# Chapter 2 Vehicle Dynamics Modeling - Virginia Tech

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Chapter 2Vehicle Dynamics ModelingThis chapter provides information on dynamics modeling of vehicle and tire.Thevehicle axis system used throughout the simulation is according to the SAE standard, asdescribed in SAE J670e . According to a brief research study of typical vehiclemodels, a nonlinear three-degree-of-freedom vehicle model will be used in this research.The derivation of that model including the tire model is discussed first. The equations ofmotion are then converted to a state space form for ease of integration and a Third OrderRunge-Kutta integration routine is used as the integration algorithm. Finally, the vehiclemodel is verified against results from Smith et al.  to show its validity.2.1 Vehicle Axis SystemThroughout this thesis, the coordinate system used in vehicle dynamics modeling will beaccording to SAE J670e  as shown in Figure 2.1. The x-axis points to the forwarddirection or the longitudinal direction, the y-axis, which represents the lateral direction,is positive when it points to the right of the driver, and the z-axis points to the groundsatisfying the right hand rule.Figure 2.1 Vehicle Axis System after SAE 10

In most studies related to handling and directional control, only the X-Y plane ofthe vehicle is considered. The vertical axis, Z, is often used in the study of ride, pitch,and roll stability type problems. The following list defines relevant definitions for thevariables associated with this research.Longitudinal direction: forward moving direction of the vehicle. There are twodifferent ways of looking at the forward direction, one with respect to the vehicle bodyitself, and another with respect to a fixed reference point. The former is often used whendealing with acceleration and velocity of the vehicle.The latter is used when thelocation information of the vehicle with respect to a starting or an ending point isdesired.Lateral direction: sideways moving direction of the vehicle. Again, there are twoways of looking at the lateral direction, with respect to the vehicle and with respect to afixed reference point. Researchers often find this direction more interesting than thelongitudinal one since extreme values of lateral acceleration or lateral velocity candecrease vehicle stability and controllability.Tire slip angle: This is equivalent to heading in a given direction but walking atan angle to that direction by displacing each foot laterally as it is put on the ground asshown in Figure 2.2. The foot is displaced laterally due to the presence of lateral forces.Figure 2.2 Walking Analogy to Tire Slip Angle after Milliken 11

Figure 2.3 shows the standard tire axis system that is commonly used in tiremodeling. It shows the forces and moments applied to the tire and other importantparameters such as slip angle, heading angle, etc.Figure 2.3 SAE Tire Axis System after Milliken Body-slip angle: is the angle between the X-axis and the velocity vector thatrepresents the instantaneous vehicle velocity at that point along the path, as shown inFigure 2.4. It should be emphasized that this is different from the slip angle associatedwith tires. Even though the concept is the same, each individual tire may have differentslip angle at the same instant in time. Often the body slip angle is calculated as the ratioof lateral velocity to longitudinal velocity.12

Figure 2.4 Body-Slip Angle after Milliken In order to simplify the vehicle model so that results of the integration can bequickly calculated, the effects of camber angle, load transfer, and aerodynamics are notincluded in this study.2.2 Vehicle ModelsThere are numerous degrees of freedom associated with vehicle dynamics. The mostsimplified vehicle dynamic model is a two-degree-of-freedom bicycle model,representing the lateral and yaw motions. The idea behind this model is that sometimesit is not necessary or desirable to include the longitudinal direction, because it does notaffect the lateral or yaw stability of the vehicle.This model, which is easier tounderstand, is often used in teaching purposes. Figure 2.5 shows the two-degree-offreedom model.13

"baθαrxδVξFξfFξrαfyFigure 2.5 Two-Degree-of-Freedom ModelA three-degree-of-freedom model adds longitudinal acceleration to the model,therefore enabling one to describe the full vehicle motion in the X-Y plane. As shown inFigure 2.6, the longitudinal velocity, U, and the longitudinal force, Ftf and Ftr, areincluded into the model. This is the model that is used throughout this research.Figure 2.6 Three-Degree-of-Freedom Model after Smith In some studies, the rotational degrees of freedom for the front and rear wheelsare added to the model to include the effects of longitudinal slip, as shown in Figure 2.7.This five-degree-of freedom model enables one to perform an in-depth study of tractionand braking forces on handling maneuvers by including the effects of wheel spin.Rotational degrees of freedom are also often used in the studies of combined braking andsteering, and braking system controller design, [2,20].14

Figure 2.7 Rotational Degree of Freedom at Wheel after Smith An eight-degree-of-freedom model no longer assumes symmetry in dynamicbehavior between right and left sides. Rotational degree of freedom for each of the fourtires is considered in this vehicle model instead of two tires. It also adds a rollingmotion, φs, between left and right sides of the vehicle. This model is often used in thesuspension design or ride comfort analysis, specifically looking at the effects of theseissues with respect to roll and side-to-side load transfer.Figure 2.8 Eight-Degree of Freedom Model after Smith 15

2.3 Three-Degree-of-Freedom Vehicle Model DerivationCapturing all the motions of a vehicle into analytical equations can be quite difficult.Although including more number of elements in the model may increase the model’saccuracy, it substantially increases the computation time. This section describes thederivation of the three-degree-of-freedom bicycle model used in this study.It alsoincludes the equations for the front and rear tire slip angles. For this research this is thefirst step in the process, before proceeding to designing an optimization algorithm. Thissection of the chapter also describes concepts such as tire slip angle, friction ellipse,longitudinal force limits, and state space representation.2.3.1 Equations of MotionThe three degrees of freedom model considered for this study is governed by thefollowing equations:I θ aPf δ bFξf bFξr(2.1) V θm( Vξη ) Pf δ Fξf Fξr(2.2) V θm( Vηξ ) Pf Pr Fξf δ(2.3)Referring to Figure 2.9, the lateral and longitudinal velocities of the vehicle withrespect to the fixed coordinate system XYZ can be described as shown in Eqs. (2.4) and(2.5)."baθαrxFηrδVηVξFξfFξrFηfyFigure 2.9 Three-Degree-of-Freedom Used for Simulation16αf

x Vξ sin θ Vη cos θ(2.4)y Vξ cos θ Vη sin θ(2.5)WhereFξf, Fξr lateral tire force on front and rear tiresa vehicle center of gravity location from front axleb vehicle center of gravity location from rear axlem vehicle massI vehicle yaw inertiaPf, Pr longitudinal force on front and rear tiresVη,Vξ longitudinal/lateral velocity of vehicle in body reference framex, y longitudinal/lateral position of vehicle in inertial reference frameδ steering angleη longitudinal axis in body reference frameθ yaw angleξ lateral axis in body reference frameThe parameters used for the simulation are shown in Table 1. These parameters arebased on vehicle specifications provided by The Goodyear Tire and Rubber Company.Table 2.1 Vehicle Parameter Used for Simulation StudyPropertyValueMass1292.2 kgInertia (about Z-axis)2380.7 kgm2Front Axle to Center of Gravity1.006 mRear Axle to Center of Gravity1.534 mHeight Center of Gravity Location0.3 m17

2.3.2 Front Tire Slip Angle DerivationFront tire slip angle is a function of front tire steering angle, the longitudinal velocityvector, the lateral velocity vector, and the lateral velocity component due to yaw. Figure2.10 illustrates the front tire slip angle, which is mathematically defined as(aθ V ) δ αfξ(2.6)Vηwhereδ steering anglea center of gravity location from vehicle rear axleθ yaw velocity of vehicleVξ lateral velocity of vehicleVη longitudinal velocity of vehicleFigure 2.10 Front Tire Slip Angle after Milliken Equation (2.6), which describes the front tire slip angle formula, assumes that theslip angle is small. It is necessary to determine the front tire slip angle because the Segellateral force tire model is a function of slip angle . This tire model will be explained indetails in a later section. The Segel lateral force model will be discussed in Section 2.5.18

2.3.3 Rear Tire Slip Angle DerivationThe rear tire slip angle is a result of a combination of the ratio of the longitudinalvelocity vector and the lateral velocity vector, and the lateral velocity component due toyaw. Figure 2.11 illustrates the rear tire slip angle.(bθ V ) αr ξ(2.7)Vηwhereb center of gravity location from vehicle rear axleθ yaw velocity of vehicleVξ lateral velocity of vehicleVη longitudinal velocity of vehicleFigure 2.11 Rear Tire Slip Angle after Milliken Equation (2.7), which describes the rear tire slip angle formula, assumes that theslip angle is small. It is necessary to determine the rear tire slip angle because the Segellateral force tire model is a function of slip angle . This tire model will be explained indetails in a later section. The Segel lateral force model will be described in Section 2.5.19

2.4 Longitudinal (Traction/Braking) ForceThe traction force limit for a front wheel drive vehicle is calculated as the following,.Wbl h1 µlµFx max, t(2.8)The maximum braking force for an independent suspension, front wheel drivevehicle is calculated as the following, .Fx max, b µW(a µh)l(2.9)whereFx max,t maximum tractive forceFxmax.bµabWlh maximum braking forcefriction coefficientcenter of gravity location from front axlecenter of gravity location from rear axlevehicle weightvehicle wheelbasecenter of gravity location from groundThe values of Eq. (2.8) and (2.9) are calculated to be 5913.73 N and -5349.33 Nrespectively, using the parameters in Table 1. Throughout this study, however, thetraction force limit and the braking force limit are set to 5000 N and -5000 N,respectively. This enables the use of one variable to describe the constraint of thelongitudinal force in the optimal control algorithm. This is explained in more details inChapter 3.20

2.5 Segel Lateral Force ModelThere are several lateral force models that have been derived in the past. They rangefrom using a collective set of tire performance data, to determining an empirical lateralforce model, to analytical tire models derived from differential equations describing thedeformation of tire structure . The lateral force model selected for this research is theSegel Model. This model was developed in the early 1970's. There are newer modelsthat can be used to describe the lateral force of a tire. However, Segel model is fairlyeasy to use and has relatively small computation time . It is a function of the slipangle, cornering stiffness, tire vertical load, friction coefficient, and longitudinal force, asdescribed below. α 3 ααP2P2 ii i 1 2 i 2 i 2 Fξi µFzi α i 327 µ Fzi c i ciα iαiµFziαf δ αri f, rξ(2.12)Vη(bθ V ) Fzf Fzr (2.10)(2.11)(aθ V ) i f, rξ(2.13)Vηmgb (Pf Pr )h(2.14)a bmga (Pf Pr )h(2.15)a bwhere21

c f , c r cornering stiffness of front and rear tiresFzf , Fzr normal tire load on front and rear tiresFξf , Fξr lateral tire force on front and rear tiresghabmµαf ,αr Pf , Pr Vξ , Vη gravitational accelerationheight of the center of gravityfront axle to center of gravity distancerear axle to center of gravity distancevehicle massfriction coefficientslip angle of front and rear tireslongitudinal force of front and rear tireslongitudinal & lateral velocity in body reference framex, y longitudinal & lateral position in inertial reference frameδ steering angleθ yaw angleθ yaw velocity2.6 Friction Ellipse ConceptThis topic is discussed in this section because it is related to the two previous topics,longitudinal force and lateral force. The friction circle concept is actually used in theoptimization portion of the research as a part of the cost function, which attempts tomaximize the tire forces.22

Figure 2.12 Friction Ellipse Diagram after Milliken Figure 2.12 illustrates a friction ellipse. The diagram combines the longitudinalforce, which is a function of slip ratio, and the lateral force, which is a function of slipangle into one plot. The elliptical shape represents the limits of longitudinal and lateralforces due to friction. Inside this ellipse, one can look up the nominal longitudinal forceat a certain slip ratio, knowing the slip angle and lateral force, and vice versa. Ashortcoming of the friction ellipse diagram is that the information is only specific to acertain load, tire pressure and temperature. Numerous diagrams are necessary to fullydescribe the performance of the tire.23

2.7 Friction Circle ConceptFriction circle is a simplified version of the friction ellipse. The limit, which is shown asa circle in Figure 2.13, is defined by the product of friction coefficient of the road andthe vertical load of the vehicle on that tire. This friction model does not take intoaccount slip angle and slip ratio. This is convenient for this research because it does notrequire extensive tire data with respect to various slip ratios, slip angles, and loads.Fx(F2x Fy2)FyµNFigure 2.13 Friction Circle Diagram2.8 Equations of Motion SolutionThe three-degree-of-freedom vehicle model equations of motion must be rewritten in firstorder differential equations, to enable using the first-order numerical integration method,such as the third order Runge-Kutta, which was used in this study.The state space representation of the dynamic equations in (2.1) - (2.5) is,24

x 1 x 2 x 3 (aP δ bFξff bFξr)(2.16)I(P δ Fξff Fξr) x x3m(P P F δ) x xfr1ξfm21(2.17)(2.18)x 4 x 2 sin x 6 x 3 cos x 6(2.19)x 5 x 2 cos x 6 x 3 cos x 6(2.20)x 6 x 1(2.21)wherex 1 θ x 2 Vξyaw ratelateral velocityx 3 Vηlongitudinal velocityx4 xx5 yx6 θlongitudinal position with respect to fixed referencelateral position with respece to fixed referenceyaw angleThere are numerous integration routines that can be used to integrate the statespace equations . Two criteria for selecting which integration routine to use are thecomputation time and accuracy of the results. One such method is the Runge-Kuttaroutine, which has an accuracy of a Taylor series integration method without requiringthe calculation of higher derivatives. The Runge-Kutta method itself has up to five levelsof accuracy where the higher the order the more accurate the integration results will be.The trade-offs in selecting which order to use are the difficulty of algorithmimplementation and computation time. After careful consideration of different routines,the third order Runge-Kutta routine was selected .2.9 Vehicle Model ValidationThe three-degree-of-freedom vehicle model used in this study is validated against aknown source, a study by Smith et al. . There are two types of steering input25

subjected by the vehicle model, saw-tooth and step. Figure 2.14 shows the saw-toothsteering input and the vehicle response corresponding to that input. Figure 2.15 showsresponse of the three-degree-of-freedom model.Figure 2.14 Smith Vehicle Response with Saw-Tooth Input after Smith 0-0.21Y (m )Steering (deg)20-1-0.4-0.6-0.8-2-1012Tim e (sec)34012Tim e (sec)3401020x (m )3040Yaw Rate, (deg/s)1050-5-10Figure 2.15 Three-Degree-of-Freedom Model Response Subjected to Saw-ToothInput26

The vehicle model is also validated against the step input from Smith's study.Figure 2.16 shows the step input and the vehicle response according to that input. Figure2.17 shows the three-degree-of-freedom model response subjected to the same input.20010-2Y (m)Steering (deg)Figure 2.16 Smith Vehicle Response with Step Input after Smith 0-10-200-4-6123Time (sec)-8041020x (m)304040Yaw Rate, (deg/s)200-20-40-6001234Time (sec)Figure 2.17 Three-Degree-of-Freedom Model Response Subjected to Step Input27

It can be concluded that the three-degree-of-freedom model is valid since theresponse of this model closely follows the response generated by Smith. Therefore, thevehicle model of this study is concluded to be acceptable and valid.28

This chapter provides information on dynamics modeling of vehicle and tire. The vehicle axis system used throughout the simulation is according to the SAE standard, as described in SAE J670e . According to a brief research study of typical vehicle models, a nonlinear three-degree-of-freedom vehicle model will be used in this research.

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