Handling Rollover Notes

Handling

Low-speed turning

High-speed turning

Understeer

Low-speed Turning

d do i

RL

t Inboard off-tracking

L2 R

2

R+t/2

TurnCenter

R+t/2d=tan-1 ----- =-----Lo

L

d=tan-1 ----- =-----R-t/2

Li R-t/2

L

For large radii, R >> t/2

d=--AckL

R

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 M

z

Pneumatic Trail, P

Slip Region

Tire Cornering Stiffness

Slip Angle, (deg)

C

Fy

Slip Angle (-)

La

tera

l F

orc

e,

F

(lb

)y

Direction of Travel

0

0

200

400

600

800

2 4 6 8 10 12

positive is

0

C

d

dFC

CF

y

y

=-=

=

Factors affecting cornering stiffness

NSL for force and moment analysis

Geometry for steer angle vs. radius

2

y f r

c

W VF F F

g R= =

R

b

c

d

f

r

f

r

dL/R

Ff

Fr

V

W Vg R

2From Newtons Second Law

From tire properties

0cg f rT F b F c= - =

Ff =WfsV 2

R gc

f =Ff

Cf=Wfs

Cf

V 2

R gc

r =Fr

Cr=Wrs

Cr

V2

R gc

d = 57.3L

R f - r

From the geometry: d = 57.3L

RWfs

Cf

V 2

R gc-Wrs

Cr

V2

R gc

d = 57.3L

R (Wfs

Cf-Wrs

Cr)V2

R gc

Understeer Gradient

Fr = WrsV 2

R gc

High-speed Turning

ry f

f

rfz

f

r

rr

ff

r

f

r

Positive understeer

Zero neutral steer

Negative oversteer

Has a critical speed

Vehicle is unstable

Oscillatory

Divergent

d = 57.3L

R (Wfs

Cf-Wrs

Cr)V2

R gc

Understeer 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/ = V2/57.3Lg(1+ KV2/57.3Lg)

Yaw velocity gain r/ = 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)

Late

ral A

cc. G

ain

(g

/deg

)

Understeer (5 Deg/g)

Neutral Steer

Oversteer (1 Deg/g)

Understeer (2 Deg/g)

108 in wheelbase

Stability limit

88 mph

SW Angle/g

5 deg

6 deg

10 deg

20 deg

40 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 f and r

6. Calculate under steer f r

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 RollOverturning moment M = Wh1 [ V

2/(Rg) + ]

M = Mf + Mr = (Kf+Kr)

Hence, = Wh1V2/[Rg(Kf+Kr-Wh1)]

Roll rate R = d/day = Wh1/[Kf+Kr-Wh1]

Where = roll angle, K = roll stiffness, h1 = distance between C.G. & roll ctr.

Vertical load difference between outside and inside wheel

(Fzof Fzif)tf = Kf* + WfhfV2/Rg and (Fzof +Fzif) = Wf

(Fzor Fzir)tr = Kr* + 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

5. Calculate Fzi and Fzo for front and rear





0 1 2 3 4 5 6 7 8 9

Camber Angle (deg)

0

50

100

150

200

La

tera

l F

orc

e (

lb) F = 1000 lb

Zero Slip Angle

z

Cg

g

Tires produce a lateral force (camber thrust) when inclined

Characterized by camber stiffness, Cg

Camber Thrust

Camber coefficient

Radials are lower

Bias-ply are higher

.01 0.02 0.03

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

20

15

10

5

0R

elat

ive

Fre

quen

cy (%

)

Bias-Ply

Radial

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

Camber Thrust

g = b +

Where

g = camber w.r.t. groundb = camber w.r.t. body = roll angle

Lateral Tire load due to camber

Fyc = C*

= C*(d/d)*(d/day)*ay

= C*(d/d)*roll rate*ay

- 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 Wr2. Assume lateral acceleration ay/g as % (g).

3. Calculate roll rate and find roll angle 4. Calculate Fzi and Fzo for front and rear

5. Calculate - 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





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

-

y

rfda

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

c

FA

d=

rrfflfcs WAWAK -=

Yaw center

Constant Radius Understeer TestS

tee

r A

ng

le/S

tee

rin

g R

atio

(d

eg

)

Lateral Acceleration (g)

Underst

eer

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 StiffnessRoll Stiffness DistributionRoll Center HeightTire CapacitySteering GeometryCamber

Mass, C.G.

Roll Inertia

Tread

Under-steer

Over-Steer

Stability

Lateral

Acceleration

Vehicle Roll-over Safety

Roll-over Forces

M*ay*h - M*g**h + Fzi*t M*g*t/2 = 0

ay/g = (t/2 + *h Fzit/Mg)/h

When =0 and ay=0, Fzi = M*g/2

When =ay/g, Fzi = M*g/2

Roll-over condition ay/g = t/2h +

Where is the cross-slope

Road super-elevation angle

Mg

Roll-over Threshold t/2h

Roll-over Forces

M*ay*h + M*g**h + Fzi*t M*g*t/2 = 0

ay/g = (t/2 - *h Fzit/Mg)/h

When =0 and ay=0, Fzi = M*g/2

When =ay/g, Fzi = M*g/2

Roll-over condition ay/g = t/2h -

Where is the vehicle roll angle

Vehicle roll angle

Mg

Roll-over Threshold

Roll-over Forces on a Suspended Vehicle

M0=0= Msayh-Msg[t/2 - (h-hr)]

= R*ay

Hence, max acceleration

ay/g = t/{2h[1+R(1-hr/h)]}

Roll-over Threshold for Suspended Vehicle

Transient Roll-over in Step Steer

I+ C + [K-Mg(h-hr)] =W ay(h-hr)/g

Where

I = Roll moment of inertia

C= Roll damping

K= Roll stiffness

h = C.G. height

hr = roll center height

W = vehicle weight

ay = lateral acceleration

Roll-over condition

ay/g = t/{2h[1+R(1-hr/h)]}

where

R = max/(ay/g)

Step Steer

L

R

V

time

Late

ral A

ccele

ration

L / V

V2/R

Roll Response to Step Steer

Effect of Damping

Transient Roll-over in Sinusoidal Steer

I+C+[K-Mg(h-hr)]=Way(h-hr)sint/g

Where

I = Roll moment of inertia

C= Roll damping

K= Roll stiffness

h = C.G. height

hr = roll center height

W = vehicle weight

ay = lateral acceleration

Roll-over condition

ay/g = t/{2h[1+R(1-hr/h)]}

where

R = max/(ay/g)

Sinusoidal Steer

V

Y0

2L

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

Sinusoidal Steer

Suspension Design to Prevent Roll-over

Vehicle

Roll Stiffness/stabilize barRoll Stiffness DistributionRoll Center HeightTire Capacity

Mass, C.G.

Roll Inertia

Tread

Roll Angle

Rollover Threshold

Step &

Sinusoidal

Steer

Documents

Handling Rollover Notes