Cornering Forces and Geometry
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Transcript of Cornering Forces and Geometry
8/17/2019 Cornering Forces and Geometry
http://slidepdf.com/reader/full/cornering-forces-and-geometry 1/16
0 of 4Conduction 3
Steady State
ornering
Timothy T. Maxwell
Advanced Vehicle Engineering Laboratory
Texas Tech University
2006
8/17/2019 Cornering Forces and Geometry
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8/17/2019 Cornering Forces and Geometry
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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
8/17/2019 Cornering Forces and Geometry
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3Steady State Cornering
Low Speed Turning
! Ideal turning angles
! Average angle is the
Ackerman Angle
! o =
L
R + t 2( )
! i =
L
R " t 2( )
! ! L
R
8/17/2019 Cornering Forces and Geometry
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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 initiallyand then decreases! Steering torque can become negative so that vehicle turns more
deeply into turn
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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 L R( )#
$ %
&
cos z = 1! z
2
2!
+ z
4
4!
!
z6
6!
+ z
8
8!
!
! =
L2
2 R
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6Steady State Cornering
High Speed Cornering
! High speed cornering produces different equations
wrt low speed cornering
! Tires must develop significant lateral forces to
counteract the lateral acceleration! Slip angles will be present at each wheel
8/17/2019 Cornering Forces and Geometry
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7Steady State Cornering
Tire Cornering Forces
! Tire slip–angle between direction of heading and
direction of travel
! Lateral or cornering force grows with slip angle
8/17/2019 Cornering Forces and Geometry
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8Steady State Cornering
Slip Angles
! Below about5˚ sliprelationshipis linear
! C ! — cornering stiffness
! Positive slip angle produces negative force (to the
left) on tire! Thus, C ! must be negative
! SAE defines C !
as the negative of the slope
F y = C ! !
8/17/2019 Cornering Forces and Geometry
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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 forces
produced by tire
8/17/2019 Cornering Forces and Geometry
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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
approximately
0.2 lbf cornering force / lbf load / deg slip angle
CC ! =
C !
F ylbf y / lbf z / deg"# $%
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11Steady State Cornering
Slip Angle Dependence
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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 "
! Sum forces in lateral direction
F y! = F yf + F yr = MV
2
R
F yf = cornering force at front
F yf = cornering force at rear
M = mass of vehicle
V = forward velocity
R = turn radius
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13Steady State Cornering
Cornering Equations
! For vehicle to be
in equilibrium
! Slip angles are
! Geometry gives
! Combining
F yf b ! F yrc = 0
! f =
W f V 2
C ! f gR
! r =W rV
2
C ! rgR
! = 57.3 L R
+" f +" r
! = 57.3 L R
+
W f
C " f
# W r
C " r
$%&
'&
()&
*&
V 2
gR
+ , -
. / 0
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14Steady State Cornering
Understeer Gradient
! Previous equation is the Understeer Gradient
! Lateral acceleration
! Magnitude and direction
of steering inputs required
! = 57.3 L R
+ ka y
V 2
gR
W f
C ! f
"
W r
C ! r
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15Steady State Cornering
Understeer Gradient
! Neutral Steer
! Understeer
! Oversteer
W f
C ! f
=
W r
C ! r
" K = 0"! f = ! r
W f
C ! f
>W r
C ! r
"K > 0"! f > ! r
W f
C ! f
<W r
C ! r
"K < 0"! f <! r