automotive dynamics and design: tire behavior

AUTOMOTIVE DYNAMICS,

Brian Paul Wiegand, B.M.E., P.E.

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TIRE BEHAVIOR

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TIRE BEHAVIOR

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TIRE BEHAVIOR

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TIRE BEHAVIOR, FRICTION

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Charles-Augustin de Coulomb (1736-1806). Although the name of this famed French physicist and engineer is the one most commonly associated with the “classic” dry friction model, Leonardo da Vinci (1452-1519), Guillaume Amontons (1663-1705), Pieter van Musschenbroek (1692-1761), and Leonhard Euler (1707-1783) all made prior contributions to the study of friction. Sometimes the “classic” friction model is referred to as “Amontons-Coulomb” friction.


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The coefficient “μ” is the Greek lower case letter “mu”.


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TIRE BEHAVIOR, TRACTION

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TIRE BEHAVIOR, TRACTION: MATERIAL

10 TIRE TRACTION: MATERIAL


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Even in this figure, the rubber stress values had to be exaggerated and the strain understated for the sake of the visual presentation. If plotted to a true scale, it would be difficult if not impossible to visually discern the rubber plot from the strain axis for most of the elongation.


12 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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In this diagram rubber (probably natural rubber or “cis-1,4-polyisoprene”) is stretched to over seven times its original length (“loading”), yet when relieved of load (“unloading”) returns to its original length with no permanent deformation. Note that there also that over no portion of this stress-strain plot is there any true linear portion which conforms to “Hooke’s Law”: “σ = E ε”. However, rubber materials can still be assigned a Modulus of Elasticity (“E”) which represents a linearized approximation of the behavior in the mid-range area; such an “E” is taken as being equal to three times the Shear Modulus (“G”), courtesy of Poisson’s ratio (“ν = 0.50”) and the equation “E = 3 × G”.


13 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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Note how on the left in the Coulomb friction case the relative rigid and smooth nature of the contact surfaces allow for the rigid block to only make contact with the supporting surface only at the high spots. The block slides along the supporting surface riding on only those high areas, and increasing normal load does not greatly change the nature of the contact. Due to the smoothness such contact tends to be very close, such that the principal resistance to tangential motion is due to attractive forces between molecules. There may be a small amount of mechanical shearing off of some of the high points (resulting in wear), but mainly mechanical shear and hysteresis forces play a very minor role. In contrast, in the traction case depicted on the right, the elastic nature of rubber allows for the rubber to conform to the roughness of the road surface. Although this mechanical “interlock” of the tire tread rubber to the road surface results in a situation much like a set of locked gears in mesh, the “gear teeth” do not necessarily “break off” (shear) in order to allow relative tangential motion between surfaces. Some rubber particles may shear off as a result of the tangential strain, which constitutes wear of the tire tread, but the extreme elasticity of rubber allows for a sort of “flow” of the rubber “teeth” up over road surface “hills” and down into the following “valleys”. Of course, such motion means a cyclical elongation-contraction occurs, resulting in a tangential resistance to motion and dissipation of energy as heat due to rubber hysteresis


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The “dividing out” is more properly referred to as “normalization” of data, i.e.: making it more general…


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The point is that this explains why traction can be so adversely affected in winter conditions, a two-fold reduction is in effect: 1) ice covered road presents a very smooth surface to the tire which limits the “gearing” effect, and 2) the cold temperatures associated with icy conditions results in the rubber being less elastic and thereby less able to conform to any road irregularity that may be left.

TIRE BEHAVIOR, TRACTION: STRUCTURE

18 TIRE TRACTION: STRUCTURE

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It is possible that the mistaken concept of how a tire works obtained credence in the public mind with the advent of “balloon tires” in the 1920’s. The term “balloon tire” referred to the lower pressure (circa 40 psi) and lower aspect ratio tires which supplanted the previous generation of tall, narrow, high pressure (circa 90 psi) tires. There were also internal changes with the tire carcass which morphed from a woven cloth construction to one of bias-ply cords, with the material tending to change from natural fibers to synthetics.


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There may well be pressure changes on a scale beyond the common pressure gauge’s sensitivity to detect, but such changes are very small and a consequence of volumetric changes resulting from tire deflection.


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That “some” seems to consist solely of some Scandinavian researchers!


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The load capacity of a tire depends upon the strength of its construction, which means the carcass material type and the number of plies of that material. The sheer size of the tire, as measured in terms of volume, is also roughly proportionate to the load that can be carried, as is the inflation pressure. However, the load capacity of a tire is also inversely proportional to the speed at which the tire runs. Load capacity was one of the earliest concerns of tire manufacturers and consumers; for over 100 years private organizations like the Tire and Rim Association (TRA) Inc. (founded in 1903) set the standards for rating tires, and load capacity was foremost. Early formulations for determining load capacity largely resulted from practical experience and group consensus, but evolved with time as theoretical approaches and empirical observations were utilized for refinement. In the United States, many TRA guidelines, such as those on tire load capacity, would eventually be incorporated into NHTSA regulations; this has proved to be a natural progression that would find its counterpart in many other nations. The traditional (since 1936) TRA load capacity formula was developed by mathematician C.G. Hoover (who later became staff director of the TRA) and has been continually refined since.


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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).


30 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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The Michelin Equation was obtained by a regression analysis preformed using data representing a wide range of commonly available tires; the equation will not function well for tires varying greatly from the tires of that regression analysis data set (this explains why the equation did not produce reasonable results for a 152/46R8 tire!). Note that the Rhynes Equation will also be subject to the same limitations as the Michelin Equation as the Rhynes Equation incorporates the Michelin Equation in its formulation.


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THE RELATION BETWEEN “e” AND THE ALIGNMENT ARM = ??? Even when rolling freely a tire has a longitudinal pressure distribution biased toward the leading edge of the contact patch, as shown. Therefore, the resultant force “Fr” of this pressure distribution is positioned at some distance “e” forward of the tire center line; the moment “Fr × e” represents the rolling resistance. When a longitudinal traction force “Fx” is generated due to an acceleration or braking torque (“Fx = T / r”) there is a distortion of the contact area, but the magnitude of the area stays essentially the same. However, when a tire is subjected to a lateral load such as to generate a reaction lateral traction force “Fy” then there is a decrease in overall contact area due to distortion causing the tread “inner” edge to “curl up” off the road surface. The extent to which this “curl up” occurs is dependent on the loading magnitude, inflation pressure, tread shape, and the internal construction; bias ply tires generally exhibit more “curl up” than radial tires as a result of the defining structural details: the orientation of the cords, the use of tread belts, etc. Note that the rolling resistance moment “Fr × e” is affected by the shear stress distribution/contact area distortion resulting from an application of acceleration/deceleration torque. The initial effect of the application of a braking torque may be a minor decrease in rolling resistance coefficient; but generally both acceleration and deceleration torques will cause significant increases in rolling resistance.


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A figure in a very authoritative of source agrees with the above figure, but similar figures in a least two derivative sources show the minimum rolling resistance as occurring under light braking. However, the more authoritative source actually says “…minimum in rolling resistance does not occur under freely rolling conditions, but rather under a small driving torque.”. In any case, the effect is minor, of short duration, may sometimes reverse itself, and in general both increasing acceleration and increasing deceleration torques cause increasing rolling resistance.

TIRE BEHAVIOR, TRACTION: LATERAL

33 TIRE TRACTION: LATERAL

μy = b – mN

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Obviously the equation only holds within certain bounds on the normal load, otherwise we would be capable of having infinite traction and zero traction…The tire’s working rrange should conservatively be taken as constituting the limits.


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Fy = (b-mN)N = bN – mN2

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A “b” value of 1.2 may be considered a good value for modern passenger car tires; values up to 1.6 might be appropriate for racing tires. For the “m” parameter a variation of 0.0002 to 0.0006 would be considered appropriate for passenger car tires.


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For such a “family” of curves, as inflation pressure “Pi” increases the tire/road contact area “Ac” will decrease, and therefore the peak lateral traction values (“Fymax”) will decrease. Also, the seeming slopes “ΔFy/ΔN” of the “near-linear” portions of the family of the parabolic “Fy = bN – mN2” plot lines will increase (grow steeper).


38 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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Famed automotive engineer and author Donald Bastow has suggested that a better term for “ψ” might be “drift angle”, an idea this author has enthusiastically endorsed but admittedly only sporadically adhered to.


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Note that here “e” represents an offset of the lateral traction force from the lateral tire contact area center, whereas earlier “e” was used to represent the offset of the vertical normal force from the tire rotation center. There is an interconnection between these two offsets because there is an interconnection between the lateral sheer stress distribution and the vertical pressure stress distribution. Some study of the matter is indicated in order to get the interaction explicit and clear as it is not properly presented in the literature. This scenario of a change in direction due a side load, possibly a wind gust or an inertial load due to an uneven road, may be called the “directional disturbance” scenario. The incidence of a disturbance force causing tire carcass distortion, “drift angle”, and consequent change in direction represents only one particular order of “cause and effect”.


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The concept of “Dynamic Equilibrium” results from the work of Jean le Rond d’Alembert (1717-1783), and is sometimes referred to as “D’Alembert’s Principle”.


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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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However, it is of importance in constructing computer simulations of dynamic effects to not only have a clear definition of cause and effect, but to have some estimation of any time lag between the two: the time lag between when the driver cranks the front tires over to the steering angle “δ” and when the tires settle into the steady state values of “ψ”, “V”, “F”, and “Mz” is generally equal to the time it takes for the tire to undergo a rotation of about 180 (π radians) to 360 degrees (2π radians).


44 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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*Hans Bastiaan Pacjeka, Professor Emeritus at Delft University of Technology in the Netherlands, has developed a tire model called the “Magic Formula” because it is based on relatively little underlying physics; instead it is mainly based on a regression analysis of reams of empirical tire data. The lateral force increases with drift angle (or vice versa) on a shallow curve, reminiscent of stress-strain curves, up to a deformation limit. At that point the situation is very unstable and will either quickly return to the safety of the “useful traction region” or result in 100% slip at the tire/road contact patch (a “skid”). If the appropriate portion of the useful region is linearized by the fitting of a straight line to the original shallow curve, then a simplified lateral force/drift angle relationship (the slope of the fitted line) can be obtained (Tangent vs Secant slope). This quantity is commonly called the “Cornering Stiffness” and symbolized as “Cs” (sometimes it is also called the “Cornering Power” of the tire, which is yet another misnomer like “shock absorber” or “slip angle”). Linearized tire relationship values such as “Cs” are often used in directional stability determinations and for various simulations and studies of automotive dynamic behavior. Reference [28], pp. 198 & 350 states that “Cs” (a.k.a. “C∝”) is the “…slope of the curve at zero slip angle…” (even though that would seem to be a poor fit). Although cars turn left and right, implying positive and negative “Cs” values, by SAE convention “Cs” is always positive. “Cs” has units such as “lb/deg”, but when normalized by the normal load “N” the “Cornering Stiffness” becomes the “Cornering Coefficient” “Cc” in units of “lb/deg/lb”.


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I say “may” because I have no definite knowledge regarding this aspect of race tires.


46 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.

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*That the lateral traction force potential is influenced by longitudinal forces is a subject yet to be dealt with, but one that would have been immediately anticipated by Poisson (Siméon Denis Poisson (1781-1840) who defined the his “Poisson’s Ratio” (“ν”, the Greek lower case letter “nu”) in terms of the orthogonal unit deformations (strains) within the proportional limit). **Campbell, Colin; The Sports Car, Cambridge, Mass.; Robert Bentley Inc., 1969, page 164, exact quote: “…cornering power (sic) increases slightly with speed...”.


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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Only the useful traction region is shown; as noted, 13 degrees is about the maximum drift angle/deformation limit for passenger car tires, which in normal driving seldom go beyond 5 degrees or so. Note that as normal load increases the lateral traction force necessary for a certain drift angle increases as well, but at a decreasing rate.


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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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This figure clearly shows how increasing normal load will increase the lateral force necessary to cause the same amount of deformation (drift angle), but at a decreasing rate and only up to a point; traction demands beyond that point are likely to result in an out-of-control skid of the vehicle.


49 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…

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The total effect is one of generalization and broadening of content. The normalization results in a value of “μ” that will be of value over a short range of “N” variation; and the extension into four quadrants shows that tire behavior is the same for both right and left turns. The really new information provided here is the indication of the effect of a simultaneous longitudinal force generation upon the lateral traction force potential; as the slip “S” (and therefore longitudinal traction force) increases the peak lateral traction coefficient (and hence the max lateral traction force available) decreases. Furthermore, if a certain level of lateral traction force is attained, and then the slip (longitudinal traction force) is increased, the lateral traction force will drop unless the slip angle is sufficiently increased.

TIRE BEHAVIOR, TRACTION: LONGITUDINAL

50 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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Pillai and Fielding-Russell, in their paper “Tire Rolling Resistance From Whole-Tire Hysteresis Ratio” (Rubber Chemistry and Technology, May 1992), present a similar formula for contact area: Ac = 1.85 d2/3 Ri1/3 tw . This formula gives results that are about halfway between gross and net contact area figures, and is generally less accurate than the Dixon-Wasson-Wiegand equation.


51 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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52 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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53 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). = Tire section aspect ratio (dimensionless). SN = Tire nominal section width (mm). 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…

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Where: tw = Tire tread width, assumed constant with load (in). = Tire section aspect ratio (dimensionless). SN = Tire nominal section width (mm). 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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55 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 Pcn

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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56 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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The first (upper) model is the constant longitudinal traction coefficient model which, while totally unrepresentative of reality, is used fairly often for vehicle acceleration/braking performance calculations. The “accuracy” of such calculations is thus very dependent on the shrewdness with which the traction coefficient value used is chosen. The second model shown is the “Gillespie” model, which does show decreasing coefficient with increasing normal load, but does so in a linear fashion and does not come close to reality in magnitude.* The third model, the one of unknown origin but associated with Formula 1 racing, is near-linear and seems to represent an improvement; however, it is still insufficient if the Koutný model is regarded as the “gold standard”. The last two models utilize the Koutný Formula, but the plot labeled “Koutný Const Area” shows how the traction coefficient would vary if the tire/road contact area stayed constant as the normal load on the tire varied. That is reasonable for a simple rubber material sample but is not realistic for a tire; constant tire/road contact area results in overvalued contact pressures with a consequent undervaluing of the longitudinal traction coefficient. If the appropriate equations are used to determine the increasing contact area “Ac” as normal load “N” increases, then the use of the resulting correct contact pressure “Pc” (equal to “N/Ac”) in the Koutný Formula gives the most accurate variation of the longitudinal traction coefficient as per the plot simply labeled “Koutný”. *Note that in his book Fundamentals of Vehicle Dynamics (Warrendale, PA; SAE R-114, 1992), on pg. 345, Figure 10.8, Prof. Gillespie illustrates his concept regarding the variation with normal load of both the static “μs” (rolling) and dynamic “μd” (skidding) tire traction coefficients as straight parallel lines, decreasing with increasing load, and with “μs > μd”. THIS PRESENTS US WITH A VERY SIGNIFICANT BUT RELATIVELY OBSCURE FACT: THE DYNAMIC OR SKIDDING FRICTION COEFFICIENT IS STILL A MATTER OF TRACTION AND NOT COULOMB FRICTION!! IT IS ALSO DEPENDENT ON CONTACT PRESSURE, AND VERY PROBABLY SPEED AND TEMPERATURE AS WELL.


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Fx = (a Pcn)N

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58 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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59 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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Note that the vertical offset “d” is depicted as being smaller in acceleration. SAE J670 2008 pg. 42 defines the acceleration force as positive and the braking force as negative.


60 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 case with increasing lateral traction force resulting in the corresponding increasing tire deformation “ψ” commonly called the “slip angle”, increasing longitudinal traction force means a corresponding increasing tire deformation “%S” (or “S”) called “percent slip” (or just “slip”). And, just as was the case with “slip angle”, “slip” is a misnomer, but no reasonable replacement has been advanced. The apparent “slip” of a rolling tire transmitting a longitudinal traction force is due to a cyclical stretching or compression of “portions” of the tire tread as those “portions” roll into contact with the road; when those “portions” roll out of contact the longitudinal stress is relieved and an expansion or contraction back to original length occurs (along with a release of energy as heat due to hysteresis). Full wheelspin “%S = ∞%”, free rolling “%S = 0%”, locked brakes “%S = -100%”. For acceleration “V < Rω”, so the “slip” equation produces a positive number for “%S”. For deceleration (braking) “V > Rω”, so the equation then produces a negative number for “%S”. This variation in sign for acceleration and braking “slip” is in accord with the SAE J670 2008 standard, although the matter is not clearly stated.


61 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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Again we have the question of how to properly identify the slope, is it tangential or secant…


62 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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This figure would appear to be of limited usefulness due to the degree of overlap and other issues, but at least it indicates that, generally, increasing normal load allows for the generation of increasing longitudinal traction for the same percent of slip


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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…

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Again an interaction between longitudinal and lateral (represented by drift angle “ψ”) traction forces is indicated; to maintain a constant longitudinal traction as slip angle increases there must be an increase in slip fraction (requiring more drive torque). Also interesting is the fact that the longitudinal force function is not identical for acceleration (positive “μx”, positive “S”) and braking (negative “μx”, negative “S”). This is due to the different contact area stress distributions engendered by these two cases as discussed earlier with regard to rolling resistance. Otherwise, most of the remarks made about lateral traction are relevant here as well.

TIRE BEHAVIOR, TRACTION: LAT + LONG

65 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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Presentation Notes

The “Magic” tire model…Prof. Hans Bastiaan Pacejka (1934- ) of the Delft University of Technology (Delft, Netherlands) has developed a series of tire behavior models over the last few decades, and introduced them in 1991. These models en total were named the 'magic formula' because there is no particular physical basis for the structure of the equations chosen; they were essentially obtained by extensive regression analysis of many tires. Each equation is characterized by 10-20 coefficients for each important force that it can produce at the contact patch, typically lateral and longitudinal force, and self-aligning torque; these coefficients are empirically obtained for a specific tire of interest, and then used in the equations to estimate how much force(s) is generated for a given vertical load, camber angle, etc. The “LuGre” tire model…The longitudinal LuGre (named after Lund, Sweden and Grenoble, France) tire friction model, initially introduced in 1995 and since continually corrected and improved upon, is based on a dynamic viscoelastoplastic friction model for point contact. These results were later extended to the combined longitudinal/lateral motion. The LuGre tire friction model for combined longitudinal/lateral motion was further refined by taking into account all aspects neglected such as the coupling of the forces in longitudinal and lateral directions, tire anisotropy, and rim rotation. In addition, a solid mathematical justification for the introduction of dynamic friction models based on fundamental physical properties of the friction forces was provided for. The major advantage of the LuGre dynamic tire friction model is that can be described by a system of three ordinary differential equations giving the forces and the aligning moment at the contact patch of the tire. The main objective of the model was to be able to capture the steady-state behavior exactly; therefore, this model did not offer any guarantees on the accuracy of the transient dynamics.


66

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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Presentation Notes


67

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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Presentation Notes


68

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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Presentation Notes

Ellipses have many wonderful properties, and have long been a subject of interest for mathematicians and astronomers such as Menaechmus (c.380-c.320 BC), Euclid (c.325-c.265 BC), Apollonius (c.262-c.190 BC), Pappus (c.290-c.350 AD), Kepler (1571-1630 AD), Newton (1642-1727 AD), and Halley (1656-1742 AD). By the way, a circle may also be considered an ellipse, although a special case wherein the focii both reside at the center.


69 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 = Fxmax

b = Fymax

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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Presentation Notes

*Wong, Jo Jung; Theory of Ground Vehicles, Hoboken, NJ; John Wiley & Sons Inc., 2008, pg.51. Brach, Raymond; and Matthew Brach; “The Tire Force Ellipse (Friction Ellipse) and Tire Characteristics”, SAE Paper 2011-01-0094; Warrendale, PA, 2011, pg. 1, states that “racing…lateral vehicle accelerations can exceed longitudinal accelerations”, which means that “Fymax” can exceed “Fxmax” for at least some racing tire types (2011). Road car tire lateral traction coefficients go as high as 1.1 (2008), and Milliken, William F., and Douglas L. Milliken; Race Car Vehicle Dynamics, Warrendale, PA; SAE R-146, 1995, on page 27 says that Formula 1 lateral traction coefficients can go as high as 1.8 (1995).


70 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 “Fxmax” (“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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Presentation Notes

This is like the previous determination of “Fx” using the circular model, only now the answer is a little different…Because of the change in traction models, a tire that could only deliver 472.9 lb (214.5 kg) of braking force, while supplying 300.0 lb (136.1 kg) lateral force in a right hand turn, is now credited with being able to deliver 502.7 lb (228.0 kg) braking force in the same turn. Still pretty simple, but now hopefully more realistic.


71 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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Presentation Notes

Although this traction ellipse is truly valid only for a normal load “N” of 661 lb (300 kg), the “normalization” of the “Fx” and “Fy” axes (“μx = Fx / N”, “μy = Fy / N”) indicates that this same graph can be used with reasonable accuracy over a short interval of normal load variation. Note that lines of constant drift angle have been superimposed over the traction ellipse as is a common practice; this increases the usefulness of the plot. Constant “slip” value lines have been superimposed as well, which is not frequently found in the literature. Hayes, D.F.; and A.L. Browne, The Physics of Tire Traction, New York, NY; Plenum Press, 1974, pg. 89 explains this last comment as “…an attempt to plot Fy versus Fx for constant values of Sx would result in a less regular set of curves as a result of the greater imprecision encountered in measuring Sx in comparison to Sy…”.


72 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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Presentation Notes

Note the similarity of the bounding Fx and Fy curves; in both cases the traction and the coefficient of traction are functions of the normal load. However, the more complex Fy function is the easiest to understand and use despite the added complexity of tread “curl-up” due to someone’s astute observation and modeling long ago. A very big oversight or drawback of this model is that it only addresses two of the big three quanties: Long & Lat Forces, but not the Aligning Moment.


73 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 + Fy2]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.

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Presentation Notes

*Useful simulations of transient maneuvering behavior have been accomplished without recourse to such complex tire models; consider: Allen, R.W.; T.J. Rosenthal, and H.T. Szostak, “Steady State and Transient Analysis of Ground Vehicle Handling”, Automotive Crash Avoidance Research, Warrendale, PA; SAE SP-699, 1987. This fine example of an automotive simulation written expressly for microcomputer utilization can be obtained as a SAE Paper (870495), or as included in the SAE Publication Automotive Crash Avoidance Research.

TIRE BEHAVIOR

74

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…

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Presentation Notes

TIRE BEHAVIOR: TEMPERATURE

75 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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76

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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77 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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Presentation Notes

Note that the passenger car tire temperature operating range is identified as being 68 °F to 194 °F; the operating range for some racing car tires can be much higher. Current (2014) Pirelli Formula 1 tires are kept warm in electric blankets at a temperature of 230 oF (110 oC) before the start of a race (the FIA may ban tire pre-race warming by 2015).


78 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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Presentation Notes

*Note that it says “the coefficient of rolling resistance”, not “the rolling resistance”, decreases with load. The total effect of increasing load is one of increased rolling resistance, but at a decreasing rate of increase. Gent, Alan N., and Joeseph D. Walter; The Pneumatic Tire, US DOT/NHTSA, DOT HS 810 561; Washington, DC; 2006, pg. 491. **Milliken, William F., and Douglas L. Milliken; Race Car Vehicle Dynamics, Warrendale, PA; SAE R-146, 1995, pg. 56.


79

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.

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Presentation Notes

Current (2014) Pirelli Formula 1 tires are kept warm in electric blankets at a temperature of 230 oF (110 oC) before the start of a race (the FIA may ban tire pre-race warming by 2015).

TIRE BEHAVIOR: INFLATION PRESSURE

80 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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Presentation Notes

* Gillespie, Thomas D.; Fundamentals of Vehicle Dynamics, Warrendale, PA; SAE R-114, 1992, pg.354: “…increasing inflation pressure results in increasing cornering stiffness for passenger car tires…tires at reduced inflation pressures arrive at lateral force saturation at substantially higher values of slip angle…” ** The pressure/deflection effect on rolling radius is large compared to a pressure/strain effect which is often neglected, but which will be considered later.


81

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

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Presentation Notes


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:

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Presentation Notes


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…

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Presentation Notes

*This is a true expansion, not a decrease in the deflection under load. Increasing inflation pressure will increase the vertical spring constant, thereby decreasing the deflection, and resulting in a larger rolling radius; this is sometimes called an “expansion” but it isn’t really. The same is true if the centrifugal force generated by rotation causes a tire to recover some of its deflection due to load, still not a true “expansion”.


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:

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85

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

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Presentation Notes

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”***.

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Presentation Notes

*Wong, Jo Jung; Theory of Ground Vehicles, Hoboken, NJ; John Wiley & Sons Inc., 2008, pg.17. This is probably meant to apply only to passenger cars. **Gillespie, Thomas D.; Fundamentals of Vehicle Dynamics, Warrendale, PA; SAE R-114, 1992, pg. 355. Also, on page 199: “Speed does not strongly influence the cornering forces produced by a tire”. Again, this only applies to road cars. ***Milliken, William F., and Douglas L. Milliken; Race Car Vehicle Dynamics, Warrendale, PA; SAE R-146, 1995, pg.41. Also, on page 27: “…lateral friction coefficient is approximately 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*:

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Presentation Notes

*Hayes, D.F.; and A.L. Browne, The Physics of Tire Traction, New York, NY; Plenum Press, 1974, pg. 81. The average reduction of dry static traction coefficient of about “-0.003/mph” (-0.002/kph) over the 20 mph (32 kph) to 40 mph (64 kph) observed range is admittedly a limited linearization of the actual relation, which raises the question as to what is the real nature of that relation.


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*:

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Presentation Notes

*Gent, Alan N., and Joeseph D. Walter; The Pneumatic Tire, US DOT/NHTSA, DOT HS 810 561; Washington, DC; 2006, pp. 434 & 456. The traction relations reveal themselves as gentle curves to which parabolic regression lines fit very well. The average reduction of the static coefficient is about “-0.008/mph” (-0.005/kph) over the 20 mph (32 kph) to 40 mph (64 kph) range, which is about 2.7 times that observed previously; however, that is probably mainly the effect of this being wet traction.


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.]

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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)”…

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Presentation Notes


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:

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Presentation Notes

Note that technically this equation is not a direct function of the normal load “N”, but of the contact pressure “Pc”. However, that is a quibble (“Pc = N/Ac”). Note that the general form of the fitted line equation to the “μx = f(Pc, V) with V held constant” plot is in agreement with the Koutný Equation for traction coefficient, while in the other “μx = f(Pc, V) with N held constant” plot the linearized variation in “μx” over the range 20 mph (32 kph) to 40 mph (64 kph) range of “-0.00345/mph” agrees well with the information of the table presented just a few slides ago. This isn’t proof of the validity of this equation, but the results seem reasonable and the equation is useful. However, it should be noted that this whole effort flies in the face of an authority’s statement that “Tire performance varies with speed…the effect is not consistent enough to be generalized” [Milliken, William F., and Douglas L. Milliken; Race Car Vehicle Dynamics, Warrendale, PA; SAE R-146, 1995, pg. 56].


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.

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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.

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Presentation Notes

Actually, there is a further complication as the decreased contact area means less tire tread being flexed, thereby ameliorating the original temperature increase tendency, but not completely offsetting it.


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).

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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.

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Presentation Notes

*On page 9 of Kasprzak, Edward M.; and David Gentz, “The Formula SAE Tire Test Consortium – Tire Testing and Data Handling”, Warrendale, PA; Society of Automotive Engineers International, SAE 2006-01-3606, 2006, there is shown a rolling vertical tire stiffness of 797 lb/in (1395 N/cm) versus a non-rolling vertical tire stiffness of 708 lb/in (1239 N/cm) and states “This trend was seen in all the tires tested”. **There are also forced vibration tests, non-rolling or rolling, which would seem to be included here.


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.

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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:

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Presentation Notes

Reference [26], pg. 10 notes: “Higher speed rated tires may feature a full (tread) width nylon cap pr plies…on top of the…belts to further restrict expansion from centrifugal forces…Nylon cap strips used in some constructions …cover only the belt edges”.


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:

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The expansion-velocity plots are both for the same 6.00×16 bias tire, just at different inflation pressures. Here by 122.5 mph (197.1 kph) the tire inflated to 23 psi (158.6 kPa) has recovered 100% of its static deflection, and the tire inflated to 45 psi (310.3 kPa) has recovered 108% of its static deflection. The exact values displayed for the 6.00×16 tire are less significant than the fact that the simulated general form seems to be in reasonable agreement with the empirical form of the 5.60×15 tire expansion shown in the previous figure. The static deflections and other tire parameters used for this figure are those of the 1984 computer simulation runs (Reference [89]); the tire methodology used by this instructor has progressed significantly since then. In 1984 the value for “Kv” @ 23 psi was 1227 lb/in, and for “Kv” @ 45 psi was 2400 lb/in, and the 1984 “E” value used was 1,000,000 psi.


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:

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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.

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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”:

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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:

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*Reference [98], pg. 4.4.1. **Only a fraction “f” of the tread cross-section area is actually resisting the expansion, which is the area of the carcass cord and tread belts. Admittedly, this is essentially a matter of “fudge factors”, but such factors are significant in computer simulations of a complex reality as the judicious use of such factors allows for a “dialing in” or “fine tuning” of the simulation. Initially “f = 0.33” may be used for just the carcass cord area, and if belts are present then the use of “f = 0.43” is a possibility.


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.

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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”)*!

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*Reference [41], pg. 4227.


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**.

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*As noted earlier, the Modulus of Elasticity for rubber exists only as an infinite number of “short interval” values of limited applicability. However, for the sort of rubber used in tires it is never more than a few hundred psi. **While in such a case the contribution of rubber is so minor as to be negligible, the Modulus of Elasticity of some carcass cord materials is high enough that there may be justification for the use of a composite “E” value.


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:

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Note that the general shape of this plot is much like the 1984 plot of the same tire as depicted earlier. More significantly, it is also similar to the earlier empirical plots. Of course, this does not constitute a validation of the methodology, but it does give some reason for confidence. Of particular interest is the point of inflection in the plot at 211 mph (340 kph) where the axle height change reaches 0.84 in (2.1 cm), indicating full initial static deflection recovery; from that point onward only true expansion of the tire periphery occurs. Ultimately, if a high enough velocity can be reached, the tire peripheral ultimate strength will be exceeded and it will suffer catastrophic destruction, if it is not destroyed by other effects (temperature/pressure, “standing wave”, etc.) first.


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).

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TIRE BEHAVIOR: CONCLUSION

108

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automotive dynamics and design: tire behavior

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