Tire Models for Vehicle Dynamic Simulation and Accident Reconstruction

2009-01-0102

Raymond M. Brach, University of Notre Dame, Brach Engineering, R. Matthew Brach, Brach Engineering

ABSTRACT simulation programs. Each model uses a different method for computing tire forces for combined Various vehicle dynamic simulation software braking and steering. Some experimentally programs have been developed for use in measured light vehicle tire properties are examined. reconstructing accidents. Typically these are used Some tire force models begin with a to analyze and reconstruct preimpact and specified level of braking force and use the friction postimpact vehicle motion. These simulation ellipse to determine the corresponding steering programs range from proprietary programs to force; this produces steering forces and a resultant commercially available packages. While the basic tire force equal in magnitude to full skidding for theory behind these simulations is Newton's laws of combined steering and braking. Comparisons are motion, some component modeling techniques presented of results from simulation programs using differ from one program to another. This is different tire models for vehicle motions involving particularly true of the modeling of tire force two types of severe yaw. The comparisons in this mechanics. Since tire forces control the vehicle paper are not of reconstructions where the user motion predicted by a simulation, the tire mechanics seeks initial conditions to match an existing model is a critical feature in simulation use, trajectory. The first comparison is a hypothetical performance and accuracy. This is particularly true postimpact motion with a given initial velocity and for accident reconstruction applications where initial angular velocity and the other is a sudden vehicle motions can occur over wide ranging steer maneuver. In some cases, the simulations kinematic wheel conditions. Therefore a thorough and their tire models predict the vehicle motion understanding of the nature of tire forces is a closely. In most cases, however, the results differ necessary aspect of the proper formulation and use significantly between simulation programs. of a vehicle dynamics program. The example simulations presented in this This paper includes a discussion of tire force paper are not intended to reflect the way vehicle terminology, tire force mechanics, the measurement dynamic simulation programs are used typically in and modeling of tire force components and accident reconstruction. combined tire force models currently used in simulation software for the reconstruction of INTRODUCTION: accidents. The paper discusses the difference between the idealized tire force ellipse and an Tire Models: Beside helping to provide a smooth actual tire friction ellipse. Equations are presented ride, the main function of an automotive pneumatic

for five tire force models from three different tire is to transmit forces (Fx, Fy, Fz) and moments in

The Engineering Meetings Board has approved this paper for publication. It has successfully completed SAE’s peer review process under the supervision of the session organizer. This process requires a minimum of three (3) reviews by industry experts. All rights reserved. No part of this publication may be reproduced, stored in a data retrieval system, or transmitted, in any form or by any means., electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of SAE. ISSN 0148-7191 Positions and opinions advanced in the paper are those of the author(s) and not necessarily those of SAE. The author is solely responsible for the content of the paper. SAE Customer Service: Tel: 877-606-7323 (inside USA and Canada) Tel: 724-776-4970 (outside the USA) Fax: 724-776-0790 Email: [email protected] SAE Web Address: http://www.sae.org Printed in USA three mutually perpendicular directions for vehicle given elsewhere [Pacejka, Allen, et al.]. Empirical directional control. This important role of tires has models employ mathematical functions capable of made tire behavior the subject of continuous study emulating the highly nonlinear behavior of the (and performance improvement) for nearly 80 forces generated by the tires. These mathematical years. functions can range from straight line segment Numerous tests have been conducted and approximations to nonlinear functions that contain mathematical models have been developed in an numerous coefficients based on experimental data attempt to understand and predict the generation of and determined by curve-fitting routines. The these forces. These models have been divided into principal use of these models is in the prediction of four different classifications [Pacejka]: 1) those that tire forces for vehicle dynamics simulation software. use a complex physical model, 2) those using a Many of these empirical models exist [Pacejka, simple physical model, 3) models using similarity Guo, Gäfvert, Hirschberg, Brach & Brach (2000), methods, and 4) models based solely on Pottinger, et al.]. This type of model is examined in experimental data, so-called empirical models. this paper. Physical models are those intended to model tire Tire forces are separated into a longitudinal performance (rather than vehicle performance). force component (braking and driving) and a lateral Physical models are concerned with such things as force component (steering/cornering). The tire wear, temperature, traction, life, cost, etc. They longitudinal tire force typically is mathematically have parameters such as construction, materials, expressed (modeled) and measured as a function loads, inflation pressure, geometry, tread design, of a variable called wheel slip. In some cases the speed, and so on. Complex physical models longitudinal force is modeled simply by a prescribed typically use finite element modeling techniques. force level, sometimes expressed as a fraction of Finite element models of the tires are of particular the normal force. The lateral tire force is use when considering the interaction between the mathematically expressed (modeled) and measured tire and road irregularities and for investigations into as a function of a variable called the slip angle. A the friction between the road and the tire within the third, distinct, feature of a tire force model is the footprint of the tire [Tonuk and Unlusoy, Hölscher, method of properly combining these two force et al.]. Models based on similarity methods were components for conditions of combined braking useful early in the tire force model development (wheel slip) and steering (slip angle). Other forces process but have found less use recently as they and moments exist at the tire-road interface that are have been superceded by the utility afforded by important for vehicle handling and design but are other models. Such methods are covered by not considered here. Effects such as self-aligning Pacejka [Pacejka]. torque, camber steer, conicity steer, ply steer, etc. The two remaining model classifications, the are usually neglected for accident reconstruction simple physical model and the empirical models, applications. are the two most prevalent models used in the Portions of this paper were presented orally understanding and prediction of tire forces. They at a conference [Brach & Brach, 2008]. relate the physical and kinematic properties of tires to the development of tractive forces at the contact Vehicle Dynamic Simulation: The use of vehicle between the tire and the roadway surface. One of dynamics models in the field of accident the most widely used simple physical models is the reconstruction to simulate vehicle motion has brush model. Brush models have been improved evolved steadily over the last few decades. Initially, and developed over the recent years [Gäfvert & the options of the reconstructionist were limited to Svedenius] but have not yet found their way into the vehicle dynamics capabilities of the variants of dynamic simulation programs applied to accident the government-funded SMAC & HVOSM reconstruction. A thorough coverage of the brush [McHenry, Segal] computer programs being the model is presented elsewhere [Pacejka]. most readily available options. Even today, The remaining tire model classification is the simulation software appears to be underutilized in empirical tire model. Such models are also referred the field as some reconstructionists continue to use to as semi-empirical tire models in many references simplified methods in attempts to address complex [Pacejka, Guo]. These models deal exclusively with motion of a vehicle based on assumptions of the steady-state behavior of a tire. Treatment of the constant deceleration [Fricke 1, Fricke 2, Orlowski, transient behavior of the tire, for example oscillatory Daily, et al., Martinez] and even concepts such as response, response lag and wheel unbalance, is point mass rotational friction [Keifer, et al. (2005) and Keifer, et al. (2007)]. Various simulation Only small differences were found for Categories 2 programs currently are available to the accident and 3. All three tire force models use the friction reconstructionist in the form of computer-based ellipse to compute combined tire forces. vehicle dynamics programs and are becoming an In all cases, the accuracy of the tire force is integral part of various accident reconstruction of considerable importance to the users of the software [PC-Crash, HVE, VCRware]. These simulation software. To a great extent, simulation vehicle dynamic programs were developed from accuracy depends on the ability of the tire model to within the accident reconstruction community and predict accurately the forces acting in the plane of are particularly suited to the needs of that field. the roadway generated by each of the vehicle’s Other, more complex vehicle dynamic software is tires. Other than aerodynamic forces, considered also available [VDANL, Car-Sim, ADAMS]. While later in the paper, it is the tire forces acting at the the latter software can be used in accident tire contact patches that control the motion of the reconstruction work, their complexity is better suited vehicle. as vehicle handling models. This paper focuses on the tire models used The basic premise behind all of the by three currently available simulation programs, variations of vehicle dynamics simulation programs PC-Crash, HVE and VCRware. These all have the is essentially the same: the user or the software capability to simulate motion in two dimensions. itself provides initial conditions (position, orientation, Some have more general capabilities such as three velocity) for the vehicle, the vehicle-specific dimensional motion but these features are not geometry, the vehicle physical parameters considered here. The tire models used by each of (including tire parameters), and any time-dependent these software programs is described in detail. This parameters (such as steering input, treatment is followed by two comparisons of braking/acceleration, etc.). The program integrates simulation results using each software package for the differential equations of motion of the vehicle the same set of tire parameters, vehicle parameters (and semitrailer) to predict the motion as a function and initial conditions. The paper concludes with a of time. The needs that the accident reconstruction discussion of the results of the simulations. The community has for a simulation program can differ topic of the tire friction ellipse is discussed. It is from other users of vehicle dynamics programs. shown that the idealized friction ellipse can differ Such needs include the ability to capture the significantly from a plot of the limit of tire forces dynamics of the vehicle through a wide range of developed by actual tires. motion and vehicle conditions such as damaged or altered wheelbase and/or track width, one or more NOTATION, ACRONYMS AND DEFINITIONS wheels that are locked, large initial yaw rates of rotation following an impact, etc. In contrast, vehicle • BNP: Bakker-Nyborg-Pajecka equations (also design and development work typically use vehicle known as the Magic Formula) [Pacejka] dynamics to study the performance of a vehicle in • Cornering stiffness: see Cα its as-designed condition and operation. • Cornering compliance: 1'Cα Comparisons have been made [Han and • EDSMAC4: simulation software [HVE], Park] between EDVAP [HVE], PC-Crash [PC-Crash] • frictional drag coefficient, μ: average, and a proprietary simulation program. These constant value of the coefficient of friction of a tire comparisons consisted of three categories of initial fully sliding over a surface under given conditions conditions that result in three different types of (wet, dry, asphalt, concrete, gravel, ice, etc.) postimpact motion. Category 1 uses initial appropriate to an application, conditions with a relatively high yaw velocity. The • friction circle: the friction ellipse when μx = μy, resulting vehicle motion showed that the yaw • friction ellipse: an idealized curve with velocity decreased to near zero and the vehicle coordinates consisting of the longitudinal and continued with a translational motion (rollout). lateral tire force components that defines the Category 2 uses initial conditions that resulted in a transition of a tire from wheel slip to the condition nonzero yaw velocity that was maintained until rest of full sliding, (spinout). Category 3 uses initial conditions that • lateral (side, cornering, steering): in the result in the vehicle experiencing a moderate yaw direction of the y axis of a tire’s coordinate velocity and translation. The results showed that the system, largest differences between EDVAP and PC-Crash • longitudinal (forward, rearward, braking, occurred for the initial conditions of Category 1. accelerating, driving): in the direction of the x axis of a tire’s coordinate system, perpendicular to the tire’s contact patch (see Fig • PC-Crash: simulation software [PC-Crash], 1), • SIMON: SImulation MOdel Nonlinear [HVE] • yaw: vehicle rotation about a vertical axis • sliding: the condition of a moving wheel and tire • α: tire slip angle (also, lateral slip angle),

locked from rotating (s = 1), or moving sideways • βp: angle of a tire’s slip velocity relative to the (α = π/2), tire’s x axis and angle of the resultant force • VCRware: simulation software [VCRware], parallel to the road plane (see Fig 2),

• Cα: lateral tire force coefficient (also cornering • β: angle relative to the x axis of the resultant tire coefficient), force_ (see Fig 2),

• Cs: longitudinal tire force coefficient, • β : nondimensional slip angle, Eq 45 & 50, • Fb: input value for the braking or acceleration SMAC, force, PC-Crash, • μx: tire-surface frictional drag coefficient for full • Fx(s): an equation with a single independent sliding in the longitudinal direction, s = 1, α = 0, variable, s, that models a longitudinal tire force for • μy: tire-surface frictional drag coefficient for full no steering, α = 0, sliding in the lateral direction, α = π/2.

• Fy(α): an equation with a single independent variable, α, that models a lateral force for no TIRE KINEMATICS braking, s = 0,

• Fx(α,s) = Fx[Fx(s),Fy(α),α,s]: an equation with Two kinematic variables typically are used with tire two independent variables, (α,s), that models a force models and with the measurement of tire longitudinal tire force component for combined forces. These are the slip angle, α, and the braking and steering, longitudinal wheel slip, s. The slip angle, is

• Fy(α,s) = Fy[Fx(s),Fy(α),α,s]: an equation of two illustrated in Fig 1 and is defined as independent variables, (α,s), that models a lateral α = tan−1 (VV / ) (1) tire force component for combined braking and yx steering, The wheel slip can have different • Fz: wheel normal force, • full sliding: a condition when the combined slip definitions [Brach & ω Brach (2000), variables (α,s) give a resultant tire force equal to Vy α Pacejka]. The one Vx μFz, see sliding, • HVOSM: Highway Vehicle Object Simulation used here is such R V Model that 0 # s # 1, V = V P V V - Rω • m-smac: simulation software [m-smac] where py y px= x y x • NCB: Nicolas-Comstock-Brach equations [Brach VRx − ω V & Brach 2000, 2005] s = z p • rollout: translational motion alone of a vehicle Vx Figure 1. Wheel/tire velocities that continues following spinout, (2) • s: longitudinal wheel slip, Figures 1 and 2 show the tire slip velocity • slip velocity: the velocity of the center of a tire components VPx = Vx - Rω and Vpy = Vy. Note that in at the contact patch relative to the ground, general the vector • slip angle: α, velocity, V, at the F Fy • SMAC: Simulation Model of Automobile wheel hub and the slip β velocity, V , at the Collisions [McHenry] p F V - Rω • spinout: motion of a vehicle that includes both contact patch center x x V β translation and yaw rotation, differ both in magnitude y p x and direction. The slip • T: an input value for the braking or acceleration y Vp force, SMAC, velocity, Vp, is the Figure 2. Tire patch velocity • wheel slip: see s, velocity of the point P and force components.

• Vx, Vy: components of the velocity of a wheel’s relative to the road hub expressed in the tire’s coordinate system, surface. Also, the direction of the resultant force, F,

• Vp: slip velocity of a tire at point P of the tire and the slip velocity, Vp, can differ. For no steering, patch. the longitudinal (braking, accelerating) tire force • x-y-z: orthogonal wheel coordinates where x is component, Fx(s), typically is expressed in the direction of the wheel’s heading and z is mathematically as a function of the wheel slip alone. Similarly, for no braking, the lateral (cornering, coefficient, Cα, (the slope of the initial linear portion steering) force component, Fy(α), typically is of the curves) depends on the normal force, Fz. A expressed mathematically as a function of the slip least square fit (using the BNP equations) angle alone. illustrating this dependence is shown in Fig 5. Figure 3 similarly shows that the slip stiffness 7000 1575 Quasi Static Braking-Driving 2 coefficient, Cs, depends on the normal force. (2007-01-0816) 6000 1350

Fz = 5407 N (1215 lb) FRICTION ELLIPSE, TIRE FORCE ELLIPSE , lb

5000 1125 x , N x F = 4131 N (929 lb) 4000 z 900 The x-y coordinate system and velocities of Cs ~ 12000 lb Fz = 3105 N (698 lb) a rotating wheel are illustrated in Fig 1. The tire 3000 675

Fz = 2068 N (465 lb) braking force, F force components Fx = Fx(α,s), Fy = Fy(α,s) and 2000 450 braking force, F resultant, F = F(α,s), are illustrated over a tire-road Fz = 1015 N (228 lb) 1000 225 contact patch in Fig 2. According to the Nicolas- Comstock theory [Brach & Brach (2000)], the force 0 0 0 20 40 60 80 100 components form a force ellipse where the abscissa longitudinal wheel slip, % is the longitudinal tire force component, Fx(α,s), and Figure 3. Experimentally measured longitudinal tire ordinate is the lateral tire force component, F (α,s). forces, P225/60R16 tire [Salaani]. y The equation of the tire force ellipse is given by Eq 3, or in a more concise form in Eq 4. The resultant EXPERIMENTALLY MEASURED TIRE FORCES 22 force is F(,)αααsFsFs=+xy (,) (,) . 20000 Experimental tire data are presented here because some of the simulation results given later α in the paper use tire parameters corresponding to 16000 measured values. The amount of data presented here is limited; more is given in a recent paper 12000 [Salaani] including a longitudinal tire force, Fx(s), as a function of wheel slip, s, and lateral tire force, sideslip coefficient, C Fy(α), as a function of slip angle α. Figure 3 shows 8000 Fx(s) for a P225/60R16 tire for different normal forces. Figure 4 shows measured values of Fy(α) for 4000 different normal forces. As indicated by the notation, 0 500 1000 1500 F (s) is measured for zero slip angle, α, and F (α) is x y normal force, Fz, N measured for zero wheel slip, s. These tire Figure 5. Measured variation (points) of Cα with Fz, properties are emulated later for use with a 2006 P225/60R16 tire [Salaani]. Ford Crown Victoria for which the P225/60R16 tire is standard. One of the conditions of the Nicolas-Comstock tire model is that the force components are aligned with 10000 2250 Quasi Static Steering the slip velocity components, that is β = βp (Fig 2). (SAE 2007-01-0816) Fz = 10240 N (2302 lb) Fz = 8193 N (1842 lb) As shown in Fig 6, the Fx(α,s) axis (abscissa) F = 6145 N (1381 lb) 5000 z 1125 represents braking alone (i.e., α = 0). The Fy(α,s) , N

Fz = 3110 N (699 lb) , lb y y axis (ordinate) represents steering alone (i.e., s = Fz = 2050 N (461 lb) 0). Each point of the friction ellipse’s interior is a 0 0 point with slip values (α,s) for combined steering and braking that represents driver control, side force, F -5000 -1125 side force, F expressed mathematically by Eq 5. A point Fx(s)|s=1 = μxFz on the abscissa represents locked wheel

-10000 -2250 skidding for braking alone. The point, Fy(α)|α= π/2 = -90 -60 -30 0 30 60 90 μ F , on the ordinate represents a vehicle tire sliding slip angle, α, deg y z laterally. Note that this formulation allows for Figure 4. Experimentally measured lateral tire forces, P225/60R16 tire [Salaani]. different frictional drag coefficients in the x and y directions, μx and μy, respectively. Full sliding of the From Fig 4 it can be seen that the slip tire under any combination of α and s occurs if the 22 FFsFxx⎡⎤⎡⎤( ), y (αα ), , s FFsF yx ( ), y ( αα ), , s ⎣⎦⎣⎦ (3) 22+=1 Fsxy() F (α ) 2 2 2 2 Fs(,)α Fsy (,)α Fs(,)α Fsy (,)α x +=1 x + <1 22 (4) 22 22 (5) Fsxy() F (α ) μμxyFFzz μxyμ μ = 22 2 2 (6) μxysinαμ+ cos α resultant tire force reaches the friction ellipse, F(α,s) SIMULATION TIRE MODELS

= μFz, where the frictional drag coefficient, μ is given by Eq 6 [Brach & Brach (2000)]. For a given normal Different tire force models exist and at least force, Fz, points outside the Friction Ellipse cannot one survey has been written [Gäfvert, M. and J. be reached because the friction force is limited by Svedenius], but the equations of most commonly

μFz. If μx = μy, then the tire force ellipse becomes a used models are not cataloged. The following is a circle and the friction ellipse becomes a friction collection of the equations of tire force models used circle. in three vehicle dynamics simulation software

Fy ( α ,s ) packages used for reconstructing accidents.

2 Friction Fy (π / ) = μyFz Ellipse VCRware Tire Model: The longitudinal and lateral tire force equations for this simulation software are modeled using a 1.2

subset of the (s) x BNP equations 1.0 0.8 Fy( α ) Tire Force [Pacejka]. Ellipse Equation 7 gives 0.6

the longitudinal 0.4 force, F (s), for F( α ,s ) x 0.2 Fy ( α ,s ) braking alone Normalized longitudinal force, F 0.0 β F ( s ) F( α ,s ) with no steering 0.0 0.2 0.4 0.6 0.8 1.0 x x Longitudinal wheel slip, s (α = 0). Figure 7 F( α ,s ) F ( 1 ) μ F Figure 7. BNP longitudinal force as a x x = x z shows anfunction of wheel slip, s, VCRware. example of a normalized plot of the longitudinal tire force with Figure 6. Diagrams of Friction (Limit) Ellipse and Tire Force Ellipse. example BNP parameter values of B = 1/15, C = 1.5, D = 1.0, E = 0.30, K = 100.0 and where the

Model equations that determine the initial slope is the braking coefficient Cs = BCDK. functions Fx(α,s) and Fy(α,s) for combined steering Equation 8 gives the lateral steering force, Fy(α), for and braking (such as shown in Fig 6 as a tire force no braking (s = 0). Figure 8 shows a sample ellipse) must be found independently from the normalized lateral force with BNP parameter values steering and braking functions Fy(α) and Fx(s). This of B = 8/75, C = 1.5, D = 1.0, E = 0.60, K = 100.0 is done later. It is important to note that the friction and the lateral stiffness coefficient is Cα = BCDK. ellipse is not a tire model. Rather, it is an idealized For a wheel with a braking force, Fx(s), and graphical display of the operating limit for resultant a lateral force, Fy(α), the longitudinal force for tire forces for any combination of steering and combined steering and braking, Fx(α,s), is braking. More than one method exists for determined in VCRware using the Nicolas- developing the resultant tire force for combined Comstock-Brach, (NCB) equations [Brach & Brach steering and braking. One is shown in the next (2000) and Brach & Brach (2005)]. It is given by Eq

Section; others are [Pottinger, et al. and Schuring, 9. For a wheel with a braking force, Fx(s), and a et al.] and [Hirschberg]. lateral force, Fy(α), the lateral force for combined steering and braking, Fy(α,s), is determined using the NCB equation and is given by Eq 10. F( s )=−+ D sin C tan−−11⎡⎤ B (1 E ) Ks E tan ( BKs ) (7) x { ⎣⎦⎢⎥} ⎧⎫−−11⎡⎤22αα FDCBEKEBKy (α )=−+ sin⎨⎬ tan⎢⎥ (1 ) tan ( ) (8) ⎩⎭⎣⎦ππ 22 2 2 2 FsFxy() (α ) s s Csax+−(1 ) cosα Fs ( ) Fsx(,)α = (9) 22 2 2 sC sFyx()αα+ F ()tan s α 222 22 FsF() (αα )tan (1−+sFC ) cosα (αα ) sin xy ys Fs(,)α = (10) y 22 2 2 C sinα sF()αα+ F ()tan s s yx F( α ,s) μFz When plotted on 1.2

axes of F (s) and z Region x F 1.0 I II III μ

s F (α), the NCB )/ 1.0 α

y e ( 0.8 Lateral Force y

equations take r 0.6 the form of a tire o

0.75 0.4 force ellipse that e

lateral force, F 0.2

depends on the T Cα functions Fx(s) 0.0 d 0.50

0.0 0.2 0.4 0.6 0.8 1.0 e

slip angle, 2α/π z and Fy(α). Three- i l 1 dimensional Figure 8. BNP lateral tire force as a a

function of normalized slip angle, m 0.25 surface plots of r Longitudinal Force these combined 2α/π, VCRware. o N α tire forces are illustrated in Appendix A. 0 αmax π_ π_ π_ 0 6 3 2 PC-Crash Linear Tire Force Model: PC-Crash allows the choice of either of two tire models, the Lateral Slip Angle,α Figure 10. Diagram of the longitudinal and lateral tire Linear Tire Force model and the TM-Easy Tire forces, PC-Crash Linear Tire Model. Force model. The Linear Tire model is as follows. Instead of Fy( α ) The lateral force becomes constant at α = using the wheel y αmax, where the lateral force reaches its maximum

, μFz 1 slip parameter, s, e value μFz. For the PC-Crash protocol, αmax = μα max,

r 1 the PC-Crash o where α max is the saturation angle for μy = 1. For

e simulation r this notation, the tire slip coefficient is computed as i C

α 1 requires an input l C = μF /α . For no longitudinal force, s = 0, (F a α z max a

e value of a t 1 = Fb = Fx = 0) the lateral tire force is defined by Eq

L constant α 11 and 12. For a wheel with braking force Fx(α,s) = 1 αmax = μα magnitude of 0 max π /2 Fb the lateral force is computed using the friction applied braking Lateral Slip Angle, α ellipse as given in Eq 13 where the longitudinal force with a force Figure 9. Lateral tire force, Linear force is adjusted for the condition of locked wheel Tire Model, PC-Crash. level, Fb, or an skidding as shown in Eq 14. For combined steering acceleration force magnitude, Fa. A force specified and braking, the PC- as a fraction of the wheel normal force can Crash Linear Tire Model can be described in three alternatively be supplied. For no steering the regions (see Fig 10). Region I is when the side longitudinal accelerating force, is specified as Fx = force increases linearly with α, Eq 15. Region II is Fa, and the longitudinal braking force is Fx = -Fb. when the side force is said to be saturated and the The PC-Crash vehicle dynamic simulation uses a lateral force is computed using the friction ellipse, bilinear lateral tire force as shown in Fig 9. The Eq 16 and Region III is for locked wheel sliding, as linear portion represents a slip coefficient of Cα. shown in Eq 17. 0 α α = μα1 : 1 (11) # # max max FFy ()αμαμα=− z / max

αmax < α < π/2: Fy ()αμ= Fz (12) ⎡⎤α 22 Fsyx(,)αμ=− min⎢⎥ Fzz ,( μ F ) F (,) α s (13) ⎣⎦αmax Fs(,)α = min⎡⎤ FF ,μα cos x ⎣⎦b z (14) α F (,)αμsFF=− ( )22 FsFy (,)αμ= z (15) y z b (16) αmax FsF(,)αμ= sinα F22(,)α sF+=μ F y z (17)y b z (18) These regions are shown in Fig 10 and are motion. However all of the following discussion is plotted on the friction ellipse in Fig 11. As the slip for zero camber and negligible contact moments. angle, α, increases from 0 to αmax, Fy(α,s) goes from According to notes on vehicle dynamics [Rill] TM- (0,0) to point A. The magnitude of the lateral force, Easy defines longitudinal slip and lateral slip

Fy(α,s), at point A is determined by Fb and Eq 17. different than above. Longitudinal slip, sx, is defined Note that in Region II, while the slip angle increases as in Eq 19. TM-Easy lateral slip is defined as in Eq from αmax to some value greater than αmax as shown 20. The consequences of normalizing slip to the in Fig 10, the resultant force at the patch does not wheel angular velocity is for TM-Easy that 0 # sx # change. Thus Region II, for which α varies from αmax 4, 0 # sy # 4 and (for combined steering and to some value greater than αmax, is concentrated at braking) that sx and sy are coupled to s (as defined a single point, B, on the tireforce diagram in Fig 11. by Eq 2) and α (Eq 1), as given in Eq 21 through 25.

In Region III Fy(α,s)goes from point B to point C (as The TM-easy model specifies that beyond a certain, α continues to increase) along the friction circle. finite value of slip sxf, full sliding occurs. The model From Eq 17 note that for Region II (point B), Eq 18 can characterize a maximum longitudinal force by holds. All of this implies that throughout Region II specifying maximum values of the force with its the PC-Crash Linear tire force model gives a lateral corresponding slip (sxm, Fxm). Figure 12 shows the force at the friction limit on the idealized friction limit longitudinal force Fx as a function of the longitudinal circle. Although the direction of Fy(α,s) is along the slip sx. A full description of the model requires that slip direction, the magnitude of the resultant tire three pieces of information be provided to define the force is equal to a fully skidding tire, μFz. A surface shape of the Fx(sx) curve: an initial slope, Cx, the plot of Fy(α,s) is given in Appendix A. maximum value of the force and its associated slip F ( α ,s ) y value (sxm, Fxm), and the value of the force at full μ F y z sliding and its associated slip value (sxf, Fxf). The III C curve for the lateral force, Fy(sy), can similarly be A B II defined using slope, Cy, maximum parameters (sym, Fym) and full-sliding parameters (syf, Fyf). Region I Fx ( s x )

μ F( α ,s ) F (s ) Fb xFz x x xm Full Sliding (0,0) Fx (s x f )

Braking Slope = Cx Friction Ellipse s s xm xf sx Figure 12. Longitudinal tire force, TM-Easy model. Figure 11. Diagram of lateral and longitudinal tire forces for combined steering and braking, PC-Crash. The process outlined above defines the shape of the curve for the longitudinal force in the

absence of lateral slip, Fx(sx), and the curve for the TM-Easy Tire Model [Hirschburg, et al.]: The TM- lateral force in the absence of longitudinal slip,

Easy model is defined for three dimensional vehicle Fy(sy). The force for combined braking and steering, V Vy px (19)s = (20) sx = y Rω Rω Vpx VRx − ω RRωωsx ss(,xy s )== (21)ss(,xy s )1=− = = (22) VVx x VRsRRpx+++ωωω x1 s x ⎛⎞vsyy s (,ss )tan−−11 tan( ) α xy==⎜⎟ (23)ssx(,α )= (24) ⎝⎠vRx ++ω 1 sx 1− s 2 2 ⎛⎞ tanα ⎛⎞sx sy ssy (,α )= (25)s =+⎜⎟ (26) 1− s xy ⎜⎟⎜⎟ ⎝⎠ssxy⎝⎠ sFCxm xm/ x sFCym ym/ y sx =+ (27)sy =+ (28) sxmymxmx++sFCFC// ymy sxmymxmx++sFCFC// ymy 2 2 2 2 ⎛⎞s ⎛⎞sym ⎛⎞ s =+xm cosϕϕ sin CCs=+()xycosϕϕ⎜⎟ Cs sin (29)m ⎜⎟⎜⎟ (30) xy⎝⎠ ⎝⎠ssmm⎝⎠ 22 ⎛⎞⎛⎞ss 2 2 fx fy (31)s =+⎜⎟⎜⎟cosϕϕ sin (32) FFmxmym=+()cosϕϕ() F sin f ⎜⎟⎜⎟ss ⎝⎠⎝⎠xy s / s 22 ss/ x y y cosϕϕ==x and sin FF=+cosϕϕ F sin (33)ss (34) fxfyf()() xyxy σ s C s Fs(),s ==≤≤m ,σ xy ,0 s s (35) x yxym⎛⎞s s ⎜⎟m m 12++σσ⎜⎟Ff − ⎜⎟F ⎝⎠m sxy− s m (36) Fs(, s )=− F ( F − F )σσσ2 (32), − = , s ≤≤ s s xy m mf ss− m xy f f m Fs(, s )=> F , s s (37) xyf xy f Fssx(,xy )= Fss (, xy )cosϕ (38)Fssyxy(, )= Fss (, xy )sinϕ (39)

F(sx,sy), is formulated by the TM-Easy model braking for the TM-Easy model are given in through the following process. A generalized slip Appendix A.

variable, sxy, which treats the longitudinal and lateral slip vectorially, is defined by Eq 26 where quantities SMAC Tire Model [HVE and m-smac]: For braking, SMAC does not use the wheel slip sx and sy are normalized slip variables and are variable, s, but the simulation user is asked to defined by Eq 27 and 28. Equations 29 through 33 specify the value of a constant braking force, T, define additional parameters. A generalized tire which also can be defined as a percentage of the force, F(s ,s ) is now described in each of the three x y available friction force at each wheel. The intervals by a broken rational function, a cubic longitudinal tire force, F , is given by Eq 40 through polynomial and a constant F and given in Eq 35, 36 x f 44 for the different variations of braking and and 37. Finally, the longitudinal and lateral force acceleration. components, Eq 38 and 39, are determined For braking: individually from the projections in the longitudinal T = 0 (s = 0), F (T) = 0 (40) and lateral directions, using n, given by Eq 34. x 0 < T # µ F , F (T) = -T (41) Three-dimensional surface plots of the longitudinal z x T > µF ,F(T) = -µ F (42) and lateral tire forces for combined steering and z x z For acceleration behind the HSRI tire model is that the tire forms a

|T| # µ Fz,Fx(T) = T (43) rectangular contact patch which can be divided into |T| > µ Fz,Fx(T) = µ Fz (44) two regions consisting of a no-slip region and a C α sliding region. The relative size of the two regions is ββα==() α (45) dependant upon the longitudinal and lateral slip μ22F z values, s and α, the sliding frictional drag ⎡⎤3 ββ β coefficient, μ, and the initial slopes, Cs and Cα, of Forβ < 3 , FFy ()αμ=−+z ⎢⎥ β (46) ⎢⎥327 the linear tire force curves. ⎣⎦ 1.2 For β ≥ 3 , FFy ()α = μ z (47) Cα/μFz = 20

z 1.0 For the lateral force, SMAC uses a F μ

)/ Cα/μFz = 15 α nondimensional variableβ , Eq 45, based on the ( 0.8 y Fiala tire model [EDSMAC, Brach & Brach (2005)] Cα/μF = 10 0.6 z and defines the lateral force Fy(α) by Eq 46 and 47. F (α) is plotted in Fig 13 for typical values of Cα/μFz = 5 y 0.4 Cα 'μFz. lateral force, F lateral force, For a wheel simultaneously steered (α > 0) 0.2 and braked (T > 0) the longitudinal tire force,

Fx(α,s), is computed by Eq 48 or 49, where the latter 0.0 0.0 0.4 0.8 1.2 1.6 case corresponds to locked wheel skidding. For slip angle, α combined braking and steering, the lateral tire force, Figure 13. Lateral tire force as a function of slip angle, α, Fy(α,s), is computed using the longitudinal force, β , SMAC. newly defined by Eq 50 and the friction ellipse. The first step in determining the SIMON tire Then forβ , Eq 51 or 52 give Fy(α,s). Equation 52 forces is to determine an equivalent frictional drag implies that forβ ≥ 3 the resultant tire force lies on coefficient, μN, that depends on the slip, s, and is the friction ellipse, as given by Eq 53 and that the calculated from the directional sliding frictional drag SMAC tire force model gives a lateral force at the coefficients, μx and μy. The coefficient μN is found friction limit for combined steering and braking using a fitting procedure whereby, (before locked wheel sliding occurs). Although the 2 as=−(1pp ) (1 + s ) (54) direction of the lateral force, Fy(α,s), is along the slip direction, the magnitude of the resultant tire force bs= (1−+−+pxp )(μμ ( s 2) pp (2 s 1) ) (55) equals that of a fully skidding tire. A three- c =−()μ μμ (56) dimensional surface plot of Fy(α,s) using Eq 51 x px through 53 is included in Appendix A. −+bb2 −4 ac For FTxx()≤=μαα Fz cos, F (,) s T (48) B = (57) 2a For FTxx()>=μ Fzz cos,ααμ F (,) s F cos α(49) (58) C α A = μx + B ββα==() α (50) 22 2 CBs=+−μ (1 ) (59) μαFFz − x (,) s x p and For β < 3 , μ ' = A − Bs (60) 22 2⎛⎞11 3 Fsyx(,)αμ=− FFsz (,) αββββ⎜⎟ −+ (51) In these equations, μp is the ratio of longitudinal tire ⎝⎠327 force Fx(s)max/Fz and sp is the slip at Fx = Fx(s)max. A β ≥ 3 For , variable Dt is defined as, 22 2 22 Fsyx(,)αμ=− FFsz (,) α (52) DCsCts=+()(sin)α α (61) 22 where s is the longitudinal tire slip and α is the slip FsFsFxy(,)α += (,)αμz (53) angle. After calculating μN, a fraction, Xs/L, representing the portion of the total contact patch SIMON Tire Model [HVE]: SIMON [EDC] uses a that is not slipping, where L is the total length of the semiempirical tire model which is based upon the rectangular tire patch, is defined as: HSRI tire model [McAdam, et al.]. The principle XXssμ 'Fz identical vehicle and tire input data and a frictional =−≤≤(1s ), 0 1 (62) drag coefficient of f = 0.75. All input data are listed LD2 t L in Appendix B. These examples are intended to The equations for combined steering and illustrate that uncertainty of simulations exists. Such braking/acceleration follow. The equations for uncertainty depends on differences in the individual steering alone and braking alone can be found by characteristics of each simulation program as well substituting s = 0 and α = 0 into the equations, as differences in the tire models. The simulation respectively. For combined braking and steering, software packages used are HVE, PC-Crash and X /L = 1: s VCRware. s FsCxs(,)α = (63) 1− s Cα sinα Fsy (,)α =− (64) 1− s A Three-dimensional surface plots of Fx(α,s) and Fx(α,s) are included in Appendix A. The sine functions in the range -π # α # π as used in the above equations for the SIMON model were changed from the tangent functions found in the original HSRI model. EDC is now investigating the full effects of this change. In addition, various empirical curves from measured tire parameters B arebuilt into the HVE software that make the tire characteristics tire specific and functions of load and speed. However, the user has the ability to enter other tire characteristics or to use setup tables based upon a specific tire tests. The SIMON tire model also considers the effects that camber stiffness has on the lateral tire forces. C

SIMULATION COMPARISONS Figure 14. Diagram of three cases A, B and C. Arrows indicate initial velocities. Gray tires indicate partial drag; black tires Comparison of Simulation Tire Force Models: indicate locked wheels. Tire forces for combined steering and braking can be compared visually using three-dimensional force First Example (Crown Victoria) The same vehicle plots. Plots are given for all of the different models and tire properties are used to compute the output in Appendix A. of the different simulations for a postimpact maneuver with specified initial conditions. The Computer Vehicle Dynamic Simulation: Two vehicle corresponds to a 2006 Ford Crown Victoria. examples are presented for comparison of the A major reason this vehicle is chosen is because it simulations and tire models. The first is a uses P225/60R16 tires with known, measured hypothetical, postimpact trajectory of a 2006 Ford lateral steering properties [Salaani] presented Crown Victoria. This example is examined for three earlier. The specifications of the vehicle are different sets of wheel conditions: A, locked wheels, contained in Appendix B. B, partial drag on each wheel with a single locked Vehicle trajectories are computed for an front wheel and C, partial drag on each wheel. initial forward speed of 34.1 mph (55 km/hr), an Results of the different simulations and tire models initial lateral speed of zero and an initial yaw are compared on a relative basis. angular velocity of 150 /s. Each trajectory is The second example is for a sudden steer E computed for three conditions of braking. First, the maneuver of a partially braked vehicle based on a output of the simulations is compared for a case test [Cliff, et al.]. Relative comparisons between the which is independent of the tire force models, that different simulation results are made. The example of locked wheel skidding, indicated as A in Fig 14. is intended to reflect a relatively rapid severe steer Then comparisons are made for the same initial with partial braking. All of the simulations use conditions for equal powertrain drag on each rear Xs /L < 1: 2 ⎛⎞ ⎛⎞μ ' Fsz ⎛⎞Xs FsCsxs(,)αμ=−+−⎜⎟ (1 s ) ' Fz 1 ⎜⎟ (65) ⎜⎟222DL⎜⎟⎜⎟ ⎝⎠t ⎝⎠⎝⎠s + sin α 2 ⎛⎞ ⎛⎞μα'sinFz ⎛⎞Xs FsCy (,)αα=− sin⎜⎟ (1 − s ) − μ ' Fz 1 − ⎜⎟ (66) α ⎜⎟222DL⎜⎟⎜⎟ ⎝⎠t ⎝⎠⎝⎠s + sin α wheel (10% of the static normal force), rolling drag of spinout differ; in particular, the angular positions on the left front wheel (0.7% of the static normal are quite different. This leads to large differences in force) and a locked right front wheel, B in Fig 14. the rest positions. For reference, the locked wheel The third case is for equal powertrain drag on each skid trajectories from the same initial conditions are rear wheels (10% of the static normal force) and shown in the same figure (note that the different equal tire rolling drag on each front wheel (0.7% of rest positions are so close that only one is shown). the static normal force), C in Fig 14. The results are Note that a sensitivity analysis to changes in as follows. initial conditions was not carried out.

A. Postimpact Motion, Locked Wheel Skidding Second Example (Honda Accord): These Table 1 lists the results of the locked wheel skid simulations use a 1991 Honda Accord with an simulations. All three software packages and all three tire models give reasonably close rest initial speed of 100 km/hr (91.13 ft/s). The driver positions, orientations and times to rest. makes a sudden, constant front wheel steer maneuver to the right of approximately 9E following B. Postimpact Motion, No Applied Braking, Power brake activation that causes a constant, equivalent, Train Drag and One Locked Front Wheel For the longitudinal deceleration of 0.273 ± 0.003 g’s. The conditions of 0.7% rolling wheel drag on the left vehicle then moves to rest. Details of the input front wheel, 10% powertrain drag on both rear vehicle and tire data are given in Appendix C. wheels and the right front wheel locked, the Since the initial vehicle speed is relatively agreement between all tire models is good, but not high, simulations were run with and without as close as the locked wheel condition. Table 2 lists aerodynamic drag where possible and, for the CG rest positions, orientations and travel times. comparison, ignoring aerodynamic drag. The

Initial motion is in the x direction and lateral travel is aerodynamic drag force, RA, in VCRware is small. VCRware and EDSMAC4 give a negative calculated using the well known equation [Hoerner] lateral travel, while PC-Crash gives a small positive 1 R = ρCAV2 (67) travel. The times to reach the rest positions are Ad2 close but not the same. The drag force depends on the density of air, ρ, a

dimensionless drag coefficient, Cd, a projected area C. Postimpact Motion, No Applied Braking with A, and a velocity relative to the wind, V. In all cases Power Train Drag and Tire Rolling Resistance treated here a wind speed of zero is used. The Results are contained in Table 3 for the same aerodynamic drag is a resultant force calculated conditions as the previous case, except with rolling using frontal and lateral components. A frontal drag drag on both front wheels (no locked wheel) and for coefficient for all simulations had a value of CdF = an additional tire model. Large differences in the 2 2 0.4 with a frontal area of AF = 25 ft (2.3 m ). The rest positions, orientations and travel times occur. corresponding lateral or side values are CdL = 0.8 The motion in this case can be divided into two 2 2 and AL = 60 ft (5.6 m ). For no aerodynamic drag components. The first is a combination of CdF = CdL = 0. In some cases, an aerodynamic translation and yaw rotation (spinout). At a point in moment (usually small) is developed since the side the travel to rest, the yaw velocity goes to zero force is not aligned with the vehicle center of (θ = 0 ); the motion that follows consists of gravity. When included, a moment arm of 0.76 ft to translation alone, or rollout, to a rest position. This the rear of the CG was used. is illustrated in Fig 15 for simulations using The front and rear tire side force EDSMAC4, VCRware and PC-Crash (two tire coefficients, Cαf and Cαr, are included as input models). The positions and orientations at the end parameters in all simulations. Stock tire size on a 0 10 20 Feet 40 EDSMAC4

0 36meters 12

REST REST POSITION POSITIONS

x VCRware y Locked Wheel Skid All Simulations PC-Crash ROLLOUT EDSMAC4

INITIAL POSITION VCRware

PC-Crash x y SPINOUT Figure 15. Diagram of results of a locked wheel skid (Case A) and rolling resistance on the front wheels and powertrain drag on rear wheels (Case C), Crown Victoria.

Critical Speed Beginning Circular Arc of Steer R = 344.2 ' v = 91.13 ft/s f = 0.75 R = 104.9 m

6 7

aerodynamic drag REST 2 3 w w/o 1 EDSMAC4 1 0 10 20Feet 40 PC-Crash Linear 23 POSITIONS PC-Crash TM-Easy 45 SIMON 6 7 0 36meters 12 4 VCRware 8 9 5

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.

1991 Honda is listed as 195-60R15. It is important values for the actual test vehicle. The values for this that these coefficients be reasonably accurate, yet example were established in the following way. tire parameter information from the open literature Engineering Dynamics Corporation [HVE] is sparse. In addition, tire properties for a given lists a value for this tire as Cα = 231.7 lb/deg sized tire can vary from manufacturer to =13275 lb/rad for a vertical load of 1230 lb. Based manufacturer. The tire parameters found and used on this, a value of Cαf = 13000 lb/rad is used for all here represent a reasonable set of values for this simulations for the static normal force at the test tire size but do not necessarily represent the exact vehicle front wheels, Wf = 932 lb. Since tire side force coefficients vary with normal force and the such as partial braking, powertrain drag, rolling static normal force for the rear of the test vehicle is wheel drag and/or the effects of an individually

approximately 660 lb, a value of Cαr must be locked wheel or wheels. It is necessary to use a estimated. An approximate formula can be vehicle dynamic simulation program for modeling of developed (for small changes in normal force) from such conditions. Despite the greater potential for an equation in a paper on tires [Salaani], as accuracy, the uncertainty due to different tire models used in the simulation software cannot be CkFα ≈ z (68) overlooked. Differences do exist. All other things This gives being equal, the more accurate the tire model, that Fzr CCαrf≈ α (69) is, the closer the tire model is to experimentally Fzf measured tire performance, the more accurate the

giving a value of Cαr = 9200 lb/rad. This combination simulation. Of course in accident reconstructions, of values of Cαf and Cαr would place the 1991 Honda accurate representation of the vehicles’ physical into a neutral steer condition (which is not the case). parameters also is a factor that influences

A second approach to estimate Cαr was taken using uncertainty. the front and rear Bundorf compliances [Milliken] for In this paper, tire models and results of a passenger car. This gives simulations for two cases that illustrate the wide ranges of s and α typically found in accident WCfrα =1.1 (70) reconstruction applications are presented. WCrfα Differences in results can be attributed to model which, in turn gives C = 10137 lb/rad. Based on uncertainty. Differences between the simulations αr using the PC-Crash Linear Tire Model and the PC- these estimates, a value of Cαr = 10000 lb/rad was chosen for the static rear tire side force coefficients Crash TM-Easy tire models are due only to the tire and used in all simulations. These values provide a models. This is not true for comparisons between static positive understeer gradient. different simulation packages because other Figure 16 shows the rest positions and modeling differences exist (such as differences in orientations from all of the simulations. suspension system models). Additional simulation comparisons need to be carried out before uncertainty due to tire models alone can discerned. DISCUSSION AND CONCLUSIONS Tire Force Models: For combined braking and The primary purpose of this paper is to steering of an individual wheel, the PC-Crash Linear demonstrate that different tire models exist, to Tire Model is based on the process of first describe them in as much detail as possible and to specifying the longitudinal (braking or accelerating) indicate which simulation programs (used in force, representing the lateral (steering) force with accident reconstruction applications) use which tire a bilinear curve and the use of the friction ellipse to models. Two example applications of these compute the resultant tire force. For combined simulation programs and tire models are presented. braking and steering of an individual wheel, the The example applications were limited to a SMAC Tire Force Model (both EDSMAC4 and m- hypothetical postimpact motion of a Ford Crown smac) is based on the process of first specifying the Victoria and to a sudden steer maneuver of a longitudinal (braking or accelerating) force, using Honda Accord. Results within the different the Fiala model for the lateral (steering) force and simulations for each example are compared. Since the use of the friction ellipse to compute the the applications are limited to only two, the resultant tire force for combined steering and conclusions that can be drawn likewise are limited. braking. The VCRware tire force model uses BNP equations with different parameters for the Alternative methods exist [Kiefer, et al., longitudinal and lateral forces and then uses the 2005, 2007] to estimate the combined effects of NCB equations for combined steering and braking. initial translational and rotational velocities on the PC-crash allows the use of the Linear Tire Model or trajectory of a vehicle to rest following impact that an alternative called the TM-Easy Model. The TM- do not use tire force models. Such methods do not Easy Model is based on a resultant wheel slip have the potential of simulating different tire vector for combined steering and braking. The properties and accident reconstruction conditions SIMON Tire Force Model is based on a modified HSRI Tire Model. For the tire models covered in this paper two the friction ellipse coordinate system. The “friction categories can be established. One category uses ellipse” corresponding to the BNP-NCB tire forces a specified level of braking (or acceleration) to is the locus of points of the curves for all values of establish the longitudinal tire force and the friction α that lie a maximum radial distance from the origin ellipse to calculate the combined longitudinal and (0,0). The friction ellipse for combined forces whose lateral tire force components for combined steering Fx(s) and Fy(α) tire force curves do not exceed μFz and braking (PC-Crash Linear and SMAC Tire is given by the dashed curve in Fig 17. As seen, the Models). The second category uses the direction of idealized friction ellipse can result in combined tire the wheel slip vector or slip velocity at the tire patch forces well below measured values. to determine the longitudinal and lateral tire force components for combined steering and braking Simulation Comparisons: More comparisons of (VCRware, PC-Crash TM-Easy and SIMON Tire the type presented and comparisons to models). Within each category, however, these experimental results are needed before any general models use different forms of equations to model conclusions concerning the influence of tire models the lateral tire forces (for no braking). on simulation accuracy can be drawn. 1.2 Different simulation models, with different o o α = 9.0 α = 13.5 tire models but the same initial conditions, have o α = 7.2 been found to produce different results for 1.0 o α = 18.0 conditions of combined steering and braking.

o However, it cannot be concluded that the observed α = 5.4 z 0.8 differences are due to the tire models alone from F

y the present work. More research is necessary to μ o determine the accuracy of the different tire models ,s)/ 0.6 α = 3.6 α and different simulation software and for different ( Idealized

y Friction F Ellipse categories of initial conditions and for different 0.4 conditions of steering input. When used for o α = 1.8 purposes of accident reconstruction, differences in simulation results can be classified as model 0.2 o uncertainty. Such uncertainty must be recognized α = 0.9 by accident reconstructionists. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ACKNOWLEDGMENTS F (α,s)/μ F x x z The authors appreciate the cooperation of MEA Figure 17. Normalized BNP-BNC combined tire forces Forensic Engineers and Scientists and for providing (solid curves) and the idealized friction ellipse (dashed information and guidance with respect the PC- curve) for μx = μy. The actual friction ellipse is the locus of points farthest from the origin that encompasses the tire Crash Linear Tire model. The assistance of Terry combined forces. Day of EngineeringDynamics Corporation is also gratefully appreciated. Finally, Prof. Dr. Georg Rill provided help and information with the formulation Friction Ellipse: It was shown that for relatively low of the TM-Easy tire model. slip angles, the use of the friction ellipse produces resultant forces equal in magnitude to a fully sliding REFERENCES tire. Some [Gäfvert & Svedenius] object to this feature. However, the use of the friction ellipse can ADAMS, actually under-predict combined tire forces. This is http://www.mscsoftware.com/products/adams.cfm because the performance of models also depends on the functions used to represent the steering- Brach, Raymond and Matthew Brach, “Tire alone and braking-alone curves, Fx(s) and Fy(α). Models used in Accident Reconstruction Vehicle Figures 3 and 4 show that experimentally measured Motion Simulation”, XVII Europäischen tire forces exceed the locked wheel skid force, μFz, Vereinigung für Unfallforschung und Unfallanalyse over some (early) regions of slip. Figure 17 is a plot (EVU) - Conference, Nice, France, 2008. of normalized BNP-NCB combined tire forces (which reflect measured characteristics) plotted on Brach, Raymond and Matthew Brach, “Tire Forces: Modeling the Combined Braking and Vehicle Systems Dynamics, Vol. 38, No. 2, pp Steering Forces”, Paper 2000-01-0357, SAE, 103-125. Warrendale, PA, 2000. Hoerner, S. F., Fluid-Dynamic Drag, Hoerner Brach, Raymond and Matthew Brach, Vehicle Fluid Dynamics, Brick Town, NJ, 1965. Accident Analysis and Reconstruction Methods, SAE, Warrendale, PA, 2005. HVE, http://www.edccorp.com/products/hve.html

Car-Sim, http://www.carsim.com/ Keifer, O., B. Reckamp, T. Heilmann and P. Layson, “A Parametric Study of Frictional Cliff, W. E., J. M. Lawrence, B. E. Heinrichs and Resistance to Vehicular Rotation Resulting From T. R. Fricker, “Yaw Testing of an Instrumented a Motor Vehicle Impact”, Paper 2005-01-1203, Vehicle with and Without Braking”, Paper 2004- SAE, Warrendale, PA, 2005. 01-1187, SAE, Warrendale, PA, 2004. Keifer, O., R. Conte and B. Reckamp, Linear and Daily, J., N. Shigemura and J. Daily, Rotational Motion Analysis in Traffic Crash Fundamentals of Traffic Crash Reconstruction, Reconstruction, IPTM, Jacksonville, FL, 2007. Volume 2 of the Traffic Crash Reconstruction Series , IPTM, Jacksonville, FL, 2006. MacAdam, C., P. S. Fancher, T. H. Garrick, T. D. Gillespie, “A Computerized Model for Simulating EDC, Engineering Dynamics Corporation, SIMON the Braking and Steering Dynamics of Trucks, Simulation Model, 5th Edition”, January 2006. Tractor-Semitrailers, Doubles and Triples Combinations”, Highway Safety Research Fricke, L., (1) Traffic Accident Reconstruction, Institute., The University of Michigan (UM-HSRI- Northwestern University , Evanston, IL, 1990. 80-58).

Fricke, L., (2) Traffic Accident Reconstruction, Martinez, J. E. and R. J. Schleuter, “A Primer on Volume 2, Traffic Accident Investigation Manual, the Reconstruction and Presentation of Rollover Northwestern University , Evanston, IL, 1990. Accidents”, Paper 960647, SAE International, Warrendale, PA, 1996 Gäfvert, M. and J. Svedenius, “Construction of Novel Semi-Empirical Tire Models for Combined McHenry, R., “Computer Program for Braking and Cornering”, ISSN 0280-5316, Lund Reconstruction of Highway Accidents”, Paper Institute of Technology, Sweden, 2003. 730980, SAE Warrendale, PA, 1973

Guo, Konghui and Lei Ren, “A Unified Milliken, W. F and D. L. Milliken, Race Car Semi-Empirical Tire Model With Higher Accuracy Vehicle Dynamics, SAE, Warrendale, PA, 1995 and Less Parameters”, Paper 1999-01-0785, SAE International, Warrendale, PA, 1999. m-smac, http://www.mchenrysoftware.com/

Han, I. and S-U Park, “Inverse Analysis of Pre- Orlowski, K. R., E. A. Moffatt, R. T. Bundorf and and Post-Impact Dynamics for Vehicle Accident M. P. Holcomb, “Reconstruction of Rollover Reconstruction”, Vehicle System Dynamics, V 36, Collisions”, Paper 890857, SAE International, 6, pp 413-433, 2001. Warrendale, PA, 1987.

Hölscher, H., M. Tewes, N. Botkin, M. Lohndorf, Pacejka, Hans, Tire and Vehicle Dynamics, SAE, K-H. Hoffman, and E. Quandt - Modeling of Warendale, PA, 2002 Pneumatic Tires by a Finite Element Model for the Development of a Tire Friction Remote Sensor, PC-Crash, preprint submitted to Computers and Structures. http://www.meaforensic.com/technical/pc_crash.ht ml Hirschberg, W., G. Rill and H.Weinfurter, "User-Appropriate Tyre-Modelling for Vehicle Pottinger, M. G., Pelz, W., and Falciola, G., Dynamics in Standard and Limit Situations," "Effectiveness of the Slip Circle, "Combinator", Model for Combined Tire Cornering and Braking Forces When Applied to a Range of Tires", SAE Paper 982747, Warrendale, PA 15096.

Rill, G, Vehicle Dynamics Lecture Notes, University of Applied Sciences, Hochschule für Technik Wirtschaft Soziales, Germany, 2007.

Salaani, K., “Analytical Tire Forces and Moments Model with Validated Data”, Paper 2007-01-0816, SAE International, Warrendale, PA, 2007.

Segal, J. Highway Vehicle Object Simulation Model, 4 Volumes (Users Manual, Programmers Manual, Engineering Manual-analysis, and Engineering Manual), 1422 pgs, Calspan Corporation, 1976.

Schuring, D. J., Pelz, W, Pottinger, M. G., "An Automated Implementation of the ’Magic Formula’ Concept", SAE Paper 931909, Warrendale, PA 15096, 1993.

Tönük, E. and Y. S. Ünlüsoy, “Prediction of automobile tire cornering force characteristics by finite element modeling and analysis”, Computers and Structures, 79 (2001), pp1219-1232.

VCRware, http://www.brachengineering.com/menu.swf

VDANL, http://www.systemstech.com/content/view/32/39/

CONTACT

Raymond M. Brach [email protected]

R. Matthew Brach [email protected] Table 1, Case A Locked Wheel Skid (SAE Coordinate System) x000===−50ft / s (15.2 m / s ), y 0,θ 150deg/ s

VCRware xyθ dt Rest 57.4 ft 2.4 ft -212E 57.4 ft 2.4 s

EDSMAC4 xyθ dt Rest 57.4 ft 2.3 ft -215E 57.4 ft 2.3 s

PC-Crash (Linear Tire Model) xyθ dt Rest 57.0 ft 2.4 ft -211E 57.1 ft 2.3 s

Table 2, Case B Locked Right Front Wheel (SAE Coordinate System) x000===−50ft / s (15.2 m / s ), y 0,θ 150deg/ s

VCRware (EBNP = 0.5) xyθ dt Rest 75.7 ft -1.1 ft -170E 75.7 ft 3.8 s

EDSMAC4 xyθ dt Rest 81.3 ft -1.4 ft -182E 82.1 ft 4.1 s

PC-Crash (Linear Tire Model) xyθ dt Rest 77.6 ft 0.3 ft -173E 77.6 ft 4.0 s Table 3, Case C Rolling Resistance and Power Train Drag (SAE Coordinate System) x000===−50ft / s (15.2 m / s ), y 0,θ 150deg/ s

VCRware (EBNP = 0.5) xyθ dt Rest 305 ft -91 ft -199E 318 ft 19.4 s θ = 0 : 85 -14 -199E 86 ft 2.5 s KE = 57373 J (42309 ft-lb)

EDSMAC4 xyθ dt Rest 242 ft -149 ft -220E 284 ft 18.5 s θ = 0 : 93 ft -22 ft -220E 96 ft 3.0 s KE = 53181 J (39226 ft-lb)

PC-Crash (Linear Tire Model) xyθ dt Rest 298 ft -69 ft -195E 307 ft 19.2 s θ = 0 : 84 ft -11 ft -195E 85 ft 2.3 s KE = 59763 J (44079 ft-lb)

PC-Crash (TM-Easy Tire Model) xyθ dt Rest 286 ft -51 ft -191E 291 ft 18.6 s θ = 0 : 75 ft -10 ft -191E 76 ft 2.1 s KE = 56765 J (41868 ft-lb) 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 different tire models are presented below.

Figures 18 through 25 are surface plots of the normalized tire forces for combined braking and steering for all of the models covered in this paper. Figures 18 and 19 are for the BNP-NCB tire model used by

VCRware. 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 for 0 # α # π/2. The longitudinal forces for PC-Crash Linear and SMAC models are not plotted since braking forces are specified directly as input to each program rather than being calculated as a function of wheel slip, 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. 1.2 1.2 1.0 0.8 1.0 0.6 0.8 z F z

μ 0.4 0.6 F μ

,s)/ 0.2 0.4 α ,s)/ ( x 0.0 α 0.2 ( F

0.0 y 1.0 F 0.0 0.8 0.4 0.0 0.6 0.8 0.2 0.4 1.2 0.4 0.2 1.2 0.8 0.6 wheel slip, s slip angle, α 0.4 0.8 0.0 1.6 slip angle, α wheel slip, s 0.0 1.0 Figure 18. Normalized longitudinal tire force for combined braking and steering, VCRware Figure 19. Normalized lateral force for combined braking and steering, VCRware.

1.2 1.0 1.2 0.8 1.0 z

F 0.6 0.8 μ

0.4 z 0.6 F ,s)/ μ

α 0.2 0.4 ( y ,s)/ 0.2 F 0.0 α

0.0 ( 1.6 y 0.0

0.2 F 0.0 1.2 0.4 1.6 0.2 0.8 1.2 0.6 braking force 0.4 0.4 0.8 0.8 0.6 slip angle, α 0.4 braking force 0.0 1.0 level, Fb/μFz slip angle, α 0.8 0.0 1.0 T / μFz Figure 20. Normalized lateral tire force for combined braking and steering, PC-Crash linear Tire Model. Figure 21. Normalized lateral tire force for combined braking and steering, SMAC

1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 Fz Fz μ

μ 0.4

0.4 ,s)/ 0.2 α ,s)/ ( α 0.2 y

( 0.0 F

x 0.0

F 0.0 0.0 1.6 0.2 1.0 1.2 0.4 0.8 0.4 0.8 0.6 0.6 0.8 0.4 0.8 0.4 1.2 slip angle, α 0.0 1.0 longitudinal slip, s longitudinal slip, s 0.2 slip angle, α 0.0 1.6 Figure 23. Normalized lateral tire force for Figure 22. Normalized longitudinal tire force for combined braking and steering, SIMON. combined braking and steering, SIMON.

1.2 1.2 1.0 1.0 0.8

0.8 z F 0.6 μ

z 0.6

F 0.4 ,s)/ μ

0.4 α ( 0.2 y ,s)/

0.2 F α

( 0.0

x 0.0 0.0 0.0 F 0.4 1.6 0.2 1.0 0.8 0.8 1.2 0.4 0.6 1.2 0.8 0.6 0.4 0.4 0.8 0.2 slip angle, α slip angle, α longitudinal slip, s 0.0 1.6 0.0 1.0 longitudinal slip, s Figure 24. Normalized longitudinal tire force for Figure 25. Normalized lateral tire force for combined braking and steering, TM-Easy. combined braking and steering, TM-Easy. Appendix B: Specifications for Crown Victoria Spinout Example

Tire Coefficients

front: Cαf = 16000 lb/rad = 279.25 lb/deg = 71171.6 N/rad = 1242.18 N/deg rear: Cαr = 14000 lb/rad = 244.35 lb/deg = 62275.1 N/rad = 1086.91 N/deg Braking Coefficient

Cs = 10000 lb = 44482.2 N Rear Wheel Drag, 0.100 μFz Front Wheel Drag, 0.007 μFz Tire-road Frictional Drag Coefficient f = 0.7

Fz: rear wheel, 892.5 lb = 3970.2 N = 404.9 kg Length 212 in., 5.38 m Wheelbase 115 in., 2.92 m Curb Weight 4057 lb, 18.05 kN Curb Weight Distribution 56% /44% Front Track 63 in., 1.60 m Rear Track 66 in.1.68 m Drive Wheels Rear Tire Size P225/60R16 Center of Gravity Ht 22.37 in., 0.57 m Yaw radius of gyration k = 4.86 ft = 1.48 m All other vehicle parameters, if any, are given by the software default parameters.

1200 1200 1200

Cα α 1000 1000 1000 ), lb ), lb α ), lb C α ( α α α y ( ( y 800 y 800 800

600 600 600 Cα α 400

lateral force, F 400 400 lateral force, F

200 200 laterl tire force, F 200

0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 sideslip angle, 2α/π sideslip angle, 2α/π sideslip angle, 2α/π Figure 26, VCRware lateral tire Figure 27. EDSMAC4 lateral tire Figure 28. PC-Crash lateral tire force, BNP: C = 1.5, E = 0.5 force force

Appendix C: Specifications for Sudden Steer Maneuver

1991 Honda Accord EX Vehicle weight, W = 3186 lb, Distribution 61%/39% Yaw Radius of Gyration, k = 4.49 ft, 1.37 m Length 185 in., 4.70 m Wheelbase 107 in., 2.72 m Front Track 58 in., 1.47 m Rear Track 58 in., 1.47 m Tire Size 195-60R15 Center of Gravity Ht 21.2 in, 0.54 m

Tire Side Force Coefficients: CαF = 13000 lb/rad, CαR = 10000 lb/rad Front Wheel Braking Force: 312.3 lb/wheel Rear Wheel Braking Force: 122.9 lb/wheel Initial Conditions: x, y, θ = 0,0,0, x,,y θ = 91.134, 0, 0 ft/s Front Wheel Steer Angle, δ: linear rise from 0E to 9E in ½ sec, constant at 9E Tire-road Frictional Drag Coefficient: 0.75 Aerodynamic Drag:

Coefficients (forward, lateral/side): CdF = 0.4, CdL = 0.8 2 2 Frontal, Lateral/side Areas: AF = 25 ft , AL = 60 ft All other vehicle parameters, if any, are given by the software default parameters (see Appendix D). Appendix D: Lists of Simulation Programs input and Output

D1: VCRware Input and Output, Crown Victoria Spinout Example:

D2: PC-Crash Linear Tire Model, w/ aero drag, Vehicle: 1991 Honda-Accord

START VALUES INPUT VALUES

Velocity magnitude (v) [ft/s] : 91.13 Vehicle : 1991 Honda-Accord Heading angle [deg] : 0.00 Length [in] : 160.80 Velocity direction (ß) [deg] : 0.00 Width [in] : 67.00 Yaw velocity [Deg/s] : 0.00 Height [in] : 53.73 Center of gravity x [ft] : 0.00 Number of axles : 2.00 Center of gravity y [ft] : 0.00 Wheelbase [in] : 107.00 Center of gravity z [ft] : 1.76 Front overhang [in] : 34.00 Velocity vertical [ft/s] : -0.00 Front track width [in] : 58.00 Roll angle [deg] : -0.00 Rear track width [in] : 58.00 Pitch angle [deg] : 0.00 Mass (empty) [lb] : 3186.00 Roll velocity [Deg/s] : 0.00 Mass of front occupants [lb] : 0.00 Pitch velocity [Deg/s] : 0.00 Mass of rear occupants [lb] : 0.00 Mass of cargo in trunk [lb] : 0.00 END VALUES Mass of roof cargo [lb] : 0.00 Distance C.G. - front axle [in] : 44.40 Velocity magnitude (v) [ft/s] : 0.53 C.G. height above ground [in] : 21.12 Heading angle [deg] : -153.51 Roll moment of inertia [lbfts^2] : 450.30 Velocity direction (ß) [deg] : 7.99 Pitch moment of inertia [lbfts^2] : 1500.90 Yaw velocity [Deg/s] : 0.54 Yaw moment of inertia [lbfts^2] : 2000.00 Center of gravity x [ft] : 222.15 Stiffness, axle 1, left [lb/in] : 121.93 Center of gravity y [ft] : -55.43 Stiffness, axle 1, right [lb/in] : 121.93 Center of gravity z [ft] : 1.77 Stiffness, axle 2, left [lb/in] : 121.93 Velocity vertical [ft/s] : -0.00 Stiffness, axle 2, right [lb/in] : 121.93 Roll angle [deg] : -0.06 Damping, axle 1, left [lb-s/ft] : 164.60 Pitch angle [deg] : -0.79 Damping, axle 1, right [lb-s/ft] : 164.60 Roll velocity [Deg/s] : -3.84 Damping, axle 2, left [lb-s/ft] : 164.60 Pitch velocity [Deg/s] : 0.11 Damping, axle 2, right [lb-s/ft] : 164.60 Linear Tire Model: SEQUENCES Max slip angle,axle 1, left [deg]: 4.11 Max slip angle,axle 1, right [deg]: 4.11 1 1991 HON : Max slip angle,axle 2, left [deg]: 3.44 Max slip angle,axle 2, right [deg]: 3.44

START VALUES Cαf = 13,000 lb/rad Cαr = 10,000 lb/rad Velocity [ft/s] : 91.13 ABS : No Friction coefficient : 0.75 SECTIONS BRAKE 1 1991 HONDA: maximum stopping distance [ft] : 300.00 Time [s], Dist. [ft], Vel. Brake force [%] [ft/s] Axle 1, left : 33.50 Axle 1, right : 33.50 Start (t=0s) -0.00 0.00 91.1 Axle 2, left : 18.60 Brake Axle 2, right : 18.60 5.15 232.60 0.4 mean brake acceleration [g] : -0.27

STEERING

Steering time [s] : 0.50 New steering angle [deg] Axle 1 : -9.00 Axle 2 : 0.00 Turning circle [ft] : -114.00 D3: SIMON, w/ aero drag, Vehicle: 1991 Honda-Accord

D4: EDSMAC4, w/ aero drag, Vehicle: 1991 Honda-Accord