Wednesday 18 May 2011

Car Dynamics in a Corner

It's important to understand what the car undergoes during a corner, or while driving  in general, so that we can understand how to drive with the car and not against it. The right technique will give us a greater reserve of grip, speed and the potential for good lap times and a reduction of mechanical wear and tear (along with gas milleage).

It all begins with Tyre Grip
For a while now I've been talking about the significance of Tyres as if they exceed the importance of the chassis and suspension. Well, they do. The reason they do is that they are the ones in contact with the ground and all the improvements and performance of the chassis, suspension, steering, engine, drivetrain and even aerodynamics, have to be realized by applying a certain force through the tires and onto the ground. What's the worth of a incredible Formula 1 Chassis if you fit it with bike tyres that can't convey it's performance to actual speed?

The tyre only contacts the ground in a little patch, about the size of an average size 9 shoe. Try stuffing two pieces of paper on both sides of the a tyre and you will see what little distance is left between them. This is the contact patch. The rubber inside the contact patch generates grip by squeezing into fybers into the little undulations of the tarmac surface. Each rubber element twists down under the weight of the car and into the surface, generating grip. The formula used to present this is:

Fs=Nμ


Fs is the force of static friction (grip) and N is the Normal force, which is the vertical loading of the rubber fyber. μ is the coefficient of friction: Softer rubber or grippier pavement hold a higher μ value. The fyber distorts and squeezes into the earth, but it also has to do things: Make the car accelerate, decelerate or turn. For the car to accelerate, the gripping elements must distort not only downward but to the back as well. For deceleration, tire rubber distorts forward, and for cornering it distorts laterally. Overall, there is so much twisting and deforming that the fyber can take, one extended too far it will start tearing apart and slide.



This can be expressed mathematically as F=√(L²+T²) and can be intuitivelly expressed like this: Every tire can only give you 100% of performance. Spend 80% on cornering and you have 20% left for either braking or accelerating. Decrease it for 50% cornering, and you've freed the extra 30% for braking/accelerating, and if you spent 100% on braking, you have zero left for steering! This is expressed in the "Friction Circle," which is an oval-shaped circle which expresses how much a car can corner, brake, accelerate and/or combined, without exceeding the limit of grip. According to this modell, you need to steer while keeping the car moving at a constant speed, which is done by sensitivelly using the throttle.


However, there is more to it than that. When the car accelerates, decelerates and/or turns, it experiences changes of balance. When we brake, we apply a "negative acceleration" vector while pulls backwards, while the force of inertia is pushing forward. The grip of the wheels when they brake and slow down resists this force. So the force tries to trick the tires. Instead of pushing straight on against the decelerating tires, it disguises itself as torque -- a force that works around in a twist and trying to roll the car forward instead of pushing it. This is why you feel the car nosediving when you brake and it is called a forward weight transfer. It changes the weight distribution of the car to cause the front axle to be pushed down against the road, while the rear axle is slightly elivated. 


Remember the first formula? Fs=Nμ? If the forward weight transfer pushes down against the tires, it increases the normal force N and increases grip in the front, on the expense of the rear. The same would happen when steering, where the wheels on the "outside" of the corner get loaded, and the rear wheels under acceleration. Weight transfers somewhat obstruct proper driveability and can reduce grip. However, a good driver can use weight transfers in a clever, accurate way which allows for faster and/or safer driving.


Weight transfers can help during transients. One transient is when we enter a corner. We turn the wheel which, through the steering mechanism, tilts the front wheels in the direction of the corner. The rubber fybers wish to continue going straight, so they twist aside in an attempt to remain perpendicular to the road. The result is a "Slip angle" which makes the tire point (and turn) at an angle slightly smaller than the angle in which you turned it. The slip angle allows the tire to use it's grip to turn the car and create a cornering force (Centripetal force), but also subjects the car to external forces.


Remember the force of inertia? It is the force that, based on Newton's first law, pushes the moving car forward. It is described as F=mv²/2. M is the car's mass; v is the car velocity or speed. The force of inertia is pushing the car forward while the grip generated by the tires is pulling it into the the corner. This turns the inertia into a side force (lateral force) which pushes the car out. The force is in part turned into torque and creates the weight transfer we talked about earlier. The side force, which originates from the inertia, is expressed as:


F=mv²/r
F=mgμ           
mv²/r = mgμ


v=√g μr


r is the radius of the corner and g is the force of gravity (downward acceleration of ~9.8m/s). If the cornering speed v exceeds the speed allowed by the variants of grip levels (μ) and the corner radius (r), the car will slide forward and out of the corner, as if it is refusing to turn. The way to reduce this possibility is:
1. Reduce speed
2. Increase grip: By investing in good tyres, tyre pressure, wheel alignment, etc...
3. Increasing radius: Taking a line through the corner (look in the article about cornering lines) that will increase the radius
4. Being smooth and percise with your steering input
5. Using a forward weight transfer 

The different ways are listed according to priority. The effect of proper corner entry speed is much larger than that of a line that allows to take a greater radius. Being smooth is also important, because the effects of tire element deformation and suspension movement takes some time, which is why we need to load the car with forces  progressively. 

A forward weight transfer is an important element in proper corner entry. When you brake, steer or accelerate, the car's center of gravity moves. When you brake, the wheels generate deceleration that slows down the car. The force of inertia (K=mv²/2) seeks to keep pushing the car forward, resisting the deceleration. This resisstence makes some of the inertia turn into torque that tries to push the car in a circular motion and basically roll it foward. This can be felt as the car nosediving during sudden braking. This movement creates an additional downforce which pushes down against the front tires, giving them more grip, while proportionally reducing grip to the "elevated" rear wheels.

It's important to understand this weight transfer: If we look at the formula for cornering speed, we see that it carries two contradicting effects: A heavier car produces a greater side force pushing it out, but also produces greater friction that keeps it in the corner. We can see that eventually the parameter M is canceled out of the formula, which might lead us to the conclusion that it carries no effect. This is not true. Because of the deformation of the tire's elements under load, it is custom to say that the coefficient of friction reduces slightly under additional load, so the net result of additional weight reduces the overal grip/turning speed.

However, with a proportional increase of tire pressure, the effect of weight on the tires' grip is reduced and the grip will, up to a certain point, be increased. The car will still suffer, though, from dynamic disfunctions like slower transients coming into the corner. 

However, this disscussion refers to actual weight or load. Weight transfers are a result of an application of torque, not actuall mass. The result is that a forward weight transfer will categorilly increase the front grip and reduce rear grip, making the rear of the car slide out. This reduces the overall cornering speed, but increases the speed that can be carried through the transient. 

When the car is turning into the corner, it does so by relying more on it's front wheels (which are the ones tilted into the corner) that on it's rear wheels. Once turned, it achieves a steady state of cornering where it relies on both outside wheels, and when it straightens out of the corner it relies on the rear wheels. Hence, turning into the corner we seek to increase the front tires' grip by gently loading them with additional downforce by slowing down very gently. Of course braking would mean using some of the tires' grip for deceleration instead of cornering, but with feel it's possible to achieve a positive influence. There is a slight trade-off is the immediate responsiveness of the steering, where the tire takes slightly longer to develop the front slip angle, so there is a slight (unnoticable) hesitation before the car responds, but than it reacts more sharply and the transient is overall quicker and can be performed at a higher speed. 

After the turn-in is finished, the braking force is smoothly removed and replaced with steady-state acceleration, where the car is not slowing down or speeding up. This allows the fastest possible speed through the corner. when coming out, we increase the throttle to accelerate, which helps the rear wheels to straighten us out. This procedure is relevant mainly to slow-speed corners. In the faster corners, there is no problem in rotating the car into the corner and "trail braking" is not nessecary.

Another point, to be mentioned after smoothness and weight transfers, is that of decisiveness. Clearly a smooth steering input is not effective if it's too slow. The subject of how quickely to turn the wheel is debated amongst many performance drivers and is perhaps the most dramatic difference in driving style betweent different drivers. It is true that being smooth is crucial, but it so happens that some professional drivers (on tarmac) are not so very smooth and reach identical and sometimes superior results as smoother drivers.

The reason is that some drivers are too smooth, while others are to "jerky" and a selected few know to achieve the right balance. The "right balance" changes depending on the corner and the car. Slow, sharp corners, where you take a very late line (the "last apex line") require a decisive steering input where the weight of the car is on the front wheels. The quick steering input is required to enable the car to change directions quickely enough, as the required line obligates.

Another situation is a fast corner, where for some reason the car has an increased understeer tendency that makes it "refuse" to take the right line. The understeer tendency can be a result of car setup (usually in powefull front-wheel drive) and of the corner (usually an uphill incline or side slope that increases understeer). In this case, we do not want to reduce the throttle input, so we compensate by a quick steering input that turns the car in the right direction.

Effect of car setup
The basic rules of effective driving are paramount for every car and every driving environment. However, slight differences of driving styles remain, and they have to fit the car being driven. A professional race car is the most extreme example. It has a low ride height and center of gravity, reduced weight and stiff suspension. The result: A sharp car that has to be driven very smoothly and without much of weight transfers. Being low and light, the car's weight transfers are smaller. A small weight transfer can occur faster, and the stiff suspension make it happen faster still. Because the car is sharp, the driver must compensate by being smooth and carefull around the limit.

Road racing cars and especially rally cars are set up differently. They are a tad softer, ride higher and with less grippy tires. They therefore generate larger weight transfers and react more progressively. These cars require a driving style which is a bit more "decisive" and not quite as smooth as a proper race car. 

Road cars portray a wide arc of handling characteristics that might be similar to one of the above and be a symbiosis of each of the prototypes. Some cars will require a smooth and accurate driving style, but others might perform better in a driving style that is a bit more "messy." A short and relativelly stiff car with a relativelly balanced weight transfer will be more responsive to a decisive driving style. Where a stiffer car with a larger wheelbase and and maybe a heavier front, will not tolerate such a style. 

It's important to understand that the difference is mainly in the approach, and the actuall changes in the driving style are much more subtle than they might appear from reading. All cars require smooth and percise inputs, which are also as decisive as necessary.