Transcript Slide 1

Handling
• Low-speed turning
• High-speed turning
• Understeer
Low-speed Turning
do
L =----L
-1 ----d=
tan
i
R-t/2
R-t/2
di
L =----L
-1 ----d=
tan
o
R+t/2 R+t/2
For large radii, R >> t/2
L
d=-Ack
R
L
R
t
 Inboard off-tracking
2
 L
2R
Turn
Center
High Speed Turning
Under Steer Path
R > R0
Original Path/
Neutral Steer Path
Over Steer Path
R < R0
R0
V
R
R R
R
R
Tire Slip Angle
F
y
M
z
Pneumatic Trail, P
Direction of Heading
Slip Angle, 
Directio
Slip Region
Contact Patch
n of Tra
vel
Tire Cornering Stiffness
Lateral Force, F y (lb)
800
Direction
of Travel
600

Slip Angle (-)
400
200
0
F
y
C
0
2
4
6
8
10
Slip Angle, (deg)
12
Fy = C 
C = C
dFy
 =0
d
is positive
Factors affecting cornering stiffness
High-speed Turning
• NSL for force and moment
analysis
• Geometry for steer angle vs.
radius
From Newton’s Second Law
W V2
 Fyy = Fff  Frr = g R
c
Tzcg = Fff  b - Frr  c = 0
From tire properties
Fff
Wfs V 2
f =
=
f
C f Cf R  gc
αf
αf
Frr Wrs V 2
rr =
=
Cr C r R  gc
αr
αr
2
V
R  gc
V2
Fr r= Wrs
R  gc
Fff = Wfs
d
d
L/R

r
f
F
f
b
V
2
WV
gR
From the geometry:
L
d = 57.3   f -  r
R
R
c
r
F
r
L Wfs V 2
Wrs V 2
d = 57.3 
R Cf R gc C r R  gc
Wfs Wrs V 2
L
d = 57.3  (
)
R Cf C r R  gc
Understeer Gradient

f
Understeer Gradient
Wfs Wrs V 2
L
d = 57.3  (
)
R Cf C r R  gc
Understeer Gradient, K
• Positive – understeer
• Zero – neutral steer
• Negative – oversteer
– Has a critical speed
– Vehicle is unstable
• Oscillatory
• Divergent
Steer Angle vs. Speed
Speeds & Gains
Characteristic speed = speed at which steer angle required to negotiate a
turn is 2 times Ackerman angle
Vchar
= √57.3Lg/K
Critical speed
= speed at which steer angle required to negotiate a
turn is 0
Vcrit
= √-57.3LgK
Lateral acceleration gain ay/δ = V2/57.3Lg(1+ KV2/57.3Lg)
Yaw velocity gain
r/δ = V/L(1+ KV2/57.3Lg)
Effect on Lateral Acceleration Gain
• Understeer – Very controlled gain with speed
• Neutral steer – Increasing gain with speed
• Oversteer – Increases dramatically with speed
Lateral Acc. Gain (g/deg)
6
Stability limit
88 mph
Understeer (5 Deg/g)
5
Neutral Steer
SW Angle/g
5 deg
Oversteer (1 Deg/g)
4
Understeer (2 Deg/g)
108 in wheelbase
3
6 deg
2
10 deg
1
20 deg
40 deg
0
0
20
40
60
Speed (mph)
80
100
120
Effect on Yaw velocity gain
Slip Angle Calculation (primary tire effect)
1. Calculate front and rear vertical wheel loads Wf and Wr
2. Assume lateral acceleration ay/g as % (g).
3. Lateral tire force (front & rear) Fyf = Wf*ay and Fyr = Wr*ay
4. From tire data find slip angles for all 4 tires, use extrapolation
5. Find average slip angle for front and rear αf and αr
6. Calculate under steer αf – αr
7. Do calculations for ay/g from 0.1 to 1.0
Effect of Body Roll
W
Fz0 > Fzi
Effect of Body Roll
No roll: For 800 lb load on each wheel 760 lb of lateral force at 5 deg slip angle
Body Roll: In hard cornering inside & outside wheel loads can be 400 & 1200 lb
with average lateral force of 680 lb, requiring more slip angle to
maintain the turn
Effect of Body Roll
Overturning moment Mφ = Wh1 [ V2/(Rg) + φ]
Mφ = Mφf + Mφr = (Kφf+Kφr) φ
Hence, φ = Wh1V2/[Rg(Kφf+Kφr-Wh1)]
Roll rate Rφ = dφ/day = Wh1/[Kφf+Kφr-Wh1]
Where φ = roll angle, Kφ = roll stiffness, h1 = distance between C.G. & roll
ctr.
Vertical load difference between outside and inside wheel
(Fzof –Fzif)tf = Kφf*φ + WfhfV2/Rg
and (Fzof +Fzif) = Wf
(Fzor –Fzir)tr = Kφr*φ + WrhrV2/Rg
and (Fzor +Fzir) = Wr
Where hf and hr = roll center height front and rear
Slip Angle Calculation (roll effect)
1. Calculate front and rear vertical wheel loads Wf and Wr
2. Assume lateral acceleration ay/g as % (g).
3. Lateral tire force (front & rear) Fyf = Wf*ay and Fyr = Wr*ay
4. Calculate roll rate and find roll angle φ
5. Calculate Fzi and Fzo for front and rear
6. From tire data find slip angles for all 4 tires, use extrapolation
7. Find average slip angle for front and rear αf and αr
8. Calculate under steer αf – αr
9. Do calculations for ay/g from 0.1 to 1.0
Camber Thrust
• Tires produce a lateral force (camber thrust) when inclined
• Characterized by camber stiffness, Cg
• Camber coefficient
Relative Frequency (%)
Lateral Force (lb)
– Radials are lower
– Bias-ply are higher
Fz = 1000 lb
Zero Slip Angle
200
g
150
100
50
0
Cg
0
1
2
15
4
5
6
7
Camber Angle (deg)
8
9
Radial
10
5
0
3
Bias-Ply
20
.01
0.02
Camber Coe fficient, Cg /Fz
0.03
(lb/lb/deg )
Camber Coefficient, Cg/Fz (lb/lb/deg)
Camber Thrust
Lateral Tire load due to camber
Fyc = Cγ*γ
= Cγ*(dγ/dφ)*(dφ/day)*ay
γg = γb + φ
Where
γg = camber w.r.t. ground
γb = camber w.r.t. body
φ = roll angle
= Cγ*(dγ/dφ)*roll rate*ay
γ-φ relationship
Lateral tire force causing tire slip = W*ay - Fyc
Slip Angle Calculation (roll/camber effect)
1. Calculate front and rear vertical wheel loads Wf and Wr
2. Assume lateral acceleration ay/g as % (g).
3. Calculate roll rate and find roll angle φ
4. Calculate Fzi and Fzo for front and rear
5. Calculate γ-φ relationship from suspension data
6. Calculate lateral tire force due to camber for each tire
7. Lateral tire force for slip (front & rear) Fyf = Wf*ay-Fycf and
Fyr = Wr*ay-Fycr
8. From tire data find slip angles for all 4 tires, use extrapolation
9. Find average slip angle for front and rear αf and αr
10. Calculate under steer αf – αr
11. Do calculations for ay/g from 0.1 to 1.0
Roll Steer
• All suspensions steer with roll
• Steer to the outside is:
– Understeer on front
– Oversteer on rear
• Solid axle on a trailing arm:
– Arm angle determines
understeer
– Angled down is oversteer
– Angled upward is understeer
K roll steer
d
= ( f -  r )
da y


-
Inclination of
Suspension Roll Axis
Roll Center
Overstee
r
Neutral Steer
Under
steer
Front of Vehicle
Lateral Force Compliance Steer
• All suspensions steer due to a
lateral force
• Minimize compliance steer
dc
Fy
K lfcs = A f W f - ArWr
Deflection Understeer
Deflection Oversteer
Turn
Turn
Yaw center
A=
Cornering
Force
Cornering
Force
Yaw center
Steer Angle/Steering Ratio (deg)
Constant Radius Understeer Test
Limit
Understeer
CONSTANT
RADIUS
Neutral Steer
er
e
t
s
r
e
Und
Ov
ers
tee
r
K (deg/g)
Limit
Oversteer
Lateral Acceleration (g)
Constant Speed Understeer Test
Process for Calculating Cornering Response
•
•
•
•
•
•
•
•
•
Decide on the lateral acceleration requirement
Calculate roll-stiffness based on the suspension properties
Calculate roll rate
Calculate left and right tire vertical loads for the max lateral acceleration
Choose tire to minimize understeer or oversteer
Determine camber vs roll angle relationship for your suspension
Make adjustments to understeer/oversteer
Calculate critical speed
Calculate yaw velocity and lateral acceleration gains
Suspension Design for Handling
Mass, C.G.
Roll Inertia
Tread
Lateral
Acceleration
Vehicle
•Roll Stiffness
•Roll Stiffness Distribution
•Roll Center Height
•Tire Capacity
•Steering Geometry
•Camber
Under-steer
Over-Steer
Stability
Vehicle Roll-over Safety
Roll-over Forces
M*ay*h - M*g*θ*h + Fzi*t – M*g*t/2 = 0
ay/g = (t/2 + θ*h – Fzit/Mg)/h
When θ=0 and ay=0, Fzi = M*g/2
When θ=ay/g, Fzi = M*g/2
Mgθ
Roll-over condition ay/g = t/2h + θ
Where θ is the cross-slope
Road super-elevation angle θ
Roll-over Threshold t/2h
Roll-over Forces
M*ay*h + M*g*φ*h + Fzi*t – M*g*t/2 = 0
ay/g = (t/2 - φ*h – Fzit/Mg)/h
When φ=0 and ay=0, Fzi = M*g/2
When φ=ay/g, Fzi = M*g/2
Roll-over condition ay/g = t/2h - φ
Mgφ
Vehicle roll angle φ
Where φ is the vehicle roll angle
Roll-over Threshold
Roll-over Forces on a Suspended Vehicle
M0=0= Msayh-Msg[t/2 - φ(h-hr)]
φ = Rφ*ay
Hence, max acceleration
ay/g = t/{2h[1+Rφ(1-hr/h)]}
Roll-over Threshold for Suspended Vehicle
Transient Roll-over in Step Steer
Iφφ”+ Cφφ’ + [Kφ-Mg(h-hr)] φ=W ay(h-hr)/g
Where
Iφ = Roll moment of inertia
Cφ= Roll damping
Kφ= Roll stiffness
h = C.G. height
hr = roll center height
W = vehicle weight
ay = lateral acceleration
Roll-over condition
ay/g = t/{2h[1+Rφ(1-hr/h)]}
where
Rφ = φmax/(ay/g)
Lateral Acceleration
Step Steer
V2/R
L/V
R
time
V
L
Roll Response to Step Steer
Effect of Damping
Transient Roll-over in Sinusoidal Steer
Iφφ”+Cφφ’+[Kφ-Mg(h-hr)]φ=Way(h-hr)sinωt/g
Where
Iφ = Roll moment of inertia
Cφ= Roll damping
Kφ= Roll stiffness
h = C.G. height
hr = roll center height
W = vehicle weight
ay = lateral acceleration
Roll-over condition
ay/g = t/{2h[1+Rφ(1-hr/h)]}
where
Rφ = φmax/(ay/g)
Sinusoidal Steer
Y = Y0 sin (π*V*t/L) and lateral accn Y” = (π*V/L)2Y0 sin (π*V*t/L)
V
2L
Y0
Sinusoidal Steer
Suspension Design to Prevent Roll-over
Mass, C.G.
Roll Inertia
Tread
Step &
Sinusoidal
Steer
Vehicle
•Roll Stiffness/stabilize bar
•Roll Stiffness Distribution
•Roll Center Height
•Tire Capacity
Roll Angle
Rollover Threshold