Driving is an action based on the laws of physics. The main aspects involved here are aspects of kinematics, friction and heat. Understanding the physics helps in understanding driving. My former article dealt with physiological aspects of driving, while this one will deal with key physical issues in handling.
It all starts with grip
Tire grip is the ultimate factor that determines a car's grip and handling characteristics. Tire grip forms a force called adhesion which is physically described as a Force of Static Friction (Fs). The tire rubber is made of little fibers ("Elements") that, upon reaching the point in which the tire contacts tire ground, deforms under the load of the car's weight and squeeze into the little undulations of the road, generating grip.
Therefore, the amount of grip depends on the amount of such downforce (physically described as the "Normal Force") and on the individual characteristics of the rubber (how soft) and the road (how rugged). This is described in the basic formula of Fs=μn, where μ stands for the "coefficient of friction" of the tire rubber and road surface.
This formula is the most basic foundation to understanding tire grip, but it is not everything. It's important to comprehend that it stands for the grip of a single rubber element. When looking at the car as a whole, things are much different. Still, even when looking at a given rubber element, we come to the conclusion that this Formula is in fact an Euclidian Vector. In English, it's not static.
Let's look at a rolling tire. Now lets pick a certain rubber element to look at. The majority of the tire circumference is not against the ground so the element is in fact airborne. It does nothing to generate grip. As the tire rolls, the element reaches the point of contact with the ground. But the transition does not occur instantly. The fiber is progressively faced with a greater and greater load until it reaches "rock bottom" when placed directly perpendicular to the road.
At this point, it carries the full load of the car and distorts as deeply as possible into the undulations of the road. As the tire continues to rotate the element is retracted from the ground. Again, not instantly but progressively. The weight of the car is eased off of the element and it gradually stops generating grip until it becomes airborne again.
Once a certain gripping element goes beyond it's natural elastic ability to bend and cram under the load, it starts sliding. Sliding is defined as "Kinetic Friction" (Fp) and consequentially holds a lower coefficient of friction. The tire in facts slides and grips (applies both sorts of friction) simultaneously, when it's best performance lies at the peak of the static friction value, at which point there is also a significant precent of sliding taking place.
The graph describing this relationship is based on the following Formula: S=(1-[r*w/V])*100. Where S stands for the percentage of slip, R stands for tire radius; W stands for the speed in which the tire is rolling and V stands for the car's speed. This graph teaches us that for maximum grip when braking or accelerating, we want the tires to rotate slightly slower/faster (accordingly) than the car itself.
The Slip Angle
What happens in a corner? The car is driving straight ahead, thus subjected to Newton's laws of motion (F=ma). This law dictates that a moving object is being pushed by a force of momentum that increases with speed or with the mass of the object. This is the greatest force working on the car and it pulls straight ahead.
When we turn the wheel, the steering system tilts the front wheels into the direction of the corner. However the car, and likewise the tread elements, seek to maintain forward motion. By turning the steering wheel we have effectively turned the wheel/rim, but not the tire itself. The rubber that is pressed against the road uses it's elastic ability to twist in order to keep on going forward. But since it does not have a mind of it's own it has to turn somewhat into the corner too. The result is that no matter how much we turn the wheel, the actual arch in which the tires are pointed will always be somewhat smaller. This angle is the slip angle.
The slip angle enables the wheels to rotate when tilted, thus harnessing the car's grip to create acceleration in a lateral plain. I.e. generate a cornering force, better known as a Centripetal force. The force of inertia, however, still exists and turns into a side force that tries to push the car aside. The tires' grip work as a anchor against this force.
Once the side force is blocked by the tires' grip, it turns in part into Torque that seeks to rotate the car's body instead of pushing it sideways. This generates a load transfer that makes the car lean against the "outside" wheels, which would thus support most grip for the cornering effort. The inside wheels become quite expendable. This allows a driver to focus on keeping his outside wheels on the grippy pavement, while he can even drop his inside wheels onto the gravel, to avoid a collision with another car in a bend.
Simultaneously, the side force pushes the front sideways, which than rotates the car's chassis, which now effectively tilts the rear wheels (which are not turned by the wheel) relative to the road, making them experience a slip angle as wheel. The rear wheel slip angle generates a side force which than works through the leverage of the chassis to multiply the slip angle of the front. This can be seen when a car is sliding it's back-end which rotates the front into the corner and even spins the car around. The side force also works on other individual car parts: The load transfer compressed the springs. The side force twists the chassis rigid frame, and tries to push the wheel axle sideways.
However, until the lateral load transfer is realized, the car is still undergoing a "transient" or "change of direction." When changing direction the car is said to be subjected to angular momentum and, as such, is still developing the potential of it's front slip angle. As such, during the transient, the car will have a natural tendency to push forward and out of the corner with it's front wheels. This would seem as the car not turning as much or not reacting to the wheel, and trying to keep on moving forward.
In fact, this tendency is maintained even after the driver finished to turn the wheel into the corner. Any force working on the car is defined as acceleration and any acceleration creates a weight transfer. When we accelerate forwards, we shift the car's weight to it's back. When we slow down (negative acceleration), some of the inertia pushing forward is translated into torque that makes the car "nosedive."
Critical cornering speed
However, assuming a car is only accelerating laterally. I.e. turning without slowing down or increasing the speed, the car will try to follow the straight-on path led by the force of inertia. This will make it push at the front end and slide forward out of the corner, rather than slide it's back end to produce a dreadful (but rare) fishtail. The highest speed in which the car can make it through the corner is called the "critical speed", and here follows the manner of calculating it:
F=MA; A=V²/t. => F=(MV²)/r.
This formula dictates that the maximal cornering speed is limited by the car's mass and speed, because increasing either of the two will increase the force of inertia and thus the side force. Likewise, tightening the corner's radius will result in a similar effect. However, we must also consider external factors like the constant effect of the earths' gravity, and the coefficient of friction. We can therefore assert that: (MV²)/r =mgμ; Which can be calculated to a finite formula of V=√g μr.
Conclusions:
- Reducing speed is the most efficient way to avoid sliding in corners, although it's also possible to try and "straighten" out the bend slightly.
- Smoothness and precision are important, as it is important to start turning the wheel a fraction of a second before the actual point where we want to start turning in, because the slip angle takes about 1/4 of a second to develop.
- The lateral weight transfer can allow cutting the corner, if required for speed (in racing) or to avoid another car in a bend.
The Traction Budget
This is defined as the car's maximal cornering potential when balanced (not speeding up or slowing down), but it can change if we apply an additional force onto the car. The tires' elements have to distort laterally to form lateral grip, and to distort forward or backwards to decelerate or accelerate the car. However, we have concluded earlier that rubber can only twist so much. The result is that if we apply several "chores" unto a tire, like accelerating and turning simultaneously, we divide the full traction potential between cornering and accelerating.
The compromise it essentially between longitudinal forces (T) and lateral forces (L) and it is described as F=√(L²+T²). When this formula is applied unto a X-Y axis system, it forms an oval shape that is described as the "Circle of Friction." You can apply 1g of forward acceleration, but you can only apply 0.5g of acceleration and 0.5 of turning combined, or 0.8g of accelerating and 0.2g of turning, etc...
This formula essentially holds true after turning into the corner. However, during the transients in which the car is turning into the corner and straightening out of it, longitudinal forces can increase a car's potential. Slowing down into corners shifts the load forward to increase grip levels to the front wheels. With feel, the overall speed at turn-in can be increased since the front wheels, which dictate this transient, will have more downforce pressing them against the road.
A weight transfer is a phenomenon of some of the forces working on the car turning into torque that tries to roll it over, either forward while braking, or sideways when turning. It's described as: Lf= (A x M x H)/w. Lf or Tf is the weight transfer, which A is the acceleration (either lateral or longitudinal), M is the car's mass, and H is the height of the car's center of gravity. W is the car's track width. Obviously a lower, wider car will be more stable than a tall and narrow one.
Sprung Mass
But what about the suspension? Surely softer springs will make a car "lean" more against the corner, right? No. The springs movement only articulates the weight transfer. It does not change the amount of load which is transfered. It does, however, change the rapidness of the weight transfer and the responsiveness of the car. Softer springs will induce a more sluggish response, but will turn more of the side force into torque (making it harder to slide).
We can point out two main prototypes of cars: The first is heavy, soft and tall -- this can has slow reactions, which makes it in-obedient but also very predicable, while the other type is light, low and stiff, which makes it more responsive, especially to the driver's inputs through the pedals, but also more quick to go over the limit.
This whole talk about handling characteristics usually evoke the fear of rolling over. It always tickles me when people fear that braking hard from a high speed will make them roll over. A car is least likely to roll under braking, because cars are naturally longer than they are wide (5 meters relative to 1.8 meters in average), which leads to a smaller weight transfer. Likewise, the car does not stop all at once, it stops over a certain stopping distance. If one was to improve his brakes and wheels to allow instant stopping, he would roll his car over.
And this brings us to the last point: The brakes are mechanical instruments that produce a stopping force (torque). The effect of this force diminishes with speed. So, while stopping quickly from a higher speed is more frightening, it is physically "easier" -- the brakes have a harder time stopping you from a higher speed, so you need to brake hard at first, when the car's speed is still very high. Unfortunately, the normal driver brakes with ease and than increase the braking force.
Rolling over when turning is also not very likely. As I said, the car is built to slide it's front wheels before anything else. Sliding is the car's way to avoid rolling over. For a car to roll over it has to generate such strong a grip force as to overcome the forces of it's suspension and the force of gravity, all without fully realizing the grip potential (without sliding). It requires a grippy car with a tall center of gravity and a short track width. The car will develop more grip and more of a cornering force than it's suspension and chassis (based on height and track width) can take, and rolls over.
Some cars lean very much on it's suspension in hard cornering, and some even momentarily lift a wheel, mostly a rear wheel. Both phenomena are a result of the function of the suspension: The former is the action of the springs and the latter is the action of an anti-roll bar -- both natural and unrelated to rolling a car over.
Rolling car depends on grip levels, ride height and track width. As we add them to the formula for critical cornering speed: mgb/2=mv²/rh =>V=√gr/2·b/h. All private sedan cars are built to slide well before this limit.
Unbelievable that driving can be broken down to the laws of physics in such a precise way! One is always learning new things everyday... and this certainly taught me something I never knew before. Thank you.
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