steady state cornering - atu587.org forces and...deviations do affect steering torques with correct...

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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


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


Low Speed Turning

❏ Ideal turning angles

❏ Average angle is theAckerman Angle

!o=

L

R + t2( )

!i=

L

R " t2( )

! !L

R


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


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


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


Tire Cornering Forces

❏ Tire slip–angle between direction of heading anddirection of travel

❏ Lateral or cornering force grows with slip angle


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!!


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


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"# $%


Slip Angle Dependence


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


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


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


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

steady state cornering - atu587.org forces and...deviations do affect steering torques with correct...

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