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The Physics of Racing, Part 10: Grip Angle
Brian Beckman, physicist and member of No Bucks Racing Club
P.O. Box 662 Burbank, CA 91503
In many ways, tire mechanics is an unpleasant topic. It is shrouded in uncertainty,
controversy, and trade secrecy. Both theoretical and experimental studies are extremely
difficult and expensive. It is probably the most uncontrollable variable in racing today.
As such, it is the source of many highs and lows. An improvement in modeling or design,
even if it is found by lucky accident, can lead to several years of domination by one tire
company, as with BFGoodrich in autocrossing now. An unfortunate choice of tire by a
competitor can lead to frustration and a disastrous hole in the budget.
This month, we investigate the physics of tire adhesion a little more deeply than in
the past. In Parts 2, 4, and 7, we used the simple friction model given by , where is the maximum traction force available from a tire; , assumed constant, is the coefficient
of friction; and is the
instantaneous vertical load, or weight, on a tire. While this model is adequate for a
rough, intuitive feel for tire behavior, it is grossly inadequate for quantitative use,
say, for the computer program we began in Part 8 or for race car engineering and set up.
I am not a tire engineer. As always, I try to give a fresh look at any topic from a
physicist's point of view. I may write things that are heretical or even wrong, especially
on such a difficult topic as tire mechanics. I invite debate and corrections from those
more knowledgeable than I. Such interaction is part of the fun of these articles for me.
I call this month's topic `grip angle.' The grip angle is a quantity that captures, for
many purposes, the complex and subtle mechanics of a tire. Most writers call this quantity
`slip angle.' I think this name is misleading because it suggests that a tire works by
slipping and sliding. The truth is more complicated. Near maximum loads, the contact patch
is partly gripping and partly slipping. The maximum net force a tire can yield occurs at
the threshold where the tire is still gripping but is just about to give way to total
slipping. Also, I have some difficulties with the analyses of slip angle in the
literature. I will present these difficulties in these articles, unfortunately, probably
without resolution. For these reasons, I give the quantity a new name.
A tire is an elastic or deformable body. It delivers forces to the car by stretching,
compressing, and twisting. It is thus a very complex sort of spring with several different
ways, or modes, of deformation. The hypothetical tire implied by with constant would be a non-elastic tire. Anyone who has driven hard
tires on ice knows that non-elastic tires are basically uncontrollable, not just because is small but because regular tires on
ice do not twist appreciably.
The first and most obvious mode of derformation is radial. This deformation is along
the radius of the tire, the line from the center to the tread. It is easily visible as a
bulge in the sidewall near the contact patch, where the tire touches the ground. Thus,
radial compression varies around the circumference.
Second is circumferential deformation. This is most easily visble as wrinkling of the
sidewalls of drag tires. These tires are intentionally set up to deform dramatically in
the circumferential direction.
Third is axial deformation. This is a deflection that tends to pull the tire off the
(non-elastic) wheel or rim.
Last, and most important for cornering, is torsional deformation. This is a
difference in axial deflection from the front to the back of the contact patch.
Fundamentally, radial, circumferential, and axial deformation furnish a complete
description of a tire. But it is very useful to consider the differencesin these
deflections around the circumference.
Let us examine exactly how a tire delivers cornering force to the car. We can get a
good intuition into the physics with a pencil eraser. Get a block eraser, of the
rectangular kind like `Pink Pearl' or `Magic Rub.' Stand it up on a table or desk and
think of it as a little segment of the circumference of a tire. Think of the part touching
the desk as the contact patch. Grab the top of the eraser and think of your hand as the
wheel or rim, which is going to push, pull, and twist on the segment of tire circumference
as we go along the following analysis.
Consider a car traveling at speed in a straight line. Let us turn the steering wheel
slightly to the right (twist the top of the eraser to the right). At the instant we begin
turning, the rim (your hand on the eraser), at a circumferential position just behind the
contact patch, pushes slightly leftward on the bead of the tire. Just ahead of the contact
patch, likewise, the rim pulls the bead a little to the right. The push and pull together
are called a force couple. This couple delivers a torsional, clockwise stress to
the inner part of the tire carcass, near the bead. This stress is communicated to the
contact patch by the elastic material in the sidewalls (or the main body of the eraser).
As a result of turning the steering wheel, therefore, the rim twists the contact patch
The car is still going straight, just for an instant. How are we going to explain a net
rightward force from the road on the contact patch? This net force mustbe there,
otherwise the tire and the car would continue in a straight line by Newton's First Law.
Consider the piece of road just under the contact patch at the instant the turn begins.
The rubber particles on the left side of the patch are going a little bit faster with
respect to the road than the rest of the car and the rubber particles on the right side of
the patch are going a little bit slower than the rest of the car. As a result, the left
side of the patch grips a little bit less than the right. The rubber particles on the left
are more likely to slide and the ones on the right are more likely to grip. Thus, the left
edge of the patch `walks' a little bit upward, resulting in a net clockwise twisting
motion of the patch. The torsional stress becomes a torsional motion. As this motion is
repeated from one instant to the next, the tire (and the eraser-I hope you are still
following along with the eraser) walks continuosly to the right.
The better grip on the right hand side of the contact patch adds up to a net rightward
force on the tire, which is transmitted back through the sidewall to the car. The chassis
of the car begins to yaw to the right, changing the direction of the rear wheels. A
torsional stress on the rear contact patches results, and the rear tires commence a
similar `walking' motion.
The wheel (your hand) is twisted more away from the direction of the car than is the
contact patch. The angular difference between the direction the wheel is pointed and the
direction the tire walks is the grip angle. All quantities of interest in tire
mechanics-forces, friction coefficients, etc., are conventionally expressed as
functions of grip angle.
In steady state cornering, as in sweepers, an understeering car has larger grip angles
in front, and an oversteering car has larger grip angles in the rear. How to control grip
angles statically with wheel alignment and dynamically with four-wheel steering are
subjects for later treatment.
The greater the grip angle, the larger the cornering force becomes, up to a point.
After this point, greater grip angle delivers less force. This point is analogous to the
idealized adhesive limit mentioned earlier in this series. Thus, a real tire behaves qualitativelylike
an ideal tire, which grips until the adhesive limit is exceeded and then slides. A real
tire, however, grips gradually better as cornering force increases, and then grips
gradually worse as the limit is exceeded.
The walking motion of the contact patch is not entirely smooth, or in otherwords,
somewhat discrete. Individual blocks of rubber alternately grip and slide at high
frequency, thousands of times per second. Under hard cornering, the rubber blocks
vibrating on the road make an audible squaling sound. Beyond the adhesive limit, squealing
becomes a lower frequency sound, `squalling,' as the point of optimum efficiency of the
walking process is bypassed.
There is a lot more to say on this subject, and I admit that my first attempts at a
mathematical analysis of grip angle and contact patch mechanics got bogged down. However,
I think we now have an intuitive, conceptual basis for better modeling in the future.
Speaking of the future, summarize briefly the past of and plans for the Physics of
Racingseries. The following overlapping threads run through it:
- Tire Physics
- concerns adhesion, grip angle, and elastic modeling. This has been covered in Parts 2,
4, 7, and 10, and will be covered in several later parts.
- Car Dynamics
- concerns handling, suspension movement, and motion of a car around a course; has been
covered in Parts 1, 4, 5, and 8 and will continue.
- Drive Line Physics
- concerns modeling of engine performance and acceleration. Has been covered in Parts 3,
6, and 9 and will also continue.
- Computer Simulation
- concerns the design of a working program that captures all the physics. This is the
ultimate goal of the series. It was begun in Part 8 and will eventually dominate
The following is a list of articles that have appeared so far:
- 1. Weight Transfer
- 2. Keeping Your Tires Stuck to the Ground
- 3. Basic Calculations
- 4. There is No Such Thing as Centrifugal Force
- 5. Introduction to the Racing Line
- 6. Speed and Horsepower
- 7. The Circle of Traction
- 8. Simulating Car Dynamics with a Computer Program
- 9. Straights
- 10. Grip Angle
and the following is a tentativelist of articles I have planned for the near
future (naturally, this list is `subject to change without notice'):
- Springs and Dampers,
- presenting a detailed model of suspension movement (suggested by Bob Mosso)
- presenting the dynamics of entering and leaving corners, chicanes, and slaloms (this one
suggested by Karen Babb)
- explaining why spins and other losses of control occur
- exploring what, exactly, is meant by smoothness
- Modeling Car Data
- in a computer program; in several articles
- Modeling Course Data
- in a computer program; also in several articles
In practice, I try to keep the lengths of articles about the same, so if a topic is
getting too long (and grip angle definitely did), I break it up in to several articles.
Next: Part 11: Braking
Previous: Part 9: Straights