Bergamo University
Italy
12th-14th June 2012
Lecture 7- Full Vehicle Modelling
Professor Mike Blundell
Phd, MSc, BSc (Hons), FIMechE, CEng
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
O1
X1
GRF
Y1

M2z Izz ω
2z
 V ω )
F2y m2 ( V
2y 2x 2z
• 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
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
Vehicle Handling
“Inertia Match” is the relationship between the CG position,
wheelbase and yaw inertia.
At the instant of turn in:
w = 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=
v
u
a
b
w
Izz / ma
Noting that Izz = m k2
Thus if c is equal to b then
1 = k2 / ab
c
Vehicle Handling
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
a
b
• DI > 1 implies c > b
• DI < 1 implies c < b
• Magic Number = 0.92
c
Vehicle Handling –
Understeer and Oversteer
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))
Roll Angle
Lateral
Acceleration (Ay )
Forward Speed
(Vx)
Typical lateral responses
measured in vehicle
coordinate frame
7
Cornering at Low-Speed
L
do
Centre of Mass
t
di
Assuming small steer angles
at the road wheels to avoid
scrubbing the wheels
L
δo 
(R  0.5 t)
L
δi 
(R  0.5 t)
R
The average of the inner and
outer road wheel angles is
Known as the Ackerman Angle
L
δ 
R
 Centre of Turn
8
Steady State Cornering
• 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
9
Steady State Cornering (continued)
• 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
ar
af
d
R

Centre of Turn
10
Bicycle Model Simplified
c
b
αr
X
αf
Fry
Y
δ
Ffy
m ay
The bicycle model can be described by the following two
equations of motion:
Ffy + Fry = m ay
Ffy b - Fry c = 0
11
Understeer and Oversteer
Understeer Path
Neutral Steer Path
Oversteer Path
Disturbing force (e.g. side gust)
Acting through the centre of mass
Olley’s Definition (1945)
12
Understeer and Oversteer
• Understeer promotes stability
• Oversteer promotes instability (spin)
Neutral Steer
Oversteer
Understeer
13
The Constant Radius Test
The constant radius turn test procedure can be use to define
the handling characteristic of a vehicle (Reference the British
Standard)
V
ay
33 m
ay = V2 / R
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
14
Understeer Gradient
• 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:
δ (deg)
 180  L
δ 
  K ay
 PI  R
Understeer
Ackerman
Angle
 180  L


PI

R
K = Understeer Gradient
Where:
Oversteer
Lateral Acceleration (g)
δ = road wheel angle (deg)
K = understeer gradient (deg/g)
Ay = lateral acceleration (g)
L = track (m)
R = radius (m)
15
Limit Understeer and Oversteer
Behaviour
δ
(deg)
Limit
Understeer
Neutral
Steer
Vehicle 1
δ
(deg)
 180  L

 PI  R
2
Understeer
Limit
Oversteer
Vehicle 2
 180  L


 PI  R
Oversteer
Characteristic
Speed
Lateral Acceleration (g)
Critical
Speed
Vehicle Speed (kph)
16
Consideration of Cornering Forces
using a Roll Stiffness Approach
-m ay
FRy
FFy
V
V
FRy
FFy
m ay
 Fy = m ay
 Fy - m ay = 0
Where ay is the centripetal
acceleration acting towards
the centre of the corner
Where –m ay is the
d’Alembert Force
17
Free Body Diagram Roll Stiffness
Model During Cornering
Representing the inertial force as a d’Alembert force consider
the forces acting on the roll stiffness model during cornering as
shown
K
Tr
RCrear
Roll Axis
FROy
FROz
FRIy
m ay
cm
h
KTf
FRIz
RCfront
FFOy
Z
X
FFIy
FFIz
Y
FFIz
18
Forces and Moments Acting at the Roll
Axis
MRRC
FRRCy
m ay
cm
Roll Axis
h
MRRC
FRRCy
MFRC
FFRCy
KTr
RCrear
FROy
FROz
FRIy
MFRC
FFRCy
FRIz
KTf
RCfront
Z
X
FFIy
FFOy
FFIz
Y
19
FFIz
Forces and Moments (continued)
• 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 = 0
MFRC + MRRC - m ay . h = 0
• The roll moment causes weight transfer to the inner and outer
wheels
20
Forces and Moments (continued)
MRRC
ΔFFzM = component of weight transfer
on front tyres due to roll moment
RCrear
ΔFRzM = component of weight transfer
on rear tyres due to roll moment
DFRzM
DFRzM
Outer Wheels
MFRC
tr
RCfront
Z
Inner Wheels
X
DFFzM
Y
DFFzM
tf
21
Forces and Moments (continued)
• Taking moments for each of the front and rear axles gives:
 K Tf
MFRC
 m a y .h 
tf
 K Tf  K Tr
1

 tf
 K Tr
MRRC
FRzM 
 m a y .h 
tr
 K Tf  K Tr
1

 tr
FFzM 
• 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
22
Forces and Moments (continued)
Consider again a free body diagram of the body roll axis and
the components of force acting at the front and rear roll centres
FRRCy
m ay
cm
Roll Axis
c
h
b
This gives:
 c 

FFRCy  m a y 
 bc 
FFRCy
 b 

FRRCy  m a y 
 bc 
23
Forces and Moments (continued)
• 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
ΔFFzL = component of weight transfer
on front tyres due to lateral force
RCrear
FRRCy
Δ FRzL = component of weight transfer
on rear tyres due to lateral force
hr
DFRzL
DFRzL
tr
Outer Wheels
RCfront
FFRCy
hr
Inner Wheels
DFFzL
DFFzL
tf
Taking moments again for each
of the front and rear axles gives:
h 
 c   hf 
  
ΔFFzL  FFRCy  f   m a y .h 
 b  c   tf 
 tf 
 b   hr 
h 
  
ΔFRzL  FRRCy  r   m a y .h 
 tr 
 b  c   tr 
24
Forces and Moments (continued)
• 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)
RCrear
cm
FROy
m ay
This gives:
Roll Axis
FROz
FRIy
FRIz
RCfront
FFOy
Z
FFIz = FFSz - DFFzM – DFFzL
FFOz = FFSz + DFFzM + DFFzL
FRIz = FRSz - DFRzM - DFRzL
FROz = FRSz + DFRzM + DFRzL
X
FFIy
FFIz
Y
FFIz
25
Loss of Cornering Force due to
Nonlinear Tyre Behaviour
• 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

ΔFy



Inner Static Outer Vertical Load Fz
Tyre Tyre Tyre
Load Load Load
26
Loss of Cornering Force
(continued)
• 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
27
The Effect of Weight Transfer on
Understeer and 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
Drift
Increase front weight
transfer - Understeer
Drift
Increase rear weight
transfer - Oversteer
28
The Effect of Weight Transfer
(continued)
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
29
Case Study - Vehicle Modelling Study
LINKAGE MODEL
LUMPED MASS
MODEL
SWING ARM MODEL
ROLL STIFFNESS
MODEL
Vehicle Handling Tests
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).
Computer Simulations
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, ...
Double Lane Change Manoeuvre
30 m
25 m
25 m
B
A - 1.3 times vehicle width + 0.25m
C - 1.1 times vehicle width + 0.25m
15 m
C
A
B - 1.2 times vehicle width + 0.25m
30 m
Lane Change Simulation
Determination of Roll Stiffness
SPH
Applied Roll
Angle Motion
Rear Roll
Centre
Rear Roll
Centre
CYL
Front Roll
Centre
CYL
Front Roll
Centre
Applied Roll
Angle Motion
SPH
INPLANE
INPLANE
INPLANE
INPLANE
Determination of Roll Stiffness
Roll Moment (Nmm)
FRONT SUSPENSION
Roll Angle (deg)
Modelling the Steering System
MOTION
Steering column
Steering
motion applied
at joint
part
Steering rack
REV
COUPLER
part
Revolute joint
to vehicle
body
TRANS
Front
suspension
Translational joint
to vehicle body
Modelling the Steering System
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
Modelling the Steering System
COUPLER
COUPLER
Modelling a Speed Controller
REV
TORQUE
Dummy transmission
part located at mass
centre of the body
COUPLER
REV
REV
FRONT
WHEELS
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
Vehicle Ballast Conditions:
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
Calibration of Dynamic Index
• Excel Spreadsheet
• ADAMS Simulation
• Prodrive Inertia Rig (Quadrifiler)
Calibration of Dynamic Index
ADAMS Quadrifiler Simulation:
Proving Ground Tests
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
Driving on Wet Basalt
Expert Driver
Non-Expert
ADAMS Simulations
ADAMS Simulations
Example Results
20
0.8
Yaw rate
0.6
Yaw rate (g)
15
0.4
10
0.2
5
0
0
0.5
-5
1.5
2.5
3.5
Time (s)
Proving Ground Results
ADAMS Results
4.5
5.5
-0.2
-0.4
Lateral Acceleration (m/s2)
0.7G Step Steer
Yaw rate vs Lateral Acceleration vs Time
Subjective Assessment
Example Questionnaire
Subjective Assessment
Example Questionnaire
Subjective Assessment
Subjective Analysis Average Results for each
DI
9
8
Score from 10
7
6
5
4
3
0.82
2
0.92
1
1.02
0
Subjective Assessment
DI 0.82 Subjective Analysis
10
9
8
Score from 10
7
6
5
4
Damian
DriverHarty
1
3
Lee
Adcock
Driver
2
2
Driver
3
David
Lapworth
1
0
Subjective Assessment
DI 0.92 Subjective Analysis
10
9
8
Score from 10
7
6
5
Damian
DriverHarty
1
Driver
2
Lee
Adcock
4
Driver
3
David
Lapworth
3
2
1
0
TURN IN
Confidence
Accuracy
Body Slip
Control
Rate of
Change
Angle
Lateral
Gain/Grip
Feel
Subjective Assessment
DI 1.02 Subjective Analysis
10
9
8
Score from 10
7
6
5
Damian Harty
4
Lee
Adcock
Driver
2
Driver 1
Driver
3
David
Lapworth
3
2
1
0
TURN IN
Confidence Accuracy
Body Slip
Control
Rate of
Change
Angle
Lateral
Gain/Grip
Feel
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