2 DOF Handling Model - Wheel...

• assumptions

• equations of motion

• steady cornering results

• stability/eigenvalue results

• frequency response results

• typical understeer/oversteer results

• Buick/Ferrari example

2 DOF Handling Model

the moving axis system, A, showing the vehicle

velocity components


• simplest representation of vehicle handling which still

captures the key elements of behaviour

• road is level and flat

• vehicle body is rigid and suspension is ignored

• the steering system is assumed to be non-flexible: the

input is assumed to be a front wheel steer angle, but it

could equally be a handwheel angle

(= front wheel angle x steering ratio)

• aerodynamic forces are ignored


assumptions

• in general, three degrees of freedom are actually needed to define

the motion of a car on a horizontal surface:

• forward velocity

• lateral velocity

• yaw velocity

• however, forward velocity will be assumed constant and it

becomes, therefore, a parameter rather than a system variable

• problem formulated in velocities

• tyre forces are dependant on velocities

• nature of the vehicle motion i.e. velocity or lateral acceleration,

is more meaningful than absolute position on the surface


assumptions

• important omissions

• roll motion of sprung mass

• suspension effects, e.g. camber

• tyre load transfer

• steering system compliance

• however, some of these effects can nevertheless be

examined indirectly using the 2 dof model


assumptions

2 DOF Handling Model - Equations

Fyr

Fyf

m (v + ur) = Fyf + Fyr

I r = a Fyf - b Fyr

b

a

Distances

a = cg to

front axle

b = cg to

rear axle

• dominant in controlling vehicle handling

• tyre side force depends primarily on slip angle

Tyre Force Properties

Resultant

motion

Slip

angle

Tyre

side force

• in the linear region, side force = cornering stiffness x slip angle

Fy = C

• lots of other secondary factors influence C - camber, load, tyre

pressure . . . .

need to calculate front and rear slip angles in order

to calculate tyre lateral forces


the final equations of motion are then:

in matrix form


m 0

0 I

v

r

Cf + Cr

U

aCf - bCr

U

aCf - bCr

U

a2Cf + b2Cr

U

mU +

+ v

r =

Cf

aCf f

• the sign convention leads to the natural results of a

rightward steer leading to a rightward tyre force

• cornering stiffnesses have positive values (some sign

conventions lead to -ve values which seems unnatural for

a stiffness)

• in the 2 dof, “bicycle model” the C refers to the

cornering stiffness of the axle - i.e. twice that of the

individual tyres

• Fy actually acts perpendicular to the wheel, but since f is

assumed small, the cos f terms which should be in the

vehicle equations are assumed to be unity

2 DOF Handling Model notes to equations of motion

• steady cornering

• fixed speed and steer angle

• consistent radius of turn

• important because it is a standard and relatively easy method of practical

testing

• stability

• straight running, no input, f = 0

• information about transient response to disturbances

• information about possible unstable conditions

• frequency response

• response to sinusoidal input applied at steering wheel for a range of

frequencies

• captures dynamic response of system to forcing inputs

2 DOF Handling Model - Results

Front tyre side force

mass x lateral

acceleration

Rear tyre side force

Tyres operate at slip angles

Vehicle Cornering Typical steady cornering condition

2 DOF Handling Model steady cornering results

• ceases to be a dynamic problem, since v and r are set to

zero

• model assumes small slip angles, so it is restricted to low

lateral accelerations (< 0.3g)

• equations manipulated to give (yaw rate output/steer

angle input):

in which l is the wheelbase, i.e. l =a+b, and the subscript

ss refers to steady state

U l Cf Cr

l2 Cf Cr + m U2(bCr - aCf)

rss

f

=


steady cornering

• lateral acceleration =

• yaw rate, r =

• path curvature, =

U2

R

U

R

1

R

r

U

R

steady state equation re-written in terms of path

curvature:

where:


1

l + K U2

ss

f

=

m (bCr - aCf)

l Cf Cr K =

Understeer / Oversteer

K=0, Neutral steer

K<0, Oversteer

K<0, Oversteer

K>0, Understeer

Critical

speed

Forward speed, U

Steady state

curvature per

unit steer angle

Steady state turning responses predicted by the two DOF

vehicle

= 1

l + KU2 l = wheelbase

K = depends on stability margin

U = forward speed

• K = 0 - NEUTRAL STEER

Path followed is same as that for a pure rolling vehicle, i.e. the Ackermann

condition

• K > 0 - UNDERSTEER

Always stable. More steering required than for the Ackermann vehicle.

Practically, vehicle is described as “running wide” as more lock than

anticipated must be applied

• K < 0 - OVERSTEER

Response increases with increasing speed. Critical speed at which response

becomes infinite is given by:

practically, vehicle wants to turn more than anticipated - rear end feels to be

swinging out. At speeds in excess of Ucrit - opposite lock solution

Steady Cornering Results

3 cases of interest (animation)

= l

-K Ucrit

underover.html


Stability Margin = bCr - aCf

+ve Understeer

-ve Oversteer

a

b

Front cornering

stiffness = Cf

Rear cornering

stiffness = Cr

v r

2 degrees of freedom:-

- sideslip, v

- yaw rate, r

The analysis of the classic 2 degree of freedom vehicle identifies

the importance of stability margin:-

• Understeer b Cr > a Cf

– car feels like it wants to run wide - not turning enough

– driver feels like he/she has to wind more lock on

– terminal understeer - can plough straight on irrespective

of how much steering is applied

Vehicle Handling - Subjective

Descriptions

Oversteer b Cr < a Cf

– car feels as if the rear end is breaking away

– driver actually has to reduce the steering input

– terminal oversteer - spin - leave the road backwards!

– not as bad as it sounds - vehicle is stable up to its

critical speed

• critical terminology in order to discuss

vehicle handling

• understeer accepted as preferable – more natural feel as limit approached

– safer - slow down to regain control

• oversteer generally regarded as dangerous – less natural to reduce steering near limit

– spin may happen too quickly in limit conditions

– rally drivers may prefer it!


• K = Understeer parameter

• constant value from linear model

• in practice it is typically a non-linear function of lateral

acceleration

• practical results are normally obtained by driving at

increasing speed around a fixed radius (typically 33m)

circle

Steady Cornering Results

Ackermann

steer angle

Handwheel

steer angle,

deg

Lateral acceleration, g

120

90

60

30

0.2 0.4 0.6 0.8

K = slope of curve (deg/g)

• effectively the free vibration response

• input i.e. steer angle is set to zero

• roots describe the natural frequency and damping of the

vehicle response

• practically, this characterises the transient response of the

vehicle lateral/yaw motion

• roots - which are typically complex numbers - can be

simply interpreted by plotting on the complex plane

2 DOF Handling Results

stability - same as eigenvalues or roots

typical eigenvalues in the complex plane and their

relationship with natural frequency and damping ratio

Stability Results

steady cornering results for an arbitrary vehicle:

Stability Results

Symbol Value

m

I

a

b

Cf

Cr

U

1

1.5

1.25

1.25

53

53

20

Parameter

Total mass

Total yaw inertia

CG to front axle

CG to rear axle

Front axle cornering stiffness

Rear axle cornering stiffness

Forward speed

Units

t

tm2

m

m

kN/rad

kN/rad

m/s

Units Understeer

m

m

kNm/rad

deg/g

m/s

1.15

1.35

+10.6

+0.85

-

Parameter

a

b

bCr - aCf

K

Ucrit

Neutral steer

1.25

1.25

0

0

Oversteer

1.35

1.15

-10.6

-0.85

41

modifications to the parameter set above:

The eigenvalues of the basic vehicle as forward speed is

increased from 10m/s (smallest symbols) to 50m/s

(largest symbols)

Stability Results

• sinusoidal input at steer wheels - linear system - results

in sinusoidal responses in yaw, sideslip and lateral

acceleration

• plot out gain and phase of outputs relative to input steer

angle - these are the frequency response functions

• describe the forced vibration characteristics of the system

- also known as the transfer function

Frequency Response Results


Yaw rate

response results


Lateral

acceleration

response results


Normalised yaw

rate and lateral

acceleration

gains

• the two most meaningful outputs are:

• yaw rate - what the driver sees

• lateral acceleration - what the driver feels

• sideslip response is small and difficult to sense

• desirable properties

• flat gain - indicates consistent response

• gain does not roll off at low frequencies

• drivers cannot provide steering inputs beyond 2-3 Hz

• phase implies a time lag between input and output, which is

generally viewed as undesirable


2 DOF Handling Model - Results Summary

Understeer

• SS - less response than Ackermann vehicle

• response 0 as U

• always stable

• damped, oscillatory response

• good transient performance - responsive

• too much understeer - possibility of very light damping

at high speeds

• good straight running properties

8

2 DOF Handling Model - Results Summary

Oversteer

• SS - more response than Ackermann vehicle

• stable below Ucrit • overdamped response

• poor transient performance

• damping increases with speed

• unstable above Ucrit • not as disastrous as it sounds!

• transition occurs gradually

• possible for driver to control an unstable system, providing rate of divergence is low

• poor straight running properties

Linear handling case study

• 2 contrasting vehicles

– 1949 Buick

– Ferrari Monza

• simple 2 dof results

– steady state

– eigenvalues

– time history to a step input

– frequency responses

Vehicle parameters

Parameter Symbol Units 1949 Buick Ferrari Monza

Mass m t 2.045 1.008

Yaw inertia I tm 2

5.428 1.031

CG to front axle a m 1.488 1.234

CG to rear axle b m 1.712 1.022

Front axle cornering stiffness C f kN/rad 77.85 117.44

Rear axle cornering stiffness C r kN/rad 76.51 144.93

Wheelbase a b + m 3.200 2.256

Stability margin bCr aCf - kNm/rad 15.15 3.20

Understeer gradient K deg/g 0.91 0.05

Steady state cornering - Buick

• 0.3g turn at 20m/s, turn radius = 136m

lateral velocity, v -0.48 m/s

steer angle, f 1.62 deg

front slip angle, f -2.37 deg

rear slip angle, r -2.09 deg

front lateral force, Fyf 3.22 kN

rear lateral force, Fyr 2.80 kN

r

v

f f

r

Fyf

Fyr

Steady state cornering - Ferrari

• 0.3g turn at 20m/s, turn radius = 136m

lateral velocity, v -0.07 m/s

steer angle, f 0.96 deg

front slip angle, f -0.66 deg

rear slip angle, r -0.64 deg

front lateral force, Fyf 1.34 kN

rear lateral force, Fyr 1.62 kN

r

v

f f

r

Fyf

Fyr

Eigenvalues, damped natural frequencies and damping ratios

Buick Ferrari

Speed Eigenvalue w d , Hz z Eigenvalue w

d , Hz z

20 m/s -3.71 1.65 i 0.26 0.91 -14.5 0.91 i 0.14 1.00

30 m/s -2.48 1.66 i 0.26 0.83 -9.68 1.45 i 0.23 0.99

50 m/s -1.49 1.67 i 0.27 0.67 -5.81 1.65 i 0.26 0.96

Root locus plots as speed increases from 10 m/s to 50 m/s

10 m/s

50 m/s

Time histories

• yaw rate response of the Buick and Ferrari at 50 m/s, following a step steer input which results in a steady state lateral acceleration of 0.3g

Frequency responses - yaw rate

• yaw rate responses for the Buick and Ferrari at two forward speeds

Ferrari

Buick

Frequency responses - lateral acceleration

• lateral acceleration responses for the Buick and Ferrari at two forward speeds

Ferrari

Buick

Buick vs Ferrari - Conclusions

• Buick

– high ratio of I:m

– low ratio of tyre force capacity (Cf, Cr) to vehicle mass

– marked understeer characteristic

– light damping at high speeds

• Ferrari

– high ratio of tyre force capacity (Cf, Cr) relative to vehicle mass/inertia

– very slight understeer - almost neutral steer characteristics

– rapid, well damped response to step steer input

– retains consistent gain properties up to much higher frequencies that the Buick

2 DOF Handling Model - Wheel...

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