As I see from comment in this thread yet clearly from first principles the relationship can't be linear for a semi rigid structure.
I'm only working with foam tires right
Fascinating (raises eyebrow a la Spock). So relative to the model car's weight these are still relatively stiff structures it appears. So this would imply that if mu falls with load that total grip per unit load also falls significantly? Did you do friction measurements on this or just contact patch area so far? Of course a real world tyre would only go through relatively small changes in load. Perhaps an order of magnitude at most for say a laden to unladen lorry or a rally car hitting the ground after a jump. An F1 car will experience downforce of about twice the static load and no doubt there will be data for changes of that sort of order at Michelin and Bridgestone. ( the current F1 suppliers)
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Even a balloon isn't a perfect example of an infinitely flexible structure of course as the internal pressure creates stiffness. Only when just the gas itself moves in response to load do we find agreement with the Universal Gas Laws.
Without knowing what empirical measurements they took to formulate their equation for fit it's hard to comment.
Also,
Even with a balloon tyre I doubt it would be a linear relationship. It was the closest analogy I could find to a flexible tyre structure in my earlier post though.
Then perhaps it would progress towards the volumetric distortion
I have a hunch, and that's all it is, that the relationship between load and contact patch area for a real tyre is not only non linear but also geometric. i.e. for small changes in load the relationship is different from that for large changes when the tyre carcass is forced to distort very significantly. As such maybe neither bulk modulus nor the Gas Laws would help model the relationship and perhaps only a completely empirical approach would suffice. Interesting though and I'm sorry I have no more answers for you. If you ever do measurements on a pneumatic tyre I'd love to know the outcome.
It may be that to develop a model car tyre that behaves somewhat closely to how a real one does that you need to do some load/area measurements on a real tyre and you might then find that the bulk modulus of the foam one is still far too stiff for the model car mass or that even any solid tyre structure behaves totally differently to an air filled one.
Dave Baker - Puma Race Engines
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"How's life Norm?" "Not for the squeamish, Coach" (Cheers, 1982)
My thought, reading this, was the instantaneous load changes as a car (race car) deals with track surface changes, etc. I'm told that the frequency of road inputs is about 100 cps, and I know that the magnitude, even on a no-downforce car like mine at moderate speeds (under 150 mph) can be fairly extreme - at least several times static and probably an order of magnitude above that, for very short periods (thank god for shock absorbers).
Also, lap times do not always follow tire width, as you know. Bridgestone only recently went to a wider front tire (F1), as they were chasing an aero advantage. But they put 14" wide rear wheels on my 1,000 pound, 250 hp car for a reason, I expect.
If mu falls with load, total grip per unit load falls too of course (same thing, perhaps that's not quite what you meant). I suspect the tires have a great deal of load sensitivity (in regards to mu), and the engineers at the RC car company I'm working with have no issues with this idea. The question for me right now is "what is the chief cause of load sensitivity?" Why does mu drop with load?
For awhile I suspected this was caused by additional stresses caused by free rolling. I.e., in a rolling tire there is a forward stress acting on the wheel in the front portion of the contact patch and a rearward stress acting in the rear. So there is a stress pattern acting in addition to lateral/longitudinal slip that would look similar to a sine wave with no stress at the center of the tire. A similar thing would occur 90 degrees to this and be influenced by tire pressure and sidewall stiffness (I just saw a PDF paper on this, give me a shout if you're interested in seeing it.) I imagined for a long time lots of little friction circles in the contact patch that detracted from lateral and longitudinal force capability as load was increased.
The nature of this lateral and longitudinal "free rolling" stress (my own phrase) is different, which I imagined might be what causes many tires to be more load sensitive in one direction than another (braking/acceleration vs. cornering). This was all purely in my imagination though, I never applied this concept to my theoretical tire model to see what would happen, and of course I've seen no measured data on it.
However, now I'm not so sure if this is the main cause. The book "Mechanics of Pneumatic Tires" shows two equations for calculating the approximate friction coefficient of rubber on a surface given contact pressure and Young's modulus of the tire. One for smooth surfaces, and one for rough ones where surface asperity size can be used.
Anyway, I played with this for awhile with both the foam RC tires and full sized car tires and was rather shocked at how close the predictions were to reality. Most noticable was that the load sensitivity also appeared to come out of it reasonably accurately. This would suggest that load sensitivity is really a property of the rubber on a small scale, not cross-patch stresses as I imagined. It's probably influenced by both really (plus the phase of the moon, etc.,), but the magnitude of the effect seemed to be predominately controlled simply by the contact patch pressure. This is still pretty rough at the moment, but next up is a tire model based on this to see how it works out. It should be interesting and I hope to learn something from it.
On the other hand, Chuck Hallund was a bit of a renegade that believed load sensitivity was caused by heat. He has some interesting ideas on transient tire behavior and makes some good points, but of course that sort of thing is difficult to test for in a lab.
So... I haven't quite sorted all this out to the level I want yet, but my thinking currently is that if contact pressure is the biggest factor in load sensitivity, then a few situations could be seen to develop:
An imaginary "balloon tire" such as you proposed that supported the weight of a car solely through air pressure would essentially have a constant contact patch pressure. I.e., the area would double when the load doubles or the pressure is cut in half as most people are taught to believe.
A solid foam tire like the RC tires. The contact patch hardly changes area at all, so as load is doubled the pressure nearly doubles. If contact pressure is the main culprit in load sensitivity, these tires should be extremely sensitive to load. Indeed it seems they are...
A real pneumatic tire. Contact patch pressure is somewhere in between as you described. In this case, I'd imagine that the stiffer the carcass is in the vertical direction, the more pronounced load sensitivity would be because the contact patch does not grow as much with load. Also, I didn't realize until yesterday that a balloon tire would bulge in a way to keep the volume constant, which clarifies what I was wondering. So I imagine a belt that inhibited the tire from expanding horizontally or prevented the sidewalls from bulging significantly with increased load would also keep contact patch change rate smaller, increasing load sensitivity.
If these behaviors are not mostly right, then perhaps contact patch pressure is not as much a contributor to load sensitivity and there are other significant things occuring (Hallund's temperature theories?)...
Did you do friction measurements on this or just contact patch
Unfortunately we have no good data on this. I made some very rough measurements for a few tires on concrete, but they were poorly controlled and did not jive with what happens on the track. These tires appear capable of producing (very roughly) 3.5g once they're hot. These RC cars can roll from excessive traction when track conditions are right. Downforce is so huge it must be fed directly to the rear wheels on 1/8th scale cars, but I've seen the cars flip on a flat track at perhaps 20-30 mph. In addition, Michael Salven ran a few laps without the body on the Serpent's new 950 to show me the difference, and could barely get the car around the track, indicating that the tires don't have so much grip after all...
This is not an easy problem of course, but when the engineers play with the simulator they usually adjust the front grip coefficient to 3.6 to get the low speed behavior right.
Ok, that was way more info than you asked for...
Of course a real world tyre would only go through relatively
I figured an F1 car produces far more than this (they can brake at 5g+), but yeah, the majority of the wheel rate is in the tire spring rate itself, so there must be far more deflection than we see in the foam tires. With foams, the ratio of contact patch growth to vertical load change is extremely small. This appears to be very different from a pneumatic tire.
"Grip" as Dave is referring to means coefficient of friction, not tractive force. Google for "lateral force vs slip angle" to see for yourself that friction coefficient drops with increased load. Of course the "grip" you are referring to (tractive force) will increase in this case.
Every lateral force vs. slip angle, longitudinal force vs. slip ratio, and combined slip force graph I've ever seen shows that the force a tire produces is pretty much the same in all directions.
The greatest variation I've ever seen, and this is from measured data done with a tire on a test stand, showed the forward/rearwards force exceeding lateral force by 20%. Most other tires fall within 5% or 10% in both directions.
If your experience is coming from drag racing where folks commonly say that a drag tire has very little sideways grip, that's a totally different phenomenon. What they have is very little stiffness laterally, so they can end up oversteering like crazy and cause you to over correct. Much like pumping up your front tires and deflating the rears. If the slip angles at the rear tires are greater than at the front, even below the limit of adhesion, the car will be oversteering (by definition!)
Like Dave said, go pick up a nice book like Milliken's "Race Car Vehicle Dynamics" and have a good read ;-)
No you don't. You want as much contact patch area per pound of vehicle weight as you can get. Subject to limitations that a pneumatic tyre must operate within a given band of tyre pressure or it will just distort all over the place and not work properly.
Dave Baker - Puma Race Engines
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"How's life Norm?" "Not for the squeamish, Coach" (Cheers, 1982)
Ummm, sorry but neither Todd nor myself need any simple lightbulbs going on here because both of us understand the fundamentals of physics and tyre behaviour. If you could approach the same level you might be able to contribute something interesting to the thread.
A starting point would be for you to grasp the fact that mu decreases with load and that pressure and force are different things.
Dave Baker - Puma Race Engines
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"How's life Norm?" "Not for the squeamish, Coach" (Cheers, 1982)
Utterly false in some cases. Dry pavement being the most common example. Why do you think NASCAR, IRL, CART, F1 run huge wide tires? To DEcrease grip in corners? Hardly. The same weight spread out over a larger area of rubber/pavement contact patch will genrally provide better overall adhesion. Its not a linear function, so there are regions where the change is very subtle.
As I stated in an earlier post, the fact that your Torino "lost grip" is in no way an indictment of wider tires. It may simply be that you didn't bother to have it re-aligned to account for the wider tires and any change to the effective scrub radius.
I can tell you definitively that my '73 Plymouth Satellite pulls more lateral G-force with P255/60 R15 tires and a suitable alignment than it EVER did with P205/75 R14s. But until I had it aligned, it was a handful.
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