale was above ground level)
The average measured value of R-H is thus 3.57 mm

Substituting in the rearranged formula, n=(R-H)/(R-Ro), we get n= 3.57/3.1 = 1.15. This is encouragingly close to 1 which is predicted if my thin membrane theory applies. In fact the difference between 1.15 and 1 could easily be accounted for by not measuring the weight of the bike, and various small experimental errors. Notice I got a range of values from the 3 pairs of measurements which could easily account for being the measured value of n differing slightly from 1.

What we can say absolutely for sure is that motorbike type of situation with n=3 does not apply to my bicycle tyre.

I conclude that the coefficient of friction between the bicycle tyre and the road is high enough in ordinary conditions to prevent the tyre in contact with the ground sliding relative to the ground. Except for possible small effects at the edge of the contact patch, the calibration constant will be independent of the coefficient of friction for normal roads. However, this may not remain true on icy roads with a very low coefficient of friction when even the thin wall of the bicycle tyre may produce effects similar to that seen in car and motor bike tyres.

**************I have now put up some pictures of this experiment at the coursemeasurement.org.uk website *******

Good data there.

The major variable in coefficient of friction will be the road surface. If one calibrates on a concrete road, and measures on asphalt, the Cf will be off. Assuming there's a significant difference between the two.

If you really want to get pointy-headed, calibrate on asphalt and concrete, and note which portions of the course are which surface.

As far as slipping goes, I've ridden enough bikes in winter to say, you'll know when a tire is slipping.

I think the asphalt vs. concrete experiment bears investigating. Concrete is far grippier; will it yield a lower constant?

Stu, I'd think it's the opposite; asphalt would seem to be "grippier."

I think Mike Sanford examined this a while ago, but that may have been comparing roughness of surfaces rather than their specific composition.

At least one science experiment begs to differ:

http://message.snopes.com/showpost.php?p=354591&postcount=8

Also, the launch area of drag strips are all concrete; the balance is usually asphalt. This leads me to believe that more traction is lent by the concrete.

In any event, it doesn't much matter who's right, just that there is a difference in Cf. Whether it applies to a rolling bicycle wheel remains to be seen. My seat-of-the-pants guess is yes, but only slightly.

I dont think I have ever had a calibration course directly on concrete. Bare concrete roads are not very common here. Here is some data on comparing different calibration courses and different tyres which I published in Certified Accurate in 1997. The table was republished in MN No 89 in May 1998. The thing to understand about the table is that a value of 100 means that the calibration constant has changed by 1m per km (ie the whole SCPF) so the numbers represent the change in calibration as a percentage of the SCPF.
N.B. All surfaces except the track were tarred surfaces containing stones of various size and sharpness as noted below. There were no loose stones.

I should like to emphasise that I think all this variation of cal constant is due to variation in surface roughness, and I do not think there is any slipping of the tyre over the surface - except possibly just at the edges of the contact patch. According to my idea of how my bicycle tyre works, when the tyre meets the road at the front edge of the contact patch, it sticks to that bit of road without sliding until, it leaves the road at the other end of the contact patch. The tyre is pressed onto the road with 80psi. This stops it sliding. Although the road imparts friction forces on the rubber parallel to the road surface, I think these do not exceed the limiting force which would be given by the coefficient of friction. As long as this is the case they tyre contact patch does not slide and it does not matter how grippy the surface is as long as it is grippy enough to stop slipping.

I think this behaviour is quite different to what happens with a typical motor car tyre where the tyre casing is so very much thicker and therefore stiffer in relation to the car weight than is the case for the bike tyre. From the papers I have looked at following my recent forum discussion with Mark Neal, it is very clear that car tyres do have significant areas where the contact patch slips. So the grippiness, or coefficient of friction, would make a difference to the calibration constant if we were measuring with car tyres.

Below is the only comparison I have of concrete and asphalt. I measured both cal courses, and both are certified. Kirkham Road course is in the street in front of my house. Cleveland Avenue is in Westerville, a nearby suburb of Columbus. It’s on a newly-poured concrete sidewalk adjacent to heavily-traveled Cleveland Avenue.

On this one comparison, I obtained a smaller constant on concrete.

A better test would be to establish a cal course on the pavement of Cleveland Avenue, adjacent to the sidewalk cal course, and do comparative rides all at once.

So, for the sake of argument, let's say Kirkham (asphalt) has a ct/km of 11000.78 (average of pre-and post runs) and Cleveland (concrete) has an ct/km of 10995.915.

The difference is 4.865/km or 1.6 counts over the length of the course. That's about 30 degrees of wheel movement, or 8 inches on the ground if I'm figuring it correctly.

Not having read the measurement procedures manual lately, I can't remember the proper braking procedure, or if it's mentioned at all. Is the front brake used while measuring? I can see where braking would induce some wheel slippage, but 8 inches of it on a cal course? Doubtful that braking alone is the cause of the discrepancy.

Interesting stuff. Might be worth fitting a counter to the rear wheel to see what effect propulsion has on the count, and how this varies on different surfaces.

In reference to automobile (or any pneumatic) tires, the carcass is only thicker because of the larger amount of fabric required to support the load. Like a bicycle tire, the load is not borne by the bottom part of the sidewall (except in the case of modern run-flat tires, and there only for short distances at reduced speeds) but rather by the top portion. The air pressure allows the tire to maintain a round shape, and the weight hangs from the top portion. Think of it like a bicycle wheel. The spokes aren't strong enough to support the load in compression, but in tension they hold it easily. The "spokes" in this case is the fabric inside the tire, and the "rim" is the air pressure supporting it.

As to contact patch slipping, it's a larger patch and tends to move around a bit, especially on the steering axle. Higher speeds and greater loads also induce more slipping than you'll find in a bicycle tire. However, it does seem that the Cf plays a significant role, as Pete's data shows.

There are indications in Pete's data of a reduction in cal constant by 50% of the SCPF when on concrete. I say indications because the data are not ideal, some being taken at different temperatures so a straight average will not be perfect.

Stu says that is because the concrete/rubber has a higher coefficient of friction so the tyre slips less.

I say it must be that, in common to all pneumatic tyres without very thick chunky treads, Pete's would show a lower constant on a rougher surface, therefore the concrete must be rougher than the asphalt. However, my simple thin elastic membrane model of the tyre has never explained why this should be so.

I think I need to do some more experimental work to measure the forces in my bike tyre casing. Then, calculating the compression and resulting shear forces on the contact patch, I can see whether or not Cf would be exceeded.

Stu: Measurers are indeed advised not to use front wheel braking.

It's possible that traction plays only a minor role in the ct/km discrepancy. If there's more friction on concrete, the tire will heat up more, and exhibit more growth.

Pete's data shows the opposite. The difference between pre- and post- runs is double on the asphalt surface, perhaps because of the darker color absorbing more heat. The data would support this, as the major discrepancy appears in the 7/21/06 data, taken 4 hours apart, from the cool morning to the hot mid-afternoon. While the 6/17/07 data varies by less than .4 ct/km, readings taken an hour apart, both in mid-morning.

Mike, I think your thin elastic membrane model is flawed, unless you're using high-end racing tyres (spelt it properly for you). Inflation should prevent the tire from behaving like a membrane, yes?

Mike,

What is the geometric derivation for the case with the perfect stick condition that leads to

effective radius = H

I'm having a little trouble with the asphalt vs. concrete arguments.

As we all have certainly observed, concrete can be finished in a number of different ways. Some of these promote traction for automobile tires, and they are the finishes we'll most likely see on road surfaces. Others do not and, I suspect, would offer less traction than typical asphalt.

But when we say "typical" asphalt, what do we mean? Asphalt pavement is laid with differing sizes of aggregate, and I suspect the friction properties differ among them. What if the asphalt is old and the aggregate is more exposed? What if the asphalt has been sealed? You don't see this much on roads, but you do in large parking lots. Is there sand in the sealer?

I think there is more than one variable here- and there may well be more unknowns than equations.

Mark: The geometric derivation is indicated in my paper 3 - The section headed "the Deformed trye effective rolling radius" starting at foot of page 5. I may have to explain it a bit more if it is not clear.

I am just working on a spreadsheet model which will enable me to plot the strain as function of longitudinal position along the contact patch. I shall then be able to model the effects of having a length at the ends of the patch where I can assume slip is occurring, and see its effect on the calibration constant. I have a hunch that it will turn out that with a 10 cm contact patch, I just need to have a region about 1 cm or so the ends where slip occurs, and have the length of this slipping region vary on different surfaces to account for my experimental measurements of pneumatic tyres. I have some mistakes in the geometry at present.

Jay: I agree there are many variations in the surface finish of asphalt and of concrete. From Pete's result I am predicting that his concrete will be rougher than the asphalt

Here’s where my two concrete calibrations fit into all the calibrations I’ve done with my tire at 100 psi initial pressure. The two big, red, circular points are concrete. The rest are asphalt.

Although the concrete constant is a bit low, it’s not quite outside the mass of the data. Still, it’s only two points. Not enough for any firm conclusion.

The key to determining the effective rolling radius of a tire is to determine the effective perimeter of the tire as it rolls. This is the only thing that matters: what is the length of the tire material that is in contact with the road during one revolution of rolling.

As weight is applied to the center of the wheel, the tire begins to flatten where it contacts the road. If there is no friction between the tire and the road, there is no external tangential force acting on the tire. So there is nothing that will cause the tire to change its length. The tire will change its shape, but its total length will remain the same. For this case, the effective radius is equal to the original radius.

In the other extreme, where friction force is very high, as soon as a point on the tire surface touches the ground it "sticks" to it. In this case, after the entire weight has been applied the tire looks like Figure 1. Because each point has "stuck" to the ground, the tire material in the arclength "s" in Figure 1 has been compressed to length "c."


Eventually every arclength "s" of the tire will be compressed to a length "c" while it is in contact with the ground, so that the effective perimeter will be given by



and therefore



Returning to the equation given by the motorcycle tire paper.

Effective radius = R-(R-H)/3

and replacing H with the expression from Figure 1.



I'm not too good at trigonometric identities, so at this point you will need to do what I did, plug in different values of alpha (make sure to use radians, not degrees) to see that both the expressions for effective radius, the one I derived and the one given in the motorcycle paper, give the same values, i.e.,



So it does appear that for the case with high friction and no slip in the contact patch, the motorcycle paper does indeed give the correct value for effective radius.

Thanks for this Mark.

Overnight I have almost convinced myself that if one has a tyre made of a thin elastic membrane, which has a high coefficient of friction, and so "sticks" to the place where it first contacts the ground until lifted off, then a wheel cant turn! But on the other hand surely a tyre made of balloon-like material will roll? More thought is required to produce a working model.

I am also rather rusty on trigonometric identities. However, there is a convenient source in Wikipedia, although there are more there than you will ever need!

I have just checked your identity . It is an approximate identity but sufficiently accurate for our purposes up to my maximum value of alpha = 0.13 radians (7 degrees) where it is in error only in the 6th decimal place! Anyhow, that is obviously how it is derived for motorbikes, but unfortunately it now seems to me that this is not for the condition of high friction, but for riding with zero coefficient of friction, ie riding on ice!

If there is no friction there is no external force acting on the tire in the direction tangent to the perimeter. So there will be no change in the length of the perimeter. For the no friction case



I had made a mistake in my earlier post in the expressions for "s" and "c" where there should have been a "2" included (Figure 1), which I have now corrected. It has no effect on the final result.

Agreed. But there must be some slip for the motorbike formula to apply. I now think that if there is no slip the overall geometry changes slightly to accommodate a shortened contact patch length. At the moment I struggling to see exactly what changes.

It seems to mean that there will slight difference of height of the axle when rolling on a sticky surface compared to just loading the static bike tyre as I did in my experiment to measure the deflection. However, I think my static measurement is still valid in demonstrating that the motor bike formula does not apply to my bike. I am still calculating...

You have demonstrated a geometrical way of deriving the motorbike formula. But this does not seem to be the whole story, since there must be slip over some portion of the contact patch to fit the tyre into the chord (or else the geometry of the simple diagram changes at the ends of the chord). I think it would be interesting to track down the actual derivation of the formula, perhaps by reading the rest of the pages not shown in the limited google book view. It would help to see if they have used anything further than the geometry you have used.

This is getting extremely scientific and pointy-headed. Just the sort of thing I enjoy, even if I can't keep up.

You may find this paper enlightening:

http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%...h5_lphmpSO3WmPpKrVqw

PDF format

I think I am moving towards a clearer understanding of the different possible situations for a rolling tyre. It is not going to be easy to write it up fully in an easily understood account, so I will restrict this post to a brief description of the different cases I have considered and explain when each is applicable.

Introduction to the problem
I first tried to think about how rolling tyres work when I was trying to understand why pneumatic tyres give a smaller constant on rougher roads compared to a smooth road, whereas a solid tyre gives a larger constant on rougher roads. My work was described in 4 papers which appeared in MN in 1998. They can be dowloaded here. The paper relevant to the present discussion is number 3, in which I tried to model the rolling pneumatic tyre. I derived a formula which enables one to calculate the effective rolling radius from the tyre radius and the axle ground separation. The formula depended on the assumption that there was no slipping between the road and the contact patch. I here call this THE NO SLIP CASE.

I did not succeed in finding the cause of the variation in calibration constant with roughness, but I did eliminate one possibility. I concluded:

quote:

Surface roughness effects probably arise in the region near the point of first contact between the wheel and the ground where they affect the amount of initial circumferential compression of the tyre. I speculate that there are three possible causes:
1) tyre deformation extending beyond the point of first contact,
2) road height irregularities modifying the geometry of initial contact,
3) varying skidding at the point of first contact.

After reconsidering the analysis which appears on page 5 on Paper 3, entitled "A Deformed Rolling Tyre:Effective Radius", I think the derivation there still stands. I would now clarify conclusion 3 and say that a cause of calibration variation may include 3) varying skidding in the region very near the point of first contact.

Prompted by Mark Neal I have now examined two other possible cases:
1. The case given by the MOTOR BIKE EQUATION for which Mark has pointed out a geometrical/trigonometrical derivation. This case must some involve slipping over part of the contact patch. The Motor Bike equation also enables one to calculate the effective rolling radius from the wheel radius and the height of the axle above the ground.
2. The case when there is NEARLY ZERO FRICTION CASE. In this case the tyre circumference does not change length, and the effective rolling radius in equal to the tyre radius.

THE NO SLIP CASE

In this case I find that the effective rolling radius is equal to the axle/ ground separation. When the tyre meets the ground at an angle at the and of the contact patch it is slightly compressed and the friction forces hold it at a constant value of strain as it passes under the axle. If the strain were to vary then the tyre would need to slip, contrary to the assumption I made for this case.

MOTOR BIKE EQUATION CASE

In this case when the tyre meets the ground it does not stick, there is a region for which it slips and where it is compressed by the forces provided by the coefficient of sliding friction. As the piece of tyre moves through the contact patch towards the axle it encounters an area where it stops sliding, the static friction is then sufficient to provide the necessary forces. In this case the effective rolling radius is intermediate between the NO SLIP CASE and the NEARLY ZERO FRICTION CASE

NEARLY ZERO FRICTION CASE

In this case the tyre slips everywhere throughout the contact patch. We assume just sufficient sliding friction to ensure that the wheel turns. Since the longitudinal circumference of the tyre is not significantly changed the rolling radius equals the tyre radius.

Conclusion of the work
In the second post of this thread I posted the results of an experiment to measure the axle ground separation. When I considered the variation of constant I find between riding and pushing the bike, I found that for my pneumatic tyre the riding calibration fits THE NO SLIP CASE, and excluded the MOTOR BIKE EQUATION CASE and the NEARLY ZERO FRICTION CASE.