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Vehicle modeling utilizing skidpad data

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Title:
Vehicle modeling utilizing skidpad data
Creator:
Garber, John Henry
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English
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xv, 107 leaves : ; 28 cm

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Subjects / Keywords:
Automobiles, Racing -- Mathematical models ( lcsh )
Automobiles, Racing -- Chassis ( lcsh )
Automobiles, Racing -- Springs and suspension ( lcsh )
Formula vee
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 106-107).
General Note:
Department of Mechanical Engineering
Statement of Responsibility:
by John Henry Garber.

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|University of Colorado Denver
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Auraria Library
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515968696 ( OCLC )
ocn515968696
Classification:
LD1193.E55 2009m G37 ( lcc )

Full Text
Vehicle Modeling Utilizing Skidpad Data
by
John Henry Garber
B.S. University of Colorado Denver, 2002
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
2009


by John Henry Garber
All rights Reserved.


This Thesis for the Master of Science
degree by
John Henry Garber
has been approved
by
Ronald Rorrer
fOtfV 30 2.GG[
Date


Garber, John Henry (M.S. Mechanical Engineering)
Vehicle Modeling Utilizing Skidpad Data
Thesis directed by Professor John Trapp
ABSTRACT
Formula Vee racing is one of the most popular forms of amateur racing in
the United States. One of the biggest challenges in any automotive racing
is the proper setup of the car. For this class of vehicle, there are only a few
tire types allowed by the sanctioning bodys rules. Regardless of its
popularity, there is no published manufacturers data for any of the
available tires. The lack of tire data can make choosing the correct type a
matter of trial and error. To compound the problem each vehicles
suspension can have different setups which can change the dynamic
balance of the vehicle and effect the forces on each tire. This also means
that setting up a car the same as a competitor is not the best way to be
competitive because every driver is different as well.
The base line of a vehicle can be determined experimentally. This
publication outlines the principles involved in analyzing the Formula Vee
race vehicle. Areas covered include suspension analysis, derivation of tire
properties, and a look at the classic vehicle roll model and its application
specifically to the Formula Vee chassis.


This abstract accurately represents the content of the candidates thesis.
I recommend its publication.
Signed


DEDICATION
To my family.


ACKNOWLEDGEMENT
My thanks to Joe Cullen for the patience and intellectual support that
helped to guide me and for which this body of work would never have been
possible.


TABLE OF CONTENTS
List of Figures....................................................xi
Preface...........................................................xiv
Chapter
1. Introduction....................................................16
1.1 Motivation for Studying Race Vehicle Dynamics..................16
1.2 Motivation for Research........................................16
1.3 Overview of Research.......................................... 17
1.4 Objectives of Research.........................................17
1.5 Benefits of Research...........................................18
1.6 Literature Search..............................................18
1.7 Outline and Chapter Summary....................................18
2. A Closer Look at the Problem....................................19
2.1 Slip Angle.....................................................20
2.2 Slip and Thrust................................................22
2.3 Low Speed Cornering............................................23
2.3.1 Off Tracking.................................................25
2.4 Initial cornering stiffness....................................26
2.5 High Speed Cornering...........................................28
2.6 Tire Data......................................................29
2.7 Tires..........................................................30
2.8 Tire Model.....................................................31
2.9 Inputs That Effect Slip Angle..................................34
2.10 Steering Geometry............................................34
2.10.1 Taking Advantage of Steering Geometry.......................34
2.11 The Steering Problem.........................................35
3. Classic Vehicle Model...........................................36
viii


3.1 Suspension Geometry and Roll Axis..............................36
3.1.1 Roll Center SLA..............................................37
3.1.2 Suspension Roll Center Height................................40
3.2 Roll Axis SLA................................................40
3.2.1 Roll Stiffness...............................................41
3.2.2 Roll Angle SLA...............................................43
3.3 Distribution of Forces During Turning..........................43
4. Formula Vee Chassis Specific Equations..........................45
4.1 Front Suspension...............................................45
4.1.1 Front Roll Center............................................47
4.1.2 Front Suspension Forces......................................47
4.2 Rear Suspension Roll Center..................................49
4.2.1 Rear Roll Center.............................................50
4.2.2 Rear Suspension Forces.......................................53
4.2.2.1 Jacking Effect.............................................54
4.3 Roll Axis......................................................55
4.3.1 Roll Angle.................................................56
5. Setup, Measurements and Procedure...............................61
5.1 Initial Setup and Static Measurements........................63
5.2 Testing Procedure..............................................64
5.3 Yaw Analysis...................................................65
5.3.1 Rear Suspension and Contact Patch Forces.....................72
5.3.2 Front Vertical Forces........................................77
5.3.3 Finding Tire Constants Cm and Wm.............................78
5.3.4 Yaw and Front Slip Angle Verification........................84
6. Results.........................................................87
6.1 Error Sources..................................................93
IX


7. Conclusion......................................................95
Appendix
A. Table of Variables.............................................96
B. Vehicle setup pictures........................................100
C. How Tires Generate Lateral Forces.............................104
References........................................................106
x


LIST OF FIGURES
Figure
2.1 TURNING ANGLES..............................21
2.2 SLIP ANGLE..................................22
2.3 CAMBER ANGLE AND THRUST.....................23
2.4 LOW SPEED TURNING...........................24
2.5 FORCE VS. SLIP ANGLE........................27
2.6 HIGH SPEED TURNING..........................29
2.7 CALSPANS TIRF TESTING MACHINE..............30
2.8 PARABOLIC CURVE.............................33
3.1 SLA SUSPENSION..............................37
3.2 HALF SIDE SLA...............................38
3.3 INSTANTANEOUS CENTER........................38
3.4 CENTERLINE VIEW.............................39
3.5 ROLL CENTER.................................40
3.6 ROLL CALCULATION VARIABLES..................41
3.7 ANTI-ROLL BAR...............................42
3.8 FBD SLA SUSPENSION..........................44
4.1 PARALLEL ARM SUSPENSION SIDE VIEW...........46
4.2 PARALLEL ARM SUSPENSION TOP VIEW............46
4.3 FRONT SUSPENSION FRONT VIEW.................47
4.4 PARALLEL ARM FBD............................49
4.5 REAR SUSPENSION VIEW........................50
4.6 REAR ROLL CENTER............................51
4.7 INSIDE WHEEL AXLE COMBINATION...............52
4.8 OUTSIDE WHEEL AXLE COMBINATION..............52
xi


4.9 VEHICLE BODY.................................53
4.10 JACKING.....................................54
4.11 FRONT AND REAR ROLL CENTERS.................55
4.12 ROLL AXIS...................................56
4.13 INNER REAR SUSPENSION.......................57
4.14 SUSPENSION GEOMETRY VARIABLES...............58
4.15 VARIABLE LABELS.............................58
4.16 FLUCRUM POSITION............................59
4.17 REAR ROLL ANGLE.............................60
5.1 PROCESS FLOWCHART............................62
5.2 CENTER OF GRAVITY LOCATION...................64
5.3 YAW GEOMETRY.................................66
5.4 FRONT AXLE STEERING GEOMETRY.................67
5.5 FRONT SLIP ANGLES............................68
5.6 REAR SLIP ANGLES.............................70
5.7 REAR INNER SLIP ANGLE........................71
5.8 REAR OUTER SLIP ANGLE........................72
5.9 FBD SUSPENSION ROCKERS.......................74
5.10 FBD INSIDE WHEEL AXLE.......................75
5.11 FBD OUTSIDE WHEEL AXLE......................76
5.12 FRONT FBD...................................78
5.13 CORNERING COEF. AND STIFFNESS PLOT..........79
5.14 BASIC PARABOLA..............................81
5.15 WM VS. CM FOR AY = 0.5......................83
6.1 BASELINE CURVES..............................89
6.2 PREDICTED CURVES.............................90
xii


6.3 TEST UNDERSTEER GRADIENT....................91
6.4 PREDICTED UNDERSTEER GRADIENT...............92
6.5 TIRE PROPERTIES DATA POINTS.................93
B.l TEST VEHICLE...............................100
B.2 DATA AQUSITION UNIT........................101
B.3 FRONT LINEAR POSITION SENSORS..............101
B.4 REAR ROCKER POSITION SENSORS...............102
B.5 STEERING SENSOR............................102
B.6 REAR G SENSOR..............................103
B.7 FRONT G SENSOR.............................103
xiii


PREFACE
I had originally started this work as an investigation into the vehicle
dynamics of the Formula Vee race car. The goal was to define the car in
terms of a multi-bodied dynamic system to learn more about race car
handling dynamics. Having defined the vehicle in mathematical terms, I
was going to make predictions about cornering properties and compare
those predictions to actual test data.
The test platform was a continuation of my under graduate senior design
project: Chassis Design and Evaluation. The purpose of this project was to
design an ultra rigid Formula Vee chassis. After my graduation the
chassis sat in a campus lab for a while. When I returned to the university
to get my graduate degree, in Mechanical Engineering, I knew that I
wanted to further investigate the unique properties of the rear suspension
of this class of race car.
During the course of this research I started finding problems with the
system of equations I had created for solving this dynamics problem. Ill
conditioned matrices were wreaking havoc on this research and caused
delays of its presentation. I started having concerns about being able to
complete my masters degree requirements on time if at all.
I was forced to re-evaluate my research from the start. Searching back
through my text books I discovered that the assumptions that I had made
about tire models were off base and too complex. I needed to find a way to
xiv


determine the properties of the tires that were on the test vehicle. This
was the beginning of a unique method for finding tire properties.
xv


1. Introduction
This chapter explains the motivation for studying race car vehicle
dynamics and this particular research. A general overview of the research
is contained within this chapter. Background information that is relevant
to race car vehicle dynamics and research objectives are introduced. A
description of how this research is intended as a tool for others is
included. Finally, an outline of the thesis and a brief description about the
contents of each chapter are also presented here.
1.1 Motivation for Studying Race Vehicle Dynamics
The technical objective when designing or tuning a race car suspension is
to create a vehicle with good handling characteristics. The configuration of
a race car vehicle suspension has to be done within a practical
interpretation of the rules. For each sanctioning body, the rules are
different for every class of race vehicle. When the rules limit the scope of
suspension modifications, finding the right tire combination becomes very
important.
1.2 Motivation for Research
The average Formula Vee racer typically does not have a budget for
research and development. Most of the testing and tuning is done at the
track. Predicting the adjustments necessary for better handling and
gaining a competitive advantage is a matter of trial and error. If you dont
have a background in engineering and understand the effect on the total
vehicle, optimizing the handling can be very hard to accomplish.
16


One of the best ways to analyze any vehicle handling properties is to
perform a series of skid pad tests. SAE has developed a set of procedures
for testing transient and steady-state handling properties for cars and
light trucks. These procedures are described in described in SAE J266 [1].
Major car manufactures have used these procedures for years to analyze
handling characteristics. Most racers rely on information discovered from
track laps. Seldom do racers ever use such simple tests to analyze their
vehicles. Skid pad testing can help to quickly develop a baseline from
which any changes can easily be evaluated.
1.3 Overview of Research
The use of approximations is common in engineering. They provide a
means to simplify a complex system and yield a useful solution to the
problem. The types of approximations used are determined by the degree
of accuracy required.
The test platform for this research is a race prepared SCCA Formula Vee.
The vehicle has been outfitted with a sensor array to measure its
suspension deflections, steering angles, lateral and longitudinal
accelerations.
1.4 Objectives of Research
The intent of this paper is to find a straight forward method to determine
the handling and tire properties for a Formula Vee that is using biased
ply non-treaded tires. There is no tire model data available for any of the
race car tires for the Formula Vee. For the part-timer or small circuit
racer it is more or less a trial and error process to find the tires and
correct chassis setup.
17


1.5 Benefits of Research
The benefits of this research are to find a low budget method for
evaluating handling performance. Laid out in the following pages is a tool
that can be used for evaluating baseline of a vehicles performance and the
effects of any suspension or tire changes made to the vehicle.
Contrary to passenger cars which have suspensions that are designed for
comfort, high performance race car suspensions are designed and tuned to
help the car stick to the road. For racing, small changes in suspension
stiffness, shock damping, roll resistance, camber and caster angle; as well
as changes in center of gravity can make a measurable difference in
vehicle performance.
1.6 Literature Search
A search of SAE Publications electronically [2] and print [3] result in no
cost effective methods in finding tire properties and using said properties
to find and predict vehicle properties.
1.7 Outline and Chapter Summary
Chapter 2 has an explanation of the problem of tuning a vehicle for racing
including a look at tire construction and force development mechanisms.
Chapter 3 contains a detailed description of the classic mathematical tools
used to analyze vehicle performance. Chapter 4 will show the adaptation
of the mathematical tools from Chapter 3 to be used on the unique
suspension of the Formula Vee. Chapter 5 is a description of the vehicle
setup, measured parameters and the methods used to find quantify
baseline performance.
18


2. A Closer Look at the Problem
Formula Vee is one of the most popular racing classes in the Sports Car
Club of America (SCCA) [4 p. 191]. The requirements of this formula car
class puts limits on engine capacity, weight, tire type, rim and size, front
suspension and steering components, and rear axle and transmission
type. However there are no limitations on the vehicle setup specifically
steering linkage, toe, camber, ride height, and rear suspension design.
Formula Vee chassis are all custom built. There are a few commercial
manufacturers that sell bare frames requiring some assembly as well as
complete kits. Through careful interpretation of the SCCA rules, a custom
chassis may be built from scratch. In either case everyone has their own
opinion on the proper vehicle setup and there is no clear opinion about
what is the best setup.
The Formula Vee, like other classes in SCCA racing, is called a restricted
class. The restrictions limit the customizing of the vehicle in order to
emphasize the preparedness and skill of the driver. The rules establish
limits to adjustments made to the vehicle to ensure homogeny across the
class [4 pp. 191-215]. Adjustments allowed are tire size and compound,
suspension spring and damping rates, anti-roll bar size, camber, toe and
caster adjustments, steering linkage and rear suspension type. All of
these adjustments effect what is called the balance of the vehicle and can
make the winning difference with the right driver.
The balance of the vehicle is critical. Because all Formula Vee chassis are
slightly different, a specific change to one may not work as well on
19


another chassis. Adjustments to the balance of the vehicle become critical
during high speed flat cornering. A suspension system, in general, is a
relatively easy system to analyze. When a vehicle is moving forward along
straight path, on a level surface, the loads on each tire can be determined
from the distribution of its mass. When a vehicle enters a flat corner the
transient condition causes the vertical loads on the tires to change. The
way that the suspension is tuned affects the magnitude of the change of
the vertical loads on the tires.
One of the most important decisions about vehicle handling is tire choice.
In order to be able to choose the correct tire one must understand the
relationship between slip angles and developed lateral forces. There are
only two manufacturers making racing tires for the Formula Vee class
with a few compound and size choices each. Neither of these
manufacturers provides any tire data that would help make choosing one
over the other less difficult.
2.1 Slip Angle
Slip is a term that is used to describe the deformation of a tire when it
changes direction. Slip does not imply that there is any sliding, skidding
or loss of traction. Slip comes from an aerodynamics term used to describe
when an airplanes nose is not in line with its forward velocity vector
during flight. In the automotive industry slip angle a is the angle
measured between the wheels heading and its velocity vector. The
velocity vector is tangential to the path of the tire, see Figure 2.1.
On a turning vehicle the velocity vector of the tire is different than that of
the vehicle. When turning, the steering wheels (tires) are turned to
20


generate the slip angle required to make the vehicle change direction.
This is called the steering angle 8 and is different for the inner and outer
tires. The measurement of steering angle is made between the
longitudinal axis of the vehicle and the wheel heading, see Figure 2.1. The
steering angle required is dependent on the vehicles velocity and the
corner radius.
Heading
FIGURE 2.1 TURNING ANGLES
A tire develops forces in response to steering inputs. When turning, tires
develop lateral forces that keep the vehicle on the path, and at the same
time the tires develop a slip angle. Deformation occurs between the
contact patch and the rolling plane of the tire. This deformation is
measured in angular degrees. During cornering the centrifugal forces from
the vehicle are counteracted by the lateral forces developed on the ground
at the contact patch of the tire. The ability of the tire to distort is what
makes cornering possible. The lateral force is assumed to be directly in
the center of the contact patch.
21


Path of Tire
Slip Angle
Wheel
Heading
Lateral Force
Velocity of Tire
Contact Patch
Distortion
Recovery Area
FIGURE 2.2 SLIP ANGLE
The slip angle is important to a vehicles ability to corner, it is key to how
much lateral force can be developed. There is a maximum slip angle that
correlates to a maximum lateral force on a tire for a given vertical load on
that tire.
2.2 Slip and Thrust
Tires generate lateral forces by two mechanisms, camber thrust and
deformation. Camber is the angle at which the wheel makes to the vertical
when viewed from the front or rear.
22


W/cos B
W tan P
W
FIGURE 2.3 CAMBER ANGLE AND THRUST
Figure 2.3 illustrates a vehicle with a negative camber. Negative camber
is a condition where the top of the tire leans towards the body of the
vehicle. Camber thrust occurs when a tire rolls at a camber angle (/>. This
lateral force component, camber thrust Fyc is always present when the
vehicle is traveling straight or cornering. Whether the top of the wheel is
tilted towards or away from the centerline of the vehicle determines if it is
negative or positive camber, respectively.
Camber thrust varies with suspension movement as well as vehicle roll
and is a function of suspension design. Its overall contribution to the total
lateral force is small if the camber angle is small.
2.3 Low Speed Cornering
At low speed cornering, 5 mph maximum, tires develop negligible slip
angles [5 p. 196]. Body roll and lateral load transfer are negligible as well.
There are lateral forces acting on the front and the rear of the vehicle and
each of the tire rolls on its own concentric circular path. The front tires
23


roll on two different radii, the front inner being smaller than the outer.
Ackerman steering assumes that the center of the turn lies on the
projection of the rear axle and a perpendicular line projected from both
the front wheels pass through the center shown in Figure 2.4.
I
R
CENTER
OF
TURN
FIGURE 2.4 LOW SPEED TURNING
A steering geometry is said to be Ackermann when the wheel angle of the
inside turning wheel is rotated a greater angle into the turn than the
outside wheel. There is a particular Ackermann angle that is dependent
on the track, the wheel base of the vehicle and the radius of the turn.
Proper setup for the front steering geometry, assuming small angles, is
determined mathematically.
24


Steer angles:
l
(2.1)
The average of the two steer angles do and Si is the Ackermann angle in
radians.
If correct Ackermann steering is not established then the tires will
experience scrub. Scrub is when a tire slides in a direction that is outside
of its rolling plane.
2.3.1 Off Tracking
Ackerman is a simple model. At low speeds the rear experiences off
tracking [5 p. 197] which is the distance off the centerline of the turn that
actually locates the rear track. This is because the rear wheels cannot
follow exactly in the same tracks as the front wheels.
(2.2)
Off tracking distance:
( l \
A = R 1 cos
\R y
(2.3)
25


Using a series expansion:
cos(z) =1

2! 4! 6!
(2.4)
Now:
(2.5)
The larger the wheel base l of the vehicle the worse the off tracking
becomes. For the purposes of this research Ackerman steering and off
track calculations are do not apply to this research. The test vehicle is not
making low-speed turns.
2.4 Initial cornering stiffness
All tires have an inherent resistance to out of plane twisting. This
resistance is called the initial cornering stiffness C measured in pounds
per degree. The initial cornering stiffness is dependent on the vertical load
on the tire: for each vertical load Fz there is a unique value of the initial
cornering stiffness. The cornering coefficient C is the initial cornering
stiffness per pound of vertical load on the tire. The units are pounds per
degree per pounds which works out to be deg1.
26


Force vs. Slip Angle
FIGURE 2.5 FORCE VS. SLIP ANGLE
The initial cornering stiffness C is the slope of the lateral force vs. slip
angle curve FY' at a 0 as shown in Figure 2.5. The value of C is
determined by the design, construction, inflation pressure, and the
vertical load on the tire. For low-speed turns the values of the slip angle
are small the initial cornering stiffness is used to determine a tires lateral
force [5 p. 198].
FY = Ca (2.6)
It is possible to derive the initial cornering stiffness for the tire
experimentally using the constant radius test procedure outlined in the
SAE J266 standards [1].
27


2.5 High Speed Cornering
At speeds greater than 5 mph the body roll, weight transfer, lateral
acceleration, and suspension and tire compliance effect handling. During
cornering the lateral acceleration increases as velocity increases or as the
radius of the turn decreases.
Lateral Acceleration:
A
(2.7)
In order for a vehicle to be able to perform a high speed cornering
maneuver the centrifugal force created by the collective masses must be
counteracted by the forces developed at the contact patch of the tire. In
response to cornering, a proportional lateral force is generated. Figure 2.5
shows, in general, the relationship between the vertical loading on the
tire, the slip angle and the lateral force that is generated. The lateral force
increases as the slip angle increases, but without low speed assumption of
a linear relationship. Instead, the slope dFy/da decreases and becomes
zero when the slip angle reaches the value am. After this point Fy
decreases as the slip angle increases. The value of am for a given a tire is
dependent on the vertical load on the tire provided that the load is the
only variable. The curves for every tire differ, and currently can only be
determined experimentally.
In Figure 2.6 each tire has a different vertical loading and slip angle. The
lateral acceleration increases the slip angles on the rear tires, causing the
rear of the vehicle to swing out. This is called yaw and is perceived by the
28


driver as if the vehicle is rotating around the inside front tire. This is
because the path of the vehicle is controlled by the front inside wheel and
any corrections to the course emanate from here.
Vehicle
Path
FIGURE 2.6 HIGH SPEED TURNING
2.6 Tire Data
The traditional way to get tire data is from a tire testing apparatus. The
machine shown in Figure 2.7 is at the Calspan Tire Research Facility
(TIRF) designed and built in 1972. TIRF can completely define and
measure the forces and moments transmitted between the tire and the
road.
29


FIGURE 2.7 CALSPANS TIRF TESTING MACHINE
Well-funded racing like NASCAR, Formula 1, and the IRL can afford to
lease time on one of these testing machines and can collaborate with the
tire manufactures directly to design a tire for their needs.
2.7 Tires
The most complicated part of vehicle dynamic analysis is modeling the
tire. Race tire compounding and construction specifications are closely
guarded of the manufacturers, as well any tire data. Only the most
prestigious and big budget race teams can work with the manufacturers to
design tires to fit their needs. As for the rest of the racing world it is left to
experimentation.
30


All tires share similar features: a bead, sidewalls, and a tread. Aside from
these similarities each series of racing tires are designed with a specific
purpose. There are drag racing, slalom, dirt, road, rock, rain tires and so
forth.
The primary function of the tire is to provide grip. Grip is developed in a
small area called the contact patch and is of major importance for the
dynamic performance of a race vehicle. Longitudinal and lateral forces
that enable a vehicle to accelerate, brake and corner are developed at the
contact patch. These forces are the tires reaction to the loads imposed on
it by the distributed masses of the vehicle and suspension.
The ability and to what limits a tire can generate grip is dependent on
many variables. Physical properties such as compounding, cord type and
angle, height, width, and side wall thickness in part determine the initial
cornering stiffness. Other variables effecting the initial cornering stiffness
are determined by chassis setup and environmental conditions: tire
pressure and temperature, track temperature, static and dynamic loading,
and suspension setup.
2.8 Tire Model
When doing a dynamic analysis of a vehicle a set of complex equations are
used to simulate the tires. Tire rubber is a polymer with viscoelastic
properties. Just by the nature of its characteristics the mathematics
involved in modeling a tire are extensive and non-linear. Typically these
equations have numerous variables many of which have little effect on the
calculation of lateral force.
31


For the purposes of simulation a tire model is required. There are many of
tire models available that range in complexity. Currently popular is the
Magic Formula developed by Pacejka [6] which is continually
undergoing refinement. The Magic Formula and others, including the
Fiala tire model, SWIFT tire model, Radt curves and various parameter
based models, are all designed to reproduce the laboratory curves created
from real data collected on tire testing machines.
For this study many of the many of these formulas are far too complex to
use. For instance, the Magic Formula can have well over 50 parameters
depending on its application [6 pp. 172-196]. Instead, a parabolic curve fit
based on a non-dimensional representation of tire data, developed by Dr.
Hugo S. Radt based on the work of Fiala, will be used. The accuracy of
this method is demonstrated in Chassis Design: Principals and Analysis [7
pp. 59-71],
Parabolic tire model:
Fy
(
= Ca
1-
V
1 Ca "
4 max y
(2.8)
The parabolic model has been found to generate curves similar to the ones
generated on tire testing apparatus [7 pp. 60 66].
32


FIGURE 2.8 PARABOLIC CURVE
In Equation (2.8) and Figure 2.8 the maximum lateral force generated is
ultimately a function of friction coefficient and the vertical force on the
tire:
Fy max = MFz (2.9)
The initial cornering stiffness C in Equation (2.10) is not only a function of
vertical force Fz, but a function of the maximum initial cornering stiffness
Cm and the maximum normal load Wm. The initial cornering stiffness is
Cm when the normal load is Wm. The typically value for ju is 1.20 for paved
roads [7 p. 69]. The values for C and /u may be determined experimentally,
which make Equations (2.8) and (2.9) well suited for this study.
33


C=Func(FCm,Wm)
(2.10)
An important property to note when looking at Figure 2.5 and Figure 2.8,
is that slip angle continues increase while the lateral force increase up to
a maximum then decreases. The value of the slip angle where this occurs
varies from tire to tire and depends on tire loading.
2.9 Inputs That Effect Slip Angle
The force that is developed at the contact patch for a particular can be
modified to peak at different load conditions. This can be done by lowering
or raising the tire air pressure, or changing the amount of camber in the
suspension. On cars that utilize aerodynamic downforce the amount of
vertical load on the suspension maybe adjusted as well.
2.10 Steering Geometry
In Section 2.3 the idea of Ackerman steering was introduced. While this is
well suited for passenger vehicles there are other steering geometries
possible for track racing; Parallel Steer and Reverse Ackerman. For
Parallel Steer the front tires remain parallel to each other through the
whole range of motion of the steering system. On Reverse Ackerman the
inside tire does not rotate as far as the outside tire. Both of these other
steering geometries can have an adverse effect on tire wear but may
provide an advantage in high-speed cornering.
2.10.1 Taking Advantage of Steering Geometry
Referring back to end of Section 2.8, there is a slip angle value am that
corresponds to the maximum generated lateral force. It is possible to alter
the amount of lateral force generated by the front tires by changing the
34


range of slip angles over which the tire operates. This is done by taking
advantage of the different steering geometries. Turning a tire too much or
too little can shift the point where the maximum lateral force peaks.
Ackerman steering works well for low-speed turns, allowing the tires to
track on their own paths, but does not produce the best conditions for
high-speed cornering. Using a Reverse Ackerman setup over steers the
outer front tire progressively as the radius of the turn decreases or as the
lateral acceleration increases. This can cause the front outer tire to reach
the maximum slip angle at a lower turning velocity and may cause the
front to plow through the turn. The Parallel Steer setup allows the front
outer tire to run at a slightly higher slip angle than the Ackerman Steer,
again reaching a higher value of slip angle sooner.
2.11 The Steering Problem
One of the problems encountered in high-speed cornering is that the
inside tire may reach the maximum lateral force that it can produce
before outside tire does. This can affect the maximum cornering speed.
Beyond a specific slip angle value am a tires maximum generated lateral
force starts to decrease. If aFi is too large then the tire is literally being
dragged along. To alleviate this problem the amount of steer angle on the
tire should be reduced. For this study the test vehicle has been setup with
Parallel Steer geometry.
35


3. Classic Vehicle Model
Automotive designs have advanced using basic analysis methods
developed in the middle of the 20th century. Tabulating values and
physical testing have driven new designs in the areas of chassis,
suspension, and tire development.
In the latter half of the 1980s computation methods for analyzing vehicle
design started emerging which allowed engineers to further fine tune
chassis, suspension, and tire designs. This was a quicker and less
expensive way to prototype designs. But without an understanding of how
and why the classical methods worked these new methods could not
evolve.
3.1 Suspension Geometry and Roll Axis
A system of forces from a vehicle suspension system, for a steady-state
turn, can be resolved in to a lateral force, a vertical force and a roll
moment at the suspension roll center. The roll center is a point attached
to the body that is determined by the suspension geometry, and can be
determined by applying a virtual displacement or rotation to the
suspension links. It turns out that the roll center is the instantaneous
center of rotation of the system. After locating the ICR the principle of
virtual work can be used to find the forces in the system. Using these two
methods in a quasi static solution, the effect of the change in vertical load
on the contact patch of the tires can be calculated.
36


Each type of racing calls for specialized suspension setups. Usually the
rules dictate the how exotic a suspension may be. There are several
different types of vehicle suspensions. One of the most common is the SLA
(Short Long Arm) suspension. As a demonstration the location of its roll
center will be detailed. The front and rear roll centers will then be used to
construct the roll axis.
FIGURE 3.1 SLA SUSPENSION
Figure 3.1 shows the SLA suspension in a basic layout. There are many
different configurations possible depending on a vehicles designed
application. In general the SLA suspension allows independent wheel
movement up and down and little change in camber along a straight path.
For turning the SLA offers good control of outside camber.
3.1.1 Roll Center SLA
Finding the instantaneous center of rotation of a suspension system
involves looking at each of the suspension links. To start, look at the
suspension from along the centerline of the vehicle and draw the links as
they exist. Consider the body as stationary and that all the connections
37


from body to the links, from the links to the wheels and from the wheels to
the ground are pin connections and have one degree of freedom.
Analyzing the links involves imposing a second order rotation at points A
and B, see Figure 3.2, then looking at the velocity vector of points C and
D.
FIGURE 3.2 HALF SIDE SLA
Drawing a right angle line to the velocity vector the ICR of the two links
can be graphically determined by the intersection of the two lines.

FIGURE 3.3 INSTANTANEOUS CENTER
Since point E is on the same solid body, the wheel, as C and D the velocity
vector of that point can be found by drawing a line from the ICR to E. The
38


velocity is perpendicular to the line between E and the ICR. Repeat this
procedure on the other side of the vehicle.
In Figure 3.4 the suspension is shown as symmetrical about the
centerline, but some vehicles may have an asymmetrical design. Typically
an asymmetrical suspension is found on vehicles designed to turn in one
direction, but the process of finding the roll center remains the same.
FIGURE 3.4 CENTERLINE VIEW
Combining the left and right hand sketches shows an intersection of the
lines drawn from the ICRs to the tire contact patches. Where these lines
meet the centerline, is the roll center of the suspension for that instance.
39


FIGURE 3.5 ROLL CENTER
Here the analysis has been more qualitative but the mathematics is just
straight forward trigonometry.
3.1.2 Suspension Roll Center Height
The distance from the suspensions ICR to the road surface is called the
suspension roll center height see Figure 3.6. The front and rear height, Hf
and h,R respectively, can be different. Most commonly the front ICR is
designed lower than the rear.
3.2 Roll Axis SLA
The roll axis is determined by the front and rear suspension roll center
height. A line is drawn through the front and rear suspension ICRs, this is
the roll axis for the whole car. As the vehicle speed increases the
centrifugal force on the sprung weight causes it to rotate about this roll
axis. Dynamic calculations are made for the roll angle based on the height
of the center of gravity from the roll axis. The front and rear suspension
tries to resist this over turning moment via their individual roll stiffness.
40


FIGURE 3.6 ROLL CALCULATION VARIABLES
3.2.1 Roll Stiffness
The roll stiffness of the suspension is a designed parameter. On a SLA
suspension that has a suspension spring and damper on each tire when
the vehicle corners, the body rolls towards the outside of the turn. The
outside suspension springs are compress resisting roll, but the opposite
happens for the inside wheels as the springs unload driving the roll. If the
cornering speed is fast enough the vehicle will become unstable and spin
out or flip over. In order to correct this problem an Anti-roll bar is added
to the system. This is usually some sort of transverse spring linking the
wheels on a single axle to each other and the chassis, and is installed on
the front suspension. The spring increases stiffness as the vehicle rolls,
which decreases the vehicles rate of roll and redistributes the vertical
loads.
The simplest of Anti-roll bars is a U-shaped piece of spring steel attached
to the suspended chassis and each end to the suspension linkages.
41


FIGURE 3.7 ANTI-ROLL BAR
The roll of the body increases the effective stiffness of the Anti-roll bar by
two mechanisms; torsional resistance and bending resistance. Here is a
typical calculation of the front suspension stiffness.
Front Roll Stiffness:
K
= K
+ K
&SF
(3.1)
The front roll stiffness is the sum of the Anti-roll Bar stiffness due to body
roll and the suspension stiffness due to the springs Equations (3.2) and
(3.3)
K
k s^
Kbar*RB
(3.2)
1 ( Grl 3EI'
R2 { lt ' l3 Lb y
(3.3)
42


(3.4)
K K t2
2 SPrinSl'F
Typically the first adjustment in roll stiffness is the installation of an
Anti-roll bar on the front suspension. A turning vehicle experiences the
greatest change in force at the front outer tire.
3.2.2 Roll Angle SLA
The roll angle is a measurement of how much the chassis rotates about
the roll axis. In Figure 3.6 the center of gravity is above the roll axis a
distance hi, which directly affects the amount of chassis roll. The greater
this distance the more exaggerated the roll angle.
Roll Angle:
\\VAy
~K^F+K^R-Wh1
(3.5)
3.3 Distribution of Forces During Turning
If all the right hand parameters in Equation (3.5) are known, chassis roll
of a vehicle traveling around a curve can be calculated. Using the value for
(j), the change in mass distribution and the amount of vertical force, the
suspension exerts on the tires can be calculated. The change in vertical
load effects the initial cornering stiffness and alters the tires ability to
generate lateral force over a range of slip angles.
43


FIGURE 3.8 FBD SLA SUSPENSION
Summing the moments around O The change in vertical loading front:
AFzf -
Wp-Ayflp
tF
Similar to the front the change in vertical loading rear:
(3.6)
^ZR ~
^4 + WRAyhR
(3.7)
This is added to the static vertical load calculation for the outside front
tire and subtracted from the inside front tires static vertical load
calculation. The process is the same for the rear of the vehicle. Equations
(3.5), (3.6) and (3.7) can be coded into a spreadsheet and tabulated for
different values of lateral acceleration.
44


4. Formula Vee Chassis Specific Equations
The test platform used in this study is a race prepared Formula Vee which
has a parallel trailing arm front suspension and a swing axle rear
suspension. One of the challenges involved in analyzing the dynamics is
adapting the classic vehicle models to this chassis. Unlike most vehicles,
the rear suspension linkage on a Formula Vee is setup to eliminate a
problem that occurs with the swing axle suspensions called jacking.
Section 4.2.2.1 covers the mechanics of jacking in more detail. The result
of this special setup is that the rear suspension is uncoupled from the
chassis which takes the rear suspension stiffness out of the roll angle
calculation, and puts a larger requirement on the front anti-roll bar to
control the entire vehicle roll.
4.1 Front Suspension
The Formula Vee uses a parallel trailing arm suspension on the front, as
shown in Figure 4.1 and Figure 4.2. It is characterized by the upper and
lower trailing arm linkages staying parallel with one another as the wheel
moves up and down.
45


Upper Trailing Arm
Lower Trailing Arm
V/////////7777/////////7//////
FIGURE 4.1 PARALLEL ARM SUSPENSION SIDE VIEW
From the top view of the chassis, the transverse tubes house torsional
springs see Figure 4.2 & Figure 4.3. In the stock configuration, both upper
and lower tubes house the springs and there is an anti-roll bar attached to
the chassis and to each of the bottom trailing arms, left and right side.
The Formula Vee is about half the mass of the standard Volkswagen
Beetle, so commonly one of the torsion springs are removed and replaced
with an anti-roll bar.
FIGURE 4.2 PARALLEL ARM SUSPENSION TOP VIEW
The spindles are attached to the trailing arms with a link bolt and the
camber is adjusted by placing shimming disks between the mating
surfaces.
46


4.1.1 Front Roll Center
This type of suspension linkage is considered to have a roll center at the
ground along the centerline of the vehicle [7 p. 332],
4.1.2 Front Suspension Forces
Similar to the SLA suspension the vertical forces can be calculated. The
front roll stiffness comes from two springs; a torsional suspension spring
and a Anti-roll bar spring.
From Section 3.2.1
K. =K
i
+ K
^SF
(4.1)
The Anti-roll bar is similar to the SLA calculation but there is no bending
arm.
47


f t \
ka =
GJ
R2
RB
(4.2)
\ "t s

1 ( f \ (ll.2xl06psi) (0.625in)4 V 32) \
(5.98m)2 36.375in

(47.24m)2
(4.3)
= 287840.8 in-lb/rad
For the Formula Vee front suspension the torsion spring is only as long as
the roll bar.
~ SRB (4.4)
K K 2 2 IYsPrinet>RB (4.5)
K^f = (33.9/6/m)(47.24m)2 2 (4.6)
= 37825.9 inlb/rad
The total roll stiffness is
=235666.7 in-lb/rad (4.7)
48


FIGURE 4.4 PARALLEL ARM FBD
The front roll height is zero which changes Equation (3.6).
AFzf -
tp COS (j)
(4.8)
4.2 Rear Suspension Roll Center
The rear suspension of this vehicle is very unique it utilizes a swing axle.
There are two short axles connected to the transmission housing called a
transaxle. At the inboard point of attachment to the transaxle there is a
pivot composed of a spade on the end of the axle, two fulcrum plates and a
slotted spider gear that engages the differential. Outboard, the brake
drum is keyed to a spline axle end and fixed with an axle nut; the rim and
tire are bolted to the brake drum. The whole axle is one rigid assembly
with a rotational axis about its length that transmits power to the tire.
The axle is constrained to rotate about the fulcrum point when the
suspension moves up and down.
49


Unlike most vehicles where the springs are attached to the body, here the
left and right linkages are attached to one another via a common spring
and damper see Figure 4.5.
The primary function of the suspension, outside of keeping of the rear of
the chassis off of the ground, is to allow the body to roll without affecting
the camber angle of the rear tires. With the correct suspension linkage
geometry the rear suspension can achieve this, in doing so the suspension
becomes uncoupled from the effects of body roll. This design eliminates
the jacking effect typically associated with the swing axle suspension. On
the other hand now the roll stiffness of the rear has been eliminated and
must be compensated by the front suspension.
4.2.1 Rear Roll Center
Figure 4.6 shows a simplified view of the rear suspension of the test
vehicle, the set up is shown in a zero camber orientation. The wheels and
axles are rigidly connected thus a line may be drawn from the contact
patch Cm and Cr0 to the fulcrum points Oi and Oo. The Line CO represents
50


a virtual link, connects the suspended body to the ground at the tire
contact patch.
Like the SLA suspension a second order rotation is introduced. The rear
roll center location is determined by the intersection of two lines drawn
from the center of the rear tire contact patch and the pivot point of the
rear suspension.
51


FIGURE 4.7 INSIDE WHEEL AXLE COMBINATION
Taking the inside wheel axle combination and imposing a second order
rotation the direction of the velocity vector at Oi can be determined.
% = rcr x % (4.9)
By similar process the outside is determined.
FIGURE 4.8 OUTSIDE WHEEL AXLE COMBINATION
52


ICR
(4.10)
(4.11)
The exact location of the roll center can be calculated using the rocker
angle measurements. These equations are detailed in Section 4.3.1 for the
rear roll angle calculations.
4.2.2 Rear Suspension Forces
The rear suspension experiences forces at the contact patch, fulcrum
points and rocker these are an effect from the vehicles share the
distributed masses supported by the rear axle. Because of the unique
geometry of the rear suspension and the fact that there is almost zero roll
stiffness. The rear roll stiffness cannot be determined using the same
methods applied on a SLA suspension.
FIGURE 4.9 VEHICLE BODY
VOJO, voa vQi
1rcr x(ro0- r0,)
= (O
aRCR x rojol
VO0 ~ RCr X rO / ICR
53


The special geometry of the rear suspension is to reduce the risk of the
rear axle to jack under hard cornering. To understand the geometric
properties of the suspension a description is required. Remove the front
tires and setup the front beam to pivot about its roll center, at the ground.
The chassis would roll all the way to the suspension stops, while at the
same time the rear suspension camber would remain unchanged. The
chassis would be free to rotate uninhibited left or right. There is
absolutely no mechanism in the rear suspension geometry to stop the
chassis from rolling. The unaffected camber angle prevents the outside
axle from developing an overturning force that would normally cause the
rear to lift.
4.2.2.1 Jacking Effect
A vehicle with a swing axle rear suspension can become unstable when
cornering. Both tires develop lateral forces, with the outside one having
the largest value of the two. The lateral force on the outside tire tries to
lift the vehicle, causing both tires to tuck under. The suspension geometry
helps drive the instability because of the location of the axle pivots and
raises the rear ICR and the center of gravity.
FIGURE 4.10 JACKING
54


The combined effect of the wheel tuck and the rise of the rear ICR is that
the lateral force on the rear axle decreases, which increases the
probability of the vehicle to spin out and over turning.
4.3 Roll Axis
The front suspension geometry results in a fixed roll center at the ground
along the centerline of the vehicle. The rear suspension roll center floats
around a stationary theoretical roll center as shown in Figure 4.11.
FIGURE 4.11 FRONT AND REAR ROLL CENTERS
The roll axis is defined by a line that intersects each of the roll centers
defined by the front and rear suspension geometry.
55


Front ICR'
FIGURE 4.12 ROLL AXIS
4.3.1 Roll Angle
Laid out in the next few pages are the geometry equations used to find the
roll angle using the suspension rocker angles Oik and 9r0. Inner suspension
variables use the subscript i , and outer suspension variables are
denoted with the subscript o.
Rocker Length:
In the following diagrams, physical links at represented by thick lines and
virtual links are represented with dashed lines.
(4.12)
Pushrod Length:
(4.13)
56


Rear Roll
wRi/ fci A :

1 i
1 T J

7777777ZW777777777777777^777777777?.
FIGURE 4.13 INNER REAR SUSPENSION
Position of Di from Oi
XDi ~ (Lb Lq ) + LRi cos 0Ri
(4.14)
yDi ~ LRi sin dR[ Hp
(4.15)
Enables definition of Idi
tan0=^
X
Di
lDl = yl(Xoi)2 +(yDif
(4.16)
(4.17)
57


C,
\b..LEl
,'O,
Pa

y'L
Ci
H.
V///7//////////////////)//?/////////,
V.
FIGURE 4.14 SUSPENSION GEOMETRY VARIABLES
The diagrams in Figure 4.14 and Figure 4.15 show the variables used in
all the geometric relations between the suspension nodes. The virtual
links were created to simplify the geometry between the axle, shock
pushrod and the suspension rocker.
FIGURE 4.15 VARIABLE LABELS
Using the Law of Cosines.
T2 T2
^Pi ~ ^Ei
+ lDi 2IjEilDl cos0Oi
(4.18)
58


(4.19)
T2 + 72 T2
COS0Ol= El a p'

Ai ~ @0i + P(a @Di
sin0 = ~~ sin 8()i
L'Pi
(4.20)
(4.21)
Figure 4.16 details the variables used to find the fulcrum height which is
used in calculating the rear roll angle.
7777777Z77777777777777777P77777777?
FIGURE 4.16 FLUCRUM POSITION
The roll angle of the rear suspension can be determined from the
suspension rocker angles Ori and 9r0. Equation (4.22) reflects the usage of
the angles formed between the virtual link from the contact patch center
to the fulcrum pivot and a line through both the fulcrums see Figure 4.17.
The vehicle is drawn in an exaggerated state of roll that is the result of
the vehicle making a left hand turn and the body rolling to the outside of
the turn.
59


La sinea LCo sin0Co
FIGURE 4.17 REAR ROLL ANGLE
Here is the specific roll angle calculation for this particular rear
suspension geometry.
Rear Roll Angle:
tan^fl =
Lasin0cl-Lcosin0co
2Lo + Lc, cos a + Lco cos Co
(4.22)
60


5. Setup, Measurements and Procedure
The test vehicle initial settings have been dialed in and remain
unchanged during the entire testing session. No modifications were done
to the steering, suspension or the tire pressures.
The baseline of the vehicle and the properties of the tires can be
determined experimentally using skidpad tests. This is accomplished by
measuring and recording real time vehicle data with a data acquisition
system, using an array of sensors to measure suspension deflections,
lateral and longitudinal acceleration, and steering angles.
The individual slip angles on the tires, when examined at collectively, can
be used to establish the baseline measurement of chassis setup and derive
the tire properties. Ideally the attitude ^of the vehicle as it moves around
the circle would be good information to record using a perpendicular radar
system or high speed photographic methods, but both are expensive and
would not be accessible by most amateur racers. Instead of measuring y/
directly, the sensor data and tire model are used in an iterative method to
find the attitude angle. The coefficient of friction is also determined using
this method. When // is chosen too high Wm will not converge and when
chosen too low Cm does not converge. In Figure 5.1 the steps for this
iterative process is described in a flow chart.
61


FIGURE 5.1 PROCESS FLOWCHART
62


In Chapter 4 the equations for rear suspension deflection, front
suspension roll stiffness, roll angle and load transfer were derived. In this
chapter the accuracy of these equations will be tested. This will be done by
finding the roll stiffness and roll angles experimentally, then evaluating
the suspension geometry equations, mathematically, for the Formula Vee
chassis and correlate those results with experimental data. The resulting
data, from the suspension geometry, will be used to determine the initial
cornering stiffness, build the tire model and construct a handling diagram.
5.1 Initial Setup and Static Measurements
The test vehicle was set up with parallel steering geometry as described in
Section 2.10. The camber angle y was set at -0.5 degree and the toe in was
initially set to +0.25 for both the front and rear axles. All tires were of the
same make, model, batch, and all tire pressures were set to 20 psi. All the
tires were balanced were balanced on matching rims.
Before the start of testing static data was taken for the test vehicle
including: static loading on each tire, tire rolling radius, suspension
deflection and spring rates, roll stiffness, and location of the center of
gravity to mention a few.
The location of the vehicles center of gravity along its longitudinal axis
was determined by measuring corner weights. Next the front of the
vehicle was raised to different heights with the rear wheels still on the
scales. The shift in wheel loading and the angle of inclination were
recorded at these different positions and used to determine the vertical
location of the center of gravity.
63


When the driver sits in the vehicle it has reached its testing weight. The
location of the C.G. for the vehicle changes slightly when a driver is
seated. Using information compiled on human factors the C.G. is corrected
for the loaded vehicle [8 pp. 705-772]. This calculation was used to find hi,
the height of center of gravity above the roll axis.
5.2 Testing Procedure
Skid pad testing is a practical and economical way to establish the
vehicles baseline performance. The baseline of a vehicle can quickly be
determined by isolating parameters. The repeatability of skidpad testing
allows variation of parameters to be used to determine vehicle baseline
performance.
The use of a skidpad is ideal for this test. The method used was to hold
the skidpad radius constant, vary the lateral acceleration and measure
the steering angle and suspension deflections. The vehicle test speeds
were increased by increments no greater than 0.10 g until the vehicle
64


could no longer hold the skidpad radius. The vehicle was run on a
constant radius circle of 15 meters [1], and the data was collected over
multiple left and right hand turns. The data acquisition system was set to
collect values for the suspension deflections, lateral and longitudinal
accelerations, wheel speeds and steering angles.
Each data pass consisted of entering the skidpad at a predetermine speed
and holding that speed for anywhere from five to seven laps. Each lap was
timed and recorded. The test vehicle was stopped and the sensor data was
downloaded to a laptop computer. The same test was run again but the
skidpad was entered from the opposite direction so that data was collected
for clockwise and counterclockwise direction of travel. Opposing runs at
the same lateral acceleration kept the suspension from developing a set,
ensuring even wear on the tires and averaging out any testing surface
irregularities.
5.3 Yaw Analysis
Finding tire properties is dependent on being able to determine the
chassis attitude angle if/. This angle, also known as yaw, can be used to
find the slip angle of the tires. As the vehicle travels around the skidpad
the driver is controlling its trajectory by keeping the front inside tire on a
predetermined path. In this test that path was an offset distance defined
by the cones that outlined the perimeter of the test circle.
65


\
Vehicle
Path
FIGURE 5.3 YAW GEOMETRY
The geometry of the steering wheels and how they rotate about the
steering knuckles on either side of the vehicle are used to compute the slip
angles for a specific value of the chassis attitude y/. The wheel rotates
around the spindle and when the car is steered the spindle/wheel
combination rotates about the king pin axis. This movement also rotates
the tire contact patch fore and aft of a virtual straight line that would
connect the inner and outer tire contact patches if the vehicle had no
steering input.
66


p
FIGURE 5.4 FRONT AXLE STEERING GEOMETRY
The angles that the wheels travel through are their respective steering
angles. Trigonometry is used to find the changes in the positions of the
contact patches. In this system the contact patch on the inner front tire is
considered to be the point about which the chassis yaws. The relative
position for the outside front contact patch is calculated from point A, see
Figure 5.4.
The chassis yaw angle y/ and the steering angle 8 are used to calculate
the front slip angles. Additionally an angle 9b is used to determine the slip
angle on the outside tire. These angles will be calculated using vector
mathematics.
xB = 2lsp sin(£; +y/) + tk sin(^)
yB = 2lSP cos(<5i +y/) + tk cos(y/)
(5.1)
(5.2)
vA = Rco
(5.3)
CO cok
(5.4)
67


(5.5)
vB a>kx rB
FIGURE 5.5 FRONT SLIP ANGLES
rB = [/SP sin(8i + y/) + tk siny/ + lsp sin(SQ + y/)]i
+ [lSP cos(St + y/ ) + tk cosy/ + lsp cos(50 +w)]j
vB = + lsp cos(Si +y/) + tk cosy/ + lsp cos(<70 +y/)\
+ft>[Zsp sin(J( +y/) + tk siny/ + lsp sin (8a + y/)]j
(5.6)
(5.7)
68


The angle Ob is the difference between the velocities uA and uB which can
be found by taking the inverse tangent of the X and Y components of uB .
vBX = R + lsp cos{Si +y/) + tk cosy/ + lSP cos(S0 +y/) (5.8)
vBY = lSpS(8l+y/) + tksiny/ + lSPsm(S0+y/) (5.9)
0B= tan-1 {^4 (5.10)
\VBX\
Now the front slip angles can be calculated.
aFl=St+y/ (5.11)
69


The slip angles for the rear are composed of the yaw plus their respective
velocity angles.
FIGURE 5.6 REAR SLIP ANGLES
70


Rear inner slip angle equations.
|YC| = R + lSP cos(c7 +y/^ + cosy/ + Lsiny/ cosy/
2 2
\XC\ = -lsp sin(£. + y/)- siny/ + Lcosy/ + siny/
2 2
a\XC
FIGURE 5.7 REAR INNER SLIP ANGLE
i \Xr\
9r = tan 11
Y
rc|
aRi = ys + @c
(5.13)
(5.14)
(5.15)
(5.16)
71


Rear outer slip angle equations.
(5.17)
(5.18)
FIGURE 5.8 REAR OUTER SLIP ANGLE
lY^I = i? + lSP cos(£ + y/} + cosi// + Lsin + cosy/
2 2
|X01 = -lSP sin ( 2 2
ix
D
(5.19)
r d\
aRo=V/ + 0D (5.20)
5.3.1 Rear Suspension and Contact Patch Forces
The data from the suspension position sensors and the accelerometers
contains enough information to calculate the forces in the rear suspension
and the lateral and vertical load transfers. The load transfers are
calculated using the sensor data and the geometric relationships
developed for the rear suspension in Chapter 4. This calculation is
possible because the rear suspension is a swing axle with no roll stiffness.
72


The forces in the suspension spring are used to calculate the pushrod
forces. In the static position, the force in the suspension spring needs to
support the rear weight of the vehicle. The spring rate ks is known and
using the geometry of the rockers the distance Ls between psi and ps0, in
Figure 5.9 is calculated. The forces in the rear suspension and the
distance Ls are used to compute an effective uncompressed length for the
spring L0. This length L0 will be used to calculate the spring force Fs
during testing.
Spring force during testing
Fs=ks(Lo~Ls) (5.21)
The suspension rocker sensor values were used for recalculating the
spring force once the chassis has rolled during the skidpad test. The
rocker sensors have 0.01 degree resolution. The rocker angles completely
define the rear suspension geometry. The FBD for the rockers are shown
in Figure 5.9 this figure also shows angles and lengths used to complete
the description of the rear suspension geometry. A spreadsheet was used
to take the two rocker arm angles, from rocker sensor data, and compute
the deflection of the rear suspension as shown in Figure 4.17.
73


FIGURE 5.9 FBD SUSPENSION ROCKERS
When the chassis rolls, the attitude of the suspension spring changes and
the change can be determined with Equation (5.22).
6S = arcsin
Lr sin(8Ro + ) Lr sin[0Ri + fiR)
2Lb + Lr cos(0Ro + PR) + Lr cos(0Rl + PR)
(5.22)
The spring attitude and force can be used to solve for inner and outer
pushrod forces by summing the moments about the pivot points Bi and B0.
After determining the suspension spring and pushrod forces, the forces at
the tire contact patches of the rear tires are computed. This is
accomplished by summing the moments about the fulcrum points, Oi and
00 and solving for the changes in the contact patch forces AFyr and AFzr.
The combined weights of the axle and wheel Wa on each side of the vehicle
are included in the summation of forces. The location of this center of
mass is Lw along the axle measured from the fulcrum (not shown in the
figures). The lateral acceleration value from the data is used for this
74


calculation. The spreadsheet is used to calculate how the geometry and
forces change when the chassis rolls.
Rear Forces inside:
xl= FPlcos(0Pl-^) +
fw,
(W 5
-fAY-AFYR
v z y
cos
+
-A F,
ZR
sin^ WaLw cos(0Pi (/>)
yi =-FPlsin(0Pi -)-

Ay A Fyr
sin^
+

A F,
ZR
cos(j> + WaAyLw cos(6pi )
(5.23)
(5.24)
75


FIGURE 5.11 FBD OUTSIDE WHEEL AXLE
Rear Forces outside:
xo =-Fpocos{Qp0+ ) +
W
\
R AY + AFra
v ^ y
cos^
+
W
+ A F
ZR
s\n + WaLw cos(dPo + (j>)
(w }
Oyo = FPo Sin {ePo + 0) + AK + hFYR J Sin
wD
+ A F.
ZR
cosif> + WaAyLw cos[6P0 + (j>)
(5.25)
(5.26)
76


Rear moments around the fulcrum inside:
AFm sin(#c; -) + AFm cos(0Cl -$) =
-^-(cos(ea -(f) + AY sin(#a - -^FPisin(ePl~(9Al-pE))
-^-(WAcos(6Al -<1>) + WaAy sin(0Ai-))
Lc
Rear moments around the fulcrum outside:
AFyr sin(#Co +

W
-y (cos(0Co +0) + Ay sin(0Co + ))
~Y-FPo sin(^Po -(Ao-Pe )) (5'28)
-^(WA cos(0Ao +)- WaAy sin(0Ao + (/>))
L'C
The resulting moment equations from both axles are used to find the A Fzr
and AFyr using Cramers Rule for solving a 2x2 matrix.
5.3.2 Front Vertical Forces
The change in the vertical force AFzf on the front axle can be easily
determined using the equations from Section 4.1 that use the front
suspension roll stiffness. The two linear suspension sensors on the front
axle measure the deflection of the suspension members. Using the
recorded difference between the two sensor readings and the fixed
distance between the sensors the roll angle of the chassis is calculated.
77


The roll angle values were also compared to those obtained with the
rocker angle data.
The change in vertical load as the chassis rolls is solved by taking the
moment about the roll center for the front axle.
FIGURE 5.12 FRONT FBD
Because the roll center is at the ground Hf- 0. Equation (3.6) can be
reduced to the following:
K,Fd>
AFzf = (5.29)
tp COS (p
5.3.3 Finding Tire Constants Cm and Wm
The iterative process for determining /i, Cm and Wm starts with choosing a
reasonable value for //.// = 1.20 was originally chosen as a reasonable
value for reasons described in Section Error! Reference source not
found.. For these skidpad tests the iterative process converged to // = 1.0.
78


After a value for jj, is selected, the two constants Cm and Wm for the tires
will be determined. The data from the skidpad has produced values for
steering angle 8, the vertical load Fz on all the tires, and the lateral forces
Fy on the rear tires for eighteen different values of the lateral acceleration
Ay The slip angles for each rear tire will be calculated after assuming a
value for the yaw angle y/. The yaw angle y/ is iterated to satisfy the slip
angle geometry and /j, is iterated to get consistent values of Cm and Wm
across the range of Ay.
It has been found, in actual tests, that the value of the cornering
coefficient C' decreases linearly with the increase of vertical load [7 p. 67].
C'is the slope of the initial cornering stiffness C and this implies that
there is a parabolic relationship between C and the vertical load Fz.
79


The resulting equation for the initial cornering stiffness is:
(5.30)
For a specific tire there is a maximum load rating called Wm, which is
related to its design and manufacture. The same tires are used on both
sides of the rear axle, which allows the use of an equality of the tire
constants from the inside to the outside to find Wm and Cm.
Assuming a parabolic relationship of slip angle to lateral force:
Fymax is the value of the maximum lateral force that a tire can generate
for a given vertical loading.
Fy = Ca 1 -
1 Ca
(5.31)
V
Fy max ~ MFz
Y max
(5.32)
80


Making a substitution for i^Ymax [9 p. 271] results in an alternative form of
Equation (5.31)
Fy = Ca
(5.33)
Rewrite Equation (5.33) and solve for Ca using the quadratic formula.
(Ca)2 4nFz (Ca)+ 4/jFzFy = 0
Ca
4mFz ^(4juFzf -1 QmFzFy
2
(5.34)
(5.35)
From the yaw analysis am and am are known from the assumed yaw
angle, and from the rear suspension analysis the value for Fz an Fy for
each rear tire are known. The initial cornering stiffness for both of the
rear tires can now be calculated.
81


c =
2fuFz
a
i*U
VFZ
(5.36)
Solving Equation (5.36) will result in a high and a low value each for Cm
and Cro.
Equation (5.30) is used to solve for Cm.
C =
C
( F ^ 1 Z ( f 1 z
W V VYm y 1*0
(5.37)
Equate Cm for the both of the rear tires. Each has its own value for C and
Fz.
C = C
'~/mRi mRo
C
Ri
c
Ro
2 F.
r 1 '
Zi
w
V ¥rm
F 2
1 Zi
' 1 ^
2 F,
f 1 '
Zo
W
V VYm
-F 2
1 Zo
( A2
w
V 7
Solve for Wmin Equation (5.39).
(5.38)
(5.39)
W
FZn2Cm-CRoFZi2
%{FZoCRi FZiCRo)
(5.40)
Due to Equation (5.36) the solution to Equation (5.40) will produce high
and low values for Wm. There are four permutations of Cr that may be
used in Equation (5.40). These values of Cri and Cr0 are from Equation
(5.37) and if the high values of Cri and Cr0 are used then the resulting
value of Wm is too low to describe the tires used in this test. In all cases
the low values for Cr from Equation (5.36) must be used.
82


Equation (5.37) can be plotted for the Cmi and Cmo to find the convergence
value for Wm.
If the value Wm is less the than l.lOWn. the tires are inadequate for their
purposes [7 p. 69] and may lead to unstable vehicle behavior in a turn.
Values below 0.5 Wr are disregarded as improbable because these values
are too low.
0 100 200 300 400 500 600 700 800 900
FIGURE 5.15 WM VS. CM FOR AY = 0.5
A check was made that the value of Wm satisfies Equation (5.41).
CBi
c
Ro
2 F,
Zi
V Wm J
-F 2
1 Zi
' 1 ^
w
V ¥ym /
2 F,
r 1 "
Zo
\WmJ
-F 2
1 Zo
' 1 ^
= 0
KWmJ
(5.41)
83


5.3.4 Yaw and Front Slip Angle Verification
Convergence to the correct yaw angle can be achieved by comparing the
calculated front slip angles from the yaw analysis in Section 5.3 to the
predicted values using Cm and Wm in a force analysis on the front
suspension. The correct yaw angle is achieved when there is no difference
in these angles. Values for the lateral acceleration Ay, front weight Wf,
and the vertical loading on each front tire are used in a force analysis on
the front suspension to compute front slip angles.
Use Equation (5.30) to compute Cn and Cfo, which results in Equations
(5.42) and (5.43).
(5.42)
(5.43)
The values of Fzi and Fzo were calculated in Section 5.3.2 The total
lateral force is Equation (5.44)
= 2 Ca
CFo = 2 C
f f A
1 Zi
w
\ VVm y
f F ^
1 Zo
w
V VYm J
-C,
-c
(F x
1 Zi
uO
(F Y
1 Zo
W
V J
Fyt = WpAy (5.44)
To solve for the outer slip angle ccf0 use Equation (5.33) to describe the
lateral forces in terms of the front slip angles, which results in Equations
(5.45) and (5.46).
Fy, CFiccFi
4 f*F.
-C2 a2
FiUFi
Zi
(5.45)
84


(5.46)
Fyo CFo(ZFo
4 mF.
-C2 a2
VjFoaFo
Zo
Combine to find Fyt
F = F +
L yt l Yi T
Yo
(5.47)
Substitute Fi.
ff =
(5.48)
A a
aFi ~ aFo = arctan
tF siny/
R + tF cos y/
(5.49)
Combining Equations (5.47) and (5.48) using Cfi, Cf0, Fzi, and Fzo will
create a parabolic equation in terms of aFo.
CFi (aFo + Aa)
C2
^ Fi
4 HF,
+FFoaFo
C2
Zi
4 fiFi
Zo
a = W A
aFo (550)
Expand and use the quadratic formula to solve for aFo, then find aFi using
Equation (5.48) and (5.50).
The front slip angles just computed are now compared to the two front slip
angles from the yaw analysis. Recall that the slip angles from the yaw
analysis were computed from an assumed value of y/. If these slip angle
values do not match, then change the value of y/ in the yaw analysis and
reiterate. Updating the yaw angle will change the values for the rear slip
angles and will in turn change the values for Cm and Wm that were
85


calculated in Section 5.3.3. The calculations are iterated until the front
slip angles computed in the yaw analysis agree with the front slip angles
computed from the values of Cm and Wm in this section.
86


6. Results
Evaluating the baseline, decisions about chassis alterations necessary to
increase performance can be made. The best method for evaluating and
understanding possible changes is to use a spreadsheet.
Evaluated in the following performance diagrams Figure 6. land Figure
6.2 is the roll angle, steering angle, yaw angle, and the averaged values of
the front and rear slip angles. All are plotted as a value dependent on
lateral acceleration vs. angular degrees.
The data points in Figure 6.1 represent the averaged date from all the test
runs. A trend line has been run through the points in order to produce a
clearer picture of the performance data. There are 4 conclusions that can
be drawn from the graph.
1. Looking at the roll angle $ it can be seen that the roll stiffness of
the vehicle is very high, rolling 1 degree over the range 0 to 1.0 g.
This is a very desirable trait in a vehicle that corners at high
speeds. A typical passenger vehicle may have roll rates of 8 deg/g or
more.
2. The yaw y/of the vehicle gets larger as the acceleration increases.
This is typical for a rear or mid-engine vehicle. Excessive yaw is an
indication of the vehicle nearing the point of being uncontrollable.
In fact during testing, on the high speed runs the driver was almost
looking at the center of the skidpad and control of the vehicle was very
difficult to maintain.
87


3. The steer angle 5 decreases as the acceleration increases. Which
gives the feeling similar to coming out of a turn as the vehicle
accelerates.
4. The difference between cif and ccr, which is called the understeer
gradient. Figure 6.3 shows how the front and rear slip angles
change with respect to each other and ultimately how predictable
the vehicle handle will be. This is a case of oversteer.
The overall balance of this vehicle is pretty good. However, there can be
improvements made to lower the amount of oversteer and yaw gain. The
goal is the boost the manageability of control at higher rates of lateral
acceleration.
An increase of the initial cornering stiffness on the rear wheels will help
to decrease the understeer gradient and the yaw. The simplest way to
achieve a gain in the initial cornering stiffness is to increase the inflation
pressure in the tires. A 25% increase in stiffness, about 5 psi will start to
have an effect on handling, please see Figure 6.2 and Figure 6.4. Note that
the yaw gain has decreased and that the steer angle is starting to flatten
out. As well, the front and rear slip angles are closer together.
88


32.0
FIGURE 6.1 BASELINE CURVES
89


32.0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10
Ar
FIGURE 6.2 PREDICTED CURVES
90


FIGURE 6.3 TEST UNDERSTEER GRADIENT
91
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00


OlOOiOOiOOlO
h 13 co co
FIGURE 6.4 PREDICTED UNDERSTEER GRADIENT
92
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00


549.0
548.0
547.0
546.0
545.0
544.0
543.0
542.0
541.0
540.0
539.0
538.0
537.0
536.0
535.0
24.92 24.94 24.96 24.98 25.00 25.02 25.04 25.06 25.08 25.10
Cm
FIGURE 6.5 TIRE PROPERTIES DATA POINTS
The data from 17 tests was evaluated to find the tire properties. Analysis
for each data set yielded a slightly different value for Cm and Wm. The
averaged values Cm 25.01 lbs/deg and Wm 542.18 lbs look appropriate
for the tires used on the vehicle. At Ay = 0.74 the vertical loading Fr0 =
493.1 lbs, which is under the Wm 542.18 lbs for the tire.
There has all ways been speculation in Formula Vee circles that all of the
specified tires maybe a little under rated for this vehicle. For this
particular size of tire it may be true when the vertical loads increase with
Ay.
6.1 Error Sources
There was early on a problem getting the values of Cm and Wm to
converge. In order to get the front slip angles to agree the chosen values of


.



25.( 1, 512.1 £ 8

* i
!





93


y/ and calculated slip angles were extremely out of line with what had
been observed during testing. Calculating the steer angles from the yaw
analysis showed that // had been over estimated From Section 2.8, The
coefficient of friction is an important part of how much lateral force can be
developed.
A final value of // = 1.0 was decided on after running some iterations
through the procedure outlined in all of Section 5. Values for /j. 0.9 and /u
= 1.1 were evaluated as well. While /j. 1.0 is probably not the exact value
of static friction, it was close enough to show a strong correlation between
actual and derived values.
94


7. Conclusion
The purpose of this study was to find tire properties Cm and Wm,
information not typically found when specifying tires for a racing vehicle.
Then use the tire properties to find the handling properties of a Formula
Vee race car and build a baseline performance diagram. Then using the
baseline and the tire properties, predict changes that would improve the
handling of the vehicle.
Outlined in this body of work is a method that will take skid pad test data
from a properly setup vehicle and produce tire property values. This
method uses basic principles for tire, roll and load transfer analysis found
in many texts, and merges them into a unique way of solving an
engineering problem.
In this paper is an outline of a practical method for discovering vehicle
handling characteristics. This method was designed to be use by the
average racer who has access to vehicle data recording equipment and a
spreadsheet program. This method could also be ported over to a program
such as Matlab to process the raw data and rapidly create performance
handling plots.
95


APPENDIX A. TABLE OF VARIABLES
Uppercase
Ay Lateral acceleration
Bo Suspension rocker pivot point outside
Bi Suspension rocker pivot point inside
C Initial cornering stiffness
C Cornering coefficient
CFi Center of inner front contact patch
Cfo Center of outer front contact patch
Cm Maximum initial cornering stiffness
Cm Center of inner rear contact patch
Cro Center of outer rear contact patch
Di Pushrod attachment point on suspension rocker inside
Do Pushrod attachment point on suspension rocker outside
E Pushrod attachment point on rear axle attachment point
Fp0 Force in pushrod outside
Fpi Force in pushrod inside
Fs Force exerted by the suspension spring
Fy Lateral force
Fyc Camber thrust
Fyf Total lateral force front axle
Fyfi Lateral force inner front tire
Fyfo Lateral force outer front tire
Fyri Lateral force inner rear tire
Fyro Lateral force outer rear tire
FYmax Maximum lateral force
Fyt Total lateral force
Fz Vertical force
Fzf Vertical force front
Fzfi Vertical force inner front tire
96


Fzfo Vertical force outer front tire
Fzm Vertical force inner rear tire
Fzro Vertical force outer rear tire
Fzr Vertical force rear
G Modulus of rigidity
H Initial rear roll height
I Moment of inertia
J Polar moment of inertia
Kspring Spring constant
K^f Front roll stiffness
K^r Rear roll stiffness
KjRB Roll bar stiffness
K^sf Front suspension spring stiffness
Moment about axle roll center
Lb Horizontal distance from centerline of vehicle to suspension
rocker pivot point
Le Distance from fulcrum to pushrod attachment point on axle
Lr Distance on suspension rocker from pivot point to pushrod
Lpi Pushrod length inner
Lp0 Pushrod length outer
Lri Rocker length inner
Lr0 Rocker length outer
L0 Horizontal distance between fulcrum pivot and longitudinal axis
Effected length of spring uncompressed
Ls Length of spring compressed static displacement
Oi Fulcrum point inner
Oo Fulcrum point outer
Oxo Reaction force at fulcrum point horizontal outside
Oxi Reaction force at fulcrum point horizontal inside
Oy0 Reaction force at fulcrum point vertical outside
Oyi Reaction force at fulcrum point vertical inside
R Radius
W Total vehicle weight
Wa Combined weight of rear half axle and wheel
97


Wf Weight on the front axle
Wm Tire normal load at maximum initial cornering stiffness
Wr Weight on the rear axle
M Moment about axle roll center
VVehicle
Lowercase
b Distance to front axle from C.G.
c Distance to rear axle from C.G.
d Diameter
g Gravity
hi Vertical distance from roll axis to C.G.
hF Vertical distance to front roll center from ground
Hr Vertical distance to rear roll center from ground
kbar Roll bar spring constant
l Wheel base
la Length of rear axle
lb Length for roll bar bending calculation
Ik Distance between steering knuckles
Isp Length of spindle
It Length for roll bar torsional calculation
srb Roll bar length
t Vehicle rack distance
tF Front track distance
tR Rear track distance
v Velocity at a point
vtire Velocity of the tire
Greek
a Slip angle
aFi Slip angle front inner tire
aFo Slip angle front outer tire
dm Slip angle maximum value
ccRi Slip angle rear inner tire
98


aRo Slip angle rear outer tire
P Camber angle
f3c angle between rear axle and \OC\
(3e Angle between Le and rear axle
Pr Angle on suspension rocker between Lr and the line from the
S Ackerman angle
Average steering angle
Sj Inner steer angle
S0 Outside steer angle
p Rotation about inner front tire contact patch
(j) Vehicle roll angle
(f>F Front roll angle
y/ Yaw angle
Attitude angle
yn Angle between |0C| inboard and ground
y/0 Angle between \OC\ outboard and ground
0Ai Angle between vehicle y-axis and inside axis
6ao Angle between vehicle y-axis and outside axis
Ob Difference between the velocities uA and vB
Oa Angle between \OC\ and vehicle y-axis inboard
Oco Angle between \OC\ and vehicle y-axis outboard
Opi Angle between Lp and Le inside
Opo Angle between Lp and Le outside
0Ri Rear inner rocker angle
Oro Rear rocker angle
0s Angle of the suspension spring WRT the vehicle y-axis
(o Rotational velocity
p Coefficient of friction
99


APPENDIX B. VEHICLE SETUP PICTURES
Provided in this appendix are some pictures of the test vehicle and the
installed sensors that were used to collect the test data.
FIGURE B.l TEST VEHICLE
Here in figure B.l is the Formula Vee used as the test platform for this
research.
100