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Bergamo University Italy12th-14th June 2012
Professor Mike BlundellPhd, MSc, BSc (Hons), FIMechE, CEng
Lecture 7- Full Vehicle Modelling
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Contents
• Underlying Theory (Bicycle Model Approach)• Understeer and Oversteer• Modelling Strategies (Lumped Mass, Swing Arm, Roll
Stiffness, Linkages)• Vehicle Body Measurements and Influences
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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
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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
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“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
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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
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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
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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
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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
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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
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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
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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)
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Understeer and Oversteer
13
• Understeer promotes stability• Oversteer promotes instability (spin)
Neutral Steer
UndersteerOversteer
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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
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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
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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
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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
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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
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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
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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
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Forces and Moments (continued)
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
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Forces and Moments (continued)
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
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Forces and Moments (continued)
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
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Forces and Moments (continued)
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
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Forces and Moments (continued)
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
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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
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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
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• 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
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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
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Case Study - Vehicle Modelling Study
LINKAGE MODELLUMPED MASS MODEL
SWING ARM MODELROLL STIFFNESS MODEL
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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
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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
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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
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Lane Change Simulation
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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
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Determination of Roll Stiffness
Roll Moment (Nmm) FRONT SUSPENSION
Roll Angle (deg)
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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
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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
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Modelling the Steering System
COUPLER
COUPLER
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Modelling a Speed Controller
Dummy transmission part located at mass centre of the body
FRONT
WHEELS
REV
REV
REVTORQUE
COUPLER
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Comparison with Track Test(Lane Change)
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Case Study – Dynamic Index Investigation
Tests Performed at the Prodrive Fen End Test Facility:
•Coordinated by Damian Harty
•Coventry University Subaru Vehicle
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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:
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Calibration of Dynamic Index
• Excel Spreadsheet• ADAMS Simulation• Prodrive Inertia Rig (Quadrifiler)
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Calibration of Dynamic Index
ADAMS Quadrifiler Simulation:
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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
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Proving Ground Tests Driving on Wet Basalt
Non-ExpertExpert Driver
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ADAMS Simulations
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ADAMS Simulations
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Example Results
Proving Ground Results
ADAMS Results
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Subjective Assessment
Example Questionnaire
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Subjective Assessment
Example Questionnaire
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Subjective Assessment
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Driver 1Driver 2Driver 3
Subjective Assessment
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Subjective Assessment
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Driver 1Driver 2Driver 3
Subjective Assessment
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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
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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