TIRE TRACTION GUIDE

AUTOMOTIVE DYNAMICS,
Brian Paul Wiegand, B.M.E., P.E.
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TIRE
BEHAVIOR

TIRE BEHAVIOR, TRACTION: MATERIAL
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TIRE TRACTION: MATERIAL

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If the elongation under load, and the subsequent unloaded contraction, of
a rubber sample were plotted on a scale more appropriate to the material’s
unique behavior, then the result would be more like:
This diagram reveals yet another curious aspect of the nature of rubber: high
“hysteresis”. Most materials subjected to cyclical stress at sufficient levels
will exhibit some conversion of mechanical energy to thermal energy, but
rubber does so abundantly at relatively low stress levels; the area between
the “loading” and the “unloading” curves represents the magnitude of this
energy loss.

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Having noted how unusual rubber is in its elasticity, hysteresis, and other
properties; we now have some clues as to the reasons for that unusual
frictional behavior known as tire traction. At the small scale level of the
contact area between tire and road the situation looks like (greatly
enlarged):

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TIRE BEHAVIOR, TRACTION: STRUCTURE
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TIRE TRACTION: STRUCTURE

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…is the circle
whose circumference is equal to
the periphery of the tire cross-
section as shown. Also shown in
the figure are various other tire
characteristics (dimensions):
S = Circumference / π
= Periphery / π

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= Tire section aspect ratio (dimensionless).
SN = Tire nominal section width (mm).
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The Rhynes Equation is very related to the Michelin
Formula which is used to estimate the width of the tire
tread “tw” (when lacking a measured value). The
Michelin Formula is:
Where:
tw = Tire tread width, assumed constant with load (in).
This equation was developed by a regression analysis
of a large selection of common road tires (and therefore
is not valid for uncommon size and type tires). Since the
Rhynes Equation incorporates the Michelin Equation
into its formulation it has the same limitation.

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TIRE BEHAVIOR, TRACTION: LATERAL
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TIRE TRACTION: LATERAL
μy = b – mN

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Fy = (b-mN)N = bN – mN2

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The presence of a lateral force on a tire not only causes a
distortion of the tire carcass diminishing the tire/road area
contact, but there are other effects as well. When a tire moving
with velocity “Vo” encounters a side load there is a
consequent sideways motion. The resultant new net motion
“V” is the combination of the original motion “Vo” and the
motion resulting from the side load. The angle “ψ” between
this new direction “V” and the original direction “Vo” is called
the “slip angle”. This term is a misnomer as it gives an
erroneous impression; the tire is not necessarily slipping or
sliding in the direction of the side load. What is actually
happening is that there are a series of small lateral
movements “dy” of the tire due to the cyclical distortion of
portions the carcass as those portions come into contact with
the road as the tire rolls forward. The combination of the
original forward rolling velocity and all those infinitesimal
“side steps” results in the new velocity direction “V”.

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“…lateral…force may be thought of as the result of slip angle, or
the slip angle as the result of lateral force…”
(Milliken, William F., and Douglas L. Milliken;
Race Car Vehicle Dynamics, Warrendale, PA;
SAE R-146, 1995, pg. 19.)

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Since the drift angle/lateral force relationship is dependent upon
quite a few parameters (inflation pressure, normal load, etc.), it is
common to look at functions which constitute only a partial
differential of the total relationship (for which no one has yet
established a complete definitive formulation based on physics*) in
order to achieve a degree of understanding. If the tire drift-
angle/lateral-force partial differential function is plotted the result
looks like:

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If matters were just as simple as the previous figure then
understanding of tire behavior would be very easy. However, the
potential or maximum lateral force that a tire can supply, and the drift
angle associated with that force, is dependent on many parameters.
The lateral force potential is primarily influenced by normal load,
longitudinal load*, camber angle (which can change with roll), roll
steer (which can be the result of normal load and camber change with
roll, but toe in/out can also change with roll), tire type (size, carcass
type and material, rubber type, tread design, aspect ratio), inflation
pressure, wheel rim width, road material and surface (smooth, rough,
dusty, etc.), weather (rain, snow, ice), temperature (road surface,
ambient, and of the tire itself), and the speed of the vehicle (all basic
tire coefficients of traction are somewhat speed dependent; the
same lateral force will produce a smaller drift angle at high speed
than at low speed**). While all these factors are significant, only tire
type, normal load, long & lat force, inflation pressure, temperature,
and speed are fundamental tire behavior and will be discussed
herein; the rest has to do with tire/suspension interaction.

47
To illustrate the effect of normal load on lateral resistance and drift angle for a
specific tire and inflation pressure a plot such as below may be used. Note
that it is essentially like the previous figure except that there are now a large
set of “Fy, ψ” functions which serve to represent an infinite variation; any
change in normal load alters the “Fy, ψ” relation, but these five example
curves may suffice as the intermediate possibilities can be approximated by
interpolation:

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Actually, the previous figure is a poor way to illustrate this behavior, which is
better shown by the following actual data plot of lateral force vs. normal load
for a 6.00x16 bias tire inflated to 28 psi (193 kPa):

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Another way of looking at tire lateral traction force behavior is presented in
this figure. This figure is somewhat like the previous, only now the lateral
traction force “Fy” is normalized by “dividing out” the now constant normal
load “N”, which of course results in the effective lateral traction coefficient
“μy” (“μy = Fy/N”). This allows for the addition of a completely new type of
extra information regarding
the interaction of the lateral
force with the longitudinal
force which is indicated by
the degree of “longitudinal
slip” symbolized as “S” :
Before this interaction can be
explored in greater detail it is
necessary to first consider
the generation and behavior
of the longitudinal traction
force…

TIRE BEHAVIOR, TRACTION: LONGITUDINAL
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TIRE TRACTION: LONGITUDINAL
The “traditional” tire lateral traction model accounts for change in
normal load and change in contact area (“curl up”) effect on the lateral
traction coefficient, but the “traditional” way of dealing with the
longitudinal coefficient of traction has been simply to choose some
seemingly appropriate constant value and make do with that. However,
since the equation giving the traction coefficient variation with contact
pressure is known, it would seem that the only info needed to relate the
longitudinal coefficient of traction “μx” to normal load “N” is an equation
relating the contact area to normal load.
There is an equation in existence which does relate the contact area
“Ac” to normal load, but it is relatively unknown. This equation was
inspired by a concept presented by Prof. Dixon (Suspension Geometry and
Computation, Chichester, UK; John Wiley & Sons Ltd, 2009, ISBN 978-0-
470-51021-6, pg. 85), then developed by Mr. J. Todd Wasson of
Performance Simulations, and finally refined by this instructor.

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That Dixon-Wasson-Wiegand tire-road gross contact area equation is:
Where:
Ac = Tire to ground plane
gross contact area (in2)
Lc = Tire to ground contact
area length (in).
tw = Tire tread width,
assumed constant with
load (in).
Ri= Tire no-load inflated
radius (in).
d = Tire vertical deflection
under load (in).

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Of course, to calculate the area “Ac” requires the vertical tire deflection
“d” under normal load “N” (“Nokian” Equation*)…
Where:
N = The normal load on the tire (lb).
KZ = Tire vertical stiffness (lb/in).
d0 = Tire deflection function
“y-intercept” value (in).
*Of course, this is known as the “Nokian” Equation to just a few
Scandinavian researchers. To everyone else this is just the old tire vertical
spring or linear deflection equation.

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Which in turn requires knowledge of the tire vertical spring constant
“Kz” (Rhynes Equation)…
Where:
KZ = Tire vertical stiffness (kg/mm).
Pi = Tire inflation pressure (kPa).
DR = Wheel rim nominal diameter (mm).
Note the “(-0.004 + 1.03) SN” term; this term when stand-alone is
actually the next equation and is known as the Michelin Formula…

Where:
tw = Tire tread width, assumed constant with load (in).
This formula was developed by a regression analysis based on data
from a large group of common passenger car tire sizes. Therefore this
formula does not work very well for unusual size tires or tires that do
not fit within the passenger car tire norm. Since the Rhynes Equation
incorporates the Michelin Formula within its formulation then it also is
subject to the same limitations.
(The “0.03937” is a Metric to English units conversion factor.)
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And knowledge of the tread width “tw” (Michelin Formula)…

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All the parametric information necessary to plug into those four
equations for determining a specific tire’s “contact area = f(normal load)”
is contained in a tire’s “P-Metric” designation as inscribed on the sidewall.
Determination of the contact area “Ac” under normal load “N” provides the
contact pressure “Pc
” (= “N/Ac”) so that the peak longitudinal coefficient of
traction can be obtained by the Koutný Formula:
μx = a Pc
n
In order to generate a realistic variation with contact pressure the
following exposition will utilize an example tire of designation
“P152/92R16”, “a” will be set to “15.7369” (English psi units, for Metric
kPa units use “58.2587”), and “n” will be set to “-0.67791” (a Koutný value,
presumed typical):
μx = 15.7369 (Pc)-0.67791

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There are a number of “rules of thumb” that also
attempt to define the relationship between the
peak longitudinal coefficient of traction and the
normal load; these may be useful for
comparison with the equation. One of these
“rules of thumb” is given by Prof. Gillespie: “…as
load increases the peak and slide (traction)
forces do not increase proportionately…in
the vicinity of a tire’s rated load… (the
traction) coefficients will decrease…0.01 for
each 10% increase in load”. There is a similar
“rule of thumb” attributed to Formula 1
competitors (source unknown) which may be
paraphrased as: “…for every +5.82% increase
in contact pressure there will be a -1.00%
decrease in the traction coefficient…”. The
variation of the peak longitudinal coefficient of
traction with normal contact pressure as per the
Koutný model and the “rules of thumb” may be
graphically presented as

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Fx = (a Pc
n)N

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As was the case with the lateral traction force potential, the
longitudinal traction force potential also varies with inflation
pressure, and will form a family of curves for a particular tire:

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Also as was the case for lateral traction, there is a tire deformation
that accompanies longitudinal traction loadings, which may be
illustrated as per the figure:
In the figure, for the conditions
of acceleration and braking, the
vertical force “Fz” (the resultant
of the contact area vertical
pressure distribution and equal
to “N”) times the offset arm “d”
constitutes the rolling
resistance. The presence of
longitudinal traction forces for
acceleration (“Fx”) and braking
(“-Fx”) both increase rolling
resistance, but not to the exact
same extent. Acceleration and
braking generate different longitudinal shear stress
distributions (compression vs. tension), which
interact with the vertical contact stress
distribution, which affects the rolling resistance

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As illustrated, as each tire tread segment rolls into contact with the
ground, there is a longitudinal stretching or compression of that
segment, followed by a contraction or expansion as the segment
rolls up out of contact. It is this cyclical distortion of the tread that
gives the appearance of “slip”, which is to say the speed of rotation
of the tire “ω” seems out of synch with the velocity “V”. “Slip” may
be represented as “%S”, or just “S”. The tire segment in contact
with the road and under traction stress is generally not in motion
with respect to the road (although some portions of the contact
area may be); on the whole the tire contact area may be regarded as
actually being “static” with respect to the road.

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Just as the drift angle “ψ” was proportional to the lateral force “Fy”,
the apparent slip “%S” is proportional to the longitudinal force “Fx”,
the potential for which depends on the normal load “N”. The
following figure shows the relationship between the longitudinal
traction force “Fx” and slip “%S” for some constant normal load
“N” (note the similarity to the previous lateral traction force “Fy” vs.
drift angle “ψ” figure):

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This next figure shows how this relationship can vary for different normal
loads; it is based on a plot made by use of the BNP (Bakker-Nyborg-
Pajecka) tire model using parameters empirically obtained for a particular
truck tire and inflation pressure; some model results are over-plotted with
their empirical counterparts to indicate the degree of model veracity. Note
that the proportional limit is indicated at around 10% slip (normal road
driving slip is generally under 3%).

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In a way analogous to the lateral
case of the lateral traction force
“Fy” variation with normal load
“N” per drift angle “ψ” family of
curves, it should be possible to
plot longitudinal traction force
“Fx” variation with normal load
“N” per percent slip “%S” to
produce a similar family of
curves. However, no such plot
could be readily found in the
literature. Therefore, the data
inherent in the previous figure
was replotted in an attempt to
construct such a figure

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Just as was done earlier for the lateral traction force vs. drift angle
functions at various normal loads, the longitudinal counterpart
can also be normalized and extended over four quadrants. Again
there is indication
of an interaction
between the long
and the lat forces,
only now it is the
lateral force effect
that is recognized
through its drift
angle “ψ”
This interaction of
the long and the lat
will now be explored
more thoroughly…

TIRE BEHAVIOR, TRACTION: LAT + LONG
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TIRE TRACTION: LONGITUDINAL & LATERAL TOGETHER
The question now presents itself: how do we “synchronize” the
longitudinal and lateral traction functions so that they coherently
represent the same tire? This is not, or should not, be a problem if all
the necessary tire coefficients are properly determined by empirical
means for use in a unified tire model like the “Magic” or “LuGre”
models, but it does become a question if the attempt is made to
construct a model for a theoretical tire using just the simple
relationships expounded on herein. Of course, the establishment of
the maximum longitudinal/lateral traction forces that a tire could
generate for a particular inflation pressure, normal load, etc., would
go a long way toward that construction, but the result can only be
used for exposition and conceptual thinking. For realistic engineering
determinations of what performance levels a detailed design could
achieve on a specific road course, only a model such as the “Magic”,
or perhaps the “LuGre”, can truly suffice.

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A tire may experience an essentially purely
longitudinal or a purely lateral traction loading under
certain limited circumstances, such as a drag racing or
skid pad simulation; but generally a tire undergoes a
simultaneous combination of lateral and longitudinal
loading. If the maximum loading a tire could undergo
were equal in either direction, then the maximum
resultant combination of longitudinal and lateral traction
forces that a tire could generate would be obliged to fall
within a “traction circle”, or at least for simplicity’s sake
the situation is often portrayed that way. Actually,
because tire traction behavior is anisotropic, the
situation is much more accurately modeled as a
“traction ellipse”, although even that is not perfect.

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When the lateral/longitudinal traction relationship is portrayed as
a circle it allows for some very simple determinations. For instance,
note that the two orthogonal traction forces “Fx” and “Fy” must
always combine to form the resultant force “Fr” as per:
When utilizing the circle model, this resultant force “Fr” can’t
exceed the circle radius or maximum traction force “R”, if a skid is
not to set in. So, using this simple relation, if “R” is 560 lb (254.0 kg),
and “Fy” is 300 lb (136.1 kg), then in order for the particular tire
considered to not go into a skid “Fx” can only go up to:
=
So simple, but so far from realistic…

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In the quest to keep things as simple as possible, but with a
closer correspondence to reality, the ellipse model was developed.
An over-plot comparison of the circle and ellipse models for the
same tire would look as follows:

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The major axis of the ellipse is “2a” in length, while the minor
axis is “2b”. In this tire traction model the “a” corresponds to the
maximum longitudinal traction available, and the “b” corresponds to
the maximum lateral traction available. Note that the ellipse properly
represents the passenger car tire relation between maximum
longitudinal and lateral traction forces in that “a > b” (*see refs):
a = Fx
max b = Fy
max
So, with “a” and “b” quantified, a property called the
“eccentricity” of the ellipse can be calculated:
The eccentricity “e” represents the ratio of the “c” dimension,
which is the distance of the “foci” from the center (origin) divided by
the “a” dimension (one half the major axis):

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The significance of the tire traction ellipse lies in the fact that no
combination of longitudinal and lateral tire traction forces, i.e. no
resultant traction force, can be so great that if plotted to scale it
would project beyond the ellipse periphery. The need for any traction
force so great that it would fall on the ellipse periphery is indicative
of impending skid. Any point on the periphery of an ellipse must
conform to the ellipse equation:
So, if “Fx
max” (“a”) = 629.4 lb (285.5 kg) and “Fy
max” (“b”) = 498.5 lb
(226.1 kg), then when “Fy” = 300.0 lb (136.1 kg) the maximum
amount of “Fx” tolerated before a skid would ensue is:

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A traction ellipse such as shown so far holds only for a particular tire
on a particular surface at specific normal load, inflation pressure,
velocity, and temperature. These limitations can be countered some-
what by normalizing the
longitudinal & lateral
forces, by assuming a
constant inflation
pressure & temperature
(due to the attainment
of thermal equilibrium),
and by placing the
“secondary” matter of
velocity effect aside for
the moment. This gives
us the slightly more
useful traction ellipse

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A 3D plot of a specific tire’s traction potential from “N = 0” to its
absolute maximum load capacity “N = Lcap” at its absolute maximum
inflation pressure “Pcap” could be constructed. If so, the result would
look something like a hemispheroid (or a grapefruit serving) as
shown:

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If enough data were available for a specific tire to construct a volume
as shown for “N = 0” to “N = Lcap” for each “Pi” increment of, say, 5
psi throughout the working pressure range, then this perhaps would
constitute enough data for utilization in a full quantitative automotive
performance simulation. Such a simulation would have to contain
interpolation routines for the determination of appropriate data
values that lie between the known “N” data levels, and further
routines to modify that interpolated data to account for the effects of
temperature, velocity, and camber. All things considered, that might
be enough to make utilization of the “Magic” or other such complex
formulations unnecessary*. Moreover, if one’s concern is on a less
sophisticated level, i.e. - “is there enough traction to complete a
particular maneuver?”, then reference to even an appropriate tire
traction ellipse may not be necessary; if the resultant of the “Fx” and
“Fy” forces (“[Fx
2 + Fy
2]0.5”) is less than “0.3 N” in value then the
answer is “yes”. It’s only when a vehicle is being driven hard,
certainly at the point of loss of control (skid), that tire traction ellipse
use becomes necessary.

TIRE BEHAVIOR
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Up to this point only the well established
basics of tire behavior have been
discussed. However, now the discussion
will involve some aspects of tire behavior
that are less well established, including
some points that are totally speculative.
Perhaps some day one of the students in
today’s class will be instrumental in
exploring and clarifying some of these
speculative areas, advancing the state of
the art…but probably not…

TIRE BEHAVIOR: TEMPERATURE
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TEMPERATURE EFFECTS
The effect of temperature is often ignored; either it is considered
inconsequential or a benign condition of thermal equilibrium is
assumed. However, there are cases when the blind discounting of
temperature can be disastrous. The variation with temperature of
material properties, mostly the properties of rubber, causes significant
variation in tire behavior. Traction, rolling resistance, and inflation
pressure are all affected, which in turn causes other effects (%slip, slip
angle, cornering stiffness, fuel economy, wear, vertical spring
constant, etc.)
Temperature and inflation pressure are very closely interrelated as per
the Ideal Gas Law:
PV = n R T

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In that equation the gas pressure “P” (in “atmospheres”) and volume
“V’ (in liters) are related to the gas temperature “T” by the factors “n”
(the amount of gas in “moles”) and “R” (the Universal Gas Constant:
0.08207 liter-atm/mole-oK). Ignoring the small changes in tire volume
“V” with inflation pressure “P” (“V” is constant) means that “P = (nR/V)
T” where “nR/V” is a constant; tire pressure varies in a direct linear
relation with tire temperature:

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Since the tire vertical spring constant “Kv” varies with pressure in
accord with the Rhynes Equation, then the deflection under load will
vary, which in turn affects the tire-road contact area. Therefore,
increased “T” means increased “Pi” and “Kv”, which in turn leads to
decreased “d” and “Ac”. And that ultimately means decreased rolling
resistance, which means improved fuel economy, but also less
traction…However, all of that has been covered by the basic tire
equations already discussed, but the tire mechanical effects caused by
temperature variation aren’t the
whole story; there are material
effects as well. Rubber energy
dissipation (hysteresis) and
traction coefficient varies
directly with temperature in a
very non-linear fashion

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The total effect of the combination of the mechanical and the
material aspects leads to somewhat puzzling statements such as*:
“…increase in energy dissipation that accompanies an
increased load causes the temperature of the tire to
rise…results in lower hysteretic loss coefficient…as a result
the coefficient of rolling resistance often decreases
somewhat with increasing load…”
There are forms of automotive endeavor in which temperature
levels play a significant, and complicated, role. For instance,
Formula 1 and Indy car tires require operation within a narrow
temperature band for optimum performance; per an authoritative
source**:
“Modern race tire compounds have an optimum temperature
for maximum grip. If too cold, the tires are very slippery; if
too hot the tread rubber will ‘melt’; in between is the correct
temperature for operation.”

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For racing tires the increase in temperature with
velocity and hard use is planned for, and if laps have to
be run at reduced speed under a safety flag, or during a
rolling start, then it is not unusual to see the cars
alternately darting hard right and left as the drivers try
to keep their tires at optimum temperature as they wait
for all out racing to recommence. Possibly one of the
worst scenarios that can occur in racing is to be caught
with rain tires installed as the track starts to dry out and
there is no chance of a pit stop for tire replacement; the
“softer” compound rain tires are certain to overheat
unless the driver commences driving far less
aggressively, which is a tactic not likely to place him on
the podium.

TIRE BEHAVIOR: INFLATION PRESSURE
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It has already been discussed how tire inflation pressure
will affect the vertical stiffness, and other measures of tire
stiffness as well, such as the “Cornering Stiffness”* or the
lateral traction coefficient “m” (which is an inverse
measure of stiffness). As noted, such changes cause a
cascade of other changes; a change in vertical stiffness will
affect the vertical deflection under load (and therefore the
rolling radius**), which in turn will affect tire-ground
contact area (and therefore traction), and that leads to
changes in rolling resistance (and therefore fuel economy),
heat generation, and temperature. And, of course, this
tends to run in a full cycle, as temperature will, in turn,
affect the inflation pressure. Here we are going to discuss
some of these consequences of inflation pressure that
have not been adequately dwelt on before, like rolling
resistance…

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The effect of inflation pressure variation on rolling
resistance as reflected in the Stuttgart Rolling Resistance
Formula Coefficients, “Static” (Cs) and “Dynamic” (CD)…

82
The variation in tire-road contact area versus inflation
pressure (note that without temperature or normal load
change the longitudinal tire traction may be considered as
varying in exact proportion to the change in area) may be
illustrated as:

83
Increasing inflation pressure will also cause some expansion of the
tire circumference, although such expansion usually is very small*.
The situation is depicted
As the inflation pressure “P”
increases the force “F” pushing the
tire semi-sections apart; the force is
equal to the pressure times the
horizontal plane area: “F = P A”.
Expressed in differential form this
relation may be expressed as:
This causes corresponding stress “dS” and strain “dε”
differentials in the tire periphery (tread):
By definition “dL/L” is substituted for “dε”, and the expression
rearranged:
The stress “dS” is equal to “2 dF/2” (“dF”) over the tire tread
cross-sectional areas “2 tt tw”, or “dS = dF/2 tt tw”, which allows for
the following substitution and simplification…

84
Remember that “dF = dP A”, and note that “A = 2 R tw”; this
allows for the following substitution and simplification:
Since the tire circumference “C” (“2 π R”) is equal to “2 L”, the
relation of the tire radius “R” to “L” is “2 π R = 2 L” or “π R = L”.
Therefore “L = π R” and “dL = π dR”; substitute for “L” and “dL”:
Remember that this is only for rough estimation as this simplified
relationship was made by ignoring the stiffness contribution of
the sidewalls, etc. However, the final relation for determining the
difference in an inflated no-load tire radius due to an inflation
pressure change is:

85
The use of this equation produces an inflated no-load tire radius
(upper plot line) versus inflation pressure plot as per:

TIRE BEHAVIOR: VELOCITY
86
An authoritative source states “In preliminary performance
calculations the effect of speed may be ignored (with respect to
rolling resistance)”*. However, the rolling resistance variation
with speed (velocity) is explicitly known via the Institute of
Technology in Stuttgart formula of circa 1938:
VELOCITY EFFECTS
CR = CS + 3.24 CD (V/100)2.5
Given coefficient values such as those of previous figures, but
appropriate for the specific tires concerned, a vehicle’s rolling
resistance can be reasonably determined for a wide range of
velocity variation, so no ignoring of the velocity effect on rolling
resistance is necessary.
However, there are other sources which state “Velocity does not
significantly affect cornering stiffness of tires in the normal
range of highway speeds”**, and “To a first order, tire forces and
moments are independent of speed”***.

87
However, there are definite decreases in longitudinal traction
coefficients with velocity which may not be well defined, but for
which there is considerable empirical data, such as that
contained in the following table for longitudinal traction*:

88
Both the static and dynamic coefficients of traction are functions
of velocity, which is contrary to the Coulomb friction model. A plot
of the static (rolling) and dynamic (skid) coefficients as a direct
function of velocity is as follows*:

89
How the variation in traction with velocity affects the longitudinal
traction/longitudinal “slip” relationship may be depicted in a
general way as follows:
[This figure is just a
“cartoon” , an ad hoc
adaptation of an
illustration found in
Harned, Johnston, and
Scharpf;
“Measurement of Tire
Brake Force
Characteristics as
Related to Wheel Slip
(Antilock) Control
System Design”, SAE
Paper 690214, 1969,
so no one should
attempt to use it for
anything quantitative.]

90
In acceleration and deceleration the variation in both the normal load
and velocity affect the maximum longitudinal traction available.
However, it is more important to account for the effect of both “N” and
“V” on “μx” in a braking simulation such as “MAXDLONG.BAS” than it
is in an acceleration simulation such as “MAXGLONG.BAS”. A braking
simulation commences at a high “V”, and the determination of the
maximum traction available for deceleration is very much dependent on
that “V”. An acceleration simulation commences at zero “V”, and as the
velocity increases the need for determination of the maximum traction
available decreases because the propulsion force available for
acceleration is also decreasing; this is contrary to the braking situation
wherein the brake force available for deceleration may actually
increase with time.
Therefore this instructor chose to develop an expression for the
variation of “μx” (a.k.a. “μpeak”, maximum longitudinal traction
coefficient) with regard to both “N” (“Pc”) and “V”: “μx = f(N, Pc)”. A
number of reference documents were utilized to develop a suitable
expression for “μx = f(V)”, then combined with the known “μx = f(Pc)”…

91
…resulted in:
The form of this equation may be generally applicable, but it
just represents the specific tire (152/92R16 @ 45 psi) for which it
was developed; a different version of this equation must be
developed to represent any other tire/inflation pressure. This
equation for determination of “μx” was incorporated in the
traction subroutine of the “MAXDLONG.BAS” program.
An indication of the validity of this equation may be gleaned
by consideration of how the equation performs as the variables
“N” and “V” are varied independently:

92
Increased velocity also means increased flexing of the tire tread
per unit time, thereby generating more heat flow and raising the
tire temperature, as indicated per the following table*:
VELOCITY and TIRE TEMPERATURE
* Woodrooffe, and Burns; “Effects of Tire Inflation Pressure and CTI on Road Life
and Vehicle Stability”, Proceedings of the International Forum for Road Transport
Technology, pp. 203-221, pg. 205.

93
The increased temperature through the material properties of rubber has
its own effect on traction, but complicating matters is the subsequent
daisy chain of structural consequences: higher inflation pressure,
increased vertical stiffness, decreased deflection under load, decreased
tire-road contact area, leading to a further decrease in traction. Given all
this action and reaction it is understandable that tire researchers have
historically had difficulty in trying to separate such things as velocity
effects from pressure effects; on the tire testing machine the two effects
go hand-in-hand; such hard to separate parameter interactions have been
the main reason for the slow progress in the understanding of tire
behavior.
Since an increase in tire temperature results from increased tread flexure,
velocity is not the only driving parameter behind that effect. Varying
longitudinal/lateral accelerations will also cause tread flexure resulting in
temperature increase; generally, the harder a vehicle is driven the higher
tire temperatures will raise. It also follows that the more underinflated a
tire is, the higher its temperature will climb; for optimum tire life it is wise
to maintain tire inflation pressures in accord with manufacturer’s
specifications.

94
Velocity also directly affects the vertical spring constant of the
tires, and seemingly in a manner that leads to all sorts of
confusion. Taylor, Bashford, and Schrock have demonstrated that
the measured value of a tire vertical spring constant “Kv” can vary
considerably depending upon the method used to do the
measuring. The most common test method, which accounts for the
vast majority of measured vertical spring constant data, is the
“Load-Deflection” (LD) method. This method, along with four other
methods, was evaluated by these researchers regarding the “Kv”
of a 260/80R20 agricultural (!) tire:
VELOCITY and TIRE VERTICAL SPRING CONSTANT
•Load-Deflection (LD).
•Non-Rolling Vertical Free Vibration (NR-FV).
•Non-Rolling Equilibrium Load Deflection (NR-LD).
•Rolling Vertical Free Vibration (R-FV).
•Rolling Equilibrium Load-Deflection (R-LD).

95
It seems that the major distinctions between the methods involve whether the
test is static/quasi-static (response to load) or dynamic (response to
impact/vibration). This instructor believes this distinction is the key to
understanding what seems to be a lot of confusing and conflicting test results,
beginning with the Taylor, Bashford, and Schrock paper and many of their cited
references, and ending with a later paper by Kasprzak and Gentz whose main
result is in seeming direct contradiction to Taylor, Bashford, and Schrock*.
This instructor’s personal resolution of the matter, based mainly on
intuition unsupported by any solid substantiation, is that there may be two types
of vertical spring rate involved. One type of stiffness may be that measured by
static/quasi-static load response methods (LD); the other type of stiffness may be
that measured by dynamic impact/vibration response methods (FV**). The
static/quasi-static stiffness decreases with increasing velocity asymptotically up
to a certain speed dependent on the particular tire/inflation pressure concerned.
The impact/vibration stiffness increases with increasing velocity (due to inertial
effects). To say more than that would require further study, but it would seem that
the static/quasi-static vertical stiffness would be suitable for use in determining
the tire rolling radius variation due to “weight transfer” as used in
acceleration/braking simulations, while the impact/vibration stiffness would be
suitable for use in suspension shock/vibration studies.

96
VELOCITY, CENTRIFUGAL FORCE, & TIRE ROLLING RADIUS
The rolling radius variation with velocity due to centrifugal force is
determinate, unlike some matters just dealt with. A tire under load tends to
“expand” back to its full no-load inflated radius dimension (“Ri” or “Di / 2”)
due to centrifugal force as the velocity increases; accompanying this is a
true expansion (stretching of the carcass) also due to centrifugal effect, but
such stretching tends to be relatively small in most cases and has already
been dealt with.
An extreme example of tire rolling radius “expansion” would
involve the huge racing slicks (usually bias, a.k.a. cross-ply, without
“belts”) that some “dragster” types use. As an “AA” fuel dragster
accelerates off the line the huge rear slicks, usually Hoosier brand,
expand until the rear of the dragster appears to be “standing on tip
toes”. This “standing on tip-toes” behavior has been witnessed and
photographically documented innumerable times. This tire behavior
is by design; the enlarged rolling radius at high speed is intended to
change the overall drive ratio of the vehicle, compensating for the
fact that such dragsters tend to be direct drive or only two-speed.

97
Such tire “expansion” is less extreme and considerably less noticeable
for passenger cars. The following figure shows a measured increase in
rolling radius with speed (to about 150 kph, or 93.2 mph) for a 5.60×5
cross-ply (0-0.43 in) and a 155SR15 radial (0-0.15 in); both tires are at 22
psi (152 kPa) pressure and under 661 lb (299.8 kg) normal load:

98
The previous figure was utilized as the inspiration for the
development of a “tire rolling radius” subroutine for use in the
“MAXGLONG.BAS” automotive acceleration program. A 1984
parameter dump during successive runs of that program
revealed the following with regard to this tire expansion
subroutine function:

99
The present methodology for the static deflection recovery is derived
as follows…
The mass of the deflected
portion of the tire periphery “m1-3”
is pushed back against the normal
force reaction “N” by the
centrifugal acceleration “V2/R”
giving rise to the radius recovery
increment “dRrecovery”:
Using weight as a measure of mass requires the substitution:
The weight “W1-3” is equal to the volume “tt tw Lc” times the
density “δ”, where “Lc” is equal to “1.24 Ri Cos-1((Ri-d)/Ri)”;
making the substitution:

100
Substitute “386.088 in/sec2” for “g”, add “mph to in/sec”
conversion constant “17.6”, and rearrange slightly:
Although the effect is usually minor, there is a real expansion
(strain) of the tire periphery that causes a further increase in the
effective tire radius with increasing velocity. This tire expansion
radius increment “dRexpand” runs concurrent with the recovery
increment, at least until the static deflection is fully counteracted
which is when “dRrecovery” ceases. The expansion model is as per
the next figure, which is a free body diagram of a half periphery
of the tire:
The periphery stress
resulting from the
centrifugal force “F”
can be determined by
a study of this free
body diagram.

101
This study begins with:
The ½ periphery length “L” is equal to half of “2 R π”, so:
Therefore “R” can be expressed as:
And the radius expansion increment as:
Since the strain “ε” is equal to “dL / L” by definition the
substitution for “dL” may be made:
Make use of the stress-strain relation “S = E ε” to substitute “S/E”
for “ε”:

102
Substitute “(F/2)/A” for “S”:
Substitute “WL ω2 / g” for “F”:
The weight “WL” of the tire tread sector “L” is equal to “L tw
tt δ”, the CG coordinate “ ” of that sector is equal to “2 R / π”*,
the angular velocity “ω” is equal to “V/Rr”, and the transverse
tread area of the tire cords is estimated as “tw tt f”**; making the
corresponding substitutions and simplifying results in:
From earlier simple circular relations the length “L” is
known to be equal to “R π”, which is now substituted for “L”:
Simplify:

103
Now some unit conversion factor (mph × 17.6 = in/sec) and
constant value (g = 386.088 in/sec2) adjustments are required, and
“R” becomes “Ri” as that radius better represents the general
condition along the periphery “C” than “Rr”:
As the velocity “V” increases the deflection recovery equation and
the centrifugal expansion of the tread equation work together (are
additive) in increasing the rolling radius “Rr” until the point is
reached when “Rr = Ri” which signifies (approximately) total
recovery of the initial static deflection. Beyond that point the
centrifugal expansion carries on alone until the tire self-destructs.
Of course, that seldom happens as velocities high enough to cause
tire destruction by centrifugal stresses are reached only by vehicles
such as LSRs.

104
For passenger car tires true centrifugal expansion is very minor,
especially if the tires in question have steel (“E ~ 29,000,000 psi”)
or other high Modulus of Elasticity material belts. Most modern
passenger car tires are radials and belted, but a bias tire dating
from the 1945-1967 period might rely on only nylon and/or rayon
cords for constraint (“E ~ 410000 psi or less”).
It should perhaps be noted at this point that the exact value of the
Modulus of Elasticity to be used for a calculation or simulation is
not necessarily a “cut and dried” matter. Even for the same tire, the
modulus value will vary depending on the use to which that value
is to be put. For example, the Krylov and Gilbert equation for
determining the critical velocity for “standing wave” formation was
derived using a model of a longitudinal tire tread segment as a
beam supported on springs. Even though a modern (c. 2003)
passenger car tire (exact type unspecified) parametric value set
served as input for their example calculation, the modulus value
was only 171,304 psi (“3·107 N/m2”)*!

105
This is a far cry from the modulus value for steel
(29,000,000 psi), or any other conceivable tread belt
material, but that is because the belts reside near the
neutral axis of the beam model, and thus have no role
in this particular calculation. The modulus value that
Krylov and Gilbert used was some combination of the
carcass cord and tread materials (Ref.: Enylon ~
410,000 psi, Erubber ~ 500 psi)*, as was suitable for
their purpose. If the calculation had instead been one
of tire peripheral expansion in response to centrifugal
force, then the modulus used would have been much
higher, probably very close to the Modulus of
Elasticity of the belt material**.

106
Consider the result of a practical application of the recovery
equations and the expansion equation as presented herein. The tire
is the 1958 Dunlop RS4 6.00×16 bias tire (with inner tube) at 45 psi
(310 kPa) under a 1238 lb (561.5 kg) normal load, resulting in a 0.84
in (21.3 mm) static deflection. The tire is presumed to be unbelted,
and largely of nylon cord/rubber construction; the Modulus of
Elasticity is taken as 450,000 psi (3·109 N/m2). An iterative
spreadsheet calculation of the subject equation results looked as
follows when plotted:

107
The symbolism for the tire expansion due to inflation pressure
through the expansion due to velocity equations is as follows:
d = Tire vertical deflection under a normal load (in).
δ = Tire tread cords/belts/rubber composite density (lb/in3)
E = Tire tread Modulus of Elasticity (lb/in2).
f = Tire tread cross-sectional area adjustment factor
(dimensionless).
Kv = Tire vertical spring rate (lb/in).
P1 = Tire initial inflation pressure (lb/in2).
P2 = Tire new inflation pressure (lb/in2).
Ri = Tire inflated no-load radius (in).
Rr = Tire rolling radius (in).
tt = Tire tread thickness (in).
tw = Tire tread width (in).
V = Vehicle velocity (mph).

TIRE TRACTION GUIDE

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TIRE TRACTION GUIDE