Understanding Parameters Influencing Tire Modeling

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Dynamics

Understanding Parameters Influencing Tire Modeling

Nicholas D. SmithColorado State University, 2004 Formula SAE Platform

Copyright 2003 Department of Mechanical Engineering, Colorado State University

ABSTRACT

The purpose of this document is to describe importanttire characteristics and their effect on vehicleperformance. Characteristics considered are those thatthe Formula SAE judges have deemed important for

discussion on tires at competition and include coefficientof friction, slip angle, slip ratio, camber angle, corneringstiffness, camber stiffness, self-aligning torque, normalload sensitivity, load transfer sensitivity and pneumatictrail. Every effort has been made to list sample values togive the reader a general idea for common values of theconsidered characteristics. Values relating to a typicalFSAE vehicle are also listed for available data.

INTRODUCTION

The importance of the tires contribution to a racingvehicle cannot be overstated. Tires are required toproduce the forces necessary to control the vehicle.Given that the tire is the only means of contact betweenthe road and the vehicle, they are at the heart of vehiclehandling and performance.

Insight into the discussed parameters will help the FSAEstudent in various ways. Knowledge of thesecharacteristics and their effects on racecar performancecan give the engineer insight into performanceoptimization. A firm grasp on what influences a tiresbehavior and what these characteristics mean in termsof vehicle dynamics terminology will better prepare thestudent to score higher during design judging atcompetition.

This document only covers one small piece of a verycomplex assembly. However, the tire itself is alsoextremely complex. The information contained withinshould allow the reader to grasp the vehicle dynamicsterminology considering tires with much less effort than ifthe tire had to be researched independently.

TIRE BACKROUND

While the wheel may have been one of mans firstinventions, the rubber tire is definitely not one of thesimplest components to analyze. The rubber tire is acomplex composite consisting mainly of vulcanized

rubber, which more specifically is an elastomer with ahigh number of sulfur cross-links between the polymerchains. The composite also contains carbon black, oiextenders and layered reinforcing strands or fabricscalled plies. These strands are normally made of NylonTerylene, Rayon or steel cords and are oriented invarious configurations (Ref 1). While much attention ispaid to the rubber itself, the reinforcing cords alsodeserve attention. These cords have a higher modulusof elasticity and less creep and therefore carry the loadwhile the rubber skin serves to seal the air. An in depthstudy of tire construction would show that cordorientation (radial or bias ply) can have a significaneffect on tire characteristics. Racing tires are set up in a

bias ply configuration, providing strength in the threeplanes simultaneously (Ref 2).

Rubber tires have been present for quite some time andhave seen many improvements such as the use ofVulcanization by Goodyear in 1839 and the addition ocarbon black by Pirelli in 1907 (Ref 1). With theseimprovements, the rubber tire has produced superiorcontrol and durability when compared with othersubstitutes that have been attempted over the years.

NOTATION

It is important first to review a tire in its most generaorientation and consider the forces acting on a tire. TheSAE tire axis terminology is shown in Figure 1. The axissystem is not the same as the axis system on the 2004Platform racecar. The SAE system (SAE J670edenotes the X being forward, Y to the right and Zdownward. The 2004 FSAE Platform racecar uses Xforward, Y left, and Z upward.


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Figure 1 - SAE Axes Terminology (from Ref 4)

There are several forces, moments and angles thatprove to be very important in tire behavior. All theseforces can be seen as the forces and moments acting onthe tire from the road. First, there are two main angles toconsider, the camber angle and the slip angle. Thecamber angle is the inclination angle from its verticalposition while the slip angle is the difference in wheelheading and direction. These two angles are associatedwith the lateral force. Forces include the longitudinalforce in the X direction, the lateral force in the Y directionand the normal force in the Z direction. Longitudinalforce (FX) is the result of the tire exerting force on theroad and becomes negative during braking. The lateralforce (FY) is the resultant of the forces produced by anon-zero camber angle and by a non-zero slip angleduring cornering. Normal force (FZ) can also be viewedas the negative of the upward vertical force. Momentsinclude the overturning moment, the rolling resistancemoment, the wheel torque and the aligning moment.The overturning moment (MX) is caused by a lateral shiftof the vertical load during cornering. Rolling resistance(MY) is created by various factors that lead to a loss ofenergy. The aligning moment (MZ), also known as theself-aligning torque, produces a restoring moment on thetire to realign the direction of travel with the direction ofheading when the slip angle is non-zero. It should alsobe noted that there is also a moment produced by the

axle on the wheel. As a final point, it may be noted thatwhen the camber angle is zero, the wheel torque (T in),points in the negative Y direction.

A list of acronyms and abbreviations is included at theend of this paper for reference.

PRINCIPLE TIRE MODELS

As stated earlier, tires are a complex compositecomprising many layers of materials (see Figure 2 (Ref3)). A tire is very anisotropic. It is for this reason that tire

behavior is not derived from the material properties andstructure of a tire. Simplifications are therefore made inorder to create empirical models for a tire. The threeforemost models used to understand tire forcesdeflection and footprint behavior through the corningprocess are the elastic foundation model, the stringmodel and the beam model (Ref 1). While none of thesemodels truly addresses the complexity of a physical tirerealistic results can be obtained when empirical stiffnessvalues are used.

Figure 2 - Section of the Common Tire (Ref 3)

Each small element in the elastic foundation model isconsidered to act independently of the other elements(see Figure 3). The aspect that each element acts as asimple spring, independent of the other elements, makesthis model the simplest of the three. It is interesting tonote that the force found under the curve of the lateraforce distribution is equal to the lateral force measured athe axle (Ref 4). This supports the belief that althoughthis model may be the simplest, it can be very useful in

predicting and illustrating various tire behaviors, as isdone by Dixon (Ref 1). The elastic foundation modealso allows for discontinuity in the distribution ofdisplacement and in the slope of the centerlineConversely, in the string model, lateral displacement isalso resisted by a tension between the elements. It alsoallows for discontinuity in slope, however, discontinuity indisplacement is not allowed. Similar to the string modeis the beam model, where each element has an effect onthe surrounding elements. In the beam model, eachelement creates bending moments on the elements nextto it. This allows no discontinuity in slope odisplacement. The beam model has been found to besuperior for radial ply or for belted bias ply tires (Ref 1)Often times a combination of these models will becombined to gain a better model (as was done by theU.S. Army Engineer Research and Development Centerin their testing (Ref 5)).


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Figure 3 - Top View of Tires Lateral Center LineDisplacement During Cornering and Corresponding

Foundation Stiffness Model (Ref 1)

COEFFICIENT OF FRICTION

The coefficient of friction is defined as a unitless ratio offriction force to normal force. It is generally consideredtrue that the resulting friction force is not proportional tothe surface area of contact. However, this is far from the

truth when a rubber tire is considered. This dissimilarbehavior is due to the viscoelastic nature of rubber.Thus, as force is applied, deformation occurs bothelastically and plastically in a non-linear fashion due tothe mechanical behavior of polymer chains (Ref 6).Viscoelasticity also explains why the coefficient of frictionof a tire is load dependent. As a tire is loaded, thesurface area grows larger increasing the total frictionforce but lowering the coefficient of friction (Ref 1).Since a tire does not follow Newtons laws of friction, acoefficient of friction above unity can be obtained. Forexample, given a 500 lb normal load on a tire, it wouldnot be uncommon for a tire to produce 800 lb of force

giving a coefficient of friction of 1.6. Under idealconditions, this would make the vehicle capable of pulling1.6 gs (Ref 2). However, ideal conditions are rarelyachieved because the coefficient of friction depends onmany transients.

The coefficient of friction can depend on many unknownvariables such as atmospheric dust, humidity,temperature, vibration and the extent of contamination(Ref 7). It may also depend on the angularity of the roadsurface, speed, and even skid duration. As a tire skidsfurther, the temperature rises above the optimal value,and the coefficient of friction begins to drop. Similarly, asthe speed increases, temperature increases and thecoefficient of friction again begins to decrease afterreaching an optimum value (plotting coefficient of frictionvs. velocity yields a curve that closely resembles anormal distribution). A final component that should notbe overlooked is the molecular bonding that occurswhich lends to total friction. While this adhesion betweenthe road and tire does not require energy to create thebond, energy is dissipated when the bonds are broken.This becomes important in dry conditions. Conversely,the tire elastically conforming to the road becomes moreimportant in wet conditions. (Ref 1)

Finally, some effects of the coefficient of frictionchanging with speed should be discussed. When slipangle become large, the rear of the footprint begins toslide and thus has a lower coefficient of frictionTherefore, the cornering force will top out at a modestslip angle and then begin to decline. This phenomenonis more dramatic in a locked wheel when braking. Sincethe wheel is locked, the local temperature rise is greaterand the relative sliding speed is greater than for arotating wheel. In either case, the decreased coefficien

of friction contributes to a negative self aligning moment(Ref 1)

For a more in-depth understanding of friction interactionswith rubber tires, one can reference items 1 and 2 underRecommended Reading.

SLIP ANGLE

If the direction of travel differs from the wheel heading (ithe wheels angular displacement is different from the

path the tire is following), the slip angle () produces acomponent of lateral force (FY). This lateral force will ac

through a point behind the center of the wheel in adirection such that it attempts to re-align the tire. Ishould be noted that the slip angle is not the same as thesteering angle.

As can be seen from the elastic foundation model, thereis a final fiction limited value of the lateral force due toslip angle that is reached (Ref 1).

c d FV

l

Equation 1

This equation, with c being the foundation stiffness, dbeing the tire centerline displacement and l being the tirefootprint length, must be satisfied for no sliding. It thenfollows that the lateral force is roughly proportional to theslip angle. This then gives the maximum non-slide force

FY1

2c l

2

Equation 2

1

2c dm l

1

2c l

FV

c l

1

2 FV

Equation 3

Therefore, the lateral force is proportional to the slipangle up to half the maximum total friction limit (Ref 1)This can be seen in Figure 4, which graphs lateral forceagainst slip angle for some example values that werechosen to be similar to a typical FSAE tire. These valuesare a 200 lbf (890 N) normal force, a coefficient of frictionof 1.5, a foundation stiffness of 725 psi (5 MPa) and acontact patch length of 8.5 in (216 mm). The actuacalculations can be seen in the appendix.


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As can be seen in Figure 4, the upper limit is reached

very quickly with the maximum force occurring at 90when the wheel is fully sideways. Generally, a racing tirewill achieve maximum lateral force at slip angles in the

range of 3-7 (Ref 4). In the figure below, the force at 3is 95% of maximum. At high slip angles, the rear of theprint actually slides laterally along the surface of theroad, which contributes to less capacity for lateral forceand reduces the stabilizing self-aligning torque (Ref 4).

Figure 4 - Lateral Force vs. Slip Angle

It may be important to realize that when not completelysliding, the lateral force is not dependent on thecoefficient of friction, although this provides the upperlimit; instead, it depends on the foundation stiffness. Analternate way to look at this is to say that the lateral forceis not dependent on coefficient of friction until the tire hasbroken away, indicating a large slip angle (Ref 4). It is

also sometimes convenient to define the lateral force dueto slip angle in terms of other coefficients as seen inequations 4 and 5.

FY F C CS FV Equation 4

C CS FV Equation 5

CS is the cornering stiffness coefficient and C is thecornering stiffness. Generally, typical values for thecornering stiffness coefficient are 0.12/deg for bias-plytires and 0.16/deg for radial ply tires. (Ref 1)

CORNERING STIFFNESS

The cornering stiffness can also be defined on a perradian basis as shown in equation 6.

C1

2c l

2

Equation 6

Cornering stiffness can also be seen to be the initialslope of the lateral force curve. Typical values for

cornering stiffness are around 195lbf/deg (867

N/deg)(Re

1). However, this value can be much higher. Foexample, an Indy Road tire and a Formula 1 tire may

have a C of 833lbf/deg (3.7

KN/deg) at 1800 lbf (8 KN

normal load and 750lbf/deg (3.4

KN/deg) at 1000 lbf (4.5 KN)

normal load respectively (Ref 4). A tire for formula SAE

would have a Caround 165lbf/deg (734

N/deg) for a 330 lb

(1.5 KN) tire load as can be seen in Figure 15 (also seeAvon Tire Curves in the Appendix). Notice how thecornering stiffness is sensitive to the range of slip angleused to find the slope on the lateral force curve. Thisfurther emphasizes that one should use caution whenusing cornering stiffness values in calculations, or aleast realize its potential inaccuracies. As an additionapoint, the cornering stiffness is normally 5-6 timesgreater than camber stiffness for traditional bias tires(Ref 4).

Since the central/drag force ratio (see Equation 7 andFigure 5) reduces with slip angle, higher corneringstiffness is desirable. The reason for this is that a givencentral force will be achieved at smaller slip angles andtherefore results in lower tire drag force. The

central/drag ratio is given as follows (Ref 1):

FS

FD

C cos ( )

R FV C sin ( )+ Equation 7

In this formula, R is the rolling resistance, FD is the dragforce in the direction of travel, and FS is the forcecomponent perpendicular to travel. FD and FS make upthe components of the resultant lateral force FY.

Figure 5 - Force Components for Undriven Tire (Ref1)

V

FS

FD

FY


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The preceding formulas are valid primarily in the linearlateral force region. However, when more aggressive,larger camber angles are employed, as with a two-wheeled vehicle, the cornering stiffness may be reduceddramatically. When camber angle is included in thecornering stiffness, a new equation for corneringstiffness can be used.

C C0 kC

Equation 8

Given thatkC

0.005

deg

This equation is virtually linear up to about 60(Ref 1).However, in a wide racing tire, large camber angles arerarely used as this would begin to lift one side of the tireoff the ground. More on camber angle will be discussedin the following sections.

When trying to model the cornering coefficientaccurately, it becomes evident that it is dependent onvertical load. The model that best fits analytical data isthe exponential equation 9 as given below (Ref 1).

C C1

FV

FV1

e

KCSFV

FV

FV1

1

Equation 9

The unitless value of KCSFV is the sensitivity of CS to FV.The maximum cornering stiffness occurs when thevertical force is equal to the reference load over thesensitivity (FV1/KCSFV).

SLIP RATIO (% SLIP)

In contrast to slip angle, which is slip in the transverseplane, the slip ratio is the slip in the longitudinal plane.The slip ratio affects acceleration and breaking andtherefore bears analogy to the slip angle in the sensethat as slip angle is related to lateral force, slip ratio isrelated to longitudinal force and traction capacity.Generally speaking, the coefficient of friction will changewith changing slip ratio. A plot of coefficient of friction vs.percent slip increases nearly linearly up to about 5% slip,peaks near 10% slip and then falls of in a nonlinearfashion (Ref 2).

The slip ratio can be defined empirically as a function ofangular velocity of the driven wheels and angular velocityof the free rolling wheels. The slip ratio is defined inequation 10 (Ref 4).

SR 0

0

Re

V cos ( )1

Equation 10

The angular velocity of the driven wheel is , the angula

velocity of the free rolling wheel is 0, Re is the effectiverolling radius and V is the velocity. Thus, for free rollingSR = 0 while when locked under breaking SR = -1. Fodynamic, real time testing, the effective radius cannot bemeasured; therefore, the loaded rolling radius iscommonly used in its place. Slip percentage is simplythe slip ratio expressed as a percentage (slip ratiomultiplied by 100).

As stated earlier, longitudinal forces, tractive andbreaking, are functions of the slip ratio. As the slip ratioincreases, the longitudinal forces rise rapidly and thenfall off after the maximum is reached in a range of 0.10to 0.15 slip ratio (Ref 4). It can also be seen fromEquation 10 that as the slip angle increases, the tractiveor breaking force will decrease.

The definition of slip ratio used here is the equivalent ofthe SAE definition. Many other definitions and variationsof slip ratio have been used and can be found in Ref 4.

CAMBER ANGLE

Camber angle is equal to inclination of the wheel from itsvertical position. Or more precisely, camber is theinclination from a plane perpendicular to the ground. Apositive camber angle is defined to be an outward leansuch that the top of the tire leans outward from thevehicle centerline. A non-zero camber angle produces acamber force directed laterally toward the low axis sideproducing another component of lateral force. Thus, anegative camber angle increases the lateral or corneringforce of the tire. Generally, the lateral force producedfrom camber is a function primarily of tire stiffnessvertical force and camber angle. While there are othe

secondary forces present, such as friction effects andpath curvature, these are small and can be neglected fomost applications. Camber force can also be affected bythe shape of the crown. A very round profile developsmaximum lateral force with negative camber angles and

a small slip angle while camber angles in the 0 - 4negative range are better when a flatter crown is used(Ref 1). For wide street radial tires, camber force tends

to fall off at camber angles above 5 (Ref 4).

With the combination of the preceding parameters, anew tire characteristic, camber stiffness, can be definedas the rate of change of camber force with change in

camber angle. The equation for the lateral forcecomponent due to camber is seen in equation 11 (Ref 1)

FY F C CC FV Equation 11

FY is the lateral force, FV is the normal force, is the

camber angle, C is the camber stiffness and CC isdefined to be the camber stiffness coefficient. A typicaCC value would be 0.018/deg for bias-ply tires and0.008/deg for radial ply tires (Ref 1). While the cambeforce is usually less than the lateral force due to slip


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angle (see Figure 6 and Figure 9), camber force canhave a significant impact on vehicle handling, especiallyas suspension geometry may change.

0

5

10

15

20

25

30

35

40

45

50

50 100 150 200 250 300 350 400 450 500

Normal Force (lbf)

LateralForce(lbf)

1 2 3 4 5

Camber Angle

Figure 6 - Lateral vs. Normal Force for VaryingCamber Angle

CAMBER STIFFNESS

The camber stiffness is the rate of change of camber

force with camber angle (FY/) (Ref 4). A commonvalue for camber stiffness is approximately 8

lbf/deg (35.6

N/deg) which may be reduced significantly at speedsabove 65 mph (104

km/hr) (Ref 1). The camber stiffness

also decreases for large slip angles near the maximumlateral force. Generally, the camber stiffness is 5-6 timesless than cornering stiffness for bias ply tires (Ref 4).

Camber force is due to lateral distortion in the contactpatch of the tire. Thus, camber stiffness is often verysmall for radial tires due to the stiffness of the belt andthe flexibility of the radial cords in the sidewall thatprevent the lateral contact patch distortion (Ref 4).

TOTAL LATERAL FORCE

Now with the components of lateral force known, thetotal later force for small angles can be computed usingsuperposition. Total lateral force is given in equations12-14.

FY F F+ C C + Equation 12

FY Cs FV

CC FV +

Equation 13

FY FV CS CC +( ) Equation 14Path curvature is not included in these equations forlateral force. Path curvature is a small contribution andis often neglected for most cases (Ref 1). It may also beconvenient to define a total lateral coefficient CY asshown in equations 15 and 16.

CY CS CC + Equation 15

FY FV CY Equation 16

It can be seen that at maximum lateral force, camberforce has only a small effect since the camber coefficienreduces for large slip angles.

COMBINED LATERAL AND LONGITUDINAL FORCES

It has been seen that the lateral force is highly correlatedwith the slip angle and the camber angle while thelongitudinal force shows correlation with the slip ratioNow it can be seen how these two forces affect thethrust or drag of the vehicle. Thrust or drag is defined inequation 17 (Ref 4).

TorD FX cos ( ) FY sin ( ) Equation 17

Formulation of an equation for rolling resistance (a dragforce) can now also be seen in equation 18 (Ref 4):

FR SR 1+( )Tin

Rl

FX

cos ( ) FY sin ( )

Equation 18

In this equation, FR is the rolling resistance, Tin is theinput torque to the wheel, and Rl is again the loadedradius of the tire. Through inspection, it can be seen tha

the term FXcos() is the traction/braking component

FYsin() is the induced drag due to lateral force, and FRis the net rolling resistance (Ref 4).

A relationship for longitudinal force can also beformulated based on wheel input torque, wheemoments, camber angle and angular acceleration. Thisrelationship is given in equation 19 (Ref 1).

=+++ IMMRFTZYlXin

)sin()cos(

Equation 19

This equation can also be useful in calculating thelongitudinal force. Note that when the camber angle is

small, as well as the acceleration and rolling resistancethen the input torque is equal to the longitudinal forcetimes the loaded radius (Ref 1).

SELF-ALIGNING TORQUE

Self-aligning torque, also known as self-aligning momentis the resultant of the lateral force and the moment armknown as pneumatic trail, t. It is a restoring moment thaattempts to return the wheels to a zero slip angle state(strait running). Essentially, the presence of the selfaligning torque exposes the fact that a tire likes to head


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in the direction it is presently running. It may beimportant to note that the self-aligning torque may beinfluenced by a mechanical trail induced fromsuspension geometry. For example, more mechanicaltrail and therefore more self-aligning torque can beinduced with the presence of caster and kingpin offset(Ref 4). Trail may also be affected by camber, which caninduce a small destabilizing force (Ref 4). However, thisis also small and often neglected. This discussionassumed no mechanical trail or trail effects due to

camber.

Empirical equations for this torque have been derivedfrom the foundation stiffness model and are as follows(Ref 1):

MZ FY tFY l

6

c l2

tan ( )

12 Equation 20

M

Z

2

FV2

4 c l tan ( )

3

FV3

6 c2 l3 tan ( )( )2

Equation 21

It should be noted that the above equations are for slipangles greater than the non-slide limit. The relationshipsdefined by Equation 21 can be seen by graphing self-aligning torque over slip angle for various values ofnormal load as is done in Figure 7.

Figure 7 - Self-Aligning Torque vs. Slip Angle ForVarying Normal Force

Many important relationships can be drawn from Figure7. Potentially the most important is that as slip angleincreases, thus increasing lateral force, the self-aligningtorque decreases. This means that the drivers feel ofthe road through steering wheel torque is in essenceremoved as lateral force begins to reach its limit prior tosliding. However, this removal of feel gives the driverwarning of front tire breakaway (Ref 2). Maximumwarning of breakaway would occur when all the steeringtorque comes from the pneumatic trail. Mechanical trailsubstantially reduces the steering wheel torque (Ref 4).

The self-aligning torque can also be influenced by tirepressure. As the pressure is decreased, the contacpatch lengthens and thus gives a longer moment armwhich in turn increases the aligning torque (Ref 4).

PNEUMATIC TRAIL

The pneumatic trail is essentially the moment armthrough which the lateral force acts. As predicted by the

foundation stiffness model, the lateral force acts behindthe centerline of the tire. This is the result of the neartriangular contact patch distribution as shown earlier inFigure 3. The model predicts that this distance t is equato the ratio of self-aligning moment to lateral force, or

1/6

of the contact patch length as seen in equation 22.

tMZ

FY

l

6Equation 22

While this model is reasonably accurate, larger valuesare commonly found when the slip angle is small. (Ref 1)

When using the foundation stiffness model to investigatefurther, Equation 23 is obtained for slip angles greaterthan the non-slide limit. This allows more relationshipsabout the pneumatic trail to be distinguished. It can beseen that the pneumatic trail decreases once sliding

beings and approaches zero when the slip angle is 90(Ref 1).

t3 FV c l

2 tan ( ) 2

2 FV

2

12 c2

l3

tan ( )( )2

6 FV c l tan ( ) Equation 23

Part of the reason for the decrease in pneumatic trailand the resulting reduction in self-aligning torque, is theincrease in rear contact patch sliding. As the slip angleincreases, more of the rear section of the contact patchbeings to slide laterally. Since the footprint is sliding, ihas less ability to stabilize the wheel. In this case, thealigning toque is reduced to near zero and may evenreverse sign (Ref 4). The effects of the change inpneumatic trail and thus the location that the force actson the contact patch can be seen in Figure 8, whichshows how the force components change depending onsteering. Figure 8 also shows a pressure distribution fothe contact patch. Notice that for normal cornering thepneumatic trail is behind the centerline on the trailingedge side (TE). For severe cornering, the pneumatictrail is actually slightly ahead of the centerline toward theleading edge (LE).


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Figure 8 - Contact Patches for Various situations(Ref 3)

NORMAL LOAD SENSITIVITY

Normal load sensitivity is defined to be the rate of

change of lateral force with the change in vertical force ata constant slip angle. The load sensitivity, CFV, isdefined in equation 24 (Ref 1):

CFV

FY

FV

Equation 24

When plotting lateral force vs. normal load, it can beseen that there is not a proportionally greater increase inlateral force for an increase in vertical load, as can be

seen in Figure 9. It should be noted that Figure 9 isbased on empirical equations that are good for lowerload ranges and actual data shows a nonlinearstabilization at higher loads and larger slip angles.

0

100

200

300

400

500

600

700

50 150 250 350 450

Normal Force (lbf)

LateralFo

rce

(lbf)

1 2 4 6 8

Slip Angle

Figure 9 - Lateral vs. Normal Load for Varying Slip

AngleThe load sensitivity tends to decrease as the normal loadincreases. This coefficient may run as high as 1.8 focurrent Grand Prix tires. Tire load sensitivity is more oless independent of speed. It can be increased by usinga tire compound that is more sticky and by keepingtemperatures in the desirable ranges. (Ref 4)

Using the 2005 CSU FSAE race car as an example, thenormal load on a tire can change by as much as 116 lb(516 N) for a 700 lbf (3.1 kN) car in a 1.5g turn as foundby Adam Skaggs (Ref 8). Reference 4 can be consultedon calculating actual load transfer for a vehicle. An

example of calculating normal load sensitivity for theAvon tire data can be found in the appendix.

LOAD TRANSFER SENSITIVITY

Load transfer can have a great impact on vehiclehandling. Increasing the amount of uneven loaddistribution laterally from load transfer (such as having anon-central center of gravity) decreases the total lateraforce that can be achieved. This effect is amplified alarge lateral accelerations. The load transfer sensitivitybetween two wheels is defined mathematically inequation 25 (Ref 1).

CFT

FY2

FVT

FV2, Equation 25

FY2 is the total side force for two wheels, and FVT is thetransferred vertical force. It is assumed that the tota

vertical load (2FV) is constant as is the slip angle ()This reduction in total lateral force is a second ordereffect since tire force varies nonlinearly with normal load(Ref 2). Even though load transfer always results in less


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total force, load transfer may not be an undesirablephenomenon. If the load transfer is from the front to therear wheels (longitudinal instead of lateral), the rear,heavier loaded pair of tires, now has an increasedtraction ability and can therefore accelerate more withless slip. This would be desirable if the car only had toaccelerate forward. For a racecar that must corner,accelerate and break, it is more desirable to have theload transferred evenly over all the wheels.

The load transfer sensitivity and the normal loadsensitivity are highly correlated. Since there are anenormous number of factors involved, it is difficult to listall relationships. However, it is generally the case thattires with a higher CFV tend to let go or break away faster(Ref 4).

EXAMPLE CALCULATIONS

A few calculations are provided in order to give anotherset of example values and illustrate how one would usetire data. The example calculations can be seen in theirentirety in the Appendix. Example calculations are

based on data from the Avon tires, a tire that one of theCSU FSAE teams has been considering for use.

The calculations begin by assuming a few known valuesor values at a point of interest. These values include a

slip angle of 1.3, a camber angle of 1 (using data fromFigure 13 and Figure 14), a 150 kg normal tire load and

a coefficient of friction of 1.5. Since the slip angle is 1.3,a point not shown in the figures, absolute values of datapoints from around this value were taken and used in aquadratic interpolation for higher accuracy. Lateral forceand self-aligning torque were then calculated to be 187lbf (831 N) and 23 ft*lbf (31.5 N*m) respectively. Using

equation 22, the pneumatic trail and contact patch lengthwere calculated to be 1.49 in (37.9 mm) and 8.95 in(227.3 mm) respectively. Equation 4 was then used forapproximating the cornering stiffness, which wasdetermined to be 143

lbf/deg (639.5

N/deg). As a final

consideration, the normal load sensitivity was found to be0.582 using equation 24. The data used for thiscalculation was taken from Figure 14 in order todetermine the lateral load at a vertical load of 150 kg and

250 kg at a 1.5 slip angle.

These values are hard to compare to other FSAE typetires due to lack of tire data. Therefore, no conclusion

can be made about the Avon tires at this time. However,these values may be helpful in dynamic analysisconcerning tires or wheels. It is also expected that asmore data becomes available, these values will behelpful in choosing the best tire for the CSU FSAEracecar.

OTHER CONSIDERATIONS

TIRE PRESSURE

Tire pressure is one of the few parameters that thevehicles operator can control. This may be part of thereason that it is one of the most common changes insetting up a racecar (Ref 4). Tire pressure can affect thetires characteristics in a variety of ways. Being that the

stiffness coefficients are essentially a measure of theelasticity of the tire, an increase in tire pressure wilincrease the stiffness of the tire. This will in turnincrease the lateral force. If an inflation pressure 70%above the design value is used, a 20% gain in tirestiffness can be obtained. Increasing pressure abovethis will then decrease stiffness (Ref 1). Pressurevariation from front to rear is often used to make minoadjustments in under/over steer balance of the vehicle(Ref 4). Tire drag is also decreased as pressure isincreased.

In contrast to lowering pressure, raising the pressure wildecrease the coefficient of friction. As mentioned earlier

lowering pressure will raise the coefficient of friction byreducing contact patch pressure and increasing the sizeof the contact patch. The increased contact patch alsoincreases the self-aligning torque, as discussed earlierLowering the pressure also decreases the tire life andreduces the vehicles handling or feel (Ref 1).

Another consideration to deal with pressure is rideThe pressure significantly affects the tire ride, which isessentially a reflection of the tires spring rate. It is oftenthe case that a specific overall spring rate for an entirecar is a design parameter (Ref 4). Therefore, the tirepressure may be constrained depending on the

adjustability of the vehicles suspension system. Springrate data for the Hoosier tire (the current FSAE tire) isgiven in the appendix.

There are tradeoffs in deciding what pressure to run atLow tire pressures may benefit lap times due to betteconformity; however, low pressures, or running with aslow leak, may lead to excessive tire temperatures. Ipressure is too high, cornering power, acceleration andbreaking may be adversely affected (Ref 2). As a resulof the many factors, there is no one right pressure to runat. Individuals must experiment with their own setup todetermine the optimal pressure for the application. As afinal comment, it may also be noteworthy to point out thapressure and temperature relations may not be asexpected depending on the moisture content of thecompressed air used to fill the tire. Therefore, dry air ispreferable (Ref 1).

TIRE TEMPERATURE

Tire temperature can change tire characteristics in avariety of ways. Specifically, temperature most heavilyinfluences the forces produced by the tire but also affectsthe life of the tire.


10/22

10

The temperature affects the tires stiffness, and thereforeforce, in at least two different ways. As the temperaturechanges, the modulus of elasticity will change thuschanging the stiffness of the tire (Ref 4). Similarly, whentires run hot for extended periods, the pressure may alsoincrease which again changes the tire stiffness. Dixonnotes that the warmed-up pressure is typically 4.3psi(30kPa) higher than cold-set values (Ref1). Thetemperature will also influence the tires force capabilitydue to change in the coefficient of friction of the tire with

changing temperature.

Pressure, speed and operating forces will affect tiretemperature. High temperatures can be achieved withexcessive camber, running with low inflation pressures(or a slow leak), or by using a tire made from acompound that is too soft for the track (such as usingrain tires on a dry track) (Ref 2).

FRICTION CIRCLES AND G-G DIAGRAMS

There is much discussion about how many lateral gs avehicle can withstand. What some may overlook is when

the driver is at the maximum number of lateral gs andbrakes or accelerates, the vehicles tires may breakaway. Breaking or accelerating produces a longitudinalcomponent of force that must be considered with thelateral force vector. Since the resultant force, oracceleration, is greater than either component alone, theintroduction of either when driving at the vehicleshandling limits may cause tire break away. The resultantforce, regardless of direction, is limited by the product ofthe vertical load and coefficient of friction. It is for thisreason that friction circles (or ellipses) and g-gdiagrams are constructed. These diagrams provide theuser with information on the vehicles limitations

(maximum force capability) over a range of possibilities.A friction circle or g-g diagram in vehicle handling isanalogous to Mohrs Circle for structures. The frictioncircle and g-g diagram could be viewed as the practicalway to use many of the tire characteristics discussedabove.

The friction circle often plots slip ratios against lateralforce while the g-g diagrams plot lateral acceleration vs.longitudinal acceleration, both for a given set ofconditions. The friction circle is limited by the coefficientof friction where as the boundary of the g-g diagramdepends on the speed of the vehicle. Some of thesediagrams may show a theoretical boundary that is further

outside the plot than the actual boundary (Figure 10 b).Vehicles cannot reach this boundary due to tractionlimitations, load transfer effects, suspension effects,limiting stability balance, and brake balance (Ref 4).Examples of g-g diagrams are shown in Figure 10.

Figure 10 Example G-G Diagrams (Ref 4)

Data can be taken to obtain actual g-g diagrams foindividual vehicles at the tested conditions. This is doneusing accelerometers and data samplinginstrumentation. Reference 4 should be consulted fomore information on the application and use of g-gdiagrams and where to obtain measurement equipment.


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11

CONCLUSION

The tire is one of the most important components on aracecar. Knowledge of how the tire operates can givethe engineer insight into design considerations and canyield information on the capabilities of a vehicle.Knowing these capabilities can be the differencebetween winning a race, or having tires break awaythrough a turn and finishing last.

Thorough knowledge of how the above characteristicsaffect a vehicles capabilities and handling can be veryuseful. Specifically, the use of tire slip curves, eitherpublished from the manufacturer or determined by theusers testing, can be very useful. Knowledge of theprinciple forces involved in tire dynamics as well asweight transfer sensitivity, self-aligning torque and g-gdiagrams can also be vitally useful tools for the engineer.

These characteristics are those deemed important by thedesign judges for formula SAE. It is the intent of theauthor that this short discussion of tire terminology willbenefit future Colorado State FSAE students in two

ways; it will allow them to quickly grasp the concepts andconsiderations dealing with tires, but it is also anticipatedthat the specific examples and numbers will givestudents a feel for what the values of the differentparameters may be for a typical FSAE vehicle. While thecapabilities were not available to the author at the time ofcomposition, it would be a worthwhile task to model theColorado State University FSAE tire in Adams/Tire andcompare the results with measured data. Modeling thetire either with simple formulas or with a computerprogram such as Adams could also become useful in tireselection if ample data is available.

REFERENCES

1. Dixon, J.C. (1996). Tires, Suspension and Handling

2nd

ed. Warrendale: Society of AutomotiveEngineers.

2. Smith, C. (1978) Tune to Win. Fallbrook: AeroPublishers.

3. Gim, G, & Choi, Y. (Nov. 2001) Role of Tire Modeling

on the Design Process of a Tire and Vehicle System.Hancook Tire Company Ltd.

4. Milliken, W.F., & Milliken, D.L. (1995) Race CarVehicle Dynamics. Warrendale: Society of

Automotive Engineers.

5. Lacombe, J. (2000) Tire Model for Simulations ofVehicle Motion on High and Low Friction Road

Surfaces. U.S. Army Engineer Research andDevelopment Center, Hanover, NH.

6. ASkelan, D, & Phule, P. (2003) The Science andEgineering of Materials. Pacific Grove: Brooks/Cole

7. Avallone, E, & Baumeister III, T. (1996). MarksStandard Handbook For Mechanical Engineers 10

th

Edition. McGraw Hill

8. Skaggs, A. (2003) Suspension Loads and Spring

Rate Analysis. Unpublished Technical SenioDesign Contribution, Colorado State University.

9. Avon Tires, Downloads http://www.avonracing.com10. Hoosier Tires, Colligate Formula SAE

http://www.hoosiertire.com/rrtire.htm

CONTACT

Nicholas D. Smith [email protected]

RECCOMENDED READING

Tire Friction

1. The Unified Theory of Tire and Rubber Frictionby H.W. Kummer and W.E. Mayer

2. The Physics of Tire Traction edited by D.F. Haysand A.L. Brooke

Various factors including horsepower loss from slip

heating

3. Understanding Racing Tires by Chuck HallumSAE Paper # 983028


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12

DEFINITIONS, ACRONYMS, ABBREVIATIONS

- Camber Angle [deg]

- Coefficient of Friction [-]

- Slip Angle [deg]

R Coefficient of Rolling Friction [-]

c Foundation Stiffness [psi]

C - Camber Stiffness [lbf/deg]

C - Cornering Stiffness [lbf/deg]

CC Camber Stiffness Coefficient [1/deg]

CFT Load Transfer Sensitivity [-]

CFV Ratio of Change in FY to FV [-]

CS Cornering Stiffness Coefficient [1

/deg]

CY Lateral Force Coefficient [-]

CSU Colorado State University

FD Tire Drag Force [lbf]

FS Tire Central Force [lbf]

FV Normal Force [lbf]

FVT Transferred Vertical Load [lbf]

FX Longitudinal Force [lbf]

FY Lateral Force [lbf]

FZ Vertical Force [lbf]

FSAE Formula SAE

KCSFV Sensitivity of CS to FV [-]

l Tire footprint length [in]

MZ Self Aligning Torque (Moment) [lbf * ft]

t Pneumatic Trail [in]

0 Angular velocity of free rolling wheel [rad

/s]

- Angular velocity of driven wheel [rad

/s]

- Angular velocity of wheel [rad/s2]

V Velocity [ft/s]

Re Effective radius of free rolling tire [ft]

Rl Loaded Tire Radius [ft]

T Thrust [lbf]

D Drag [lbf]

FR Rolling Resistance [lbf]

Tin Input torque to wheel [lbf * ft]

SAE Society of Automotive Engineers.


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13

APPENDIX

INFORMATION CONTAINED IN APPENDIX:

1. Avon Tire Data

2. Hoosier Tire Data

3. Example Calculations Using Given Avon Tire Data

4. Example Calculations Using A Few Selected Equations

AVON TIRE CURVES

Presented here is technical data given for Avon Tires (Ref 9). This tire is very similar to that used by many FSAE Teams

The CSU 2004 FSAE team will be using Hoosier tires with a 2 camber (available data is shown below in Table 1). Whilethe tires are not the same, the following information should give the reader an estimate of what the lateral force and self-aligning moments could be.

Project: RC353STB Size: 7.0/20.0-13 Camber: 2

Spec: 10998 Tyre: 3 ply Pro Series Pressure:

26

P.S.I.

Tested: 31/07/01 Rim: 6 x 13 Speed:

20

KPH

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0 0.

01.0

2.0

3.0

4.0

5.0

6.0

7.0

Slip Angle (deg)

SelfAligningTorque(N

m

150 kg

250 kg

350 kg

Figure 11 - Self-Aligning Torque vs. Slip Angle (Ref 9)


14/22

14

-5

-4

-3

-2

-1

0

1

2

3

4

5

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0 0.

01.0

2.0

3.0

4.0

5.0

6.0

7.0

Slip Angle (deg)

CorneringForce(KN

150 kg

250 kg

350 kg

Figure 12 - Cornering Force vs. Slip Angle (Ref 9)


15/22

15

Project: RC353STB Size: 7.0/20.0-13 Camber: 1

Spec: 10998 Tyre:

3 ply Pro

Series Pressure:

26

P.S.I.

Tested: 31/07/01 Rim: 6 x 13 Speed:

20

KPH

-120

-100

-80

-60

-40-20

0

20

40

60

80

100

120

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Slip Angle (deg)

SelfAligningTorque(Nm

150 kg

250 kg

350 kg

Figure 13 - Self-Aligning Torque vs. Slip Angle (Ref 9)


16/22

16

-5

-4

-3

-2

-1

0

1

2

3

4

5

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0 0.

01.0

2.0

3.0

4.0

5.0

6.0

7.0

Slip Angle (deg)

CorneringForce(KN

150 kg

250 kg

350 kg

Figure 14 - Cornering Force vs. Slip Angle (Ref 9)

Lateral Force vs Slip Angle At 1deg Camber and 330 lbf (150kg) Normal Load

y1 = -164.37x - 1.5432

y2 = -139.03x + 5.4779

-400

-300

-200

-100

0

100

200

300

400

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

Slip Angle (deg)

Lateralforce

(lbf)

-1 to 1 (y1)

-2 to 2 (y2)


17/22

17

Figure 15 - Lateral Force vs. Slip Angle for Different Ranges of Slip Angle (Data taken from Ref 9)

HOOSIER TIRE DATA

TIRE SIZE: 20.0 x 7.0 - 13

COMPOUND = R25A

RIM WIDTH = 8"

PRELOAD = 0

AIR =

18 PSI

DEFLECTION

(in)

ACTUAL

LOAD (lbs)

SPRING RATE

(lbs/in)

0.2 144 720

0.4 314 785

0.6 490 816.67

0.8 655 818.75

1 821 821

AIR =

16 PSIDEFLECTION

(in)ACTUAL

LOAD (lbs)SPRING RATE

(lbs/in)

0.2 123 615

0.4 288 720

0.6 451 751.67

0.8 618 772.5

1 783 783

AIR =

14 PSI

DEFLECTION

(in)

ACTUAL

LOAD (lbs)

SPRING RATE

(lbs/in)0.2 123 615.00

0.4 275 687.50

0.6 430 716.67

0.8 582 727.50

1 749 749.00

Table 1 Hoosier Tire Data (Ref 10)


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18

FY 1.3 deg( ) 831.323N=

Self Aligning Torque: MZ 1.3 deg( ) 23.233lbf ft= MZ 1.3 deg( ) 31.499N m=

Pneumatic Trail tMZ 1.3 deg( )

FY 1.3 deg( ):= t 1.492in= t 37.89mm=

Contact Patch Length l 6 t:= l 8.951in= l 227.343mm=

Approximate Cornering Stiffness C

FY 1.3 deg( )

1.3 deg:= C 143.761

lbf

deg= C 639.479

N

deg=

Normal Load Sensitivity:

Tire Data: 1 1.5 deg:= FY1 203.64lbf:= FV1 150kg g:= FV1 330.693lbf=

2 1.5 deg:= FY2 332lbf:= FV2 250kg g:= FV2 551.156lbf=

CFV

FY2 FY1

FV2 FV1:= CFV 0.582=

Example Calculations for the Avon Tire

Point of interest: Slip angle: 1.3 deg:=

Camber Angle: 1 deg:=

Load on Tire FV 150kg g:= FV 330.693lbf=

Coefficient of Friction 1.5:=

Known Lateral Load and Moments:Slip angle: 1 1.5 deg:= Lateral Force: FY1 203.64lbf:=

2 1 deg:= FY2 156.59lbf:=

3 0.5 deg:= FY3 92.3 lbf:=

Self Aligning Torque:

MZ1 23.32lbf ft:=

MZ2 22.516lbf ft:=

MZ3 19.76lbf ft:=

Quadratic Spline Interpolation:

FY ( ) 2( ) 3( )

1 2( ) 1 3( )FY1

1( ) 3( )

2 1( ) 2 3( )FY2+

1( ) 2( )

3 1( ) 3 2( )FY3+:=

MZ ( ) 2( ) 3( )

1 2( ) 1 3( )MZ1

1( ) 3( )

2 1( ) 2 3( )MZ2+

1( ) 2( )

3 1( ) 3 2( )MZ3+:=

Lateral Force: FY 1.3 deg( ) 186.889lbf =


19/22

19

FY 3 deg( )

FY 90 deg( ) 0.945=

0 5 10 15 200

100

200

300

Lateral Force Vs. Slip Angle

Slip Angle (deg)

LateralForce(lbf)

FY ( )

lbf

deg

0 0.001, 20 deg..:=Slip Angle Range:

FY ( ) FV

2FV

2

2 c l2

tan ( ):=Lateral Force:

l 215.9mm=l 8.5 in:=Contact Patch Length

c 725.189psi=c 5 106

Pa:=Foundation Stiffness

Coefficient of Friction 1.5:=

FV 889.644N=FV 200lbf=FV800 lbf

4:=Vertical Load:

Lateral Force From Slip angle


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20

Camber Angle

Camber Stiffness Coefficient CC0.018

deg:=

Normal Load Range Fv 50 50.1, 500..:=

Lateral Force FY Fv,( ) CC Fv :=

100 200 300 400 5000

10

20

30

40

50

Lateral Force Vs. Normal Force

Normal Force (lbf)

LateralForce(lbf)

FY 1 deg Fv,( )

FY 2 deg Fv,( )

FY 3 deg Fv,( )

FY 4 deg Fv,( )

FY 5 deg Fv,( )

Fv

Self Aligning Torque

MZ FV,( )

2FV

2

4 c l tan ( )

3

FV3

6 c2

l3

tan ( )( )2

:= 0 0.001,

2

..:=

0 10 20 30 400

20

40

60

500 lbf

200 lbf

100 lbf

Self Aligning Torque Vs. Slip Angle

Slip Angle ( deg )

SelfAligningTorque(lbf*ft)

MZ 500 lbf,( )

MZ 200 lbf,( )

MZ 100 lbf,( )

deg


21/22

21

Pneumatic Trail

FY ( ) FV

2FV

2

2 c l2

tan ( ):=

Normal Load FV 200 lbf:=

Pneumatic Trail (Equation 21) t ( )MZ FV,( )

FY ( ):=

0 1 2 3 40

5

10

15

20

Pneumatic Trail With Equation 21

Slip Angle (deg)

Trail(in) t ( )

in

deg

Pneumatic Trail (Equation 22)t ( )

3 FV c l2

tan ( ) 2 2

FV2

12 c2

l3

tan ( )( )2

6 FV c l tan ( )

:=

0 1 2 3 40

5

10

15

20Pneumatic Trail With Equation 22

Slip Angle (deg)

Trail(in) t ( )

in

deg


22/22

Load Transfer Sensitivity

CS0.14

deg:=

FY Fv,( ) CS Fv :=

Fv 50 50.1, 500..:=

100 200 300 400 5000

200

400

600Lateral Vs. Normal Load

Lateral Force (lbf)

N

ormalForce(lbf)

FY 1 deg Fv,( )

FY 2 deg Fv,( )

FY 4 deg Fv,( )

FY 6 deg Fv,( )

FY 8 deg Fv,( )

Fv

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Understanding Parameters Influencing Tire Modeling

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