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sprior

Original Poster:

96 posts

130 months

[news] 
Thursday 29th January 2004 quote quote all
Sorry if this is off-topic, but I reckoned there must be clever people, and people who know about tyres, on this forum....

The boredom of unemployment got me doing an OU applied maths course. When I got to the unit on friction I got massively confused:

Frictional resistance is Coef-of-friction x pressure x contact area.
The area and the area part of pressure cancel, giving:
Friction is Coef-of-friction x force.
Note that the contact area makes no difference to the frictional resistance!

Now the bit that confuses me, is that this means that I might as well buy 145 tyres when I replace my 255's, because they're cheaper. (Though I don't think you get N rated 145's).
Can anyone explain why wide tyres work?

Simon

JamieBeeston

8,825 posts

151 months

[news] 
Thursday 29th January 2004 quote quote all
More rubber in contact with road = more frictionable area = can take more force (either acceleration or brakeing) before the maximum transfer (either way) of power is breached, and skin / spin / slide occurs.


how easy is it to cut bread with a knife... now try cutting bread with a knife turned sideways....

Opposite sides of the coin.. but

paulmc

127 posts

140 months

[news] 
Thursday 29th January 2004 quote quote all
I'm not 100% sure of the physics of this myself but aren't the things you're refering to as cancelling out not just the units of measurement. I.e. In this example where the coefficient of friction for the material is say 1, you get:

Friction = 1 x 20 kg/cm2 x 10 cm2 = 200 kg

The units of measurement cancel out, but you still use the actual value of the contact patch area.

MOD500

2,548 posts

136 months

[news] 
Thursday 29th January 2004 quote quote all
Does this work:

Pressure = Force / Area

Hence F = P x A

Make the F above into your 'frictional resistence', and so the frictional resistence is proportional to the contact area, hence increase your contact area, and so increase your frictional force to 'move' it. So (hopefully) wider tyres give more grip.

The m2 from the A in F= P x A would not cancel out in your initial equation, so meaning the area influences the forces though it is 'hidden' within other units so as to simplify the equation down.

Not 100% sure though......long time since I did all that.

MOD.

sprior

Original Poster:

96 posts

130 months

[news] 
Thursday 29th January 2004 quote quote all
The trouble is, the frictional resistance for something with (now this next bit might be relevant) a constant coef of friction (mu) is independant of the contact patch.

with a bit of general approximation, an example for the outside loaded rear tyre on my 996 on hard cornering might be:
Contact patch on the rear wheel is about 100mm front-back x 255 wide x (say) 75% non grooved = about 20,000mm^2.

downwards force on that tyre is whatever it is, but let's say 40/60 fr/rr weight dist, and say 70/30 loaded/non-loaded dist, giving for kerb weight of about 1400Kg, a downward force on the tyre of about 5,000N.

That gives a contact patch pressure of about 0.25 N/mm^2

To work out a rough mu, and to keep the sums simple, pretend the cornering peaks at 1.0 lateral g, and that (to be consistent with the assumed weight dist) this tyre is doing 40% of the work of keeping the car from sliding.
The lateral force on this tyre is then 1 x 1,400 x 9.81 x 0.4 = about 5,500 KN.
This gives a mu close enough to 1 to pretend it is 1 for this discussion.

So, in this case, with 255 tyres, the max frictional resistance to opose lateral force on this wheel is:
mu x pressure x area
= 1 x 0.25 x 20,000
= 5000KN

Now, pretend the same car has 145 section tyres:
contact patch on the rear wheel is about 100mm front-back (for consistency) x 145 wide x (for consistency) 75% non grooved = about 10,875 mm^2.

Pressure = 0.46 N/mm^2

and so, in this case, with 145 tyres, the frictional resistance to opose lateral force on this wheel is:
mu x pressure x area
= 1 x 0.46 x 10,875
= 5000KN

So the maths says narrow tyres are as good as wide ones.

It all might be something to do with rubber having a higher mu at lower contact pressure - but I'd love to hear from someone who really can explain this...

In the meantime - next time I get pulled over for wild sliding I am going to claim mathematical research.

simon.

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james

1,362 posts

170 months

[news] 
Thursday 29th January 2004 quote quote all
Simon,

You're actually attacking this from the wrong direction. Look at what you first said. You're assuming that the tyres let go at 1 lateral g. Therefore you're actually working how much work the tyres have to do in order for the car to take that 1g lateral load. You could work this out for a bike tyre, and you'd still get the same answer. This tyre has to be able to cope with 5000KN of lateral load in order to stop the car sliding. The size of the tyre is irrelevant for the calculation that you are doing. But it isn't irrelevant if you are trying to work out how much grip a tyre will actually provide before it lets go.

The actual answer to your question is that wider tyres give you better grip up to a certain point. The tyres offer a certain amount of grip per square mm. Multiply this up, and you get to the conclusion that (for the same compound) a wider tyre is more grippy (which we already knew from experience).

Ok. So now we're all going to go out and but 500mm wide tyres for our cars. Maybe not. In practice, there is another limitation. The way that a tyre grips a road surface is that the rubber deforms around imperfections in the road surface. This gives you the grip. However, a wider tyre will have less downforce per square mm, so you need to use a softer rubber. This is why our expensive wide tyres wear out so quickly. It isn't just the power we're putting through them. It's also the fact that the rubber has to be softer on a wide tyre. If you used the same compound for a 285 tyre as for a 145, you'd probably end up sliding around all over the place.

I hope this helps.

James

911fan

434 posts

146 months

[news] 
Thursday 29th January 2004 quote quote all
Isn't the most important equation:

Porsche + licence + full tank = Big Grin?

sprior

Original Poster:

96 posts

130 months

[news] 
Thursday 29th January 2004 quote quote all
with reference to the comment from James:

the bit about 1 g was just to come up with some sort value for mu, which I assumed would be constant for rubber of a certain compound.
I can almost go along with your argument that wider tyres allow softer compound, and it is the softer compound that provides a highe mu, and more grip, but I am a bit doubtful that the same brand and designation tyre has a range of compounds for different widths.

Still confused. Think I'll 'ask goodwin'.

simon

911newbie

422 posts

146 months

[news] 
Thursday 29th January 2004 quote quote all
In fact friction isn;t quite as simple as student text books make it out to be.
The coefficient of friction as described here (the static coeff of friction) describes the lateral frictional force up to the point of slipping. The lateral force is at a peak just before slipping starts and then it reduces to a second lower level.
This second lower level of lateral friction during slipping is much more complicated than the static coeff and is sensitive to contact area.

The calculation for contact area for the two cases (narrow and wide tyres) is not straight forward. A simple balance of forces would tend to show both had similar contact areas but I think tyre wall stresses etc may play a part.

Additionally, the wider tyre would present a great resistance to lateral forces tending to lift tyre off from the road surface when corenering. In other words because the moment arm is greater with a wider tyre it wont lift off so readily, and thus maintina a higher contact area.

Hope this is clear.

No doubt this has been the subject of considerable (unpublished) study by the tyre manufacturers.

willr

363 posts

139 months

[news] 
Friday 30th January 2004 quote quote all
Slightly O/T, but perhaps interesting for those on this thread:

www.toyo.com.au/tech_info.html

http://images-eu.amazon.com/images/P/1560915269.02.LZZZZZZZ.jpg

So perhaps we should consider lateral grip as the advantage of wide tyres.

sprior

Original Poster:

96 posts

130 months

[news] 
Friday 30th January 2004 quote quote all
911newbie said:
In fact friction isn;t quite as simple as student text books make it out to be.


A little surfing found plenty to support this.
Found a very interesting and math rich set of articles at www.pelicanparts.com/techarticles/physics_racing/
part 22 has loads of equations for slip angle, which probably explain everything, but it will take me a while to understand it.
Interestingly - this article is hosted by Pelican Parts - seemingly a Porsche part outfit.

The Toyo link was interesting, but whilst stated that larger contact patch = more grip, didn't go anywhere to explain it.

Did find ads for a book:
THE RACING & HIGH-PERFORMANCE TIRE
Using the Tires to Tune for Grip and Balance
by Paul Haney
Chapter 6, Tire Behavior, explains how a tire produces lateral force and turns a car. An interview with Jim Hall tells the story of tire development leading to wider treads. Learn the real reason why wide tires produce more grip.
Myth 1: The classic equation for friction is Cf = Ff/Fvert. Contact area doesn't matter. Wrong!
The truth is rubber generates friction force in at least three ways, the major components of friction being adhesion, momentary molecular bonding, and deformation, mechanical keying.
Will buy this, read and try and understand.

Now I'm worried that other things aren't as simple as student text books say.
In fact, the Earth is tetrahedral, but the jump from the flat earth model was too much, and it was much easier to pretend it's round.

simon

ChrisW.

1,820 posts

141 months

[news] 
Friday 30th January 2004 quote quote all
I go with Sprior. Goodwin deserves this question ..

JohnM993c4

27 posts

137 months

[news] 
Saturday 31st January 2004 quote quote all
That same overly-simplified frictional analysis tends to get used to "prove" that cars cannot accelerate, brake or corner at greater than 1g (in the absence of aerodynamic downforce) as the frictional force cannot exceed the weight on the contact patches and mu (by that simplified friction model) cannot exceed 1...yet a Top Fuel Dragster can launch at almost 8g and averages 4g for the whole run. Applied Maths is about using mathematical models to approximate real systems and processes, the closer the approximation has to be the more complex the model has to get.

GreenV8S

23,471 posts

170 months

[news] 
Saturday 31st January 2004 quote quote all
Warning - long!

The school physics books tell you that the coefficient of friction is a constant, but in reality this is an oversimplification. In the case of tyres, the coefficient of friction drops off as you increase the weight on the wheel. This is why you can tune a car's handling by adjusting the anti-roll bar, and also why you *can't* tune with anti-roll bars if the load on the tyres is too small (the vehicle is overtyred).

Because of this, the bigger the contact patch the more grip you can get. In a drag race, dropping the tyre pressure increases the contact patch area and increases grip. Even on road bikes you will see people dropping the tyre pressure to almost nothing for the absolute maximum grip down the strip.

BUT, when you look at lateral grip (side force) other factors start to matter. The tyre develops side force because of the slip angle between the tyre and the road. This slip angle means the tread is being pulled sideways by the road surface. At the front of the contact patch the deflection is relatively small. As you move back along the contact patch the deflection increases steadily. At some point, the sideways forces in the tyre exceed the friction between the tread and the road and the tread starts to slip relative to the road. When the tread is slipping like this it produces less grip on the road. Now imagine increasing the slip angle and imagine what effect this has on the side force. As the slip angle increases the sideways deflection builds up quicker so the front of the contact patch works harder. But more and more of the back of the contact patch is sliding and losing grip. At some point you reach a maximum point where more slip angle means less side force because you are losing more grip at the rear of the contact patch that you are gaining at the front. This is often referred to as 'breaking away' where you ask the tyre for more grip and end up getting less.

The longer the contact patch is, the more gradually this break away occurs. If you shorten the contact patch, the break away occurs more abruptly but you get more absolute grip at the peak. This is because there is less variation in sideways distortion between the front and back of the contact patch, more of the contact patch reaches maximum grip and starts to slide at the same point. Having a shorter contact patch also means you get less self-aligning torque so there is less feedback through the steering about how close the tyres are to breaking away.

When you fit wider tyres, what you're doing is making the contact patch wider and shorter for the same tyre pressure. This means you get a more abrupt breakaway but more grip right on the limit. The disadvantage is more expensive tyres, more tramlining and steering kickback, more wind and rolling resistance and noise, less grip in slippery conditions, a more abrupt breakaway to catch out the unwary driver and less warning through the steering about how close the tyres are to breaking away.

This probably explains why manufacturers tend to put wide tyres on high performance cars and narrower ones on ordinary family saloons.

dazren

22,336 posts

147 months

[news] 
Sunday 1st February 2004 quote quote all
Makes sense to member of the ignorenti such as myself.

DAZ

ChrisW.

1,820 posts

141 months

[news] 
Sunday 1st February 2004 quote quote all
Mr GreenV8S,

What a great reply. I believe I understand it !

That also explains why, for maximum feel and feedback, it is desireable for a given tyre width, to fit wheels offering the longest possible contact patch i.e. larger diameter wheels.

This also provides more room for the larger brakes which should both retard and cool better.

sprior

Original Poster:

96 posts

130 months

[news] 
Wednesday 11th February 2004 quote quote all
Having read and, I believe, understood Paul Haney s excellent book I now feel up to commenting on the excellent post from greenv8s.

The big mistake I had been making was to assume that there was some truth in that oft quoted motoring journalists line about more rubber on the road when talking about cars with wide tyres.
Contact patch area is a function of vertical load on the wheel and the tyre internal pressure the tyre deforms to achieve this area (pressure x area = force). The tyre construction is such that the tread width is essentially fixed, so the deformation manifests itself as lengthening or shortening the contact patch length.

Thought that greenv8s explanation of slip angle and effect of contact patch length was great. The book helped take me through this a bit more slowly.

After understanding this, my next question was why doesn t increasing pressure in the tyre further improve grip, since more pressure = less contact patch area = shorter contact patch = better grip?
The answer is to do with the fact that more rubber on the road does help grip this is due to the hysteresis properties of rubber. As rubber expands to fill a depression in the road (or a gap between aspersions), it takes some (small) time to do so. When a tyre is sliding (and due to the slip angle, the rear most portion of the contact patch slides at even low cornering forces), this means that the upward rise of the depression to which the tyre is moving has more rubber acting on it that does the upwards rise on the other side. This allows a pressure differential in the lateral plane, providing frictional resistance over and above that offered by simple friction. (Pictures would help here).
Now, as the tyre vertical load increases, the rubber is forced more fully, and more quickly into the depressions, overcoming the hysteresis and reacting on both sides of the upward rise from the depression more evenly giving less pressure differential and less grip.
Hence, low tyre pressure is better for grip from deformation and hysteresis.
Tuning the pressure is about balancing the contact patch length (which is better as pressure goes up), and the contact patch pressure (which is better as tyre pressure goes down). Of course high pressure gives harsh ride, and low pressure increases tyre deflection, which increases heat (promoting wear and potentially catastrophic failure), reduces responsiveness, and is less fuel efficient (like I care?), and these factors are also considered when optimising tyre pressure. In fact, track work with road tyres creates so much heat that the tyres become severely damaged (as I know from SCCA autotesting with Hertz rentals in the US, sorry Hertz). Even though the optimum grip is achieved at low pressures (because the hysteresis element is significant), use of road tyres on the track is better at highish pressures (40PSI+), for this reason. In fact, the best way to increase grip (without going into suspension or aerodynamic changes) is to widen the tyre this gives a shorter contact patch for the same inflation pressure.

The hysteresis effect also explains why slicks work saying that the contact patch area is a function of the tyre pressure is true for the tread carcass area on the inside of the tyre. The tread carcass is stiff enough to mean that grooves in the tread create increased contact pressure at the rubber to road interface, reducing hysteresis effect and reducing grip.

This has been a fantastically interesting diversion from doing any maths for a couple of weeks, and now I m getting a bit behind, but I feel that it was worth it to understand one of the great mysteries of (my) life.

I did ask Goodwin before the post from greenv8s, and it will be very interesting to see if he comes up with the same answer since to do so would be to admit that every time motoring journo s come out with that more rubber on the road.. line, they are demonstrating complete ignorance of their subject

Regarding the last post from ChrisW the OD of the tyre isn t really significant in determining contact patch length, the construction of the tyre means the tread width is pretty much fixed, and the tread carcass deflects to give whatever area is required to support the vertical load.
Shame this, as I d love to convince myself my 17 s had more ultimate grip than 18 s.

Simon
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