In many ways, tyre 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 modelling or design, even if it is found by lucky
accident, can lead to several years of domination by one tyre company, as with
BFGoodrich in autocrossing now. An unfortunate choice of tyre by a competitor
can lead to frustration and a disastrous hole in the budget.
This month, we investigate the physics of tyre adhesion a little more deeply
than in the past. In Parts 2, 4, and 7, we used the simple friction model given
by F µW, where F is the maximum
traction force available from a tyre; µ, assumed constant, is the
coefficient of friction; and W is the instantaneous vertical load,
or weight, on a tyre. While this model is adequate for a rough, intuitive feel
for tyre behaviour, 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 tyre 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 tyre 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
tyre. Most writers call this quantity "slip angle." I think this name
is misleading because it suggests that a tyre 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 tyre can yield occurs at
the threshold where the tyre 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 tyre 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
tyre implied by F µW with constant µ
would be a non-elastic tyre. Anyone who has driven hard tyres on ice knows that
non-elastic tyres are basically uncontrollable, not just because µ
is small but because regular tyres on ice do not twist appreciably.
The first and most obvious mode of deformation is radial. This deformation is
along the radius of the tyre, the line from the centre to the tread. It is
easily visible as a bulge in the sidewall near the contact patch, where the tyre
touches the ground. Thus, radial compression varies around the circumference.
Second is circumferential deformation. This is most easily visible as
wrinkling of the sidewalls of drag tyres. These tyres are intentionally set up
to deform dramatically in the circumferential direction.
Third is axial deformation. This is a deflection that tends to pull the tyre
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 tyre. But it is very useful to consider the
differences in these deflections around the circumference.
Let us examine exactly how a tyre 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 tyre. 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 tyre circumference as
we go along the following analysis.
Consider a car travelling at speed v 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 tyre. 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 tyre 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 clockwise.
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
must be there, otherwise the tyre 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 tyre (and the
eraser-I hope you are still following along with the eraser) walks continuously
to the right.
The better grip on the right hand side of the contact patch adds up to a net
rightward force on the tyre, 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 tyres 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 tyre walks is the grip angle. All quantities of
interest in tyre 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 tyre behaves qualitatively like an ideal tyre, which grips until
the adhesive limit is exceeded and then slides. A real tyre, 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 other
words, 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 squalling
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 modelling in the future.
Speaking of the future, summarizing briefly the past of and plans for the Physics
of Racing series. The following overlapping threads run through it:
Tyre Physics
concerns adhesion, grip angle, and elastic modelling. 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 modelling 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 discussion.
The following is a list of articles that have appeared so far:
Weight Transfer
Keeping Your Tyres Stuck to the Ground
Basic Calculations
There is No Such Thing as Centrifugal Force
Introduction to the Racing Line
Speed and Horsepower
The Circle of Traction
Simulating Car Dynamics with a Computer Program
Straights
Grip Angle
and the following is a tentative list 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)
Transients,
presenting the dynamics of entering and leaving corners, chicanes, and
slaloms (this one suggested by Karen Babb)
Stability,
explaining why spins and other losses of control occur
Smoothness,
exploring what, exactly, is meant by smoothness
Modelling Car Data
in a computer program; in several articles
Modelling 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.
µW, where F is the maximum
traction force available from a tyre; µ, assumed constant, is the
coefficient of friction; and W is the instantaneous vertical load,
or weight, on a tyre. While this model is adequate for a rough, intuitive feel
for tyre behaviour, 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.