lecture 7- full vehicle modelling

Bergamo University Italy12th-14th June 2012

Professor Mike BlundellPhd, MSc, BSc (Hons), FIMechE, CEng

Lecture 7- Full Vehicle Modelling

Contents

• Underlying Theory (Bicycle Model Approach)• Understeer and Oversteer• Modelling Strategies (Lumped Mass, Swing Arm, Roll

Stiffness, Linkages)• Vehicle Body Measurements and Influences

Underlying Theory - Bicycle Model

• The simplest possible representation of a vehicle manoeuvering in the ground plane (bicycle model)

• Weight transfer• Tyre lateral force characteristics as a function of tyre

load

GRF

O1

X1

Y1

O2 G2

Y2

X2

Vx2

Fy

Fy

Vy2

z22z

ωzzI2zM

)2z

ω2x

V2y

V( 2

m2yF

3

Vehicle Handling

“Handling” is

–different to maximum steady state lateral acceleration (“grip”)

–much less amenable to a succinct definition

–“a quality of a vehicle that allows or even encourages the operator to make use of all the available grip”

–(Prodrive working definition)

–Emotional definitions like:

“Confidence” (Consistency/Linearity to Inputs)

“Fun” (High Yaw Gain, High Yaw Bandwidth)

“Fluidity” (Yaw Damping Between Manoeuvres)

“Precision” (Disturbance Rejection)

Courtesy of www.drivingdevelopment.co.uk

“Inertia Match” is the relationship between the CG position, wheelbase and yaw inertia.

At the instant of turn in:

= Caf af a t / Izz

v = Caf af t / m

Combining these velocities gives an

“instant centre” at a distance c behind the CG:

c = Izz / ma

Noting that Izz = m k2

Thus if c is equal to b then

1 = k2 / ab

a

b

c

Vehicle Handling

uv

k2/ ab therefore describes the distance of the centre of rotation

with respect to the rear axle.

a

b

c

• It is referred to as the “Dynamic Index”

• DI fraction is length ratio c / b

• DI > 1 implies c > b

• DI < 1 implies c < b

• Magic Number = 0.92

Vehicle Handling

Vehicle Handling – Understeer and Oversteer

7Forward Speed

(Vx)

Lateral Acceleration (Ay )

Roll Angle

• Lateral Acceleration (g)• Yaw Rate (deg/s)• Roll Rate (deg)• Trajectory ( Y (mm) vs. X (mm))

For pure cornering (Lateral Response) the following outputs are typically studied:

Typical lateral responses measured in vehicle

coordinate frame

Cornering at Low-Speed

8 Centre of Turn

Centre of Masso

i

R

t

L

Assuming small steer angles at the road wheels to avoid scrubbing the wheels

The average of the inner and outer road wheel angles is Known as the Ackerman Angle

t)0.5(R

Lδo

t)0.5(R

Lδi

R

Lδ

Steady State Cornering

9

• Start with a simple ‘Bicycle’ model explanation• The model can be considered to have two degrees of

freedom (yaw rotation and lateral displacement) No roll! • In order to progress from travelling in a straight line to

travelling in a curved path, the following sequence of events is suggested:

1. The driver turns the hand wheel, applying a slip angle at the front wheels

2. After a delay associated with the front tyre relaxation lengths, side force is applied at the front of the vehicle

3. The body yaws (rotates in plan), applying a slip angle at the rear wheels

Steady State Cornering (continued)

10

r

f

Centre of Turn

c b

R

• After a delay associated with the rear tyre relaxation lengths, side force is applied at the rear of the vehicle

• Lateral acceleration is increased, yaw acceleration is reduced to zero

Bicycle Model Simplified

11

The bicycle model can be described by the following two equations of motion:

c b

αr

δαf

X

Y

m ay

Fry Ffy

Ffy + Fry = m ay

Ffy b - Fry c = 0

Understeer and Oversteer

12

Neutral Steer Path

Disturbing force (e.g. side gust)

Acting through the centre of mass

Understeer Path

Oversteer Path

Olley’s Definition (1945)

Understeer and Oversteer

13

• Understeer promotes stability• Oversteer promotes instability (spin)

Neutral Steer

UndersteerOversteer

The Constant Radius Test

14

The procedure may be summarised as:

•Start at slow speed, find Ackerman angle•Increment speed in steps to produce increments in lateral acceleration of typically 0.1g•Corner in steady state at each speed and measure steering inputs•Establish limit cornering and vehicle Understeer / Oversteer behaviour

The constant radius turn test procedure can be use to definethe handling characteristic of a vehicle (Reference the BritishStandard)

ay

V

33 m

ay = V2 / R

Understeer Gradient

15

At low lateral acceleration the road wheel angle d can be found using:

Where:δ = road wheel angle (deg)K = understeer gradient (deg/g)Ay = lateral acceleration (g)L = track (m)R = radius (m)

δ (deg)

Lateral Acceleration (g)

Ackerman Angle

Understeer

Oversteer

K = Understeer Gradient

R

L

PI

180

• It is possible to use results from the test to determine Understeer gradient

• Use steering ratio to establish road wheel angle d from measured hand wheel angles

yaKR

L

PI

180δ

Limit Understeer and Oversteer Behaviour

16

δ(deg)

Lateral Acceleration (g)

LimitUndersteer

LimitOversteerVehicle 1

Vehicle 2

Neutral Steer

Vehicle Speed (kph)

Understeer

Oversteer

Critical Speed

2

Characteristic Speed

δ(deg)

R

L

PI

180

R

L

PI

180

Consideration of Cornering Forces using a Roll Stiffness Approach

17

m ay

V

FRy FFy

-m ay

V

FRy FFy

Fy = m ay

Where ay is the centripetal acceleration acting towards

the centre of the corner

Fy - m ay = 0

Where –m ay is the d’Alembert Force

Free Body Diagram Roll Stiffness Model During Cornering

18

Representing the inertial force as a d’Alembert force consider the forces acting on the roll stiffness model during cornering as shown

cmm ay

RCrear

h

Y

Z

X

Roll Axis

RCfront

FFIy

FFIz

FFOy

FFIz

FROy

FROz

FRIy

FRIz

KTr

KTf

Forces and Moments Acting at the Roll Axis

19

FFOy

RCrear

Y

Z

X

RCfront

FFIy

FFIz

FFIz

FROy

FROz

FRIy

FRIz

KTr

KTfFFRCy

FRRCy

MRRC

MFRC

cmm ay

h

Roll Axis

FRRCy

FFRCy

MRRC

MFRC

Forces and Moments (continued)

20

• Consider the forces and moments acting on the vehicle body (rigid roll axis)

• A roll moment (m ay .h) acts about the axis and is resisted in the model by the moments MFRC and MRRC resulting from the front and rear roll stiffnesses KTf and KTr

FFRCy + FRRCy - m ay = 0MFRC + MRRC - m ay . h = 0

• The roll moment causes weight transfer to the inner and outer wheels


21

RCrear

Y

Z

X

RCfront

DFFzM

DFFzM

DFRzM

DFRzM

MRRC

MFRC

Inner Wheels

Outer Wheels

tf

tr

ΔFFzM = component of weight transfer on front tyres due to roll moment

ΔFRzM = component of weight transfer on rear tyres due to roll moment


22

• Taking moments for each of the front and rear axles gives:

• It can be that if the front roll stiffness KTf is greater than the rear roll stiffness KTr there will be more weight transfer at the front (and vice versa)

• It can also be seen that an increase in track will obviously reduce weight transfer

fTrTf

Tfy

f

FRCFzM t

1

KK

Kh.am

t

MF

rTrTf

Try

r

RRCRzM t

1

KK

Kh.am

t

MF


23

Consider again a free body diagram of the body roll axis and the components of force acting at the front and rear roll centres

cm

m ay

h

Roll Axis

FRRCy

FFRCy b

c

This gives:

cb

camF yFRCy

cb

bamF yRRCy


24

• From this we can see that moving the body centre of mass forward would increase the force, and hence weight transfer, reacted through the front roll centre (and vice versa)

• We can now proceed to find the additional components, DFFzL and DFRzL, of weight transfer due to the lateral forces transmitted through the roll centres

RCrear

RCfront

DFFzL

DFFzL

DFRzL

DFRzL

Inner Wheels

Outer Wheels

ΔFFzL = component of weight transfer on front tyres due to lateral force

Δ FRzL = component of weight transfer on rear tyres due to lateral force

tf

tr

FFRCy

FRRCy

hr

hr

f

fy

f

fFRCyFzL t

h

cb

ch.am

t

hFΔF

Taking moments again for eachof the front and rear axles gives:

r

ry

rRRCyRzL t

h

cb

bh.am

t

hFΔF

r


25

• It can be that if the front roll centre height hf is increased there will be more weight transfer at the front (and vice versa)

• We can now find the resulting load shown acting on each tyre by adding or subtracting the components of weight transfer to the front and rear static tyre loads ( FFSz and FRSz)

FFOy

RCrear

Y

Z

X

RCfront

FFIy

FFIz

FFIz

FROy

FROz

FRIy

FRIz

cmm ay

Roll Axis This gives:

FFIz = FFSz - DFFzM – DFFzL

FFOz = FFSz + DFFzM + DFFzL

FRIz = FRSz - DFRzM - DFRzL

FROz = FRSz + DFRzM + DFRzL

Loss of Cornering Force due to Nonlinear Tyre Behaviour

26

• At this stage we must consider the tyre characteristics• The tyre cornering force Fy varies with the tyre load Fz but

the relationship is not linear

Lateral Force

Fy

Vertical Load Fz

ΔFy

Inner Tyre

Load

Static TyreLoad

Outer TyreLoad

Loss of Cornering Force (continued)

27

• The figure above shows a typical plot of tyre lateral force with tyre load at a given slip angle

• The total lateral force produced at either end of the vehicle is the average of the inner and outer lateral tyre forces

• From the figure it can be seen that DFy represents a theoretical loss in tyre force resulting from the averaging and the nonlinearity of the tyre

• Tyres with a high load will not produce as much lateral force (in proportion to tyre load) compared with tyres on the vehicle

• More weight transfer at either end will tend to reduce the total lateral force produced by the tyres and cause that end to drift out of the turn

• At the front this will produce Understeer and at the rear this will produce Oversteer

The Effect of Weight Transfer on Understeer and Oversteer

28

Drift

Increase front weight transfer - Understeer

Drift

Increase rear weight transfer - Oversteer

In summary the following changes could promote Understeer: •Increase front roll stiffness relative to rear.•Reduce front track relative to rear.•Increase front roll centre height relative to rear.•Move centre of mass forward

The Effect of Weight Transfer (continued)

29

Case Study - Vehicle Modelling Study

LINKAGE MODELLUMPED MASS MODEL

SWING ARM MODELROLL STIFFNESS MODEL

The following are typical of the tests which have been performed on the proving ground:

(i) Steady State Cornering - where the vehicle was driven around a 33 metre radius circle at constant velocity. The speed was increased slowly maintaining steady state conditions until the vehicle became unstable. The test was carried out for both right and left steering lock.

(ii) Steady State with Braking - as above but with the brakes applied at a specified deceleration rate ( in steps from 0.3g to 0.7g) when the vehicle has stabilised at 50 kph.

(iii) Steady State with Power On/Off - as steady state but with the power on (wide open throttle) when the vehicle has stabilised at 50 kph. As steady state but with the power off when the vehicle has stabilised at 50 kph.

(iv) On Centre - application of a sine wave steering wheel input (+ / - 25 deg.) during straight line running at 100 kph.

(v) Control Response - with the vehicle travelling at 100 kph, a steering wheel step input was applied ( in steps from 20 to 90 deg. ) for 4.5 seconds and then returned to the straight ahead position. This test was repeated for left and right steering locks.

(vi) I.S.O. Lane Change (ISO 3888) - The ISO lane change manoeuvre was carried out at a range of speeds. The test carried out at 100 kph has been used for the study described here.

(vii) Straight line braking - a vehicle braking test from 100 kph using maximum pedal pressure (ABS) and moderate pressure (no ABS).

Vehicle Handling Tests

Following the guidelines shown performing all the simulations with a given ADAMS vehicle model, a set of results based on recommended and optional outputs would produce 67 time history plots. Given that several of the manoeuvres such as the control response are repeated for a range of steering inputs and that the lane change manoeuvre is repeated for a range of speeds the set of output plots would escalate into the hundreds.

This is an established problem in many areas of engineering analysis where the choice of a large number of tests and measured outputs combined with possible design variation studies can factor the amount of output up to unmanageable levels.

MANOEUVRES - Steady State Cornering, Braking in a Turn, Lane Change, Straight Line Braking, Sinusoidal Steering Input, Step Steering Input,

DESIGN VARIATIONS - Wheelbase, Track, Suspension, ...

ROAD SURFACE - Texture, Dry, Wet, Ice, m-Split

VEHICLE PAYLOAD - Driver Only, Fully Loaded, ...

AERODYNAMIC EFFECTS - Side Gusts, ...

RANGE OF VEHICLE SPEEDS - Steady State Cornering, ...

TYRE FORCES - Range of Designs, New, Worn, Pressure Variations, ...

ADVANCED OPTIONS - Active Suspension, ABS, Traction Control, Active Roll, Four Wheel Steer, ...

Computer Simulations

Double Lane Change Manoeuvre

30 m 25 m 25 m 30 m 15 m

A

B

C

A - 1.3 times vehicle width + 0.25m

B - 1.2 times vehicle width + 0.25m

C - 1.1 times vehicle width + 0.25m

Lane Change Simulation

Determination of Roll Stiffness

Rear Roll Centre

Front Roll Centre

Applied RollAngle Motion

CYL

SPH

INPLANE

INPLANE

Rear Roll Centre

Front Roll Centre

Applied RollAngle Motion

CYL

SPHINPLANE

INPLANE

Determination of Roll Stiffness

Roll Moment (Nmm) FRONT SUSPENSION

Roll Angle (deg)

Modelling the Steering System

Steering column

part

Revolute joint to vehicle body

Steering motion applied at joint

REV COUPLER

MOTION

Steering rack

part

TRANS Translational joint to vehicle body

Front

suspension


Motion on the steering system is ‘locked’ during the initial static analysis

Downward motion of vehicle body and steering rack relative to suspension during static equilibrium

Connection of tie rod causes the front wheels to toe out


COUPLER

COUPLER

Modelling a Speed Controller

Dummy transmission part located at mass centre of the body

FRONT

WHEELS

REV

REV

REVTORQUE

COUPLER

Comparison with Track Test(Lane Change)

Case Study – Dynamic Index Investigation

Tests Performed at the Prodrive Fen End Test Facility:

•Coordinated by Damian Harty

•Coventry University Subaru Vehicle

Calibration of Dynamic Index

High DI = 1.02 Mid DI = 0.92 Low DI = 0.82

Front Ballast 48 kg 27 kg 5.5 kg

Rear Ballast 57 kg 29 kg 0 kg

Central Ballast 40.5 kg 90 kg 140 kg

Vehicle Ballast Conditions:


• Excel Spreadsheet• ADAMS Simulation• Prodrive Inertia Rig (Quadrifiler)


ADAMS Quadrifiler Simulation:

Tests Performed at the Prodrive Fen End Test Facility:

•Basalt Strip X2

•Lane change (50MPH)

•0.3g and 0.8g Step steer

•Sine wave steering input increased frequency (50MPH)

•Lift off and turn in

•Lane change (60MPH)

•3 Expert Drivers (Prodrive)

•1 Experienced Automotive Engineer (Coventry University)

•5 Non-Expert Student Drivers (Coventry University)

•3 Settings of Dynamic Index (0.82, 0.92 and 1.02)

Proving Ground Tests

Proving Ground Tests Driving on Wet Basalt

Non-ExpertExpert Driver

ADAMS Simulations

Example Results

Proving Ground Results

ADAMS Results

Subjective Assessment

Example Questionnaire

Driver 1Driver 2Driver 3


Conclusions

• Dynamic index (DI) is an important modifier of vehicle handling performance.

• Subjective assessment indicates a DI of 0.92 is desirable.

• Experienced drivers may prefer a more “agile” vehicle with a low DI.

•Non-expert drivers may prefer a more “forgiving” car with a high DI.

• A detailed validated multi-body systems model of a vehicle allows in depth analysis of responses that may be difficult to measure on the proving ground.

•Subjective/objective correlation remains a challenge in vehicle dynamics

Tutorial 8 – Planning Full Vehicle Models

• Demonstration of Roll Stiffness Model in Solver File

• Fiala and Road Data Files

• AView Demonstrations of Lane Change

• Parameter changes such as CM height

lecture 7- full vehicle modelling

Documents

yaw acceleration

distance c

yaw inertia

lateral displacement

vehicle manoeuvering

ayffy b fry c

c bmagic number

c brf x y