Handling

• Low-speed turning

• High-speed turning

• Understeer

Low-speed Turning

o i

RL

t Inboard off-tracking

L2 R

2

R+t/2

TurnCenter

R+t/2tan-1L

o L

tan-1R-t/2

Li R-t/2

L

For large radii, R >> t/2

AckLR

High Speed Turning

V

RRR

RR

R0

Original Path/Neutral Steer Path

Under Steer Path

R > R0

Over Steer Path

R < R0

Tire Slip Angle

Direction of Heading

Direction of TravelContact Patch

Slip Angle,

Fy Mz

Pneumatic Trail, P

Slip Region

Tire Cornering Stiffness

Slip Angle, (deg)

C

Fy

Slip Angle (-)

La

tera

l Fo

rce

, F

(

lb)

y

Direction of Travel

0

0

200

400

600

800

2 4 6 8 10 12

positive is

0

Cd

dFC

CF

y

y

Factors affecting cornering stiffness

• NSL for force and moment analysis

• Geometry for steer angle vs. radius

2

y f rc

W VF F F

g R

R

b

c

f

r

f

r

L/R

Ff

Fr

V

W Vg R

2From Newton’s Second Law

From tire properties

0cg f rT F b F c

Ff Wfs

V 2

Rgc

f FfC f

Wfs

Cf

V 2

Rgc

r FrCr

WrsCr

V2

Rgc

57.3L

R f r

From the geometry: 57.3L

RWfs

Cf

V 2

RgcWrsCr

V2

Rgc

57.3L

R (Wfs

Cf

WrsCr

)V2

RgcUndersteer Gradient

Fr WrsV 2

Rgc

High-speed Turning

ry f

f

rfz

f

r

rr

f αf

αr

αf

αr

• Positive – understeer

• Zero – neutral steer

• Negative – oversteer

– Has a critical speed

– Vehicle is unstable

• Oscillatory

• Divergent

57.3L

R (Wfs

Cf

WrsCr

)V2

RgcUndersteer Gradient, K

Understeer Gradient

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)

• Understeer – Very controlled gain with speed• Neutral steer – Increasing gain with speed

• Oversteer – Increases dramatically with speed

0

1

2

3

4

5

6

0 20 40 60 80 100 120

Speed (mph)

Lat

eral

Acc

. Gai

n (

g/d

eg)

Understeer (5 Deg/g)

Neutral Steer

Oversteer (1 Deg/g)

Understeer (2 Deg/g)

108 in wheelbase

Stability limit88 mph

SW Angle/g5 deg

6 deg

10 deg

20 deg40 deg

Effect on Lateral Acceleration Gain

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 lbwith average lateral force of 680 lb, requiring more slip angle to maintain the turn

Effect of Body RollOverturning 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

0 1 2 3 4 5 6 7 8 9

Camber Angle (deg)

0

50

100

150

200

Late

ral F

orce

(lb

) F = 1000 lb

Zero Slip Anglez

C

• Tires produce a lateral force (camber thrust) when inclined• Characterized by camber stiffness, C

Camber Thrust

• Camber coefficient– Radials are lower

– Bias-ply are higher

.01 0.02 0.03

Camber Coefficient, C /F (lb/lb/deg)z

20

15

10

5

0R

ela

tive

Fre

qu

en

cy (

%)

Bias-Ply

Radial

Camber Coefficient, C/Fz (lb/lb/deg)

Camber Thrust

γg = γb + φ

Where

γg = camber w.r.t. ground

γb = camber w.r.t. body

φ = roll angle

Lateral Tire load due to camber

Fyc = Cγ*γ

= Cγ*(dγ/dφ)*(dφ/day)*ay

= Cγ*(dγ/dφ)*roll rate*ayγ-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 data6. 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

Roll Center Inclination of Suspension Roll Axis

Neutral Steer

Oversteer

Understeer

Front of Vehicle

yrf da

dK

)(steer roll

Lateral Force Compliance Steer

• All suspensions steer due to a lateral force

• Minimize compliance steer

Yaw center

CorneringForce

Deflection Understeer

Turn

CorneringForce

Deflection Oversteer

Turn

y

cFA

rrfflfcs WAWAK

Yaw center

Constant Radius Understeer TestS

tee

r A

ng

le/S

tee

rin

g R

atio

(d

eg

)

Lateral Acceleration (g)

Understeer

Neutral Steer

Oversteer

Limit Understeer

Limit Oversteer

CONSTANT RADIUS

K (deg/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

Vehicle

•Roll Stiffness•Roll Stiffness Distribution•Roll Center Height•Tire Capacity•Steering Geometry•Camber

Mass, C.G.Roll Inertia

Tread

Under-steerOver-SteerStability

LateralAcceleration

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

Roll-over condition ay/g = t/2h + θ

Where θ is the cross-slope

Road super-elevation angle θ

Mgθ

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

Where φ is the vehicle roll angle

Vehicle roll angle φ

Mgφ

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 accelerationRoll-over condition

ay/g = t/{2h[1+Rφ(1-hr/h)]}

where

Rφ = φmax/(ay/g)

Step Steer

L

R

V

time

Late

ral A

ccel

erat

ion

L / V

V2/R

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 accelerationRoll-over condition

ay/g = t/{2h[1+Rφ(1-hr/h)]}

where

Rφ = φmax/(ay/g)

Sinusoidal Steer

V

Y0

2L

Y = Y0 sin (π*V*t/L) and lateral accn Y” = (π*V/L)2Y0 sin (π*V*t/L)

Sinusoidal Steer

Suspension Design to Prevent Roll-over

Vehicle

•Roll Stiffness/stabilize bar•Roll Stiffness Distribution•Roll Center Height•Tire Capacity

Mass, C.G.Roll Inertia

Tread

Roll AngleRollover Threshold

Step &SinusoidalSteer