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0 of 4Conduction 3
Steady StateCornering
Timothy T. MaxwellAdvanced Vehicle Engineering Laboratory
Texas Tech University
2006
1Steady State Cornering
Steady State Cornering
❏ Handling — responsiveness of vehicle to driverinputs or ease of control
❏ Handling is measure of the driver–vehicleclosed–loop system
❏ Vehicle only must be characterized asopen–loop system
❏ Vehicle response to steering input ordirectional response
❏ Most common measure of open–loop response is theUndersteer Gradient
❏ Understeer Gradient only valid for steady state
2Steady State Cornering
Low Speed Turning
❏ At low speed (parking lot maneuvers) tires need not developlateral forces❏ Tires roll with no slip angle — center of turn must lie off projection
of rear axle❏ Perpendiculars from front wheels pass through same turn center
3Steady State Cornering
Low Speed Turning
❏ Ideal turning angles
❏ Average angle is theAckerman Angle
!o=
L
R + t2( )
!i=
L
R " t2( )
! !L
R
4Steady State Cornering
Ackerman Angle (Low Speed)
❏ Ackerman angle or Ackerman Geometry❏ Exact geometry of front wheels as on previous slide❏ Correct angles depend on vehicle wheelbase & turn angle❏ Deviations from Ackerman angles for the right or left
steer angles can significantly affect tire wear❏ Deviations do not significantly affect directional response
for low speed turns❏ Deviations do affect steering torques❏ With correct Ackerman geometry, steering torques
increase with steer angle and provide feedback to driver❏ With parallel steering, steering torque increases initially
and then decreases❏ Steering torque can become negative so that vehicle turns more
deeply into turn
5Steady State Cornering
Off–Tracking at Low Speed
❏ Off–tracking at rear wheels❏ Off–tracking distance
❏ Recall series form of cos
❏ Then
❏ Off–tracking is primarily a concern for longwheelbase vehicles like trucks and buses
❏ For articulated vehicle — Tractrix equations
! = R 1" cos LR( )#
$%&
cos z = 1!z2
2!+z4
4!!z6
6!+z8
8!!
! =L2
2R
6Steady State Cornering
High Speed Cornering
❏ High speed cornering produces different equationswrt low speed cornering
❏ Tires must develop significant lateral forces tocounteract the lateral acceleration
❏ Slip angles will be present at each wheel
7Steady State Cornering
Tire Cornering Forces
❏ Tire slip–angle between direction of heading anddirection of travel
❏ Lateral or cornering force grows with slip angle
8Steady State Cornering
Slip Angles
❏ Below about5˚ sliprelationshipis linear
❏ Cα — cornering stiffness❏ Positive slip angle produces negative force (to the
left) on tire❏ Thus, Cα must be negative❏ SAE defines Cα as the negative of the slope
Fy = C!!
9Steady State Cornering
Slip Angle
❏ Cornering stiffness depends on several variables❏Tire size ❏ Number of plies❏Tire type (radial or bias ply) ❏ Cord angles❏Wheel width ❏ Load❏Tread design ❏ Inflation pressure
❏ Speed not a strong influence on cornering forcesproduced by tire
10Steady State Cornering
Cornering Coefficient
❏ Cornering stiffness divided by load
❏ Cornering force — strong dependence on load❏ Cornering coefficient largest at light load and
diminishes as load reaches rated value❏ Tire & Rim Association rated load❏ At 100% load — cornering coefficient
approximately0.2 lbf cornering force / lbf load / deg slip angle
CC! =C!
Fylbfy / lbfz / deg"# $%
11Steady State Cornering
Slip Angle Dependence
12Steady State Cornering
Cornering Equations
❏ Apply NSL and description of geometry in turns❏ Bicycle model❏ At high speed turn radius >> wheelbase
assume small angles – inner and outer slip angles same❏ Both front wheels represented by one with steer angle δ❏ Sum forces in lateral direction
Fy! = Fyf + Fyr =MV
2
R
Fyf = cornering force at front
Fyf = cornering force at rear
M = mass of vehicle
V = forward velocity
R = turn radius
13Steady State Cornering
Cornering Equations
❏ For vehicle to bein equilibrium
❏ Slip angles are
❏ Geometry gives❏ Combining
Fyf b ! Fyrc = 0
! f =WfV
2
C! f gR
! r =WrV
2
C!rgR
! = 57.3LR+" f +" r
! = 57.3LR+
Wf
C" f
#Wr
C"r
$%&
'&
()&
*&
V2
gR
+,-
./0
14Steady State Cornering
Understeer Gradient
❏ Previous equation is the Understeer Gradient
❏ Lateral acceleration
❏ Magnitude and directionof steering inputs required
! = 57.3LR+ kay
V2
gR
Wf
C! f
"Wr
C!r
15Steady State Cornering
Understeer Gradient
❏ Neutral Steer
❏ Understeer
❏ Oversteer
Wf
C! f
=Wr
C!r
" K = 0"! f = ! r
Wf
C! f
>Wr
C!r
"K > 0"! f >! r
Wf
C! f
<Wr
C!r
"K < 0"! f <! r