Various vehicle dynamic simulation softwareprograms have been developed for use inreconstructing accidents. Typically these are usedto analyze and reconstruct preimpact andpostimpact vehicle motion. These simulationprograms range from proprietary programs tocommercially available packages. While the basictheory behind these simulations is Newton's laws ofmotion, some component modeling techniquesdiffer from one program to another. This isparticularly true of the modeling of tire forcemechanics. Since tire forces control the vehiclemotion predicted by a simulation, the tire mechanicsmodel is a critical feature in simulation use,performance and accuracy. This is particularly truefor accident reconstruction applications wherevehicle motions can occur over wide rangingkinematic wheel conditions. Therefore a thoroughunderstanding of the nature of tire forces is anecessary aspect of the proper formulation and useof a vehicle dynamics program.
This paper includes a discussion of tire forceterminology, tire force mechanics, the measurementand modeling of tire force components andcombined tire force models currently used insimulation software for the reconstruction ofaccidents. The paper discusses the differencebetween the idealized tire force ellipse and anactual tire friction ellipse. Equations are presentedfor five tire force models from three different
simulation programs. Each model uses a differentmethod for computing tire forces for combinedbraking and steering. Some experimentallymeasured light vehicle tire properties are examined.
Some tire force models begin with aspecified level of braking force and use the frictionellipse to determine the corresponding steeringforce; this produces steering forces and a resultanttire force equal in magnitude to full skidding forcombined steering and braking. Comparisons arepresented of results from simulation programs usingdifferent tire models for vehicle motions involvingtwo types of severe yaw. The comparisons in thispaper are not of reconstructions where the userseeks initial conditions to match an existingtrajectory. The first comparison is a hypotheticalpostimpact motion with a given initial velocity andinitial angular velocity and the other is a suddensteer maneuver. In some cases, the simulationsand their tire models predict the vehicle motionclosely. In most cases, however, the results differsignificantly between simulation programs.
The example simulations presented in thispaper are not intended to reflect the way vehicledynamic simulation programs are used typically inaccident reconstruction.
INTRODUCTION:
Tire Models: Beside helping to provide a smoothride, the main function of an automotive pneumatictire is to transmit forces (Fx, Fy, Fz) and moments in
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three mutually perpendicular directions for vehicledirectional control. This important role of tires hasmade tire behavior the subject of continuous study(and performance improvement) for nearly 80years.Numerous tests have been conducted andmathematical models have been developed in anattempt to understand and predict the generation ofthese forces. These models have been divided intofour different classifications [Pacejka]: 1) those thatuse a complex physical model, 2) those using asimple physical model, 3) models using similaritymethods, and 4) models based solely onexperimental data, so-called empirical models.Physical models are those intended to model tireperformance (rather than vehicle performance).Physical models are concerned with such things astire wear, temperature, traction, life, cost, etc. Theyhave parameters such as construction, materials,loads, inflation pressure, geometry, tread design,speed, and so on. Complex physical modelstypically use finite element modeling techniques.Finite element models of the tires are of particularuse when considering the interaction between thetire and road irregularities and for investigations intothe friction between the road and the tire within thefootprint of the tire [Tonuk and Unlusoy, Hölscher,et al.]. Models based on similarity methods wereuseful early in the tire force model developmentprocess but have found less use recently as theyhave been superceded by the utility afforded byother models. Such methods are covered byPacejka [Pacejka].
The two remaining model classifications, thesimple physical model and the empirical models,are the two most prevalent models used in theunderstanding and prediction of tire forces. Theyrelate the physical and kinematic properties of tiresto the development of tractive forces at the contactbetween the tire and the roadway surface. One ofthe most widely used simple physical models is thebrush model. Brush models have been improvedand developed over the recent years [Gäfvert &Svedenius] but have not yet found their way intodynamic simulation programs applied to accidentreconstruction. A thorough coverage of the brushmodel is presented elsewhere [Pacejka].
The remaining tire model classification is theempirical tire model. Such models are also referredto as semi-empirical tire models in many references[Pacejka, Guo]. These models deal exclusively withthe steady-state behavior of a tire. Treatment of thetransient behavior of the tire, for example oscillatoryresponse, response lag and wheel unbalance, is
given elsewhere [Pacejka, Allen, et al.]. Empiricalmodels employ mathematical functions capable ofemulating the highly nonlinear behavior of theforces generated by the tires. These mathematicalfunctions can range from straight line segmentapproximations to nonlinear functions that containnumerous coefficients based on experimental dataand determined by curve-fitting routines. Theprincipal use of these models is in the prediction oftire forces for vehicle dynamics simulation software.Many of these empirical models exist [Pacejka,Guo, Gäfvert, Hirschberg, Brach & Brach (2000),Pottinger, et al.]. This type of model is examined inthis paper.
Tire forces are separated into a longitudinalforce component (braking and driving) and a lateralforce component (steering/cornering). Thelongitudinal tire force typically is mathematicallyexpressed (modeled) and measured as a functionof a variable called wheel slip. In some cases thelongitudinal force is modeled simply by a prescribedforce level, sometimes expressed as a fraction ofthe normal force. The lateral tire force ismathematically expressed (modeled) and measuredas a function of a variable called the slip angle. Athird, distinct, feature of a tire force model is themethod of properly combining these two forcecomponents for conditions of combined braking(wheel slip) and steering (slip angle). Other forcesand moments exist at the tire-road interface that areimportant for vehicle handling and design but arenot considered here. Effects such as self-aligningtorque, camber steer, conicity steer, ply steer, etc.are usually neglected for accident reconstructionapplications.
Portions of this paper were presented orallyat a conference [Brach & Brach, 2008].
Vehicle Dynamic Simulation: The use of vehicledynamics models in the field of accidentreconstruction to simulate vehicle motion hasevolved steadily over the last few decades. Initially,the options of the reconstructionist were limited tothe vehicle dynamics capabilities of the variants ofthe government-funded SMAC & HVOSM[McHenry, Segal] computer programs being themost readily available options. Even today,simulation software appears to be underutilized inthe field as some reconstructionists continue to usesimplified methods in attempts to address complexmotion of a vehicle based on assumptions ofconstant deceleration [Fricke 1, Fricke 2, Orlowski,Daily, et al., Martinez] and even concepts such aspoint mass rotational friction [Keifer, et al. (2005)
and Keifer, et al. (2007)]. Various simulationprograms currently are available to the accidentreconstructionist in the form of computer-basedvehicle dynamics programs and are becoming anintegral part of various accident reconstructionsoftware [PC-Crash, HVE, VCRware]. Thesevehicle dynamic programs were developed fromwithin the accident reconstruction community andare particularly suited to the needs of that field.Other, more complex vehicle dynamic software isalso available [VDANL, Car-Sim, ADAMS]. Whilethe latter software can be used in accidentreconstruction work, their complexity is better suitedas vehicle handling models.
The basic premise behind all of thevariations of vehicle dynamics simulation programsis essentially the same: the user or the softwareitself provides initial conditions (position, orientation,velocity) for the vehicle, the vehicle-specificgeometry, the vehicle physical parameters(including tire parameters), and any time-dependentparameters (such as steering input,braking/acceleration, etc.). The program integratesthe differential equations of motion of the vehicle(and semitrailer) to predict the motion as a functionof time. The needs that the accident reconstructioncommunity has for a simulation program can differfrom other users of vehicle dynamics programs.Such needs include the ability to capture thedynamics of the vehicle through a wide range ofmotion and vehicle conditions such as damaged oraltered wheelbase and/or track width, one or morewheels that are locked, large initial yaw rates ofrotation following an impact, etc. In contrast, vehicledesign and development work typically use vehicledynamics to study the performance of a vehicle inits as-designed condition and operation.
Comparisons have been made [Han andPark] between EDVAP [HVE], PC-Crash [PC-Crash]and a proprietary simulation program. Thesecomparisons consisted of three categories of initialconditions that result in three different types ofpostimpact motion. Category 1 uses initialconditions with a relatively high yaw velocity. Theresulting vehicle motion showed that the yawvelocity decreased to near zero and the vehiclecontinued with a translational motion (rollout).Category 2 uses initial conditions that resulted in anonzero yaw velocity that was maintained until rest(spinout). Category 3 uses initial conditions thatresult in the vehicle experiencing a moderate yawvelocity and translation. The results showed that thelargest differences between EDVAP and PC-Crashoccurred for the initial conditions of Category 1.
Only small differences were found for Categories 2and 3. All three tire force models use the frictionellipse to compute combined tire forces.
In all cases, the accuracy of the tire force isof considerable importance to the users of thesimulation software. To a great extent, simulationaccuracy depends on the ability of the tire model topredict accurately the forces acting in the plane ofthe roadway generated by each of the vehicle’stires. Other than aerodynamic forces, consideredlater in the paper, it is the tire forces acting at thetire contact patches that control the motion of thevehicle.
This paper focuses on the tire models usedby three currently available simulation programs,PC-Crash, HVE and VCRware. These all have thecapability to simulate motion in two dimensions.Some have more general capabilities such as threedimensional motion but these features are notconsidered here. The tire models used by each ofthese software programs is described in detail. Thistreatment is followed by two comparisons ofsimulation results using each software package forthe same set of tire parameters, vehicle parametersand initial conditions. The paper concludes with adiscussion of the results of the simulations. Thetopic of the tire friction ellipse is discussed. It isshown that the idealized friction ellipse can differsignificantly from a plot of the limit of tire forcesdeveloped by actual tires.
NOTATION, ACRONYMS AND DEFINITIONS
• BNP: Bakker-Nyborg-Pajecka equations (alsoknown as the Magic Formula) [Pacejka]• Cornering stiffness: see Cα• Cornering compliance: 1'Cα• EDSMAC4: simulation software [HVE],• frictional drag coefficient, μ: average,constant value of the coefficient of friction of a tirefully sliding over a surface under given conditions(wet, dry, asphalt, concrete, gravel, ice, etc.)appropriate to an application,• friction circle: the friction ellipse when μx = μy,• friction ellipse: an idealized curve withcoordinates consisting of the longitudinal andlateral tire force components that defines thetransition of a tire from wheel slip to the conditionof full sliding,• lateral (side, cornering, steering): in thedirection of the y axis of a tire’s coordinatesystem,• longitudinal (forward, rearward, braking,accelerating, driving): in the direction of the x
axis of a tire’s coordinate system,• PC-Crash: simulation software [PC-Crash],• SIMON: SImulation MOdel Nonlinear [HVE]• sliding: the condition of a moving wheel and tirelocked from rotating (s = 1), or moving sideways(α = π/2),• VCRware: simulation software [VCRware],• Cα: lateral tire force coefficient (also corneringcoefficient), • Cs: longitudinal tire force coefficient, • Fb: input value for the braking or accelerationforce, PC-Crash,• Fx(s): an equation with a single independentvariable, s, that models a longitudinal tire force forno steering, α = 0,• Fy(α): an equation with a single independentvariable, α, that models a lateral force for nobraking, s = 0,• Fx(α,s) = Fx[Fx(s),Fy(α),α,s]: an equation withtwo independent variables, (α,s), that models alongitudinal tire force component for combinedbraking and steering,• Fy(α,s) = Fy[Fx(s),Fy(α),α,s]: an equation of twoindependent variables, (α,s), that models a lateraltire force component for combined braking andsteering,• Fz: wheel normal force,• full sliding: a condition when the combined slipvariables (α,s) give a resultant tire force equal toμFz, see sliding,• HVOSM: Highway Vehicle Object SimulationModel• m-smac: simulation software [m-smac]• NCB: Nicolas-Comstock-Brach equations [Brach& Brach 2000, 2005]• rollout: translational motion alone of a vehiclethat continues following spinout,• s: longitudinal wheel slip,• slip velocity: the velocity of the center of a tireat the contact patch relative to the ground,• slip angle: α,• SMAC: Simulation Model of AutomobileCollisions [McHenry]• spinout: motion of a vehicle that includes bothtranslation and yaw rotation,• T: an input value for the braking or accelerationforce, SMAC,• wheel slip: see s,• Vx, Vy: components of the velocity of a wheel’shub expressed in the tire’s coordinate system,• Vp: slip velocity of a tire at point P of the tirepatch.• x-y-z: orthogonal wheel coordinates where x isin the direction of the wheel’s heading and z is
perpendicular to the tire’s contact patch (see Fig1),• yaw: vehicle rotation about a vertical axis• α: tire slip angle (also, lateral slip angle),• βp: angle of a tire’s slip velocity relative to thetire’s x axis and angle of the resultant forceparallel to the road plane (see Fig 2),• β: angle relative to the x axis of the resultant tireforce (see Fig 2),• β_
: nondimensional slip angle, Eq 45 & 50,SMAC,• μx: tire-surface frictional drag coefficient for fullsliding in the longitudinal direction, s = 1, α = 0,• μy: tire-surface frictional drag coefficient for fullsliding in the lateral direction, α = π/2.
TIRE KINEMATICS
Two kinematic variables typically are used with tireforce models and with the measurement of tireforces. These are the slip angle, α, and thelongitudinal wheel slip, s. The slip angle, isillustrated in Fig 1 and is defined as (1)1tan ( / )y xV Vα −=The wheel slip canh a v e d i f f e r e n tdefinitions [Brach &B r a c h ( 2 0 0 0 ) ,Pacejka]. The oneused here is suchthat 0 # s # 1,where
x
x
V RsV
ω−=
(2)Figures 1 and 2 show the tire slip velocitycomponents VPx = Vx - Rω and Vpy = Vy. Note that ingeneral the vectorvelocity, V, at thewheel hub and the slipvelocity, Vp, at thecontact patch centerdiffer both in magnitudeand direction. The slipvelocity, Vp, is thevelocity of the point Prelative to the roadsurface. Also, the direction of the resultant force, F,and the slip velocity, Vp, can differ. For no steering,the longitudinal (braking, accelerating) tire forcecomponent, Fx(s), typically is expressedmathematically as a function of the wheel slip alone.
=Vyyp -V Rωx
x
V P Vy
z
=px
Vp
αω
V
R
y Vx
V
Figure 1. Wheel/tire velocities
xF
y
Vy
pV
-V Rωx
x
βF Fy
βp
Figure 2. Tire patch velocityand force components.
Similarly, for no braking, the lateral (cornering,steering) force component, Fy(α), typically isexpressed mathematically as a function of the slipangle alone.
EXPERIMENTALLY MEASURED TIRE FORCES
Experimental tire data are presented herebecause some of the simulation results given laterin the paper use tire parameters corresponding tomeasured values. The amount of data presentedhere is limited; more is given in a recent paper[Salaani] including a longitudinal tire force, Fx(s), asa function of wheel slip, s, and lateral tire force,Fy(α), as a function of slip angle α. Figure 3 showsFx(s) for a P225/60R16 tire for different normalforces. Figure 4 shows measured values of Fy(α) fordifferent normal forces. As indicated by the notation,Fx(s) is measured for zero slip angle, α, and Fy(α) ismeasured for zero wheel slip, s. These tireproperties are emulated later for use with a 2006Ford Crown Victoria for which the P225/60R16 tireis standard.
From Fig 4 it can be seen that the slip
coefficient, Cα, (the slope of the initial linear portionof the curves) depends on the normal force, Fz. Aleast square fit (using the BNP equations)illustrating this dependence is shown in Fig 5.Figure 3 similarly shows that the slip stiffnesscoefficient, Cs, depends on the normal force.
FRICTION ELLIPSE, TIRE FORCE ELLIPSE
The x-y coordinate system and velocities ofa rotating wheel are illustrated in Fig 1. The tireforce components Fx = Fx(α,s), Fy = Fy(α,s) andresultant, F = F(α,s), are illustrated over a tire-roadcontact patch in Fig 2. According to the Nicolas-Comstock theory [Brach & Brach (2000)], the forcecomponents form a force ellipse where the abscissais the longitudinal tire force component, Fx(α,s), andordinate is the lateral tire force component, Fy(α,s).The equation of the tire force ellipse is given by Eq3, or in a more concise form in Eq 4. The resultantforce is .2 2( , ) ( , ) ( , )x yF s F s F sα α α= +
One of the conditions of the Nicolas-Comstock tiremodel is that the force components are aligned withthe slip velocity components, that is β = βp (Fig 2).As shown in Fig 6, the Fx(α,s) axis (abscissa)represents braking alone (i.e., α = 0). The Fy(α,s)axis (ordinate) represents steering alone (i.e., s =0). Each point of the friction ellipse’s interior is apoint with slip values (α,s) for combined steeringand braking that represents driver control,expressed mathematically by Eq 5. A point Fx(s)|s=1= μxFz on the abscissa represents locked wheelskidding for braking alone. The point, Fy(α)|α= π/2 =μyFz, on the ordinate represents a vehicle tire slidinglaterally. Note that this formulation allows fordifferent frictional drag coefficients in the x and ydirections, μx and μy, respectively. Full sliding of thetire under any combination of α and s occurs if the
Figure 5. Measured variation (points) of Cα with Fz,P225/60R16 tire [Salaani].
(3)2 2
2 2
( ), ( ), , ( ), ( ), ,1
( ) ( )x x y y x y
x y
F F s F s F F s F sF s F
α α α αα
⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦+ =
(4) (5)22 ( , )( , )
12 2( ) ( )
F sF s yxF s Fx y
αα
α+ =
22 ( , )( , )12 2 2 2
F sF s yxF Fz zx y
αα
μ μ+ <
(6)2 2 2 2sin cos
x y
x y
μ μμ
μ α μ α=
+
resultant tire force reaches the friction ellipse, F(α,s)= μFz, where the frictional drag coefficient, μ is givenby Eq 6 [Brach & Brach (2000)]. For a given normalforce, Fz, points outside the Friction Ellipse cannotbe reached because the friction force is limited byμFz. If μx = μy, then the tire force ellipse becomes acircle and the friction ellipse becomes a frictioncircle.
Model equations that determine thefunctions Fx(α,s) and Fy(α,s) for combined steeringand braking (such as shown in Fig 6 as a tire forceellipse) must be found independently from thesteering and braking functions Fy(α) and Fx(s). Thisis done later. It is important to note that the frictionellipse is not a tire model. Rather, it is an idealizedgraphical display of the operating limit for resultanttire forces for any combination of steering andbraking. More than one method exists fordeveloping the resultant tire force for combinedsteering and braking. One is shown in the nextSection; others are [Pottinger, et al. and Schuring,et al.] and [Hirschberg].
SIMULATION TIRE MODELS
Different tire force models exist and at leastone survey has been written [Gäfvert, M. and J.Svedenius], but the equations of most commonlyused models are not cataloged. The following is acollection of the equations of tire force models usedin three vehicle dynamics simulation softwarepackages used for reconstructing accidents.
VCRware Tire Model: The longitudinal and lateraltire force equations for this simulation software aremodeled using asubset of theBNP equations[ P a c e j k a ] .Equation 7 givesthe longitudinalforce, Fx(s), forbraking alonewith no steering(α = 0). Figure 7s h o w s a nexample of anormalized plot of the longitudinal tire force withexample BNP parameter values of B = 1/15, C =1.5, D = 1.0, E = 0.30, K = 100.0 and where theinitial slope is the braking coefficient Cs = BCDK.Equation 8 gives the lateral steering force, Fy(α), forno braking (s = 0). Figure 8 shows a samplenormalized lateral force with BNP parameter valuesof B = 8/75, C = 1.5, D = 1.0, E = 0.60, K = 100.0and the lateral stiffness coefficient is Cα = BCDK.
For a wheel with a braking force, Fx(s), anda lateral force, Fy(α), the longitudinal force forcombined steering and braking, Fx(α,s), isdetermined in VCRware using the Nicolas-Comstock-Brach, (NCB) equations [Brach & Brach(2000) and Brach & Brach (2005)]. It is given by Eq9. For a wheel with a braking force, Fx(s), and alateral force, Fy(α), the lateral force for combinedsteering and braking, Fy(α,s), is determined usingthe NCB equation and is given by Eq 10.
( )α,syF
μ FzxxF ( )1 =
yF π/2( ) μ Fzy=
xF s( ) ( )α,s
xF
yF ( )α
( )α,sF
( )α,sx
F
FrictionEllipse
β
Tire ForceEllipse
( )α,syF
Figure 6. Diagrams of Friction (Limit) Ellipse and TireForce Ellipse.
Longitudinal wheel slip, s0.0 0.2 0.4 0.6 0.8 1.0
Nor
mal
ized
long
itudi
nal f
orce
, Fx(
s)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 7. BNP longitudinal force as afunction of wheel slip, s, VCRware.
(7){ }1 1( ) sin tan (1 ) tan ( )F s D C B E Ks E BKsx− −⎡ ⎤= − +⎢ ⎥⎣ ⎦
(8)2 21 1( ) sin tan (1 ) tan ( )F D C B E K E BKyα ααπ π
⎧ ⎫⎡ ⎤− −= − +⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
(9)2 2 2 2 2( ) ( ) (1 ) cos ( )
( , )2 2 2 2( ) ( ) tan
F s F s s C s F sx y a xF sx sCs F F sy x
α αα
αα α
+ −=
+
(10)
2 2 2 2 2(1 ) cos ( ) sin( ) ( ) tan( , )
sin2 2 2 2( ) ( ) tan
s F CF s F y sx yF sy Cs F F s sy x
α α αα αα
αα α
− +=
+
When plotted onaxes of Fx(s) andFy(α), the NCBequations takethe form of a tireforce ellipse thatdepends on thefunctions Fx(s)and Fy(α). Three-d i m e n s i o n a lsurface plots ofthese combinedtire forces are illustrated in Appendix A.
PC-Crash Linear Tire Force Model: PC-Crashallows the choice of either of two tire models, theLinear Tire Force model and the TM-Easy TireForce model. The Linear Tire model is as follows.
Instead ofusing the wheelslip parameter, s,the PC-Crashs i m u l a t i o nrequires an inputv a l u e o f ac o n s t a n tmagnitude ofapplied brakingforce with a forcelevel, Fb, or anacceleration force magnitude, Fa. A force specifiedas a fraction of the wheel normal force canalternatively be supplied. For no steering thelongitudinal accelerating force, is specified as Fx =Fa, and the longitudinal braking force is Fx = -Fb.The PC-Crash vehicle dynamic simulation uses abilinear lateral tire force as shown in Fig 9. Thelinear portion represents a slip coefficient of Cα.
The lateral force becomes constant at α =αmax, where the lateral force reaches its maximumvalue μFz. For the PC-Crash protocol, αmax = μα1
max,where α1
max is the saturation angle for μy = 1. Forthis notation, the tire slip coefficient is computed asCα = μFz /α1
max. For no longitudinal force, s = 0, (Fa= Fb = Fx = 0) the lateral tire force is defined by Eq11 and 12. For a wheel with braking force Fx(α,s) =Fb the lateral force is computed using the frictionellipse as given in Eq 13 where the longitudinalforce is adjusted for the condition of locked wheelskidding as shown in Eq 14. For combined steeringand braking, the PC-Crash Linear Tire Model can be described in threeregions (see Fig 10). Region I is when the sideforce increases linearly with α, Eq 15. Region II iswhen the side force is said to be saturated and thelateral force is computed using the friction ellipse,Eq 16 and Region III is for locked wheel sliding, asshown in Eq 17.
slip angle, 2α/π0.0 0.2 0.4 0.6 0.8 1.0
late
ral f
orce
, Fy(
α)/μ
F z
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 8. BNP lateral tire force as afunction of normalized slip angle,2α/π, VCRware.
(17) (18)( , ) sinF s Fzy α μ α= 2 2( , )F s F Fzy bα μ+ =
These regions are shown in Fig 10 and areplotted on the friction ellipse in Fig 11. As the slipangle, α, increases from 0 to αmax, Fy(α,s) goes from(0,0) to point A. The magnitude of the lateral force,Fy(α,s), at point A is determined by Fb and Eq 17.Note that in Region II, while the slip angle increasesfrom αmax to some value greater than αmax as shownin Fig 10, the resultant force at the patch does notchange. Thus Region II, for which α varies from αmaxto some value greater than αmax, is concentrated ata single point, B, on the tireforce diagram in Fig 11.In Region III Fy(α,s)goes from point B to point C (asα continues to increase) along the friction circle.From Eq 17 note that for Region II (point B), Eq 18holds. All of this implies that throughout Region IIthe PC-Crash Linear tire force model gives a lateralforce at the friction limit on the idealized friction limitcircle. Although the direction of Fy(α,s) is along theslip direction, the magnitude of the resultant tireforce is equal to a fully skidding tire, μFz. A surfaceplot of Fy(α,s) is given in Appendix A.
TM-Easy Tire Model [Hirschburg, et al.]: The TM-Easy model is defined for three dimensional vehicle
motion. However all of the following discussion isfor zero camber and negligible contact moments.According to notes on vehicle dynamics [Rill] TM-Easy defines longitudinal slip and lateral slipdifferent than above. Longitudinal slip, sx, is definedas in Eq 19. TM-Easy lateral slip is defined as in Eq20. The consequences of normalizing slip to thewheel angular velocity is for TM-Easy that 0 # sx #4, 0 # sy # 4 and (for combined steering andbraking) that sx and sy are coupled to s (as definedby Eq 2) and α (Eq 1), as given in Eq 21 through 25.The TM-easy model specifies that beyond a certain,finite value of slip sxf, full sliding occurs. The modelcan characterize a maximum longitudinal force byspecifying maximum values of the force with itscorresponding slip (sxm, Fxm). Figure 12 shows thelongitudinal force Fx as a function of the longitudinalslip sx. A full description of the model requires thatthree pieces of information be provided to define theshape of the Fx(sx) curve: an initial slope, Cx, themaximum value of the force and its associated slipvalue (sxm, Fxm), and the value of the force at fullsliding and its associated slip value (sxf, Fxf). Thecurve for the lateral force, Fy(sy), can similarly bedefined using slope, Cy, maximum parameters (sym,Fym) and full-sliding parameters (syf, Fyf).
The process outlined above defines theshape of the curve for the longitudinal force in theabsence of lateral slip, Fx(sx), and the curve for thelateral force in the absence of longitudinal slip,Fy(sy). The force for combined braking and steering,
bF
EllipseFriction
CIII
IIB
μyFz
Fy ( )α,s
μxFz ( )α,sx
F
Braking
(0,0)
IRegion
A
Figure 11. Diagram of lateral and longitudinal tire forcesfor combined steering and braking, PC-Crash.
(36)2( , ) ( ) (3 2 ), ,s sxy mF s s F F F s s sx y m m m xyf fs smf
σ σ σ−
= − − − = ≤ ≤−
(37)( , ) ,F s s F s sx y xyf f= >
(38) (39)( , ) ( , )cosF s s F s sx x y x y ϕ= ( , ) ( , )sinF s s F s sy x y x y ϕ=
F(sx,sy), is formulated by the TM-Easy modelthrough the following process. A generalized slipvariable, sxy, which treats the longitudinal and lateralslip vectorially, is defined by Eq 26 where quantities
and are normalized slip variables and arexs ysdefined by Eq 27 and 28. Equations 29 through 33define additional parameters. A generalized tireforce, F(sx,sy) is now described in each of the threeintervals by a broken rational function, a cubicpolynomial and a constant Ff and given in Eq 35, 36and 37. Finally, the longitudinal and lateral forcecomponents, Eq 38 and 39, are determinedindividually from the projections in the longitudinaland lateral directions, using n, given by Eq 34.Three-dimensional surface plots of the longitudinaland lateral tire forces for combined steering and
braking for the TM-Easy model are given inAppendix A.
SMAC Tire Model [HVE and m-smac]: Forbraking, SMAC does not use the wheel slipvariable, s, but the simulation user is asked tospecify the value of a constant braking force, T,which also can be defined as a percentage of theavailable friction force at each wheel. Thelongitudinal tire force, Fx, is given by Eq 40 through44 for the different variations of braking andacceleration.For braking:
T = 0 (s = 0), Fx(T) = 0 (40)0 < T # µ Fz, Fx(T) = -T (41)T > µFz, Fx(T) = -µ Fz (42)
For acceleration|T| # µ Fz, Fx(T) = T (43)|T| > µ Fz, Fx(T) = µ Fz (44)
(45)( )2 2C
Fz
ααβ β αμ
= =
For , (46)3β <3
( )3 27
F Fzyβ β βα μ β
⎡ ⎤= − +⎢ ⎥
⎢ ⎥⎣ ⎦
For , (47)3β ≥ ( )F Fzy α μ=
For the lateral force, SMAC uses anondimensional variable , Eq 45, based on theβFiala tire model [EDSMAC, Brach & Brach (2005)]and defines the lateral force Fy(α) by Eq 46 and 47.Fy(α) is plotted in Fig 13 for typical values ofCα 'μFz.
For a wheel simultaneously steered (α > 0)and braked (T > 0) the longitudinal tire force,Fx(α,s), is computed by Eq 48 or 49, where the lattercase corresponds to locked wheel skidding. Forcombined braking and steering, the lateral tire force,Fy(α,s), is computed using the longitudinal force, ,βnewly defined by Eq 50 and the friction ellipse.Then for , Eq 51 or 52 give Fy(α,s). Equation 52β
implies that for the resultant tire force lies on3β ≥the friction ellipse, as given by Eq 53 and that theSMAC tire force model gives a lateral force at thefriction limit for combined steering and braking(before locked wheel sliding occurs). Although thedirection of the lateral force, Fy(α,s), is along the slipdirection, the magnitude of the resultant tire forceequals that of a fully skidding tire. A three-dimensional surface plot of Fy(α,s) using Eq 51through 53 is included in Appendix A.For (48)( ) cos , ( , )F T F F s Tzx xμ α α≤ =For (49)( ) cos , ( , ) cosF T F F s Fz zx xμ α α μ α> =
(50)( )2 2 2( , )
C
F F sz x
ααβ β αμ α
= =−
For ,3β <
(51)1 12 2 2 3( , ) ( , )3 27
F s F F szy xα μ α β β β β⎛ ⎞= − − +⎜ ⎟⎝ ⎠
For ,3β ≥
(52)2 2 2( , ) ( , )F s F F szy xα μ α= −
(53)2 2( , ) ( , )F s F s Fzx yα α μ+ =
SIMON Tire Model [HVE]: SIMON [EDC] uses asemiempirical tire model which is based upon theHSRI tire model [McAdam, et al.]. The principle
behind the HSRI tire model is that the tire forms arectangular contact patch which can be divided intotwo regions consisting of a no-slip region and asliding region. The relative size of the two regions isdependant upon the longitudinal and lateral slipvalues, s and α, the sliding frictional dragcoefficient, μ, and the initial slopes, Cs and Cα, ofthe linear tire force curves.
The first step in determining the SIMON tireforces is to determine an equivalent frictional dragcoefficient, μN, that depends on the slip, s, and iscalculated from the directional sliding frictional dragcoefficients, μx and μy. The coefficient μN is foundusing a fitting procedure whereby,
(54)2(1 ) (1 )a s sp p= − +
(55)( )(1 ) ( 2) (2 1)b s s sp x p p pμ μ= − + − +
(56)( )c x p xμ μ μ= −
(57)2 4
2b b acB
a− + −
=
(58)A Bxμ= +(59)(1 )C B sx pμ= + −
and(60)' A Bsμ = −
In these equations, μp is the ratio of longitudinal tireforce Fx(s)max/Fz and sp is the slip at Fx = Fx(s)max. Avariable Dt is defined as,
(61)2 2( ) ( sin )t sD C s Cα α= +where s is the longitudinal tire slip and α is the slipangle. After calculating μN, a fraction, Xs/L,representing the portion of the total contact patchthat is not slipping, where L is the total length of therectangular tire patch, is defined as:
slip angle, α0.0 0.4 0.8 1.2 1.6
late
ral f
orce
, Fy(
α)/μ
F z
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Cα/μFz = 5
Cα/μFz = 10
Cα/μFz = 15
Cα/μFz = 20
Figure 13. Lateral tire force as a function of slip angle, α,SMAC.
(62)' (1 ), 0 1
2X XFs sz sL D Lt
μ= − ≤ ≤
The equations for combined steering andbraking/acceleration follow. The equations forsteering alone and braking alone can be found bysubstituting s = 0 and α = 0 into the equations,respectively. For combined braking and steering,Xs /L = 1:
(63)( , )1sF s Cx s s
α =−
(64)sin
( , )1C
F sy sααα = −
−
Three-dimensional surface plots of Fx(α,s) andFx(α,s) are included in Appendix A. The sinefunctions in the range -π # α # π as used in theabove equations for the SIMON model werechanged from the tangent functions found in theoriginal HSRI model. EDC is now investigating thefull effects of this change. In addition, variousempirical curves from measured tire parametersarebuilt into the HVE software that make the tirecharacteristics tire specific and functions of loadand speed. However, the user has the ability toenter other tire characteristics or to use setup tablesbased upon a specific tire tests. The SIMON tiremodel also considers the effects that camberstiffness has on the lateral tire forces.
SIMULATION COMPARISONS
Comparison of Simulation Tire Force Models:Tire forces for combined steering and braking canbe compared visually using three-dimensional forceplots. Plots are given for all of the different modelsin Appendix A.
Computer Vehicle Dynamic Simulation: Twoexamples are presented for comparison of thesimulations and tire models. The first is ahypothetical, postimpact trajectory of a 2006 FordCrown Victoria. This example is examined for threedifferent sets of wheel conditions: A, locked wheels,B, partial drag on each wheel with a single lockedfront wheel and C, partial drag on each wheel.Results of the different simulations and tire modelsare compared on a relative basis.
The second example is for a sudden steermaneuver of a partially braked vehicle based on atest [Cliff, et al.]. Relative comparisons between thedifferent simulation results are made. The exampleis intended to reflect a relatively rapid severe steerwith partial braking. All of the simulations use
identical vehicle and tire input data and a frictionaldrag coefficient of f = 0.75. All input data are listedin Appendix B. These examples are intended toillustrate that uncertainty of simulations exists. Suchuncertainty depends on differences in the individualcharacteristics of each simulation program as wellas differences in the tire models. The simulationsoftware packages used are HVE, PC-Crash andVCRware.
First Example (Crown Victoria) The same vehicleand tire properties are used to compute the outputof the different simulations for a postimpactmaneuver with specified initial conditions. Thevehicle corresponds to a 2006 Ford Crown Victoria.A major reason this vehicle is chosen is because ituses P225/60R16 tires with known, measuredlateral steering properties [Salaani] presentedearlier. The specifications of the vehicle arecontained in Appendix B.
Vehicle trajectories are computed for aninitial forward speed of 34.1 mph (55 km/hr), aninitial lateral speed of zero and an initial yawangular velocity of 150 E/s. Each trajectory iscomputed for three conditions of braking. First, theoutput of the simulations is compared for a casewhich is independent of the tire force models, thatof locked wheel skidding, indicated as A in Fig 14.Then comparisons are made for the same initialconditions for equal powertrain drag on each rear
A
B
CFigure 14. Diagram of three cases A, Band C. Arrows indicate initial velocities.Gray tires indicate partial drag; black tiresindicate locked wheels.
Xs /L < 1:
(65)2
'( , ) (1 ) ' 12 2 2sin
XF sszF s C s s Fzx s D Lt s
μα μα
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ +⎝ ⎠
(66)2
' sin( , ) sin (1 ) ' 12 2 2sin
XF szF s C s Fzy D Lt s
μ αα α μαα
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ +⎝ ⎠
wheel (10% of the static normal force), rolling dragon the left front wheel (0.7% of the static normalforce) and a locked right front wheel, B in Fig 14.The third case is for equal powertrain drag on eachrear wheels (10% of the static normal force) andequal tire rolling drag on each front wheel (0.7% ofthe static normal force), C in Fig 14. The results areas follows.
A. Postimpact Motion, Locked Wheel SkiddingTable 1 lists the results of the locked wheel skidsimulations. All three software packages and allthree tire models give reasonably close restpositions, orientations and times to rest.
B. Postimpact Motion, No Applied Braking, PowerTrain Drag and One Locked Front Wheel For theconditions of 0.7% rolling wheel drag on the leftfront wheel, 10% powertrain drag on both rearwheels and the right front wheel locked, theagreement between all tire models is good, but notas close as the locked wheel condition. Table 2 liststhe CG rest positions, orientations and travel times.Initial motion is in the x direction and lateral travel issmall. VCRware and EDSMAC4 give a negativelateral travel, while PC-Crash gives a small positivetravel. The times to reach the rest positions areclose but not the same.
C. Postimpact Motion, No Applied Braking withPower Train Drag and Tire Rolling ResistanceResults are contained in Table 3 for the sameconditions as the previous case, except with rollingdrag on both front wheels (no locked wheel) and foran additional tire model. Large differences in therest positions, orientations and travel times occur.The motion in this case can be divided into twocomponents. The first is a combination oftranslation and yaw rotation (spinout). At a point inthe travel to rest, the yaw velocity goes to zero( ); the motion that follows consists of0θ =translation alone, or rollout, to a rest position. Thisis illustrated in Fig 15 for simulations usingEDSMAC4, VCRware and PC-Crash (two tiremodels). The positions and orientations at the end
of spinout differ; in particular, the angular positionsare quite different. This leads to large differences inthe rest positions. For reference, the locked wheelskid trajectories from the same initial conditions areshown in the same figure (note that the differentrest positions are so close that only one is shown).
Note that a sensitivity analysis to changes ininitial conditions was not carried out.
Second Example (Honda Accord): Thesesimulations use a 1991 Honda Accord with an
initial speed of 100 km/hr (91.13 ft/s). The drivermakes a sudden, constant front wheel steermaneuver to the right of approximately 9E followingbrake activation that causes a constant, equivalent,longitudinal deceleration of 0.273 ± 0.003 g’s. Thevehicle then moves to rest. Details of the inputvehicle and tire data are given in Appendix C.
Since the initial vehicle speed is relativelyhigh, simulations were run with and withoutaerodynamic drag where possible and, forcomparison, ignoring aerodynamic drag. Theaerodynamic drag force, RA, in VCRware iscalculated using the well known equation [Hoerner]
(67)212A dR C AVρ=
The drag force depends on the density of air, ρ, adimensionless drag coefficient, Cd, a projected areaA, and a velocity relative to the wind, V. In all casestreated here a wind speed of zero is used. Theaerodynamic drag is a resultant force calculatedusing frontal and lateral components. A frontal dragcoefficient for all simulations had a value of CdF =0.4 with a frontal area of AF = 25 ft2 (2.3 m2). Thecorresponding lateral or side values are CdL = 0.8and AL = 60 ft2 (5.6 m2). For no aerodynamic dragCdF = CdL = 0. In some cases, an aerodynamicmoment (usually small) is developed since the sideforce is not aligned with the vehicle center ofgravity. When included, a moment arm of 0.76 ft tothe rear of the CG was used.
The front and rear tire side forcecoefficients, Cαf and Cαr, are included as inputparameters in all simulations. Stock tire size on a
1991 Honda is listed as 195-60R15. It is importantthat these coefficients be reasonably accurate, yettire parameter information from the open literatureis sparse. In addition, tire properties for a givensized tire can vary from manufacturer tomanufacturer. The tire parameters found and usedhere represent a reasonable set of values for thistire size but do not necessarily represent the exact
values for the actual test vehicle. The values for thisexample were established in the following way.
Engineering Dynamics Corporation [HVE]lists a value for this tire as Cα = 231.7 lb/deg=13275 lb/rad for a vertical load of 1230 lb. Basedon this, a value of Cαf = 13000 lb/rad is used for allsimulations for the static normal force at the testvehicle front wheels, Wf = 932 lb. Since tire side
ROLLOUT
y Locked Wheel Skid
SPINOUT
All Simulations
y
INITIALPOSITION
PC-Crash
EDSMAC4
VCRware
0 Feet10 20 40
0 3 6 12meters
POSITIONREST
x
x
PC-Crash
POSITIONS
REST
EDSMAC4
VCRware
Figure 15. Diagram of results of a locked wheel skid (Case A) and rolling resistance on the front wheels andpowertrain drag on rear wheels (Case C), Crown Victoria.
2 aerodynamic drag w w/oEDSMAC4 PC-Crash LinearPC-Crash TM-EasySIMONVCRware 8 9
Figure 16. Diagram of sudden steer maneuver simulation. “w” indicates aerodynamic drag is taken into account and“w/o” is with no aerodynamic drag. The circular arc is the path according to the critical speed formula.
force coefficients vary with normal force and thestatic normal force for the rear of the test vehicle isapproximately 660 lb, a value of Cαr must beestimated. An approximate formula can bedeveloped (for small changes in normal force) froman equation in a paper on tires [Salaani], as (68)zC kFα ≈This gives
(69)zrr f
zf
FC CFα α≈
giving a value of Cαr = 9200 lb/rad. This combinationof values of Cαf and Cαr would place the 1991 Hondainto a neutral steer condition (which is not the case).A second approach to estimate Cαr was taken usingthe front and rear Bundorf compliances [Milliken] fora passenger car. This gives
(70)1.1f r
r f
W CW C
α
α
=
which, in turn gives Cαr = 10137 lb/rad. Based onthese estimates, a value of Cαr = 10000 lb/rad waschosen for the static rear tire side force coefficientsand used in all simulations. These values provide astatic positive understeer gradient.
Figure 16 shows the rest positions andorientations from all of the simulations.
DISCUSSION AND CONCLUSIONS
The primary purpose of this paper is todemonstrate that different tire models exist, todescribe them in as much detail as possible and toindicate which simulation programs (used inaccident reconstruction applications) use which tiremodels. Two example applications of thesesimulation programs and tire models are presented.The example applications were limited to ahypothetical postimpact motion of a Ford CrownVictoria and to a sudden steer maneuver of aHonda Accord. Results within the differentsimulations for each example are compared. Sincethe applications are limited to only two, theconclusions that can be drawn likewise are limited.
Alternative methods exist [Kiefer, et al.,2005, 2007] to estimate the combined effects ofinitial translational and rotational velocities on thetrajectory of a vehicle to rest following impact thatdo not use tire force models. Such methods do nothave the potential of simulating different tireproperties and accident reconstruction conditions
such as partial braking, powertrain drag, rollingwheel drag and/or the effects of an individuallylocked wheel or wheels. It is necessary to use avehicle dynamic simulation program for modeling ofsuch conditions. Despite the greater potential foraccuracy, the uncertainty due to different tiremodels used in the simulation software cannot beoverlooked. Differences do exist. All other thingsbeing equal, the more accurate the tire model, thatis, the closer the tire model is to experimentallymeasured tire performance, the more accurate thesimulation. Of course in accident reconstructions,accurate representation of the vehicles’ physicalparameters also is a factor that influencesuncertainty.
In this paper, tire models and results ofsimulations for two cases that illustrate the wideranges of s and α typically found in accidentreconstruction applications are presented.Differences in results can be attributed to modeluncertainty. Differences between the simulationsusing the PC-Crash Linear Tire Model and the PC-Crash TM-Easy tire models are due only to the tiremodels. This is not true for comparisons betweendifferent simulation packages because othermodeling differences exist (such as differences insuspension system models). Additional simulationcomparisons need to be carried out beforeuncertainty due to tire models alone can discerned.
Tire Force Models: For combined braking andsteering of an individual wheel, the PC-Crash LinearTire Model is based on the process of firstspecifying the longitudinal (braking or accelerating)force, representing the lateral (steering) force witha bilinear curve and the use of the friction ellipse tocompute the resultant tire force. For combinedbraking and steering of an individual wheel, theSMAC Tire Force Model (both EDSMAC4 and m-smac) is based on the process of first specifying thelongitudinal (braking or accelerating) force, usingthe Fiala model for the lateral (steering) force andthe use of the friction ellipse to compute theresultant tire force for combined steering andbraking. The VCRware tire force model uses BNPequations with different parameters for thelongitudinal and lateral forces and then uses theNCB equations for combined steering and braking.PC-crash allows the use of the Linear Tire Model oran alternative called the TM-Easy Model. The TM-Easy Model is based on a resultant wheel slipvector for combined steering and braking. TheSIMON Tire Force Model is based on a modifiedHSRI Tire Model.
For the tire models covered in this paper twocategories can be established. One category usesa specified level of braking (or acceleration) toestablish the longitudinal tire force and the frictionellipse to calculate the combined longitudinal andlateral tire force components for combined steeringand braking (PC-Crash Linear and SMAC TireModels). The second category uses the direction ofthe wheel slip vector or slip velocity at the tire patchto determine the longitudinal and lateral tire forcecomponents for combined steering and braking(VCRware, PC-Crash TM-Easy and SIMON Tiremodels). Within each category, however, thesemodels use different forms of equations to modelthe lateral tire forces (for no braking).
Friction Ellipse: It was shown that for relatively lowslip angles, the use of the friction ellipse producesresultant forces equal in magnitude to a fully slidingtire. Some [Gäfvert & Svedenius] object to thisfeature. However, the use of the friction ellipse canactually under-predict combined tire forces. This isbecause the performance of models also dependson the functions used to represent the steering-alone and braking-alone curves, Fx(s) and Fy(α).Figures 3 and 4 show that experimentally measuredtire forces exceed the locked wheel skid force, μFz,over some (early) regions of slip. Figure 17 is a plotof normalized BNP-NCB combined tire forces(which reflect measured characteristics) plotted on
the friction ellipse coordinate system. The “frictionellipse” corresponding to the BNP-NCB tire forcesis the locus of points of the curves for all values ofα that lie a maximum radial distance from the origin(0,0). The friction ellipse for combined forces whoseFx(s) and Fy(α) tire force curves do not exceed μFzis given by the dashed curve in Fig 17. As seen, theidealized friction ellipse can result in combined tireforces well below measured values.
Simulation Comparisons: More comparisons ofthe type presented and comparisons toexperimental results are needed before any generalconclusions concerning the influence of tire modelson simulation accuracy can be drawn.
Different simulation models, with differenttire models but the same initial conditions, havebeen found to produce different results forconditions of combined steering and braking.However, it cannot be concluded that the observeddifferences are due to the tire models alone fromthe present work. More research is necessary todetermine the accuracy of the different tire modelsand different simulation software and for differentcategories of initial conditions and for differentconditions of steering input. When used forpurposes of accident reconstruction, differences insimulation results can be classified as modeluncertainty. Such uncertainty must be recognizedby accident reconstructionists.
ACKNOWLEDGMENTS
The authors appreciate the cooperation of MEAForensic Engineers and Scientists and for providinginformation and guidance with respect the PC-Crash Linear Tire model. The assistance of TerryDay of EngineeringDynamics Corporation is alsogratefully appreciated. Finally, Prof. Dr. Georg Rillprovided help and information with the formulationof the TM-Easy tire model.
Brach, Raymond and Matthew Brach, “TireModels used in Accident Reconstruction VehicleMotion Simulation”, XVII EuropäischenVereinigung für Unfallforschung und Unfallanalyse(EVU) - Conference, Nice, France, 2008.
Brach, Raymond and Matthew Brach, “Tire
Fx(α,s)/μxFz
0.0 0.2 0.4 0.6 0.8 1.0 1.2
F y(α
,s)/μ
yFz
0.0
0.2
0.4
0.6
0.8
1.0
1.2
α = 0.9o
α = 1.8o
α = 3.6o
α = 5.4o
α = 7.2oα = 9.0o
α = 13.5o
α = 18.0o
IdealizedFriction
Ellipse
Figure 17. Normalized BNP-BNC combined tire forces(solid curves) and the idealized friction ellipse (dashedcurve) for μx = μy. The actual friction ellipse is the locus ofpoints farthest from the origin that encompasses the tirecombined forces.
Forces: Modeling the Combined Braking andSteering Forces”, Paper 2000-01-0357, SAE,Warrendale, PA, 2000.
Brach, Raymond and Matthew Brach, VehicleAccident Analysis and Reconstruction Methods,SAE, Warrendale, PA, 2005.
Car-Sim, http://www.carsim.com/
Cliff, W. E., J. M. Lawrence, B. E. Heinrichs andT. R. Fricker, “Yaw Testing of an InstrumentedVehicle with and Without Braking”, Paper 2004-01-1187, SAE, Warrendale, PA, 2004.
Daily, J., N. Shigemura and J. Daily,Fundamentals of Traffic Crash Reconstruction,Volume 2 of the Traffic Crash ReconstructionSeries , IPTM, Jacksonville, FL, 2006.
EDC, Engineering Dynamics Corporation, SIMONSimulation Model, 5th Edition”, January 2006.
Fricke, L., (1) Traffic Accident Reconstruction,Northwestern University , Evanston, IL, 1990.
Fricke, L., (2) Traffic Accident Reconstruction,Volume 2, Traffic Accident Investigation Manual,Northwestern University , Evanston, IL, 1990.
Gäfvert, M. and J. Svedenius, “Construction ofNovel Semi-Empirical Tire Models for CombinedBraking and Cornering”, ISSN 0280-5316, LundInstitute of Technology, Sweden, 2003.
Guo, Konghui and Lei Ren, “A UnifiedSemi-Empirical Tire Model With Higher Accuracyand Less Parameters”, Paper 1999-01-0785, SAEInternational, Warrendale, PA, 1999.
Han, I. and S-U Park, “Inverse Analysis of Pre-and Post-Impact Dynamics for Vehicle AccidentReconstruction”, Vehicle System Dynamics, V 36,6, pp 413-433, 2001.
Hölscher, H., M. Tewes, N. Botkin, M. Lohndorf,K-H. Hoffman, and E. Quandt - Modeling ofPneumatic Tires by a Finite Element Model for theDevelopment of a Tire Friction Remote Sensor,preprint submitted to Computers and Structures.
Hirschberg, W., G. Rill and H.Weinfurter,"User-Appropriate Tyre-Modelling for VehicleDynamics in Standard and Limit Situations,"
Vehicle Systems Dynamics, Vol. 38, No. 2, pp103-125.
Keifer, O., B. Reckamp, T. Heilmann and P.Layson, “A Parametric Study of FrictionalResistance to Vehicular Rotation Resulting Froma Motor Vehicle Impact”, Paper 2005-01-1203,SAE, Warrendale, PA, 2005.
Keifer, O., R. Conte and B. Reckamp, Linear andRotational Motion Analysis in Traffic CrashReconstruction, IPTM, Jacksonville, FL, 2007.
MacAdam, C., P. S. Fancher, T. H. Garrick, T. D.Gillespie, “A Computerized Model for Simulatingthe Braking and Steering Dynamics of Trucks,Tractor-Semitrailers, Doubles and TriplesCombinations”, Highway Safety ResearchInstitute., The University of Michigan (UM-HSRI-80-58).
Martinez, J. E. and R. J. Schleuter, “A Primer onthe Reconstruction and Presentation of RolloverAccidents”, Paper 960647, SAE International,Warrendale, PA, 1996
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Milliken, W. F and D. L. Milliken, Race CarVehicle Dynamics, SAE, Warrendale, PA, 1995
m-smac, http://www.mchenrysoftware.com/
Orlowski, K. R., E. A. Moffatt, R. T. Bundorf andM. P. Holcomb, “Reconstruction of RolloverCollisions”, Paper 890857, SAE International,Warrendale, PA, 1987.
Pacejka, Hans, Tire and Vehicle Dynamics, SAE,Warendale, PA, 2002
Schuring, D. J., Pelz, W, Pottinger, M. G., "AnAutomated Implementation of the ’Magic Formula’Concept", SAE Paper 931909, Warrendale, PA15096, 1993.
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Table 1, Case ALocked Wheel Skid (SAE Coordinate System)
0 0 050 / (15.2 / ), 0, 150deg/x ft s m s y sθ= = = −
VCRwarex y θ d t
Rest 57.4 ft 2.4 ft -212E 57.4 ft 2.4 s
EDSMAC4x y θ d t
Rest 57.4 ft 2.3 ft -215E 57.4 ft 2.3 s
PC-Crash (Linear Tire Model)x y θ d t
Rest 57.0 ft 2.4 ft -211E 57.1 ft 2.3 s
Table 2, Case BLocked Right Front Wheel (SAE Coordinate System)
0 0 050 / (15.2 / ), 0, 150deg/x ft s m s y sθ= = = −
VCRware (EBNP = 0.5)x y θ d t
Rest 75.7 ft -1.1 ft -170E 75.7 ft 3.8 s
EDSMAC4x y θ d t
Rest 81.3 ft -1.4 ft -182E 82.1 ft 4.1 s
PC-Crash (Linear Tire Model)x y θ d t
Rest 77.6 ft 0.3 ft -173E 77.6 ft 4.0 s
Table 3, Case CRolling Resistance and Power Train Drag (SAE Coordinate System)
0 0 050 / (15.2 / ), 0, 150deg/x ft s m s y sθ= = = −
VCRware (EBNP = 0.5)x y θ d t
Rest 305 ft -91 ft -199E 318 ft 19.4 s: 85 -14 -199E 86 ft 2.5 s KE = 57373 J (42309 ft-lb)0θ =
EDSMAC4x y θ d t
Rest 242 ft -149 ft -220E 284 ft 18.5 s: 93 ft -22 ft -220E 96 ft 3.0 s KE = 53181 J (39226 ft-lb)0θ =
PC-Crash (Linear Tire Model)x y θ d t
Rest 298 ft -69 ft -195E 307 ft 19.2 s: 84 ft -11 ft -195E 85 ft 2.3 s KE = 59763 J (44079 ft-lb)0θ =
PC-Crash (TM-Easy Tire Model)x y θ d t
Rest 286 ft -51 ft -191E 291 ft 18.6 s: 75 ft -10 ft -191E 76 ft 2.1 s KE = 56765 J (41868 ft-lb)0θ =
Appendix A. Three-dimensional plots of Tire Forces of Different Models
Three-dimensional surface plots of the tire forces (for combined braking and steering) from the differenttire models are presented below.
Figures 18 through 25 are surface plots of the normalized tire forces for combined braking and steering forall of the models covered in this paper. Figures 18 and 19 are for the BNP-NCB tire model used byVCRware. Figure 20 shows the lateral force from PC-Crash Linear Tire Model for values for 0 # Fb/μFz #1 and for 0 # α # π/2. Figure 21 shows the normalized lateral force from SMAC for 0 # T/μxFz # 1 and for0 # α # π/2. The longitudinal forces for PC-Crash Linear and SMAC models are not plotted since brakingforces are specified directly as input to each program rather than being calculated as a function of wheelslip, s. Figures 22 and 23 are the longitudinal and lateral tire forces from the SIMON model, respectively.Finally, Fig 24 and 25 are plots of the TM-Easy tire forces.
wheel slip, s slip angle, α
F x(α
,s)/μ
F z
0.00.2
0.40.6
0.81.0 0.0
0.40.8
1.21.6
0.00.2
0.40.6
0.8
1.0
1.2
Figure 18. Normalized longitudinal tire force forcombined braking and steering, VCRware
slip angle, α wheel slip, s0.0
0.40.8
1.2
0.00.2
0.40.6
0.81.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
F y(α
,s)/μ
F z
Figure 19. Normalized lateral force for combinedbraking and steering, VCRware.
slip angle, αbraking force
0.00.4
0.81.2
1.6 0.00.2
0.40.6
0.81.0
0.00.2
0.4
0.6
0.8
1.0
1.2
F y(α
,s)/μ
F z
level, Fb/μFz
Figure 20. Normalized lateral tire force for combined braking and steering, PC-Crash linear Tire Model.
slip angle, α braking force0.0
0.40.8
1.21.6 0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.8
1.0
1.2
T / μFz
F y(α
,s)/μ
F z
Figure 21. Normalized lateral tire force forcombined braking and steering, SMAC
longitudinal slip, s slip angle, α
F x(α
,s)/μ
Fz
0.00.2
0.40.6
0.81.0
0.00.4
0.81.2
1.6
0.00.20.4
0.6
0.8
1.0
1.2
Figure 22. Normalized longitudinal tire force forcombined braking and steering, SIMON.
slip angle, α longitudinal slip, s
F y(α
,s)/μ
Fz
0.00.4
0.81.2
1.60.0
0.20.4
0.60.8
1.0
0.00.20.4
0.6
0.8
1.0
1.2
Figure 23. Normalized lateral tire force forcombined braking and steering, SIMON.
longitudinal slip, s slip angle, α
F x(α
,s)/μ
F z
0.00.20.40.60.81.00.0
0.40.8
1.21.6
0.00.20.40.60.81.0
1.2
Figure 24. Normalized longitudinal tire force forcombined braking and steering, TM-Easy.
slip angle, α longitudinal slip, s
F y(α
,s)/μ
F z
0.00.4
0.81.2
1.60.0
0.20.4
0.60.8
1.0
0.00.20.4
0.6
0.8
1.0
1.2
Figure 25. Normalized lateral tire force forcombined braking and steering, TM-Easy.
Appendix B: Specifications for Crown Victoria Spinout Example
