# my relative cornering capacity ready reckoner

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### #1 mariner

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Posted 22 June 2012 - 18:52

This one is so basic its a ready reckoner not a model!

As I am always designing cars that I will never build I needed a simple way of guessing what the relative cornerng power of different types of layout may be and to help trade offs between width ( i.e tyre size), weight and power.

- total car weight in kg
- total width of tyres on one side on mm
- type of tyre - 1.0 = std road tyre 1.1 = road legal trackday , 1.2 = full slick
- track in mm - average of front and rear ( caters for three wheelers)
- likely C of G in mm.

Basic formula (a)

- weight in kg/ total tyre width - kg per outer tyre mm.
- first " improvement " - adjust value (a) by tyre factor to give value (b)
- second " improvement" - adjust (b) by rolling moment of ^2(average track/C of G height). to give value ( c )

so to take two extremes

1) basic ford focus (a) = 1334kg /(215+215mm = 430mm) = 3.10 kg per mm.
(b) = 3.10 * 1.0 tyre factor = 3.10
© = 3.10 * ^2(1554/ 500)= 5.5

2) bugatti veyron (a) = 1838kg/(265+305) = 3.22
(b) = 3.22*1.1 = 3.54
© = 3.54 * ^2(1690/400) = 7.22

I think the formula gives too much value to track/CG height as it suggests that tyre-for-tyre a Veyron won't out-corner a Focus but it will by 20% due to track width

Anyways its not meant to give an absolute cornering capacity but to allow different cars and weight/tyre size/ width ( ie drag) trade off's to be checked.
Please note the word " capacity" it assumes spring rates, camber curves etc will all be optimised for each car!

Edited by mariner, 22 June 2012 - 18:56.

### #2 Greg Locock

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Posted 22 June 2012 - 22:38

If you plot the max lateral mu (Fy/Fz) vs vertical load on a tire you get rather a surprising shape. For modern production tires the trend is linear as much as anything.

So with no load transfer, ie cgz at 0, Fy1+Fy2=2*Fz*mu

with load transfer f total lateral force =(Fz-f)(mu+dm)+(Fz+f)(mu-dm) QED

so cg/track is less important than you might think.

The maximum mu of a tire as a function of tire width is complicated becasue the load capacity of the tire increases with width, and you will probably change the sidewall height at the same time. And if you put the wider tire on a wider rim the mu increases again. But overall a 10% increase in tire width is NOT going to give you anything like 10% increase in max Fy for a given Fz.

To further complicate things, production cars not optimised for max Ay, or anything like. So your database is going to be rather small.

### #3 CSquared

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Posted 24 June 2012 - 03:33

If you plot the max lateral mu (Fy/Fz) vs vertical load on a tire you get rather a surprising shape. For modern production tires the trend is linear as much as anything.

So with no load transfer, ie cgz at 0, Fy1+Fy2=2*Fz*mu

with load transfer f total lateral force =(Fz-f)(mu+dm)+(Fz+f)(mu-dm) QED

so cg/track is less important than you might think.

The maximum mu of a tire as a function of tire width is complicated becasue the load capacity of the tire increases with width, and you will probably change the sidewall height at the same time. And if you put the wider tire on a wider rim the mu increases again. But overall a 10% increase in tire width is NOT going to give you anything like 10% increase in max Fy for a given Fz.

To further complicate things, production cars not optimised for max Ay, or anything like. So your database is going to be rather small.

Interesting. So we've defeated load sensitivity?

### #4 Wolf

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Posted 24 June 2012 - 09:35

Interesting. So we've defeated load sensitivity?

I'm not sure that's what Greg's post implies- if it indeed was so, it would read constant instead of linear. For comparison, here's older graph:

### #5 Greg Locock

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Posted 24 June 2012 - 22:13

Wot he said. The trick is to look at the part of the graph you are actually using, where it is linear, rather than going down to zero vertical load and up to irrelevant Fz. Incidentally when somebody pointed this out I had a hard time believing it, but then checked out all of our recent tires and yes, it is right, especially for the most important axle.

Having said that I looked on the optimum G website and the tire they have there has a higher mu at operating load than lower or higher, so you have to be careful.

If load sensitivity didn't exist then it would be a horizontal line. It isn't, it falls more or less gently.

Edited by Greg Locock, 26 June 2012 - 03:33.

### #6 Kelpiecross

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Posted 25 June 2012 - 04:59

[quote name='mariner' date='Jun 23 2012, 05:52' post='5782038']

A very interesting idea - however:

It seems a bit unfair on three-wheelers which will have a very narrow apparent track.

If you are designing a car (especially of the "sporty" variety) the c of g will probably be as low as practical (and always much the same). Same applies to tyre size/ track etc. - you can't really propose a track of ten feet for example.

I wonder if the "ready reckoner" could be reduced to cornering power is proportional to overall weight. Meaning a go-kart has probably the best potential cornering power.

I seem to recall that Laurence Pomeroy Junior had many "empirical" (if that is the correct word) formulas to describe a car's performance - even relating the cylinder bore diameter to the lap time.
One of his ( I think it was LPJ) formulas also predicted that of two cars of equal weight and equal horsepower but one having half the maximum torque of the other - the car with the bigger torque figure would accelerate much faster. I hasten to add that I am not sure this is a correct prediction.

### #7 mariner

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Posted 25 June 2012 - 20:51

thanks for the expert info.

What I am trying to make is a simple ready reckoner on cornering potential of different car sizes purely in terms of how tyre size/rim width can be traded off versus drag ( and some weight) to see if widenng a design is worth it.

The ultimate cornering force isn't the question so good geometry, spring rates etc etc and optimum rim width are all assumed for any tyre width.

Racing cars definitely got faster round corners as tyre width increased and tracks generally increased on single seaters unless the regs said otherwise, so track and tyre size do seem to improve lat. G based purely on real world trends.

IF track and tyre width DONT improve lateral G then you would design any car to be as narrow as possible for low drag and probably lower weight.

In terms of that question my relative tyre grip ( 1 = std road tyre , 1.1 = trackay tyre 1.2 = race tyre) is only a second order adjustment so I'm not sure if the shape of the lateral force curve is that significant.

If, as Greg says track vs C of G is also less significant then maybe the initial metric of kg weight / mm of tread is all thats needed.

I know Im asking a question with a thousand variables and a million answers etc. but if you were designing from scratch how would you make the trade off ( or will any car of any tyre width generate the same lateral g in reality?

### #8 Greg Locock

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Posted 25 June 2012 - 23:56

All I'm saying is that grip is not proportional to tire width. It increases as tire width increases, but it is not directly proportional. In practice you can't just dial up a new width, what you do is increase the width, reduce the aspect ratio and change the rim width, and possibly the tire pressure. So there are at least 4 factors other than construction, for a given OD and rim dia.

You could look at whichever magazine quotes skidpan g and try and see if there is a useful trend of weight vs tire width vs lateral g for sporty cars. If you can find results for the same car on different width tires your job will be much easier.

Trouble is, construction. In general sporty tires have much lower life requirements than economy tires, so the hairdresser's Mustang will have some sort of low rolling resistance thing that'll last 30000 miles whereas the second youth one will have a grippy soft wide tire that'll last 10000 miles. Even if they are both called the same thing by the manufacturer their internal construction and compounds will be different.

### #9 Greg Locock

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Posted 26 June 2012 - 03:36

For instance, suppose you were to replace one of the tires wolf posted above, at 1000 lb vertical loading, with two, at 500 lb.

The mm of tire width has grown by a factor of 2, the grip has increased from 1250 to 1400 lb, at most.

There is no reason to believe that two narrow tires are much less effective than one tire of the same overall width, is there?

### #10 munks

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Posted 27 June 2012 - 02:21

Having said that I looked on the optimum G website and the tire they have there has a higher mu at operating load than lower or higher, so you have to be careful.

Thanks for pointing out the optimum G website, but I haven't found the data you're talking about yet. In any case, I assume that those results are probably due to this:

Trouble is, construction.

One can say tires have load sensitivity ... they certainly do, but some portion of it is due to rubber load sensitivity (which, for every graph I've ever seen, goes down with increased load in a somewhat logarithmic function), and some is due to the construction. So if the tire is constructed to be optimized for a certain load, I could certainly see the mu being worse for a lighter load, even if you've got slightly better rubber grip.

### #11 Greg Locock

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Posted 02 July 2012 - 03:27

Incidentally this data may be somehwat useful for trying to construct a tire size based model of cornering stiffness, and possibly grip
http://www.hoosiertire.com/spring.htm
At first sight vertical tire rate vs psi is not much related to cornering stiffness. Have a think about it.

Edited by Greg Locock, 02 July 2012 - 03:28.