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Lecture 7- Full Vehicle Modelling

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Bergamo University

Italy

12th-14th June 2012

Lecture 7- Full Vehicle Modelling

Professor Mike Blundell

Phd, MSc, BSc (Hons), FIMechE, CEng

- Underlying Theory (Bicycle Model Approach)
- Understeer and Oversteer
- Modelling Strategies (Lumped Mass, Swing Arm, Roll Stiffness, Linkages)
- Vehicle Body Measurements and Influences

O1

Vy2

X1

- 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

Vx2

Y1

Y2

X2

O2 G2

Fy

wz2

Fy

“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

w

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

At the instant of turn in:

w= Cafaf a t / Izz

v = Cafaf t / m

Combining these velocities gives an

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

c = Izz/ ma

Noting that Izz = mk2

Thus if c is equal to b then

1 = k2 / ab

u

v

a

b

c

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

with respect to the rear axle.

- 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

a

b

c

Roll Angle

Lateral Acceleration (Ay )

Forward Speed (Vx)

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

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

Typical lateral responses measured in vehicle coordinate frame

L

do

Centre of Mass

t

di

R

Centre of Turn

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

- 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:
- The driver turns the hand wheel, applying a slip angle at the front wheels
- After a delay associated with the front tyre relaxation lengths, side force is applied at the front of the vehicle
- The body yaws (rotates in plan), applying a slip angle at the rear wheels

c

b

ar

af

d

R

Centre of Turn

- 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

c

b

αr

X

αf

δ

Fry

Ffy

Y

m ay

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

Ffy + Fry = m ay

Ffy b - Fry c = 0

Understeer Path

Neutral Steer Path

Oversteer Path

Disturbing force (e.g. side gust)

Acting through the centre of mass

Olley’s Definition (1945)

Neutral Steer

Understeer

Oversteer

- Understeer promotes stability
- Oversteer promotes instability (spin)

V

ay

33 m

The constant radius turn test procedure can be use to define

the handling characteristic of a vehicle (Reference the British

Standard)

- 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

ay = V2 / R

δ (deg)

Understeer

Ackerman Angle

K = Understeer Gradient

Oversteer

Lateral Acceleration (g)

- 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

- 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)

Limit

Understeer

Neutral Steer

Limit

Oversteer

Vehicle 1

Vehicle 2

Lateral Acceleration (g)

2

Understeer

Oversteer

Critical Speed

Characteristic Speed

Vehicle Speed (kph)

δ

(deg)

-m ay

FRy

FFy

V

V

FFy

FRy

m ay

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

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

KTr

RCrear

Roll Axis

FROy

m ay

cm

FROz

FRIy

h

KTf

FRIz

RCfront

FFOy

Z

X

FFIy

FFIz

Y

FFIz

MRRC

m ay

FRRCy

cm

Roll Axis

h

MRRC

MFRC

FFRCy

FRRCy

KTr

RCrear

FROy

FROz

FRIy

MFRC

FFRCy

KTf

FRIz

RCfront

Z

FFOy

X

FFIy

FFIz

Y

FFIz

- 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 stiffnessesKTf and KTr
- FFRCy+ FRRCy - m ay = 0
- MFRC + MRRC - m ay . h= 0
- The roll moment causes weight transfer to the inner and outer wheels

MRRC

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

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

RCrear

DFRzM

Outer Wheels

MFRC

tr

DFRzM

RCfront

Z

Inner Wheels

X

DFFzM

Y

tf

DFFzM

- 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

FRRCy

m ay

cm

Roll Axis

c

h

FFRCy

b

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

This gives:

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

RCrear

FRRCy

hr

DFRzL

Outer Wheels

tr

RCfront

DFRzL

FFRCy

hr

Inner Wheels

DFFzL

tf

DFFzL

- 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

Taking moments again for each

of the front and rear axles gives:

RCrear

cm

FROy

m ay

Roll Axis

FROz

FRIy

FRIz

RCfront

FFOy

Z

X

FFIy

FFIz

Y

FFIz

- 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)

This gives:

FFIz = FFSz - DFFzM – DFFzL

FFOz = FFSz + DFFzM + DFFzL

FRIz = FRSz - DFRzM - DFRzL

FROz = FRSz + DFRzM + DFRzL

Lateral Force Fy

ΔFy

Outer

Tyre

Load

Inner

Tyre Load

Static Tyre

Load

Vertical Load Fz

- At this stage we must consider the tyre characteristics
- The tyre cornering force Fy varies with the tyre load Fzbut the relationship is not linear

- 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 DFyrepresents 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

Drift

Increase front weight transfer - Understeer

Drift

Increase rear weight transfer - Oversteer

- 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

- 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

LINKAGE MODEL

LUMPED MASS MODEL

SWING ARM MODEL

ROLL 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).

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, ...

30 m

25 m

25 m

30 m

15 m

C

A

B

A - 1.3 times vehicle width + 0.25m

B - 1.2 times vehicle width + 0.25m

C - 1.1 times vehicle width + 0.25m

Roll Moment (Nmm) FRONT SUSPENSION

Roll Angle (deg)

Steering motion applied at joint

MOTION

COUPLER

Steering column

part

Steering rack

part

REV

Revolute joint to vehicle body

Translational joint to vehicle body

TRANS

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

REV

TORQUE

Dummy transmission part located at mass centre of the body

COUPLER

REV

REV

FRONT

WHEELS

- Tests Performed at the Prodrive Fen End Test Facility:
- Coordinated by Damian Harty
- Coventry University Subaru Vehicle

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)

Driving on Wet Basalt

Non-Expert

Expert Driver

Proving Ground Results

ADAMS Results

Example Questionnaire

Example Questionnaire

Driver 1

Driver 2

Driver 3

Driver 1

Driver 2

Driver 3

- 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

- Demonstration of Roll Stiffness Model in Solver File
- Fiala and Road Data Files
- AView Demonstrations of Lane Change
- Parameter changes such as CM height