Simulating tyre drag when cornering

[spacer]

Hello all,

I'm currenly in the process of creating a race simulation using Matlab Simulink (yes, another one of those) for our Shell eco-marathon car, which will be competing in the European race in may 2011. One of the important components in this simulation is the tyre drag, and mainly the increase in this drag while cornering.

Although I've spend quite some time searching through the forums for topics like slip angle, pacejka's formulae and fsae-tyre-data-threads, I'm somewhat lost in the info and probably missing the big picture.

I was hoping some of you guys could help me out by looking at where my thinking goes wrong, and help me put the final pieces of the puzzle together.

This is what I've got so far. Please feel free to shoot at any of these assumptions:

1. The track data is properly available, so I know the corner radius, verhicle speed (calculated throughout the simulation), track inclination and corner banking at any given point in the simulation.
2. Our car is a three-wheel prototype-class car. It has two wheels at the front, both having a 8* negative camber, and has one wheel at the back. The rear wheel is used for steering. For the ease of this simulation (assuming rigid fasteners), the entire car (full carbon firbe btw) can be assumed fully rigid.

3. Since our car has no movable suspension whatsoever and the tyres are incredibly stiff, the vehicle pitch and roll are dictated by the track inclination and corner banking (?).
4. The track width, wheelbase, CoG coordinates and vehicle mass are known. Will item 1+3+4 combined allow me to directly calculate the vertical load (Fz) on each wheel through trigoniometrics? I don't need any tyre data or slip angles for this one right?

5. Michelin has supplied us with some Pacejka data for our tyres, allowing me to calculate the cornering stiffness (Cα) for each wheel at any load (Fz) through this formula:
Cα = (a30 + a31 P) sin (2 tan -1 (Fz / a40 + a41 P))

6. Now this is where I get stuck. I suspect I need to get my slip angles next, but those slip angles are a function of steering input, generating lateral force, moving the car into a corner (right? ). So I need to work the other way around, finding out what (max) steering angle is needed within the car for a given turn radius. But the prob is I only seem to find the formulae regarding single-wheeled models. Any help here?

One of the formulae I found: α = (m v2 / R )/(Cα - Cr Fz π/180)

7. Apart from the above, what is the influence the front camber has? Both the front wheels are rigid (steering input 0* at all times), does this mean the camber thrust of both wheels cancels eachother out? And does this create some additional drag?

Hope some of you guys can help me out, thanks in advance for reading all of this!

Kind regards,
Tom

[Ciro Pabón]

Regarding assumption 6, isn't your vehicle a single wheeled one? Frontal wheels are not "producing" slip angles.

As for point 7, I would say you need to know scrub radius. The friction while steering depends on it, AFAIK. I have NO idea if this is true, but I think intuitively that any force that produces friction, like the one induced by camber cannot "cancel out" with the opposite one.

Out of curiosity, do you have the maximum banking data or do you have also data about the transitions? (that is, where the banking starts and ends). The difference between the two should complicate your model a lot (and give you around 2-4 percent difference in lateral "effort").

[Jersey Tom]

Even if your front wheels don't steer mechanically, they most certainly are going to "steer" themselves by generating slip angles.

Regarding "equations" for solving out slip angles... don't bother. Hint: You will need to use some sort of 'goal seek'.

[Ciro Pabón]

Of course! Why didn't I think about it. Now everything is clear. Goal seek. So the answer to number 7 is "Lionel Messi".

[Jersey Tom]

Landon Donovan.

[xpensive]

Don't forget to involve "Planck's constant" and "Avogadro's number" now Ciro, as JT advised us before!

[spacer]

Thanks for the replies guys! I gave Lionel a call this weekend, but he said something about tarmac having a less slippery surface than a soccer pitch, before hanging up on me .

Out of curiosity, do you have the maximum banking data or do you have also data about the transitions? (that is, where the banking starts and ends). The difference between the two should complicate your model a lot (and give you around 2-4 percent difference in lateral "effort").

The event organisation have provided us with a complete geographical blueprint of the track, this includes the track height at every 50cm of the track or so. So yes, the banking and jaw gradient of the track can be calculated. We even could be able to vary those depending on which "racing line" one uses on the track (so varying the data with lateral position on the track in stead of just the longitudinal distance travelled). But that would make the model unnecessary complicated.

Even if your front wheels don't steer mechanically, they most certainly are going to "steer" themselves by generating slip angles.

Regarding "equations" for solving out slip angles... don't bother. Hint: You will need to use some sort of 'goal seek'.

Ok does this mean that at low slip angles the longitudinal component of the tyre force is somekind of a lineair relation to corner speed and radius; allowing me to know the tyre drag without knowing the slip angle?

[747heavy]

not that I know much (anything) about eco marathon &/or three wheeled vehicles, but just some thoughts/questions.

which kind of tires do you use?, because 8° camber seems quite excessive even by "touring car standards", so is it fair to assume, that your tires are very narrow, and probably closer to a motorcycle tire then a car tire?

Why do you run such high camber settings?
Don´t get me wrong, I´m not questioning your design, just beeing curious

Do you run any toe in/toe out on the front tires?, This will lead to slip angles even when you run in a straight line.

the formula which you (try to) use, would only apply for very slow (low speed) corners (to define steering angle by assumed curvature from speed and lateral accel.), because it neglets slip angles at the non steered axle. Due to the slip angles your corner radius changes.

what is the lateral accel. you reach normally with your vehicle?
Do you run on a flat or banked track?

on a flat road slip angles will be a function of lateral accel.

at which axle, do you "drive" your vehicle?
front or rear?
if it is at the front, do you use an indepentend drive for each wheel, a differential or a "solid drive"/spool between the two wheels?

[spacer]

not that I know much (anything) about eco marathon &/or three wheeled vehicles, but just some thoughts/questions.

which kind of tires do you use?, because 8° camber seems quite excessive even by "touring car standards", so is it fair to assume, that your tires are very narrow, and probably closer to a motorcycle tire then a car tire?

Why do you run such high camber settings?
Don´t get me wrong, I´m not questioning your design, just beeing curious

Do you run any toe in/toe out on the front tires?, This will lead to slip angles even when you run in a straight line.

the formula which you (try to) use, would only apply for very slow (low speed) corners (to define steering angle by assumed curvature from speed and lateral accel.), because it neglets slip angles at the non steered axle. Due to the slip angles your corner radius changes.

what is the lateral accel. you reach normally with your vehicle?
Do you run on a flat or banked track?

on a flat road slip angles will be a function of lateral accel.

at which axle, do you "drive" your vehicle?
front or rear?
if it is at the front, do you use an indepentend drive for each wheel, a differential or a "solid drive"/spool between the two wheels?

Thanks for your input 747, don't be afraid to question our design, this'll only help us improve our ideas and concepts!

Since the car we're building only weighs ~35kg without the driver, we use Michelin tyres specifically designed for this eco-marathon contest. They are very narrow and look very much like BMX-bike tyres (dimension 45/75R16). These tyres are used by nearly all of the teams in our class, therefore lots of test data are available.
One of these is the "straight-line" tyre drag plotted against camber degree. When plotting this graph, the drag slowly lineairly increases up to just over 8* camber, after that the drag starts increasing at a far greater amount. So by running the tyres at 8* camber we trade in some extra drag, but this allows us to gain a far greater advantage in reduction of the overall body frontal area while still meeting the minimum trackwidth regulations ; thus decreasing aero drag.

We will not run any toe in or out.

Highest lateral accel will be very low: 0.06G (7m/s at 80m radius corner). We run at a track with some banked corners (banking data available). Although we could neglect this in our first simulation, and assume a flat track just to get things started. By "on a flat road slip angles will be a function of lateral accel." do you mean that by knowing the lateral accel and cornering stiffness of the tyre, the required slip angle will result?

We both steer and drive our car at the rear wheel, so front wheels are completely independent.

[Jersey Tom]

Even if your front wheels don't steer mechanically, they most certainly are going to "steer" themselves by generating slip angles.

Regarding "equations" for solving out slip angles... don't bother. Hint: You will need to use some sort of 'goal seek'.

Ok does this mean that at low slip angles the longitudinal component of the tyre force is somekind of a lineair relation to corner speed and radius; allowing me to know the tyre drag without knowing the slip angle?

Kind of... I mean at low acceleration and slip angle where everything is more or less linear in response, I guess you could say that. But really, the longitudinal drag is a function of slip angle. Fy * sin(slip angle) = drag.

If you know how much Fy you need to get through a corner, and if you have a F&M curve showing how much slip angle is needed to get that Fy, then you can get Fx pretty easy.

[747heavy]

Hi Spencer,

Thanks for coming back and explain some things.
As I said before, I have zero expirience with your type of vehicle, so I will just stay quite, as they are too far away, from what I normaly deal with.
Also your tyre, seem to be rather special, therefore I´m not sure if "normal" assumptions and methods used with (car) racing tyres apply.

Lateral Force and Tire Slip Angle
While cornering, a vehicle undergoes lateral acceleration, (3), (1). As the tires provide the only contact of the vehicle with the road, they must develop forces which result in this lateral acceleration. When a steering input is given, the successive treads of the tires that come in contact with the road are displaced laterally with respect to the treads already in contact with the road. Thus an angle is created between the angle of heading and the direction of travel of the tire. This angle is known as the tire slip angle which gives an estimate of twist of the treads of the tire. It can also be defined as the ratio of the lateral and forward velocities of the wheel. The twisted treads try to get back to their original positions, thus producing the force required for lateral acceleration (2) (Figure 2.5).
This force is known as the Lateral Force (Fy) or the Cornering Force.
At a given load, the cornering force grows with slip angle. At low slip angles (5 degrees or less) the relationship is linear. In this region, cornering force is often described as Fy =Cαα. The proportionality constant Cα is known as cornering stiffness and is defined as the slope of the curve for Fy versus α at α = 0 (3).

Maybe you find something of interest for you here

I (think I) understand your idea behind your (high) camber angles on the front.
I just looked up some photo´s of similar vehicles, it seem´s that others opt for a different approach, with less camber.
But I´m sure, you have done your tests and maths, and have your reasons behind your approach.

If you keep your camber angles, maybe it is worth a test to run some toe out at the front wheels, and see how that affects your rolling resistance.
(if it is not to much hassle, as I don´t know you front axle construction)

Good luck in any case

[Kristian89]

Hey Tom

I'm a new user in this board, so I'm not authorized to send you a private message...
I'm currently develop a Matlab tool to calculate the tire force during cornering. In your first post in this topic you wrote, that Michelin has send you some Pacejka data for the Prototype tires 45/75 R16. Could you send me this data?

Kind regards
Kristian

5. Michelin has supplied us with some Pacejka data for our tyres, allowing me to calculate the cornering stiffness (Cα) for each wheel at any load (Fz) through this formula:
Cα = (a30 + a31 P) sin (2 tan -1 (Fz / a40 + a41 P))