Chapter 4 Vehicle Dynamics - Virginia Tech

1 31 Chapter 4 Vehicle IntroductionIn order to design a controller, a good representative model of thesystem is needed. A Vehicle mathematical model, which is appropriate for bothacceleration and deceleration, is described in this section. This model will beused for design of control laws and computer simulations. Although the modelconsidered here is relatively simple, it retains the essential Dynamics of thesystem. System DynamicsThe model identifies the wheel speed and Vehicle speed as statevariables, and it identifies the torque applied to the wheel as the input two state variables in this model are associated with one-wheel rotationaldynamics and linear Vehicle Dynamics .

2 The state equations are the result ofthe application of Newton s law to wheel and Vehicle Wheel DynamicsThe dynamic equation for the angular motion of the wheel is&[]/ wTeTbRwFtRwFwJw= ( )where Jw is the moment of inertia of the wheel, w is the angular velocity of thewheel, the overdot indicates differentiation with respect to time, and the otherquantities are defined in Table Wheel ParametersRwRadius of the wheelNvNormal reaction force from the groundTeShaft torque from the engineTbBrake torqueFtTractive forceFwWheel viscous frictionRwNvTeTbgroundFt + FwMvgdirection of Vehicle motionwheel rotating clockwiseFigure Wheel Dynamics (under the influence of engine torque.)

3 Braketorque, tire tractive force, wheel friction force, normal reaction force from theground, and gravity force)The total torque acting on the wheel divided by the moment of inertia of thewheel equals the wheel angular acceleration (deceleration). The total torqueconsists of shaft torque from the engine, which is opposed by the brake torqueand the torque components due to the tire tractive force and the wheel viscousfriction force. The tire tractive (braking) force is given byFtNv= ()( )where the normal tire force (the reaction force from the ground to the tire), Nv,depends on Vehicle parameters such as the mass of the Vehicle , location of thecenter of gravity of the Vehicle , and the steering and suspension a driving torque or a braking torque to a pneumatic tire producestractive (braking) force at the tire-ground contact patch.

4 The driving torqueproduces compression at the tire tread in front of and within the contact , the tire travels a shorter distance than it would if it were freerolling. In the same way, when a braking torque is applied, it produces tensionat the tire tread within the contact patch and at the front. Because of thistension, the tire travels a larger distance than it would if it were free rolling. Thisphenomenon is referred as the wheel slip or deformation slip (Wong, 1978).The adhesion coefficient, which is the ratio between the tractive (braking) forceand the normal load, depends on the road-tire conditions and the value of thewheel slip (Harnel,1969).

5 Figure shows a typical () , wheel slip is defined as = ()/,wv0( )where vVRw= is the Vehicle angular velocity of the wheel which is defined asbeing equal to the linear Vehicle velocity, V, divided by the radius of the variable is defined as =max(,)wv( )which is the maximum of the Vehicle angular velocity and wheel adhesion coefficient () is a function of wheel slip . For variousroad conditions, the () curves have different peak values and slopes, asshown in Figure In our simulation (see Chapter 5), the function ()=+222ppp is used for a nominal curve, where p and p are the peakvalues.

6 For various road conditions, the curves have different peak values andslopes (see Figure and Table ). The adhesion coefficient slipcharacteristics are also influenced by operational parameters such as speed34and vertical load. The peak value for the adhesion coefficient usually hasvalues between (icy road) and (dry asphalt and concrete).( )Wheel Slip00-11 1(Acceleration)(Deceleration)Adhesion Coefficient ( )Linear portionof the curvePeak Figure Typical - curve. ( )Wheel Slip( )Adhesion CoefficientDry PavementWet AsphaltUnpacked Figure - Curves for Different Road Average peak values for friction coefficient for different PeakAsphalt and concrete (dry) (wet) (wet) road (dry) road (wet) (hard packed) Vehicle DynamicsThe dynamic equation for the Vehicle motion is&[]/.

7 VNwFtFvMv= ( )where Fv = wind drag force (function of Vehicle velocity), Mv = Vehicle mass, Nw =number of driving wheels (during acceleration) or the total number of wheels(during braking), and Ft = tire tractive force, which is the average friction force ofthe driving wheels for acceleration and the average friction force of all wheelsfor deceleration. The linear acceleration of the Vehicle is equal to the differencebetween the total tractive force available at the tire-road contact and theaerodynamic drag on the Vehicle , divided by the mass of the Vehicle .

8 The totaltractive force is equal to the product of the average friction force, Ft, and thenumber of wheels, Nw. The aerodynamic drag is a nonlinear function of thevehicle velocity and is highly dependent on weather conditions (Kachroo 1992).It is usually proportional to the square of the Vehicle ForceWind DragFtFvMvFigure Vehicle Combined SystemThe dynamic equation of the whole system can be written in statevariable form by defining convenient state variables. Using equations ( )and ( ), and defining the state variables asxVRw1=( )xw2= ( )and denoting xxx=max(,)12, we obtain&()()xfxbN1111= + ( )&()()xfxbNbT22223= + ( )whereTTeTb= = ()/xxx21fxFvRwxMvRw111()[()]/()=bNNvNwMv Rw1=/()fxFwxJw222()()/=bNRwNvJw2=/bJw31= /( )

9 The combined dynamic system can be represented as shown in Figure control input is the applied torque at the wheels, which is equal to thedifference between the shaft torque from the engine and the braking acceleration, engine torque is the primary input while duringdeceleration, the braking torque is the primary System Dynamics in Terms of Wheel SlipWheel slip is chosen as the controlled variable for braking controlalgorithms because of its strong influence on the braking force between the tireand the road.

10 By controlling the wheel slip, we control the braking force toobtain the desired output from the system. In order to control the wheel slip, wecan have system dynamic equations in terms of wheel slip. Duringdeceleration, condition xx21 , ()x10 is satisfied, and therefore wheel slipis defined as: = ()/xxx211( )Differentiating this equation, we obtain &[&()&]/ = +xxx2111( )Substituting equations ( ) ( ) and ( ) into equation ( ), we obtain&[()()()][()]]/ =+ +++1112221131fxfxbNbNbTx ( )This gives the wheel slip dynamic equation for deceleration.