"But when the wheel turns a full 360 degrees, the distance the vehicle moves is 2piR where R is the unloaded radius of the tyre. While the wheel is rotating, the effective force would appear to be T/R."

No it isn't.

1) In order for the tyre to create grip it has to move relative to the road surface.

2) The tractive force is t/r2 where r2 is the laden radius, not the unladen.

Cheers

Greg Locock

Ok Greg
Then does the vehicle move 2piR or 2pir2?

Does the tyre have to move relative to the road surface to create grip? Railway wheels don't. (The rail and wheel have the same elasticity and presumably do not move relative to each other at the contact patch)

I can see that the tyre and the road must move relative to each other to solve the paradox, or perhaps the tyre structure (including the tread)deforms while the contact patch retains its momentary shape.

Jeff Stanton

Distance traveled is somewhere in between what the free radius and the loaded radius predict.  There's a derivation given in "Mechanics of Pneumatic Tires", that identifies the tread as being compressed in the contact patch and also over the zones immediately before and after the contact patch.  It goes on to talk in terms of effective radius and effective deflection that differ from the loaded radius and actual deflection respectively.  And it mentions that very little longitudinal slip occurs within the contact patch (assuming no acceleration/braking), hence there's little wear under conditions of rolling in a straight line.  

Eventually the discussion gets around to providing some experimental data, with the actual distance traveled measured for a bias tire given at 96% of what the free radius predicts but the loaded radius being only 94% of the free radius (yes, this dates my reference material somewhat).  For radials, it gives 98% distance traveled with the loaded radius being only 92% of loaded radius.  Data for more recent tires may differ in the specific percentages, but I'd certainly expect the general relation to hold.

Norm

Thanks Norm. My "thought experiments" ran along those lines. Nice to see some data,

jeff

There is a very easy way to see "what happens to the extra rubber".  Just push down on the tread of a flat, or unmounted tire, and observe the the gap between the individual tread blocks.  You will see that the gap in the tread becomes smaller as a flat "contact patch" is created by you pushing a section of the tire flat.  So if you were to measure each gap between the blocks when it is round, and when it is flat, you will see where the "extra rubber" went...it just got closer together.

Just seen this thread - thought I would make a comment:

"Does the tyre have to move relative to the road surface to create grip? Railway wheels don't. (The rail and wheel have the same elasticity and presumably do not move relative to each other at the contact patch)".

Actually, a railway wheel does a very similar thing, although the effect is very small. The rail and wheel do not have the same "elasticity" since they are different shapes, although of course the E values are effectively the same. Relative sliding within the contact patch is one of the sources of rolling resistance, which you can read about in texts on ball bearing analysis for example.

Fair enough English Muffin, but is relative movement necessary to create grip? Or is it an unwanted effect of grip?

Jeff

Grip is created by several mechanisms, off the top of my head:

1) chemical grip - bonding between the rubber and the road - doesn't need slip

2) cogging grip - intermeshing of asperities on the surface of the tyre and road - doesn't need slip

3) hysteresis grip - deformation of the rubber as it drags over the surface needs energy, so it need sa force and a velocity. Needs slip

I don't know the relative proportions of those in a given situation, but it is worth pointing out that standard tyre results are presented with slip on the x axis - implying that slip is a very important factor.

Cheers

Greg Locock

No, although it may well be a contributory factor, you do not have to have relative motion to produce grip. For example, if you had two steel rollers of identical diameter pressed against each other, with parallel axes of rotation, you would have no significant relative sliding in the contact patch, but you would still have grip. You would also still have some rolling resistance, stemming from the internal hysteresis produced by the internal cyclic deformation of the rollers, although it would be small. But elastomers behave rather differently from steel objects - for example, friction is no longer independent of surface area - and spinning dragster tyres produce more grip than non spinning ones. But this particular aspect of the subject is probably better understood by Mr. Locock, in view of his automotive experience.

OK, actually, after consulting my "extensive" library (grin), I find that I actually do have a book which goes into all this - the increase in friction of elastomers with sliding speed, the pressure distribution on the contact patch of rubber tyres under braking and accelerating conditions etc etc. It is called "The Friction and Lubrication of Elastomers", by D.F. Moore. The same guy has also published an even more relevant book called "The Friction of Pneumatic Tyres", but I don't have that one. I can't find either of them on Amazon or Abe Books, so I guess it doesn't help much - there is too much to reproduce here.

Thanks Greg and EnglishMuffin. Interesting and productive replies. Mind you, I only asked the questions out of idle curiosity, and got more than I expected.

Jeff

One last thing: When writing a program to calculate acceleration performance, it's tempting to use the wheel revolutions per mile figure in the calculation of thrust. This would be an error, of course, but at least one of the "big 3" has included this small error for many, many years. In fact, I'm only assuming that it was eventually "caught" and eliminated.

dont forget about tire roll out (circumfrence) increases due to centrifugual forces at higher speeds.    
                                                
                                                

Very little with radial tires, but considerably with bias ply. About 0.2 wheel revolutions per mile per mile per hour, as I recall. (It's been a while.)