Round Black Donuts, or in Other Words, Tires Part 2 Bob White ...

Round Black Donuts, or in Other Words, Tires
Part 2
Bob White, December 2009
Now Some Tire and Vehicle Parameters That Affect the Story
The non-tech or non-auto cross/racing types may want to quit at this point since the
following is a bit more technical but don’t let that deter you. For sport car handling
performance and feel there are a few basic concepts that determine how the tire and car
work together. For the car these are its weight, polar moment of inertia about the vertical
axis through the center of gravity, and the ratio of the two. Aerodynamic effects are
neglected here since for street driven cars they are generally not of much effect, except
for gas mileage, at the speed most people drive. The weight is self evident; however, the
moment of inertia depends on the location, or distribution if you like, of the weight. The
farther heavy parts such as the engine, battery, and fuel tank are from the center of
gravity the greater the polar moment of inertia. A good example of this that is well
known to almost everyone is a spinning (yaw rate as it is called for cars) ice skater
(professionals mainly). When they are spinning fast they have their arms at their sides
and when they want to slow down they stick their arms out, redistributing the weight
farther out from the spin axis, which increases the rotational inertia and they slow down
and vice versa.
A careful examination of the data for the weight and polar moment of inertia of hundreds
of cars of all sizes and shapes leads to an interesting, if not surprising, observation. When
the weight and polar moment of inertia are plotted on a graph, see the figure below,
almost ALL of the good handling cars in the world fall along single line. These range
from Formula 1 , Indy, and great sports cars such as the Porsche 908/3 to a MB model
and the 986 Porsche and even the Ford Taurus.

Figure. 2 Polar Moment of Inertia of selected cars.
Data exists for many more vehicles but they would be scattered all over the figure on
either side of the dark line, mostly to the left, which passes through the points for
accepted good handling cars. To convert kg m^2 to lb ft^2 multiply by 23.67. To convert
kg to lbm multiply by 2.2.
Cars falling to the left of the dark line in figure no. 2 such as the Olds 98 have the “turn
the steering wheel and wait for something to happen “ type of handling and those to the
right have a “ jittery” type of feeling and are easily affected by grooves in the road such
as those in blacktop caused by heavy trucks. Mid engine cars tend to have the latter
characteristic but manufacturers have learned to compensate for this by moving certain
heavy components such as batteries and radiators out to the corners to get the correct
relationship, the Boxster being a good example. This does not mean that low polar
moment of inertia cars are bad handling. It means that it must also be associated with an
appropriate weight as in figure no. 2. Cars with outstanding response and handling such
as in F1 and the Porsche 908/3 have low moments of inertia but also low weight.
Weight is the big factor in acceleration and braking and the weight to polar moment of
inertia ratio for handling and response accelerations. The tire at the road contact area
determines the forces to accomplish these accelerations. This brings up the need to
discuss specific tire properties. In the following the emphasis will be on those that are the
most important for handling and feel.

Let’s start with friction. The coefficient of friction is defined as the ratio of the force to
pull or push something along a surface to its weight. In equation form this is shown
below.
The coefficient of friction mu = (pulling force) / weight
The coefficient of friction for rubber and a tire is a function of almost everything i.e. mu
is a function of load, area, temperature, tire pressure, speed, sliding speed, road surface,
contaminants and lubrication, tire construction, the rubber composition and a few more.
The numerical value of mu varies from almost zero (0.02 wet ice and hydroplaning) to
numbers like 3.0 for a pink pearl eraser being pulled along a sheet of glass. Without
going into details, hydroplaning is a serious event where the tire rides up on a layer of
water resulting in complete loss of control. We won’t talk about it further, but you should
know that the speed at which hydroplaning occurs (there are NASA reports documenting
the phenomena and I can supply them for those who wish to know more) is given by
Hydroplaning speed in mph = 10.35 x square root (tire pressure in psi).
Thus, if the tire pressure is 36 psi the hydroplaning speed is = 10.35 x sqrt(36) = 10.35 x
6 = 62 mph. This is basically independent of the tread which basically determines the
depth of water in which hydroplaning occurs but not the speed.
Typical maximum coefficient of friction values for some common types of tires can
usually be well correlated by a linear function such that, mu = mu at zero load - C1 *
(load in pounds). Figures no. 3 and 4 show the maximum coefficient of friction for some
street and race type tires respectively.
Figure no. 3 Maximum Coefficient of Friction vs Load for Some Street Tires

Figure no. 4 Maximum Coefficient of Friction vs Load for Some Race Tires
The following table is a key to the race tire legend in Figure no. 4.
Table no. 1 Race Tire legend key
Letter Tire Type and Size
A 165 HR 15 (5 ½ J x 15)

B 245/575 – 15 slick (12 K x 15)
C 265/575 – 15 slick (12 K x 15)
D 215/60 VR XWX (7 J x 15)
E 245/575 – 15 intermediate (12 K x 15)
F 340/575 – 15 slick (15 K x 15)
Notice in comparing the friction values in Figures 3 and 4 that it is important to use the
same load values since the weight scales are different. For the worst race type tires the
friction values are 20% or so higher than the best street values and the best race tire
values are 60% or more better than the street values! For the same radius of a turn the
speed increases with the square root of the mu value which means sqrt(1.6) = 1.26 or a
26% increase in speed, i.e. a 50 mph maximum turn speed becomes 60 mph with race
tires.
For straight line acceleration or braking the change is directly proportional to the friction
value assuming the engine has sufficient torque (and correct gearing) to keep the tire at
its maximum friction value. This leads to another characteristic of tires, there must be
some slip between the tire and the road to produce any braking, acceleration, or turning
force. For braking and acceleration the slip is ratio of the actual rotation speed of the tire
and the ideal rotational speed being the rotation speed if you let the tire roll down a hill
by itself. We will return to what slip is for a cornering tire a bit later. Thus, a free rolling
tire has a slip of zero and a locked up tire has 100% slip. Figure no. 5 below shows how
the friction varies with slip.

Figure no. 5 The Relationship Between Tire Slip at the Road Interface and Coefficient of
Friction. Note the mu max = 1.0 is used here is arbitrary to show the typical relation to
slip but may be much larger as seen in Fig. no. 3 and 4
Note how the coefficient of friction has a maximum at around 20% slip except for new
snow. The same is true of sand as on a beach, where the plowing up of the snow or sand
in front of the tire leads to an increase in resistance. On normal road surfaces if the brakes
are applied hard enough to generate slip greater than the maximum, the coefficient of
friction drops and the tire spins down to 100% slip or locked up tire. At that point the tire
has about 25% less stopping capability. This is why people are taught to pump the brakes
to try to stay at or near the peak coefficient of friction. ABS systems do the this by
changing brake line pressure to keep the friction at its peak. The ABS software knows on
which side of the peak the tire slip is, since the braking effect decreases as speed
decreases if the slip is too low and the tire starts to spin down if it is too high.
Under acceleration and braking a sensitive driver can tell if the tire is past its maximum
friction point since at the maximum all of the friction is being used for traction, so none is
left to control side forces acting on the car and the car begins to move around sideways.
For rear drive cars during acceleration, this was sometimes referred to as fish tailing. If
you don’t believe the latter it is easy to demonstrate now that winter is here. Just step too
hard on the gas on the ice and snow and note how the rear end immediately begins to
slide sideways. For front wheel drive cars if you step on the gas in a turn the car just goes
straight.
Now let’s return to the cornering friction problem referred to earlier. The question in this
case is what is meant by slip. For cornering, the definition is the difference in degrees
between the direction the tire is pointing and the actual direction it is moving. Most
drivers have experienced this, for example, when driving on a slippery surface. If the
steering wheel is turned too quickly the car only slowly begins to turn despite the fact that
the tire is at a significant angle to the car’s direction. The difference in angle is called,
rather imaginatively, the slip angle. The cornering force generated is usually measured
by placing the tire on a rotating drum or conveyor-like flat moving surface and measuring
the side force exerted on the rim as the angle of the tire to the direction of motion is
changed. Figure no. 6 below shows the value of side force as a function of slip angle for
six different race tires at three different tire loads.

Figure no. 6 Tire Cornering Side Force as a Function of Slip Angle for Some Race Tires
Street tires exhibit the same characteristics except with lower maximum values, a smaller
rate of increase from zero slip, and about twice as large of a slip angle at the maximum
value. The maximum value compared to the tire load determines the maximum mu value,
see figures no. 3 and 4.
The rate of increase (or slope) of the cornering force from zero slip is essentially the side
force at a slip angle of 1 degree and is called the cornering force coefficient Cf. Figures
no. 7 and 8 show values of the cornering force coefficient for typical street tires and the
race tires of Figure no. 6 respectively.

Figure no. 7 Typical Cornering Force Coefficients Cf for Street Tires

Figure no. 8 Cornering Force Coefficients Cf for the Race Tires of Figure no. 6 and it’s
Legend.
Notice that the cornering force coefficients of the race tires are substantially larger than
for street tires and in many cases are almost twice that of everything but the P7 which
was the first of the so called street super tires.
Even good street tires seldom exceed 350 lbf./deg, while the larger, wider race tires can
easily be in the 500 to 700 lbf/deg. range. Also keep in mind that the load on the tire
when turning is not the static load but the static load plus or minus the weight transfer.
For a Corvette in a 1 gee turn the total weight transfer is about 1150 pounds which is
distributed between the front and rear axles approximately in proportion to the front to
rear weight distribution depending on the shock absorbers, springs, and anti-roll bars. To
be more exact it is according to the front and rear axle roll resistance values.
Knowing the cornering force information, it is now possible to determine the basic level
of vehicle steering response. Figure no. 9 defines a few basic vehicle parameters that are
needed to understand a cars response.

Figure no. 9 Some Vehicle Definitions
With these definitions, it is possible to determine how a car turns and what its initial
response will be to a steering input. So let’s look at figure no. 10. It also includes the
definition of the lateral velocity acceleration and yaw velocity acceleration.

Figure no. 10 Definition of how a Car Turns and it’s Accelerations on Steering Input
From the definitions above for the accelerations generated, the v and r with the dots
above them in Figure no. 10, on initial steering wheel input a total of five factors
determine the rate at which the vehicle responds. Other than the steering input there is
only one parameter that the owner can control without serious race car type
modifications. That is the front tire cornering force coefficient by his or her choice of the
tires. The car mass i.e. weight, polar moment of inertia, and distance between the center

of gravity and front axle are determined during manufacture. Thus, an auto-x/super tire
set with front tires having 375 lbf/deg of cornering force compared to just a good tire
with 275 lbf./deg. results in about a 36% increase in the rate of vehicle initial response.
This article is about tires and how they come to be selected for your car. However, how
the tires effect the vehicles response and feel also depends on the properties of those
tires.That is why the driving tests by both the tire and car manufacturer are so important
to the feel that are associated with Porsche or Corvette or Mercedes-Benz and Lexus. So
lets hear it for the engineers, expert drivers, and their companies that understand the
importance of that feel to their customers.
A final note on a related but some what different aspect of tires. Earlier it was pointed
out that the average car owner only has a flat tire on average every 60,000 miles. In some
countries, Germany for example, a tire may not be repaired, no plugs for a nail in a tire.
The tire must be replaced regardless of whether it has been driven 10 or 100 or 10,000
miles. The figure below is a photograph of the inside of a tire that has had a nail hole
repaired with a plug. There is an almost one inch high by one quarter inch in diameter
piece of the rubbery filament used for the plug sticking out of the carcass of the tire.
Figure no. 11 An Inside view of a Plugged Tire
It is about as resilient as a piece of hot dog although stronger. So when you are driving it
is bent by the centrifugal force and flops around every time the brakes are applied,
steering inputs change the car’s direction or bumps or tar strips flex the tire. Think about
that the ext time you are going over the speed limit with a tire on your car that has a plug

in it. Seems like the Germans may have it right, especially with many parts of the
autobahn still without speed limits.