Maximum cornering speed (Lotus vs Lada)?

[Phisp]

Possibly posted on the wrong forum, but I'm sure there are a few people here that will know the answer

There are numerous videos around the Net showing how a standard Elise can out corner many more powerful cars. It has sparked my interest to try and understand why from a maths/physics point of view.

Currently I've reached the point where the couple of equations I'm using imply that any car, be it a Lotus or a Lada, can corner at the same speed as long as their tyres have the same coefficient of friction (CoF) - which seems wrong.

So far, I am using the frictional force offered by the tyres as:
F = µmg ; where µ=CoF, m=mass of vehicle and g=force of gravity

I'm also using the equation for Centripetal force to model the force exerted as the car travels around a fixed radius corner at a constant velocity:
F = mv²/r ; where m=mass of vehicle, v=velocity and r=raduis of corner.

The max velocity a car can travel around a corner is reached when the Centripetal force equals the Frictional force offered by the tyres, i.e. µmg = mv²/r. Rearranging, to represent in terms of velocity gives:
v = √(µrg)

So, given a corner of the same radius (r), tyres with the same coefficient of friction (µ) and travelling on the same planet (g) then, no matter the vehicle, it will be able to acheive the same cornering velocity.

I have got to be missing something (a lot)! Could anyone enlighten me or point me in the right direction?

Edited by Phisp on Thursday 14th May 15:17

Edited by Phisp on Thursday 14th May 15:17

[RenesisEvo]

You are missing a lot.

The main thing is this: tyres do not have a constant value of µ; it varies with camber for a start. Suspension tends to have a lot of compliant and kinematic effects that cause the geometry to vary in cornering - camber, toe (bump steer), castor and track if non-parallel. This affects exactly how much grip is available at the contact patch. Poor suspension (the Lada) will not maximise the tyres for cornering.

µ also varies with tyre compound and temperature, and most importantly - the load placed on the tyre. Generally, more weight on a tyre = more grip, but the value of µ does not increase linearly with load - +10% weight gives +9.5% more grip, etc. The result is that weight transfer is a bad thing from the point of view of grip, thus a car that minimises weight transfer laterally and longitudinally, by controlling body roll (anti-roll bars can help) and having the correct centre of gravity and roll centre heights, will have more grip than a car with poor body roll. Note that weight transfer is unavoidable (sounds like you've got the maths that would prove it), but it can be controlled to balance out the grip between all four tyres.

In your model, the vehicle does not roll, and IMO that is the fundamental parameter missing. With the Lada, it will have a higher centre of gravity (CG) than the Lotus, combined with softer roll stiffness, there will be more lateral weight transfer, thus even if the Lada had exactly the same tyres and suspension geometry (which it certainly does not!) the greater weight transfer would rob the Lada of some (loads?) grip.

That is a VERY simplistic view. Have a Google for vehicle dynamics, or get yourself to a library - book called Fundamentals of Vehicle Dynamics (T. Gillespie) is fairly easy to follow.

HTH

[TonyHetherington]

I've no input, but just to say that was very interesting to read. Thanks

[p1esk]

TonyHetherington said:

I've no input, but just to say that was very interesting to read. Thanks

Ditto, but just wait until we get the Stressed input, then all will become clear - or maybe not.

Best wishes all,
Dave.

[StressedDave]

RenesisEvo said:

The main thing is this: tyres do not have a constant value of µ; it varies with camber for a start. Suspension tends to have a lot of compliant and kinematic effects that cause the geometry to vary in cornering - camber, toe (bump steer), castor and track if non-parallel. This affects exactly how much grip is available at the contact patch. Poor suspension (the Lada) will not maximise the tyres for cornering.

µ is invariant with camber I'm afraid and the sensitivity with respect to vertical load is actually rather small - in the data I've seen it's around 1-2%. Important if it's an F1 car, but relatively unimportant for the rest of the world. Tyre compound is much more important and the trend these days is for sporty cars to use compounds with up to 20% more grip than the average tyre.

RenesisEvo said:

µ also varies with tyre compound and temperature, and most importantly - the load placed on the tyre. Generally, more weight on a tyre = more grip, but the value of µ does not increase linearly with load - +10% weight gives +9.5% more grip, etc. The result is that weight transfer is a bad thing from the point of view of grip, thus a car that minimises weight transfer laterally and longitudinally, by controlling body roll (anti-roll bars can help) and having the correct centre of gravity and roll centre heights, will have more grip than a car with poor body roll. Note that weight transfer is unavoidable (sounds like you've got the maths that would prove it), but it can be controlled to balance out the grip between all four tyres.

You're confusing the coefficient of friction (i.e. the grip between the tyre and the road) with cornering stiffness (the amount of cornering force per degree of slip angle). Weight transfer isn't a function of body roll either - it's a function of track and centre of gravity height and nothing else for a given speed and radius. How that weight transfer is split between front and rear wheels is a function of the suspension design (the position of the linkages) and the relative stiffness of the suspension. Now having a stiffer suspension may help with the former (to quote Colin Chapman, the suspension works great if you don't let it). The more weight you transfer at one end compared to the other, the less cornering force you get for a given slip angle - effectively it's less efficient at converting slip angle into cornering and you end up wasting some of it as tyre drag.

Careful control of those stiffnesses can unlock more cornering potential, but you're still limited by the dimensions of the vehicle and the tyres fitted. The lower and wider the car, the greater the potential, so even with everything else equal the LOtus will most probably be capable of more lateral g, but you can get a lot closer than you think.

[Martin A]

Hi All

RenesisEvo said:

You are missing a lot.

The main thing is this: tyres do not have a constant value of µ; it varies with camber for a start. Suspension tends to have a lot of compliant and kinematic effects that cause the geometry to vary in cornering - camber, toe (bump steer), castor and track if non-parallel. This affects exactly how much grip is available at the contact patch. Poor suspension (the Lada) will not maximise the tyres for cornering.

µ also varies with tyre compound and temperature, and most importantly - the load placed on the tyre. Generally, more weight on a tyre = more grip, but the value of µ does not increase linearly with load - +10% weight gives +9.5% more grip, etc. The result is that weight transfer is a bad thing from the point of view of grip, thus a car that minimises weight transfer laterally and longitudinally, by controlling body roll (anti-roll bars can help) and having the correct centre of gravity and roll centre heights, will have more grip than a car with poor body roll. Note that weight transfer is unavoidable (sounds like you've got the maths that would prove it), but it can be controlled to balance out the grip between all four tyres.

In your model, the vehicle does not roll, and IMO that is the fundamental parameter missing. With the Lada, it will have a higher centre of gravity (CG) than the Lotus, combined with softer roll stiffness, there will be more lateral weight transfer, thus even if the Lada had exactly the same tyres and suspension geometry (which it certainly does not!) the greater weight transfer would rob the Lada of some (loads?) grip.

That is a VERY simplistic view. Have a Google for vehicle dynamics, or get yourself to a library - book called Fundamentals of Vehicle Dynamics (T. Gillespie) is fairly easy to follow.

HTH

There is also a question here of what is meant by maximum cornering speed; highest peak g force, highest average corner speed or highest minimum speed? As has been said a higher c of g and less roll stiffness can have an effect but not necessarily always to the benefit of steady state cornering g, it may even reduce it and be compromised for improved transient response in some vehicles. Driving technique can also vary the results of the interpretations that I have offered to the idea of maximum cornering speed.

Talking of technique, does anyone else find it strange that Stressed Dave advocates being on the power before turning in whereas other luminaries such as Mark Hales and John Lyon advocate turning in on the brakes. Surely they can't all be right and so one or more persons are charging big bucks for teaching the wrong thing.

Even 10p per hour is big bucks to be taught the wrong thing imho. £100 per hour for the right info on the other hand is money well spent. But who is right?

Hope this helps stimulate debate.

Best regards

Martin A

[RenesisEvo]

StressedDave said:

You're confusing the coefficient of friction (i.e. the grip between the tyre and the road) with cornering stiffness (the amount of cornering force per degree of slip angle).

I think you're right!

[7db]

In addition the weight differential may help -- the lighter vehicle will have a greater proportion of non-Coulomb friction in its grip (it sticks as well as grips, but it's stickiness is the same as the heavier car, sticking less masS)).

It will give this up disproportionately where surface adhesion is impaired, which is why the 7 is so much fun in the wet/diesel/air, giving up a large percentage of its grip.

Furthermore, Lotus drivers tend to be keener at finding the limit of grip than Lada drivers, and I suspect this is the key component, but then when did you last see a Lada going careening off arse first into the barriers?

[WhoseGeneration]

7db said:

but then when did you last see a Lada going careening off arse first into the barriers?

Oh, probably back in the days of the Lada Challenge.
Them Rally folks, always trying out obscure vehicles.

[StressedDave]

Martin A said:

Talking of technique, does anyone else find it strange that Stressed Dave advocates being on the power before turning in whereas other luminaries such as Mark Hales and John Lyon advocate turning in on the brakes. Surely they can't all be right and so one or more persons are charging big bucks for teaching the wrong thing.

Actually we are all right - we're just working in different regimes. I'm not averse to using the brakes to control the contact patch forces. Working on turning a car in under power has a couple of useful side-effects. Firstly you have a better balanced car normally (my experience is that trail braking is normally poorly done) and secondly it forces you to concentrate on exit speed rather than entry speed to a corner and IMHO exit speed is key to having fun on road and track and also gives lower lap times in the main.

[Joe911]

StressedDave said:

Martin A said:

Talking of technique, does anyone else find it strange that Stressed Dave advocates being on the power before turning in whereas other luminaries such as Mark Hales and John Lyon advocate turning in on the brakes. Surely they can't all be right and so one or more persons are charging big bucks for teaching the wrong thing.

Actually we are all right - we're just working in different regimes. I'm not averse to using the brakes to control the contact patch forces. Working on turning a car in under power has a couple of useful side-effects. Firstly you have a better balanced car normally (my experience is that trail braking is normally poorly done) and secondly it forces you to concentrate on exit speed rather than entry speed to a corner and IMHO exit speed is key to having fun on road and track and also gives lower lap times in the main.

Also - the type of corner, and the type of car, will affect the technique that works best. Some techniques are very difficult to learn/master.

[Martin A]

Hi All

Joe911 said:

StressedDave said:

Martin A said:

Talking of technique, does anyone else find it strange that Stressed Dave advocates being on the power before turning in whereas other luminaries such as Mark Hales and John Lyon advocate turning in on the brakes. Surely they can't all be right and so one or more persons are charging big bucks for teaching the wrong thing.

Actually we are all right - we're just working in different regimes. I'm not averse to using the brakes to control the contact patch forces. Working on turning a car in under power has a couple of useful side-effects. Firstly you have a better balanced car normally (my experience is that trail braking is normally poorly done) and secondly it forces you to concentrate on exit speed rather than entry speed to a corner and IMHO exit speed is key to having fun on road and track and also gives lower lap times in the main.

Also - the type of corner, and the type of car, will affect the technique that works best. Some techniques are very difficult to learn/master.

What is meant by 'different regimes'? Not entirely convinced by the explanation as Messrs Hales and Lyon are also keen on corner exit speed and their lap times seem fairly on the pace. That's not to say I think they are completely correct or not, just looking for an explanation. When could the brakes be used to control the contact patch forces?

Are there any general rules about type of corner and car, how they affect technique and how the difficulty of learning can be reduced to prevent trail braking being poorly done?

Best regards

Martin A

[p1esk]

Puntopower99 said:

lotus has the engine in the middle, which is the best weight distribution which means faster cornering.

That reminds me that some BMW advertisements have claimed that (at least some of) their models have the ideal weight distribution of 50:50 front to rear. This puzzled me because I would have thought that it was very difficult to avoid a front engined car being nose heavy, because as far as major mechanical assemblies are concerned there doesn't seem to be enough at the rear to keep it in balance, but I might be wrong about that.

In any case, is it true to say that a weight distribution of 50:50 front to rear is the optimum for cornering performance? I imagine there may be other factors that would suggest not.

Best wishes all,
Dave.

[james_gt3rs]

p1esk said:

In any case, is it true to say that a weight distribution of 50:50 front to rear is the optimum for cornering performance? I imagine there may be other factors that would suggest not.

Best wishes all,
Dave.

No, because slight rear weight bias allows the car to turn faster and more responsively (lower polar moment of inertia I think), and also having more rear weight bias allows more traction at the end of the corner, hence why F1 cars vary between 45:55 and 40:60 front to rear.

[p1esk]

james_gt3rs said:

p1esk said:

In any case, is it true to say that a weight distribution of 50:50 front to rear is the optimum for cornering performance? I imagine there may be other factors that would suggest not.

Best wishes all,
Dave.

No, because slight rear weight bias allows the car to turn faster and more responsively (lower polar moment of inertia I think), and also having more rear weight bias allows more traction at the end of the corner, hence why F1 cars vary between 45:55 and 40:60 front to rear.

The ability to 'turn faster' could be a mixed blessing, couldn't it? For example, I'm given to understand that the Toyota MR2 (a tail heavy device - my son has a Mark 2) can be a swine if you ask too much of it when cornering, because they spin very suddenly, and are then extremely difficult to control.

Best wishes all,
Dave.

[RT106]

james_gt3rs said:

No, because slight rear weight bias allows the car to turn faster and more responsively (lower polar moment of inertia I think), and also having more rear weight bias allows more traction at the end of the corner, hence why F1 cars vary between 45:55 and 40:60 front to rear.

I would imagine that the weight distribution in F1 is down to the relative sizes of the front and rear wings and tyres, which are controlled by the regs. Big rear tyres plus more downforce at the back means it's more effective to have more weight at the back.

50/50 weight distribution doesn't necessarily mean a low polar moment of inertia; that depends on as much mass as possible being centred around the car's vertical axis. The lower the polar moment of inertia the more responsive the car, but also the harder it is to control in extremis.

Why do some cars corner faster than others...? Driving ability, tyre compoound, and damping control are probably the greatest factors affecting steady-state cornering speed (on a less than perfect surface), and in that order. In my opinion, of course.

[StressedDave]

Martin A said:

What is meant by 'different regimes'? Not entirely convinced by the explanation as Messrs Hales and Lyon are also keen on corner exit speed and their lap times seem fairly on the pace. That's not to say I think they are completely correct or not, just looking for an explanation. When could the brakes be used to control the contact patch forces?/quote]

By different regimes, I meant the difference between cornering on the road where vision rather than grip controls speed and cornering on the track where the size of one's cojones combined with the coefficient of friction controls things. There is more of a read across than many people think, especially away from the rarified atmosphere of competition and the bleeding tip of the performance envelope. If you were to go out with Lyon on the road, you'll see that his road cornering style is markedly different to his track driving style.

When there is cornering capacity spare then you can control the relationship between slip angle and cornering force by the application of torque, be that engine torque or braking torque. So you can do it everywhere on the road, as limited by vision or on track when you aren't using all of the capabilities of the car in any given direction

Martin A said:

Are there any general rules about type of corner and car, how they affect technique and how the difficulty of learning can be reduced to prevent trail braking being poorly done?

There are no hard or fast rules whatsoever. As for the difficulty of learning, the issue I find is one of consistency. I get a lot of clients who have what they believe to be all the tools in the toolbox (heel-and-toe, left foot braking, trail braking etc.) but cannot consistently corner quickly, or to be more specific, they're really good at finding the limit of grip in a corner, but not necessarily at the most efficient point and end up flailing at the steering like Tiff Needell after 18 pints.

By concentrating on being able to set up the car properly every time, you'll have a decent footing to build upon. The first stage for that is having the car at right speed, right gear, right suspension attitude before you start to steer and having the power properly set is one of the key things to work on. Once you can consistently do that then it's time to bring the other toys to play.

[StressedDave]

james_gt3rs said:

No, because slight rear weight bias allows the car to turn faster and more responsively (lower polar moment of inertia I think), and also having more rear weight bias allows more traction at the end of the corner, hence why F1 cars vary between 45:55 and 40:60 front to rear.

It's very little to do with polar moment of inertia (and a 50:50 mid-engined car would have a lower polar moment than a 45:55 car anyway) and more to do with the fact that the front tyres are further away from the centre of gravity and cause a larger yaw moment for a given amount of cornering force.

There's a bit on it (if you don't mind algebra) here: http://www.vehicledynamics-expo.com/06vdx_conf/pre...

[7db]

StressedDave said:

but not necessarily at the most efficient point and end up flailing at the steering like Tiff Needell after 18 pints.

Or in other words, the limit of grip is a lot easier when you are driving up to it from below than scrabbling down to it from above.

[ajcj]

7db said:

StressedDave said:

but not necessarily at the most efficient point and end up flailing at the steering like Tiff Needell after 18 pints.

Or in other words, the limit of grip is a lot easier when you are driving up to it from below than scrabbling down to it from above.

I like that explanation. It has also been explained to me that cornering quickly is difficult (any fool can do 150mph in a straight line) and so you want to have as few distractions as possible. If you turn in having already lost enough speed for the corner you only need to worry about the steering wheel and the accelerator all the way round, whereas a trail braking approach demands that you think about brakes and possibly gear change as well.