Design of a Suspension for a Formula Student Race Car

Design of a Suspension for a
Formula Student Race Car
Adam Theander
VEHICLE DYNAMICS
AERONAUTICAL AND VEHICLE ENGINEERING
ROYAL INSTITUTE OF TECHNOLOGY
TRITA-AVE-2004-26
ISSN 1651-7660
Postal Address
Visiting Address
Internet
Telephone
Telefax
KTH
Vehicle Dynamics
SE-100 44 Stockholm
Sweden
Teknikringen 8
Stockholm
www.ave.kth.se
+46 8 7906000
+46 8 7909290
Abstract
In July of 2004 KTH Racing will attend at the Formula Student event in England. The
Formula Student event is a competition between schools that has built their own
formula style race cars according to the Formula SAE rules. In January of 2004 the
Formula Student project started at KTH involving over seventy students. The aim of
this thesis work is to design the suspension and steering geometry for the race car being
built. The design shall meet the demands caused by the different events in the
competition. The design presented here will then be implemented into the chassis being
built by students participating in the project. Results from this thesis work shows that
the most suitible design of the suspension is a classical unequal length double A-arm
design. This suspension type is easy to design and meets all demands. This thesis work
is written in such a way that it can be used as a guidebook when designing the
suspension and steering geometries of future Formula Student projects at KTH.
Acknowledgements
This master thesis has been conducted at the Division of Vehicle Dynamics, Department
of Aeronautical and Vehicle Engineering at the Royal Institute of Technology, KTH, in
Stockholm, Sweden. The work has been carried out from December 2003 to May 2004.
There are a few persons to whom I would like to especially express my gratitude.
Professor Annika Stensson, my examiner, who gave me the opportunity to carry out this
thesis work; research engineer Mats Beckman, my supervisor, for his engagement and
time spent helping me; Fredrik Westin, Ph.D student at Division of Internal Combustion
Engines, who has had a major role in KTH Racing as project leader and spent almost all
of his spare time working with the project; all students participating in the KTH
Formula Student project, without all of you there wouldn’t have been any KTH Racing.
Finally I would like to take the opportunity to give a special thank to the students
known as “Järngänget”, you know who you are. Without their effort the last couple of
weeks there would never have been a car to show on the 14th of May.
Stockholm, May 2004
Adam Theander
Table of Contents
1
Introduction ..................................................................................... 12
1.1
Background..................................................................................................12
1.2
Aim of the work...........................................................................................12
1.3
Competition Objective .................................................................................12
1.3.1 Vehicle Design Objectives .......................................................................13
1.4
Competition Events and Judging of the Cars ................................................13
1.4.1 Acceleration Event...................................................................................14
1.4.2 Skid-Pad Event ........................................................................................14
1.4.3 Autocross Event .......................................................................................15
1.4.4 Endurance and Fuel Economy Event ........................................................15
1.4.5 Judging of the cars ...................................................................................16
1.5
Rules Relevant to the Chassis Design...........................................................17
1.5.1 Wheelbase and Vehicle Configuration......................................................17
1.5.2 Vehicle Track Width ................................................................................17
1.5.3 Ground Clearance ....................................................................................17
1.5.4 Wheels and Tires......................................................................................17
1.5.5 Suspension ...............................................................................................17
2
Suspension Design Aspects.............................................................. 18
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Wheelbase ...................................................................................................18
Track Width.................................................................................................19
Kingpin and Scrub Radius............................................................................20
Caster and Trail............................................................................................21
Instant Centre and Roll Centre .....................................................................21
Tie Rod Location .........................................................................................22
Anti Features ...............................................................................................23
Ackerman steering .......................................................................................25
Camber ........................................................................................................26
Toe ..............................................................................................................27
3
Benchmark....................................................................................... 28
4
Methods............................................................................................ 30
4.1
Track width and wheelbase ..........................................................................30
4.2
Front Suspension Design..............................................................................33
4.2.1 The Rims .................................................................................................33
4.2.2 The Brakes...............................................................................................34
4.2.3 Front View Geometry...............................................................................34
4.2.4 Side View Geometry ................................................................................35
4.2.5 Control Arm Pivot Axis ...........................................................................36
4.2.6 Tie Rod Location and Ackermann Geometry............................................38
4.3
Rear Suspension Design...............................................................................38
5
Model Building ................................................................................ 40
5.1
5.2
5.3
5.4
5.5
5.6
6
Front Suspension Modelling.........................................................................41
Rear Suspension Modelling..........................................................................42
Steering Modelling.......................................................................................43
Wheels Modelling........................................................................................44
Body ............................................................................................................44
Simulation ...................................................................................................44
Parameter Study.............................................................................. 46
6.1
The Taguchi Methods ..................................................................................46
6.2
Parameters of Interest...................................................................................47
6.2.1 Parameter Levels......................................................................................47
6.3
Results .........................................................................................................48
6.3.1 Parameter Study of Front Suspension .......................................................48
6.3.2 Parameter Study of Steering Geometry.....................................................53
6.3.3 Parameter Study of Rear Suspension ........................................................57
7
Discussion and Design Results ........................................................ 62
7.1
Track Width and Wheelbase ........................................................................62
7.2
Front Suspension Geometry .........................................................................64
7.2.1 Camber Characteristics ............................................................................65
7.2.2 Anti Features............................................................................................65
7.2.3 Roll Centre Characteristics.......................................................................66
7.3
Steering........................................................................................................66
7.3.1 Bump Steer ..............................................................................................66
7.3.2 Ackerman ................................................................................................67
7.4
Rear Suspension Geometry ..........................................................................68
7.4.1 Camber Characteristics ............................................................................69
7.4.2 Anti Features............................................................................................69
7.4.3 Roll Centre Characteristics.......................................................................69
8
Future Work .................................................................................... 70
9
Nomenclature................................................................................... 72
10 References ........................................................................................ 74
10.1
10.2
Literature .....................................................................................................74
Oral References ...........................................................................................74
1 Introduction
1.1 Background
In the autumn of 2003 a group of students started a project at KTH. The objective of the
project was to build a race car according to the Formula Student rules and compete in
the event at Bruntingthorpe Proving Ground, Leicestershire, England, in July 2004.
There were three different courses given involved in the Formula Student project, one
project course in Internal Combustion Engines, one project course in Advanced
Machine Elements and a small project course in Machine Design. Soon there were over
70 students involved, either participating in one of the three courses or as volunteers.
1.2 Aim of the work
The aim of this thesis work is to design the suspension geometry for a Formula Student
race car. The design shall meet the demands caused by the different dynamic events in
the competition. The aim of the work can be separated into:
•
•
•
•
•
•
•
Identifying relevant design parameters.
Investigate the design parameters influences on the whole car and the
interactions between them.
Identifying the packaging issues for the suspension together with the Machine
Elements students, MME-students, who builds the frame
Identifying different driving conditions.
Investigate the adjustment levels needed.
Optimize the primary set-up.
Purpose further work.
The work carried out will be used by the students participating within the KTH Racing
project for year 2004 and hopefully be of much interest for upcoming years projects.
1.3 Competition Objective
The objective of the competition is for students to conceive, design, fabricate and
compete with small formula-style racing cars. The design of the car frame and engine
are restricted in order to challenge the knowledge, creativity and the imagination of the
students [1].
1.3.1 Vehicle Design Objectives
For the purpose of the competition the students are to assume that a manufacturing firm
has engaged them to produce a prototype race car for evaluation as a production item.
The intended market is the non-professional weekend autocross racer. Therefore the car
must have high performance in terms of acceleration, braking and handling qualities.
The car must also be low in cost, easy to maintain and reliable. The production rate is
estimated to four cars per day for a limited production run and the prototype vehicle
should cost below $25000. The challenge for the students is to design and build a
prototype car that meets these goals. Each car will be compared and judged with other
competing cars to determine the best overall car [1].
1.4 Competition Events and Judging of the Cars
The competition is divided into static events and dynamic events. The static events are:
•
•
•
Presentation
Engineering Design
Cost Analysis
A presentation is held for the imaginary manufacturing firm who ordered the prototype.
The purpose of the presentation event is to evaluate the team’s ability to sell their
product. The presentation judges evaluate the organization, content and delivery of the
presentation. An engineering design event is held to evaluate the effort put into the
design process and how the design meets the intent of the market. The purpose of the
cost analysis event is to teach the students participating that cost and budget are very
important and must be taken into account in every engineering process.
The dynamic events are:
•
•
•
•
Acceleration
Skid-Pad
Autocross
Endurance and Fuel Economy
1.4.1 Acceleration Event
The objective of the acceleration event is to evaluate the car’s acceleration in a straight
line on flat pavement. The cars will be staged 0.3m behind the starting line and when
the cars cross the starting line the time will start. The goal is located 75m ahead of the
starting line. Each team will have two drivers, who can do two runs each, a total of four
runs. This is the event were the suspension design is of least importance among the
dynamic events, but not negligible.
1.4.2 Skid-Pad Event
The objective of the skid-pad event is to measure the cornering ability of the car on a
flat surface while making a constant-radius turn. The skid-pad layout will consist of two
circles with a diameter of 15.25m separated by 18.25m. The driving path will be 3m
wide. The layout of the skid-pad and driving directions are showed in Figure 1.1.
3m
Finish
15
.2
5m
18.25m
Start
Figure 1.1. Skid-Pad layout and driving directions.
The procedure of the event is as follows: the cars will start by entering the right circle
completing one lap. Next lap will be timed and immediately after the left circle is
entered for the third lap. The fourth lap will be timed. Then the driver has the option to
make a second run immediately after the first. Each team will have two drivers who can
do two runs each. The design of the suspension and steering geometry will influence the
performance much.
1.4.3 Autocross Event
The objective of the autocross event is to evaluate the car’s manoeuvrability and
handling qualities on a tight course. The autocross course will combine the performance
features of acceleration, braking and cornering. The layout of the autocross track is
made to keep the speeds from being dangerously high, average speeds should be
between 40km/h and 48km/h. The layout is specified as follows:
•
•
•
•
•
•
Straights – No longer than 60m with hairpins at both ends or no longer than 45m
with wide turns on the ends.
Constant Turns – 23m to 45m in diameter.
Hairpin Turns – Minimum of 9m outside diameter.
Slaloms – Cones in a straight line with 7.62m to 12.19m spacing.
Miscellaneous – Chicanes, multiple turns, decreasing radius turns, etc. The
minimum track width will be 3.5m
Length – Approximately 0.805km.
Each team will have two drivers entering the event. Each driver will drive two timed
laps and the best time for each driver will stand as the time for that heat.
1.4.4 Endurance and Fuel Economy Event
To evaluate the overall performance and to test the car’s reliability an endurance event
is performed. This event is combined with a fuel economy event implying that the fuel
economy will be measured during the endurance event. A single 22km heat is made
during which the teams will not be allowed to work on their cars. A driver change must
be made during a three-minute period at the mid point of the event. The layout of the
endurance track is similar to the layout of the autocross track:
•
•
•
•
•
Straights – No longer than 77m with hairpins at both ends or no longer than 61m
with wide turns on the ends. There will be passing zones at several locations.
Constant Turns – 30m to 54m in diameter.
Hairpin Turns – Minimum of 9m outside diameter.
Slaloms – Cones in a straight line with 9m to 15m spacing.
Miscellaneous – Chicanes, multiple turns, decreasing radius turns etc. The
minimum track width will be 4.5m.
In both the autocross event and the endurance event the suspension design and steering
geometry is of major importance. A well working design helps the drivers perform at
the edge of their capacity. The layout of the 2003 event endurance track can be viewed
in Figure 1.2.
Figure 1.2. GPS plot of the 2003 endurance track.
[Courtesy of Honda Research and Development Europe Ltd.]
1.4.5 Judging of the cars
The cars are judged based on the performance in the static and dynamic events. These
events are scored to determine how well the car performs. In each event the
manufacturing firm has specified a minimum acceptable performance. The following
points are possible [1]:
Static Events
Presentation
Engineering Design
Cost Analysis
Dynamic Events
Acceleration
Skid-Pad
Autocross
Fuel Economy
Endurance
Total Points
75
150
100
75
50
150
50
350
1000
Table 1.1. Scoring points in each of the events.
1.5 Rules Relevant to the Chassis Design
The major part of the Formula SAE rules concerns the safety of the drivers. But there
are a few rules that will have to be taken into consideration when designing a chassis.
1.5.1 Wheelbase and Vehicle Configuration
Rule 3.1.2: “The car must have a wheelbase of at least 1525mm. The wheelbase is
measured from the centre of ground contact of the front and rear tires with wheels
pointing straight ahead. The vehicle must have four wheels that are not in a straight
line” [1].
1.5.2 Vehicle Track Width
Rule 3.1.3: “The smaller track of the vehicle (front or rear) must be no less than 75% of
the larger track” [1].
1.5.3 Ground Clearance
Rule 3.2.1: “Ground clearance must be sufficient to prevent any portion of the car
(other than the tires) from touching the ground during track events” [1].
1.5.4 Wheels and Tires
Rule 3.2.2: “The wheels of the car must be 203.2mm (8.0 inches) or more in diameter”
[1].
1.5.5 Suspension
Rule 3.2.3: “The car must be equipped with a fully operational suspension system with
shock absorbers, front and rear, with usable wheel travel of at least 50.8mm (2 inches),
25.4mm (1 inch) jounce and 25.4mm (1 inch) rebound, with driver seated” [1].
2 Suspension Design Aspects
The purpose of the suspension is to make the job easier for the tires and give a
predictable behaviour so that the driver will have control of the car. The suspension
shall help to keep the tires in constant contact with the ground so that the tires can be
used to the limit of their capacity. When designing a suspension there are a number of
factors that influence the behaviour of the suspension and a lot of these factors also
interacts in one way or another. Therefore much work is put into making a compromise
that will function well in all the driving events at the competition. The factors taken into
this work are as follows below.
2.1 Wheelbase
The wheelbase, l, is the distance between the centre of the front axle and the centre of
the rear axle. The wheelbase has a big influence on the axle load distribution. A long
wheelbase will give less load transfers between the front and rear axles than a shorter
wheelbase during acceleration and braking according to Equation 2.1 and Figure 2.1.
Figure 2.1. Side view parameters for longitudinal load transfer calculations.
Fz1 = (1 − λ ) ⋅ mg + κ ⋅ a x ⋅ m
Fz 2 = λ ⋅ mg + κ ⋅ a x ⋅ m
(2.1)
A longer wheelbase will therefore be able to be fitted with softer springs and will
increase the level of comfort for the driver. On the other hand a shorter wheelbase have
the advantages of smaller turning radius for the same steering input, see section 2.8 [3].
A car with too short wheelbase may act nervously on corner exits and in straight line
driving. Anti features can be built into a suspension and these will also affect the
longitudinal load transfer, see section 2.7.
2.2 Track Width
The track width is of major importance when designing a vehicle. It has influence on the
vehicle cornering behaviour and tendency to roll. The larger the track width is the
smaller the lateral load transfer is when cornering and vice versa according to Equation
2.2 that shows the load transfer for a rear axle [3].
∆Fz 2 =
µ lat ⋅ hCG
tw2
(2.2)
A larger track width has the disadvantage that more lateral movement of the vehicle is
needed to avoid obstacles. According to the regulations the smallest section of the SkipPad may not be smaller than 3m and the Autocross and Endurance tracks no smaller
than 3.5m [1]. The amount of lateral load transfer wanted depends on tires fitted on the
car, see section 2.9. If the car has anti-roll bars these will also affect the load transfer.
Kingpin Inclination
Kingpin Axis
Wheel Offset
Spindle Length (+)
UBJ
UBJ
+
Wheel Flange Plane
Side View
Kingpin Offset
+
LBJ
LBJ
Caster (+)
Mechanical Trail
Scrub Radius (-)
FORWARD
Figure 2.2. Kingpin geometry, side view and front view.
2.3 Kingpin and Scrub Radius
The Kingpin axis is determined by the upper ball joints, UBJ, and lower ball joints, LBJ,
on the outer end of the A-arms. This axis is not necessary centred on the tire contact
patch. In front view the angle is called Kingpin inclination and the distance from the
centre of the tire print to the axle centre is called Scrub or Scrub radius. The distance
from the kingpin axis to the wheel centre plane measured horizontally at axle height is
called Spindle length. Figure 2.2 shows the kingpin geometry. There are numerous of
effects due to the values of these factors, the effects considered to in this work are found
in [2], [3]:
•
•
•
If the spindle length is positive the car will be raised up as the wheels are turned
and this results in a increase of the steering moment at the steering wheel. The
larger the kingpin inclination angle is the more the car will be raised regardless
of which way the front wheels are turned. If there is no caster present this effect
is symmetrical from side to side. The raise of the car has a self-aligning effect of
the steering at low speeds.
Kingpin inclination affects the Steer camber. When a wheel is steered it will
lean out at the top, towards positive camber if the kingpin inclination angle is
positive. The amount of this is small but not to neglect if the track includes tight
turns.
If the driving or braking force is different on the left and right side this will
introduce a steering torque proportional to the scrub radius, which will be felt by
the driver at the steering wheel.
2.4 Caster and Trail
In the side view the kingpin inclination is called Caster angle. If the kingpin axis
doesn’t pass through the centre of the wheel then there is a side view Kingpin offset
present. The distance from the kingpin axis to the centre of the tire print on the ground
is called Trail or Caster offset. See Figure 2.2 for the side view geometry. The caster
angle and trail is of importance when designing the suspension geometry. The effects
considered in this work are [2], [3]:
•
•
•
•
The larger the trail is the higher steering torque is needed.
Caster angle will cause the wheel to rise and fall with steer. This effect is
opposite from side to side and causes roll and weight transfer. Leading to an
oversteering effect.
Caster angle has a positive effect on steer-camber. With positive caster angle the
outside wheel will camber in a negative direction and the inner wheel in a
positive direction, causing both wheels to lean into the turn.
The size of the mechanical trail due to caster may not be too large compared to
the Pneumatic trail from the tire. The pneumatic trail will approach zero as the
tires reaches the slip limit. This will result in lowering the self-centring torque
that is present due to the lever arm between the tires rotation point at the ground
and the point of attack for the lateral force. This will be a signal to the driver that
the tire is near breakaway. This “breakaway signal” may be lost if the
mechanical trail is large compared to the pneumatic trail.
2.5 Instant Centre and Roll Centre
Instant centre is the momentary centre which the suspension linkage pivot around. As
the suspension moves the instant centre moves due to the changes in the suspension
geometry. Instant centres can be constructed in both the front view and the side view. If
the instant centre is viewed in front view a line can be drawn from the instant centre to
the centre of the tire’s contact patch. If done for both sides of the car the point of
intersection between the lines is the Roll centre of the sprung mass of the car. The
position of the roll centre is determined by the location of the instant centres. High
instant centres will lead to a high roll centre and vice versa. The roll centre establishes
the force coupling point between the sprung and the unsprung masses of the car. When
the car corners the centrifugal force acting on the centre of gravity can be translated to
the roll centre and down to the tires where the reactive lateral forces are built up. The
higher the roll centre is the smaller the rolling moment around the roll centre is. This
rolling moment must be restricted by the springs.
Another factor is the horizontal-vertical coupling effect. If the roll centre is located
above the ground the lateral force generated by the tire generates a moment about the
instant centre, which pushes the wheel down and lifts the sprung mass. This effect is
called Jacking. If the roll centre is below the ground level the force will push the sprung
mass down. The lateral force will, regarding the position of the roll centre, imply a
vertical deflection. If the roll centre passes through the ground level when the car is
rolling there will be a change in the movement direction of the sprung mass.
Centre of Car
Instant Centre
+
Roll Centre
+Roll Centre Height
Centre of Contact Patch
fvsa length
Figure 2.3. SLA front view geometry.
The camber change rate is a function only of the front view swing arm length, fvsa
length. Front view swing arm length is the length of the line from the wheel centre to
the instant centre when viewed from front. The amount of camber change achieved per
mm of ride travel would be as described in Equation 2.3 and Figure 2.3.
degrees
mm
= arctan
1
fvsa length
(2.3)
The camber change is not constant throughout the whole ride travel since the instant
centre also moves with wheel travel.
2.6 Tie Rod Location
The location of the tie rods is of major importance. The location shall be such that Bump
steer effects are kept at a minimum. Bump steer is the change in toe angle due to wheel
travel. A car with much bump steer will have a tendency to change its movement
direction when the front wheels runs over an obstacle. The affects of this can be
hazardous when running on an uneven track. The simplest way to minimize bump steer
is to locate the tie rod in the same plane as either the upper or lower A-arms. Another
factor to keep in mind is the camber compliance under lateral force. If the tie rods are
located either above and behind or below and in front of the wheel centre the effect on
the steering will be in understeer direction. If the A-arms are stiff enough the effects
will be small and thereby minimize the risk of oversteering effects due to compliance in
the A-arms. The length of the lever arm from the outer tie rod end to the upper ball joint
determine together with the steering rack ratio the total ratio from the steering wheel’s
angle to the wheel’s steering angle.
Braking Force
CG
l
h
+ ∆FZ
− ∆FZ
Braking Force=
W(ax/g)
CG
svsa length
Moment=
W(ax/g)(% front braking)(svsa height)
IC
φF
svsa
height
Anti Dive Force=
W(ax/g)(% front braking)(svsa height)/(svsa length)
Figure 2.4. Braking anti features with outboard brakes.
2.7 Anti Features
The anti effect in a suspension describes the longitudinal to vertical force coupling
between the sprung and unsprung masses. It results from the angle of the side view
swing arm, svsa. Anti features do not change the steady-state load transfer at the tire
patch; it is only present during acceleration or braking. The longitudinal weight transfer
during steady acceleration or breaking is a function of wheelbase, CG height and
acceleration or breaking forces as described in Figure 2.4.
The anti features changes the amount of load going through the springs and the pitch
angle of the car. Anti features are measured in percent. A front axle with 100% anti dive
will not deflect during braking, no load will go through the springs, and a front axle
with 0% anti dive will deflect according to the stiffness of the springs fitted; all load is
going through the springs. It is possible to have negative anti effects. This will result in
a gain of deflection. Equation 2.4 gives the percent of anti-dive in the front of a car with
outboard brakes.
( h)
Anti Dive = (% front braking )(tan φ F ) l
(2.4)
By substituting % front braking with % rear braking and tan φF tan with φR in Equation
2.4 the amount of anti lift can be calculated. The way that brake and drive torque is
reacted by the suspension alters the way to calculate the amount of anti present. If the
control arms react torque, either from the brakes or from drive torque, the anti’s are
calculated by the IC location relative to the ground contact point. If the suspension
doesn’t react drive or brake torque, but only the forward or rearward force, then the
“anti’s” are calculated by the IC location relative to the wheel centre. For a rear-wheel
driven car there are 3 different types of anti features:
•
•
•
Anti dive, which reduces the bump deflection during forward braking.
Anti lift, which reduces the droop travel in forward braking.
Anti squat, which reduces the bump travel during forward acceleration.
Figure 2.5 shows the configuration for calculating the anti features for a car with
outboard front brakes and inboard rear brakes.
CG
l
IC
h
φF
IC
φR
svsa
heights
svsa length
Figure 2.5. Anti features during braking with outboard front brakes and inboard rear brakes.
2.8 Ackerman steering
At low speed turns, where external forces due to accelerations are negligible, the
steering angle needed to make a turn with radius R is called the Ackermann steering
angle, δa, and can be calculated by using Equation 2.5.
δa =
l
R
(2.5)
If both front wheels are tangents to concentric circles about the same turning centre,
which lays on a line trough the rear axle, the vehicle is said to have Ackermann steering.
This results in the outer wheel having a smaller steering angle than the inner. If both
wheels have the same steering angle the vehicle is said to have Parallel steer and if the
outer wheel has a larger steering angle than the inner it is called Reverse Ackermann.
Passenger cars have a steering geometry somewhere between Ackermann steering and
parallel steering while it’s common among race cars to use reverse Ackermann. By
using Ackermann steering on passenger cars, or other vehicles only exposed to low
lateral accelerations, it is ensured that all wheels roll freely with no slip angles because
the wheels are steered to track a common turn centre. Race cars are often operated at
high lateral accelerations and therefore all tires operate at significant slip angles and the
loads on the curve inner wheels are much less than the curve outer wheels due to the
lateral load transfer. Tires under low loads require less slip angle to reach the peak of
the cornering force. Using a low speed steering geometry on a race car would cause the
curve inner tire to be dragged along at much higher slip angles than needed and this
would only result in raises in tire temperature and slowing down the car due to the slip
angle induced drag. Therefore race cars often use parallel steer or even reverse
Ackermann. The different types of Ackerman are shown in Figure 2.6.
l
l
R
Ackermann
R
l
Parallel
R
Figure 2.6. Ackermann steering, parallel steer and reverse Ackermann.
Reverse
Ackermann
2.9 Camber
Camber angle is the angle between a tilted wheel plane and a thought vertical plane.
Positive camber is defined as when the wheel is tilted outwards at the top relative to the
car. The camber angle has influences on the tires ability to generate lateral forces. A
cambered rolling pneumatic wheel produces a lateral force in the direction of the tilt.
This force is referred to as Camber thrust when it occurs at zero slip angles. Camber
also affects the aligning torque due to distortion of the tire print. The effect of this is
rather small and tends to be cancelled with increasing slip angle. Cambering the wheel
also leads to a raise in the lateral force produced by the wheel when cornering. This is
true in the linear range of the tire. If the linear range is exceeded the additive effects of
the camber inclination decreases, this effect is called Roll-off. Therefore the difference
in lateral force when comparing a cambered wheel and a non-cambered wheel is small,
around 5-10% at maximum slip angle. The difference is much larger at zero degrees slip
angle due to the camber thrust. The effects of cambering the tyre are bigger for a bias
ply tyre than a radial ply tyre. For radial tyre the camber forces tends to fall of at camber
angles above 5° while the maximum force due to camber for a bias ply racing tyre
occurs at smaller angles.
Camber Thrust
Lateral Force
Negative Cambered
No Roll-Off
Negative Cambered
With Roll-Off
0° Camber
Slip Angle
Figure 2.7. The camber thrust effect principle.
2.10 Toe
Toe adjustment can be used to overcome handling difficulties in the car. Rear toe-out
can be used to improve the turn-in. As the car turns in the load transfer adds more load
to the outside wheel and the effect is in an oversteer direction. The amount of static toe
in the front will depend on factors such as Ackermann steering geometry, ride and roll
steer, compliance steer and camber. Minimum static toe is desirable to reduce rolling
resistance and unnecessary tyre heating and tyre wear caused by the tires working
against each other.
3 Benchmark
2004 is the first year the Formula Student project is held at KTH there are no previous
experience from cars to evolve from. To get a rough estimation about the dimensions
and weights of Formula Student race cars data was collected from the 2003-year event.
The data collected was wheelbase, track width and weight and can be viewed in Table
3.1.
Car Nr.
Wheelbase Track Width
[mm]
front [mm]
11
1650
1260
26
1920
1360
17
1670
1280
4
1750
1290
35
1730
1320
21
1620
1250
16
1560
1250
14
1600
1219
33
1610
1200
12
1760
1080
1687
1250,9
Average
Track Width
Rear [mm]
1200
1360
1200
1240
1240
1060
1200
1194
1000
1160
1185,4
Axle Weight Axle Weight
Front [kg] Rear [kg]
140
157
139
157
98
120
99
121
108
162
114
176
95
116
141
147
120
130
90
110
114,4
139,6
Result
2003
6
35
8
19
7
30
21
17
10
20
Table 3.1. Average dimensions and weights of Formula Student race cars at the 2003 event.
A major part of the cars participating in the 2003 event didn’t have all data available so
the data listed in Table 3.1 are only from the cars where all dimensions and weights
where available.
Based on the literature survey, knowledge about the cars from the 2003 event and
discussions with persons with good knowledge in vehicle dynamics and racing a
guideline for the race car was set up. The purpose of this guideline is to have a defined
goal for the work. The guidelines set up was:
•
•
•
•
•
•
•
•
•
•
•
Kingpin inclination angle between 0° and 8°
Scrub radius between 0mm and 10mm
Caster angle between 3° and 7°
Static camber adjustable from 0° to -4°
Camber gain 0.2-0.3 degrees/roll angle at front axle
Camber gain 0.5-0.8 degrees/roll angle at rear axle
Maximum roll angle about 2°
Roll centre height between 0mm and 50mm in front and slightly higher at rear
Well controlled and predictable movement of the roll axle
Minimize bump steer
50% - 65% of the roll stiffness on the rear axle
The kingpin inclination is kept below 8° since too much kingpin inclination causes a lot
of rising of the front axle when steering. Keeping scrub radius small will make the car
easier to handle at low speeds and reduces the risk that a sudden lost of traction for one
of the front wheels during braking causes the car to change direction and reduces the
steering moment disturbance. The caster angle has positive effects during cornering but
too much caster causes weight transfer that will have an oversteering effect. The
possibility to adjust the camber angle from 0° to about -4° will be very helpful during
the testing of the car. During the competition this also allows to set the camber to 0°
during the acceleration event minimizing the rolling resistance. The camber gain is to
compensate for the lost of camber due to the roll angle during cornering. The reason for
having a much larger camber gain at the rear axle is to have as big contact patch
between the rear tyre and the ground during corner exits as possible. This will allow the
driver give more and earlier throttle. Having a slightly higher roll centre at the rear has
at least two advantages. The first is that softer springs can be used at the rear axle since
less rolling moment will appear here. The second is to keep the roll axle as parallel to
the cars main inertia axle.
4 Methods
4.1 Track width and wheelbase
The track width and wheelbase will have influences on the amount of load transfer
between the front and rear axle during acceleration and braking and the load transfer
from curve inner to curve outer wheels during cornering. To investigate the interactions
between longitudinal load transfer and wheelbase a MATLAB code was used. This code
doesn’t take the alternation of the centre of gravity’s height due to the pitch-attitude into
consideration since this is small and can be neglected. In Figure 4.1 the load transfer
from the front axle to the rear axle during acceleration is presented for two different
wheelbases, 1525mm and 1700mm as an example of the output from the MATLAB
code. The methods for determine the wheelbase is primary based on the packaging as it
will decide how small the wheelbase can be. The wheelbase should be as short as
possible, but not shorter than 1525mm, to optimize the ability to make sharp corners.
Figure 4.1. Longitudinal load transfer for two different wheelbases.
The load distrubation between curve inner and outer wheels during high speed cornering
is among other things a function of the front and rear trackwidth. Due to the lateral
acceleration there will be a load transfer from the curve inner to the curve outer wheels
present. In Figure 4.2 the MATLAB output for a steady state cornering simulation can
be viewed.The load on the curve inner wheels can be viewed as a function of trackwidth
and lateral acceleration. These plots are then compared with tyre data obtained from
Goodyear Racing showing how the vertical tyre loads influnces on the tyre’s ability to
produce lateral forces. An example of a tyre plot obtained from Goodyear Racing can be
viewed in Figure 4.3.
Figure 4.2. Lateral weight transfer as a function of track width and lateral acceleration.
Figure 4.3. Lateral force as a function of tire load and slip angle
for a 20x8-13” Goodyear racing tire for light vehicles [6].
Another factor that will determine the track width is the size of the race track.
According to the regulations no part of the race track may be smaller than 3.5m and the
tightest hairpin may not have an outside diamter smaller than 9m. The track will be 3m
wide in the Skid-Pad event but here the track width is of less importance. The turning
radius for a car having ackermann steering geometry is at low speed proportional to the
wheelbase and the steering angle. To make a hairpin that have an outside diameter of
9m the turning radius of the cars centerline need to be 9m minus half the trackwidth.
This results in an Ackermann angle of:
δa =
l
r−
1
2
⋅ tw
(4.1)
A larger trackwidth will have the disadvantage of a narrower A-arm angle to allow the
required wheel angle, causing the A-arm taking up more forces in the longitudinal
directions.
4.2 Front Suspension Design
The design of the front suspension is primary based on packaging. Track width, wheel
size, tyre size, the brakes, the dampers, etc., will all have to be kept in mind when
determine the locations available for the lower ball joint. The front suspension type
designed is an SLA suspension. SLA stands for Short-Long Arm and refers to the
different length of the upper and lower control arms.
4.2.1 The Rims
One of the first things to consider is what kind of rim that will be used. The dimensions
of the rims determine together with the brakes the space available in the rim for
placement of the ball joints. The rims to be used are Tecnomagnesio 13x6” in front and
13x8” at the rear. The rims where measured and a CAD-drawing was made in
Pro/DESKTOP. Figure 4.4 shows the CAD-drawings.
Front View
Side View
Rendered CAD-model
1 .
Figure 4.4. CAD-model of a Tecnomagnesio 13x6" rim.
4.2.2 The Brakes
Special designed brake discs was ordered from ISR Brakes due to packaging conditions.
ISR Brakes are specialists in “one of a kind” brake systems for high performance
motorcycles. With the dimensions known, the space left in the rim and the location of
the lower ball joint can be estimated. The final placement of the lower ball joint is given
by the packaging of the rims.
4.2.3 Front View Geometry
The possible locations for the lower ball joint is now set by the space left after fitting
the brake system. To obtain proper roll camber characteristics the front view swing arm
length, fvsa length, is calculated by using Equation 4.2.
fvsa =
tw 2
tw 2
=
1 − roll camber 1 − wheel camber angle chassis roll angle
(4.2)
A line is projected from the ground contact patch through the wanted roll centre and to
the instant centre located at the desired fvsa length distance from the ground contact
patch. From the instant centre one line is projected back to the location of the lower ball
joint and one line to the location of the upper ball joint.
The length of the lower control arm shall be made as long as possible but is limited by
packaging. The driver’s legs will have to be fitted in between the lower control arms in
order to keep the centre of gravity height as low as possible. The length of the upper
control arm will determine the curvature of the camber curve. If upper and lower control
arms are of the same length the camber curve will be a straight vertical line and if the
upper arm is shorter than the upper the curve will be concave toward negative camber
which is preferable. The shorter the upper arm is the more concave the camber curve
will be. It is possible to design geometry that will have progressive camber in bump
with much less in drop.
+
+
tw/2
fvsa length
Figure 4.5. Front view swing arm geometry for establishing locations of ball joints.
4.2.4 Side View Geometry
The design of the side view geometry is based on the desired anti-features. For the front
suspension of a rear wheel driven car the only anti feature is anti dive. The amount of
anti feature is calculated with Equation 2.4.
This gives the wanted angle of the side view swing arm, φF. The length of the arm, svsa
length, determines the amount of longitudinal wheel travel during bump and drop. The
geometries for establishing anti features are seen in Figure 4.6.
ax
CG
IC
φF
svsa
height
h
svsa length
l
Figure 4.6. The side view swing arm geometry for establishing wanted amount of ant feature.
4.2.5 Control Arm Pivot Axis
The locations of the inner ball joints, the ball joints located on the frame, are also
geometrically designed. The method used is the one described in Race Car Vehicle
Dynamics written by Milliken/Milliken. The method is a projection technique that can
be used for SLA and other suspension types.
See Figure 4.7. First, in the front view, the upper control arm inner pivot point is
marked as point #1 and the upper ball joint as point #2. Point #2 projection onto the
longitudinal plane is marked as point #3. Corresponding points for the lower control
arm is #11, #12 and #13. These points are then transferred into the side view. A line is
drawn from the side view instant centre and through point #3 and a bit further. This
arbitrary point is marked as point #4. The same procedure is made for point #13 giving
the location of point #14. These points are then projected into the front view. In both the
front view and the side view a line is drawn from point #4 through point #2 and beyond
point #1. This is repeated for the lower control arm using points #14, #12 and #11. The
inner pivot points are wanted to be parallel with the car’s centreline. This is made by
drawing a vertical line from point #1. This line’s intersection with the line from point #4
through point #2 is the desired location of point #5 in the front view. This is repeated
for the lower control arm with the corresponding points, #11, #14, #12, giving point
#15. Points #5 and #15 are then projected into the side view and lines are drawn trough
point #1 and #5 and #11 and #15 giving the axis of the inner pivot points. The points
can be placed anywhere wanted as long as they still are on the lines.
1
15
5
5
2
12
3
13
3
1
11,12
13
2
14
4
xis
a
LCA
4
UCA axis
LCA axis
14
UCA axis
Car Centreline
11
IC
IC
15
13
3
4
13
14
12
2
UCA axis
LCA axis
Car Centreline
Figure 4.7. Control arms pivot axis construction layout.
1
5
15
11
4.2.6 Tie Rod Location and Ackermann Geometry
To minimize bump steer the placement of the tie rods are critical. There are several
packaging issues to considerate. The placement of the steering rack gives the height of
the inner pivot points for the tie rods. Since the steering rack was to be mounted in the
upper part of the frame the solution was to have the tie rods in the same plane as upper
control arms. Theoretically this solution will give zero bump steers if the tie rods are
exactly in plane with the upper control arms. It is desired to have adjustable Ackermann
to optimize the car for different driving events. So a span from 0% Ackermann to 100%
Ackermann is wanted, negative Ackermann is used on high-speed race cars only as
described in section 2.8, and due to this the tie rods have to be placed in front of the
upper control arms as seen in Figure 4.8. This will cause the compliance effects to work
in a negative way. But if the tie rod were to be located behind the upper control arms it
would result in too less adjustability of the Ackermann geometry.
Car
Centreline
Tie Rod
UCA
axis
Forward
Control
Arms
Figure 4.8. Tie rod location.
4.3 Rear Suspension Design
The design of the rear suspension geometry is performed in a similar way as the front
suspension. It is possible to use the same design for the rear suspension as for the front
suspension with the difference that front left will be placed rear right and vice versa to
give the right toe compliance effects. To simplify the design even further the toe links
can be connected to the control arms instead of to the frame. This can be made since the
outer pivot point of the toe link is at the same height as the outer ball joint. This
configuration is called “ungrounded” toe link.
5 Model Building
In order to evaluate the suspension system when designing according to the discussions
in chapter 4 a model is created to make it possible to simulate the car’s dynamic
behaviour during different conditions.
All vehicle simulations are carried out in ADAMS. ADAMS is a simulation software
for dynamic simulation of mechanical systems. There are several subprograms included
in ADAMS for different simulation applications. One of these subprograms,
ADAMS/Car, is especially designed for use with vehicle simulations. In ADAMS/Car
all subsystems of a car can be modelled one at a time and then put together to a
complete vehicle. This also gives the advantage that it is very simply to change from
one model of a front suspension to another or between different tyre models. There are
also built in test rigs for testing suspensions or whole cars. All simulations can be
visualized, written to files or viewed as graphs with the included tools. A complete
vehicle is built up by subsystems. To simulate a vehicle the required subsystems are:
•
•
•
•
•
•
Front Suspension Subsystem
Rear Suspension Subsystem
Steering Subsystem
Front Wheel Subsystem
Rear Wheel Subsystem
Body Subsystem
If a more detailed model is wanted, powertrains, brakes, anti-roll bars, and differentials
etc. can be added to the vehicle model. There are several pre-made subsystems included
and the opportunity to build your own subsystems. The subsystems interact with each
other via communicators. There are input communicators that read information into a
subsystem with data and output communicators that send information from a subsystem
to another.
5.1 Front Suspension Modelling
Front suspension design is of SLA type. The model used is obtained from MSC
Software, the company who develop ADAMS, and is modified to fit the purpose of the
modelling. The wheel is connected to the upright via a hub bearing which is a revolute
joint. On the upright four control arms are connected with spherical joints, two lower
ones and two upper ones. The other ends of these control arms are connected to the cars
frame via revolute joints. On each upright there also is a tie rod connected with a
spherical joint. The other end of the tie rod is to be connected to the steering rack via a
spherical joint. Springs and dampers are connected to the uprights via pull-rods with
spherical joints. The pull-rods are connected to the rockers with hooke joints. The
rockers are connected to the chassis with revolute joints and to the dampers with hooke
joints. The communicators in the front suspension act between the front suspension and
the body, the steering, the front anti-roll bar and the front wheels.
Figure 5.1. The ADAMS/Car model of the front suspension.
5.2 Rear Suspension Modelling
The rear suspension is also of SLA type. The model consists of uprights upon which the
wheel hubs are connected with revolute joints, see Figure 5.2. The control arms are
connected to the upright with spherical joints. To the body the control arms are
connected via revolute joints. The drive shafts are connected to the spindles with convel
joints. The spindles are connected to the uprights with revolute joints. The drive shafts
are also connected to the body with convel joints. The push rods are connected to the
upright with spherical joints and to the rockers with hooke joints. The rockers are
connected to the body with revolute joints and to the dampers with hooke joints. The toe
linkage is connected to the uprights with spherical joints and with hooke joints to the
lower control arms. The communicators in the rear suspension act between the rear
suspension and the body, the steering, the rear anti-roll bar and the rear wheels.
Figure 5.2. The ADAMS/Car model of the rear suspension.
5.3 Steering Modelling
The model of the steering system is a rack and pinion configuration, see Figure 5.3.
There is no need for a complete steering system to be modelled since the steering input
can be applied directly on the uprights via a rod. But to prepare for future simulations a
complete steering subsystem was modelled. This gives the possibilities to have a
steering wheel input instead of just a movement of the rod connecting the uprights. The
steering rack is connected to the tie rods via spherical joints. The steering rack is
connected to the steering rack house with a translational joint and to the steering shaft
with a revolute joint. The communicators in the steering system act between the steering
system and the front suspension and the steering system and the body.
Figure 5.3. The ADAMS-model of the steering system.
5.4 Wheels Modelling
The wheel model is based on data provided by Goodyear Racing. Goodyear Racing had
ADAMS-data available for their 13” racing slicks. The representation used is the 94
Pacejka Magic Formula. Unfortunately the tyre data provided by Goodyear doesn’t take
camber angles into consideration. Communicators act between the wheels and the
suspension.
5.5 Body
The body subsystem used consists only of a point mass in the centre of gravity.
Communicators act between the body subsystem and steering subsystem, front
suspension subsystem and rear suspension subsystem.
5.6 Simulation
A simulation is very easily done with ADAMS. There are a number of different pre-set
simulation modes that can be used such as a suspension analyser where different kinds
of wheel travel and steering simulations can be run. There are also full vehicle
simulation models available such as ISO lane change and steady state cornering. A
problem when simulating with ADAMS is the sensitivity for equilibrium problems with
ill-defined models.
6 Parameter Study
Studies were made to investigate the influences of different settings of system
parameters such as kingpin inclination angle, caster angle, roll centre height. Knowing
how these parameters influences and interacts made it possible to improve the models
until the guidelines were fulfilled. The parameter study also reveals the interactions
between parameters. This information is lost when performing tests using the “one
parameter at the time” approach. This kind of information is very useful when changing
one parameter as this can lead to unexpected effects on other parameters.
6.1 The Taguchi Methods
Since there are a lot of different parameters influencing the dynamics of a suspension it
is a difficult and time-consuming task to analyse. The Japanese engineer Ganichi
Taguchi introduced a new approach in quality engineering using orthogonal
experiments. The benefit of using Taguchi’s methods is the information obtained about
the parameters interactions. This information is not revealed when using methods where
the “one parameter at the time” approach is used [5].
The Taguchi methods are very powerful ways to investigate a large set of parameters.
By using orthogonal matrices the number of experiments can be reduced while still
producing results of interest. It also gives the possibility to study the interactions
between the parameters studied. The resolution of the experiments is a function of the
number of experiments performed. Few experiments will lead to mixed interaction
results and therefore makes it harder to correctly identify interactions that have high
influences. The resolution of an orthogonal matrix is its possibility to isolate the main
effects from the interactions. Resolution for orthogonal matrixes are defined in levels as
follows:
•
•
•
V: All main effects are isolated and two factor interactions are isolated from
other two factor interactions.
IV: All main effects are isolated from two factor interactions, but two factor
interactions can be mixed with other two factor interactions.
III: Main effects and two factor interactions can be mixed.
The size of the orthogonal matrix to obtain a certain resolution can be viewed in Table
6.1.
Resolution
Matrix
III
IV
V
7
4
3
L8
15
8
5
L16
31
16
6
L32
Table 6.1. Maximum of two level factors as a function of resolution and matrix size.
Trial no.
1
2
3
4
5
6
7
8
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
Parameters
AxB
C
1
-1
-1
-1
-1
-1
1
-1
1
1
-1
1
-1
1
1
1
AxC
1
-1
1
-1
-1
1
-1
1
BxC
1
1
-1
-1
-1
-1
1
1
Table 6.2. L8 Orthogonal matrix with resolution V.
The orthogonal matrix should be filled with low levels replacing the minus signs and
high levels replacing the plus signs. Trials are then performed with accurate settings for
the parameters of each row in the orthogonal matrix. The desired quantify is measured
for every attempt. When all trials are performed the column for each parameter is
multiplied with the result column and then divided by the number of plus signs. This
gives the influence of the parameter. The influence is a measure of the product of the
selected parameter interval and the quantity measured in the trials. Table 6.2 shows the
L8 matrix with resolution V which were used for most of parameter studies conducted.
The Taguchi methods are often used in tests generating one result per trial. It is possible
to use the Taguchi method in tests generating more than one result per trial such as time
dependent tests. This gives the possibility to study how the parameters influence the
sought variable over a specified interval. This approach is presented by Beckman and
Agebro in their thesis work, “A Study of Vehicle Related Parameters Influencing the
Initial Phase of Ramp Rollovers”, and is called “the Continuous Taguchi Method” [5].
By using the Taguchi method the design parameters are investigated to find important
interactions and which parameters that influences on the behaviour of the system
analyzed.
6.2 Parameters of Interest
There are many parameters that are of interest when designing the suspension geometry.
Due to time there was no possibility to perform tests including all parameters of interest.
During the model building phase certain parameters were found to be more important
than others and those parameters are used in the parameter studies. Also parameters that
will be adjustable on the car are investigated in the study. The parameters chosen to
investigate are the A-arm’s mounts to the frame as the design of the frame allows the Aarms joints to be moved in vertical direction at the frame. Some important parameters,
such as tyre characteristics, could not be investigated due to lack of information.
6.2.1 Parameter Levels
The levels chosen for the studies have been chosen so they can be used as the
adjustment levels for the race car. The low levels then correspond to one endpoint of the
wanted adjustment possibility and the high levels to the other endpoint. Several
iterations had to be performed until the wanted span of adjustability was found.
6.3 Results
The parameter study was an iterative process. After each iteration the results were
analysed and the models used improved until the guidelines were fulfilled. The results
presented below, Figure 6.2 to Figure 6.24, are from the final design and the adjustment
levels are the wanted. The upper part of each figure shows how the adjustment levels
affect the studied parameter and the lower part of the figure shows the unchanged
design. By adding the results presented in the upper part of each figure to the unchanged
set-up in the lower part the same figure the characteristics after made changes is
received.
6.3.1 Parameter Study of Front Suspension
The major purpose of the parameter study of the front suspension design was to analyse
the camber gain and the effects of adding anti dive. Also the behaviour of the roll centre
was investigated as the roll centre height and movement affects the handling qualities of
the car. The design of the front suspension is made in such way that the joints at the
frame can be changed in Z-direction to make it possible to change the cambergain
characteristics and add anti features. When changing camber gain or anti feature other
parameters are also affected. The parameters used in the studies can be viewed in Table
6.3 and in Figure 6.1.
Parameter
A
B
C
Explanation
Height to the inner joints for the upper control arms [mm]
Height to the inner joints for the lower control arms [mm]
Anti-Dive adjustment of the rear lower control arm [mm]
Table 6.3. Explanation of parameters used in parameter study.
B
A
C
+
Figure 6.1. Parameters used in the parameter study of the front suspension.
In Table 6.4 the chosen low and high levels of the parameters are shown. The mounts
on the frame will also have levels between the low and high levels used in the parameter
study.
Parameter
A
B
C
Low Level
300.0
119.0
0.0
High Level
320.0
139.0
20.0
Unit
[mm]
[mm]
[mm]
Table 6.4. Levels used in the parameter study of the front suspension.
Figure 6.2. Results from parameter study of Anti Dive characteristics during roll.
Figure 6.3. Results from parameter study of the camber angle of left wheel during roll.
Figure 6.4. Results from parameter study of roll centre vertical travel during roll.
Figure 6.5. Results from parameter study of the lateral travel of roll centre during roll.
Figure 6.6. Results from the parameter study of toe angle variation during roll.
Figure 6.7.Results from parameter study of lateral wheel travel at track during roll.
Figure 6.8. Results from parameter study of scrub radius variation during roll.
A
C
B
Figure 6.9. Parameters used in steering geometry parameter study.
6.3.2 Parameter Study of Steering Geometry
The design of the steering geometry allows the outer end of the tie rods to be connected
to the uprights at three different locations in the Y-direction and the steering rack can be
moved back and forth in the cars longitudinal direction. To minimize Bump steer the tie
rods are located in the plane made up by the two upper A-arms of the front suspension.
The parameter study is made to investigate the adjustability of the Ackermann geometry
and the bump steer. The parameters used can be viewed in Table 6.5 and the levels in
Table 6.6. Two different test sessions were performed, one testing the bump steer
characteristics and one testing the Ackerman geometries. The results from the two tests
are shown in Figures 6.10 to 6.15.
Parameter
A
B
C
Explanation
Location of outer tie rod end in Y-direction [mm]
Location of inner tie rod end in X-direction [mm]
Location of inner tie rod end in Z-direction [mm]
Table 6.5. Explanation of parameters used in parameter study.
Parameter
A
B
C
Low Level
572.0
-40.0
325.0
High Level
592.0
-60.0
335.0
Unit
[mm]
[mm]
[mm]
Table 6.6. Levels used in the parameter study of the front suspension.
Figure 6.10. Results from parameter study showing camber as function of wheel travel.
Figure 6.11. Results from parameter study of toe angle variations as function of wheel travel.
Figure 6.12. Results from parameter study of camber characteristics for curve outer wheel as function
of steering rack displacement.
Figure 6.13. Results from parameter study of camber characteristics for curve inner wheel as function
of steering rack displacement.
Figure 6.14. Results from parameter study of outside turn diameter as function of steering rack
displacement.
Figure 6.15. Result from parameter study of percent Ackerman as function of steering rack
displacement.
6.3.3 Parameter Study of Rear Suspension
The design of the rear suspension is made in similar way as the front suspension, the
joints at the frame can be changed in Z-direction to make it possible to change the
camber gain characteristics and add anti features. The influences of this can be read out
from Figures 6.17 to 6.24. The parameters used in the studies can be viewed in Table
6.7 and in Figure 6.1. Table 6.8 shows the parameter levels used.
Parameter
A
B
C
Explanation
Height to the inner joints for the upper control arms [mm]
Height to the inner joints for the lower control arms [mm]
Anti-Dive adjustment of the rear lower control arm [mm]
Table 6.7. Explanation of parameters used in parameter study.
Parameter
A
B
C
Low Level
300.0
119.0
0.0
High Level
320.0
139.0
20.0
Unit
[mm]
[mm]
[mm]
Table 6.8. Levels used in the parameter study of the rear suspension.
Figure 6.17. Result from parameter study of anti-squat as function of roll angle.
Figure 6.18. Result from parameter study of camber for curve inner wheel during roll.
Figure 6.19. Results of parameter study of camber for curve outer wheel during roll.
Figure 6.20. Result from parameter study of roll centre vertical travel during roll.
Figure 6.21. Result from parameter study of roll centre lateral travel during roll.
Figure 6.22. Result from parameter study of toe angle variation of curve inner wheel during roll.
Figure 6.23. Result from parameter study of toe angle variation of curve outer wheel during roll.
Figure 6.24. Result from parameter study of lateral wheel travel at wheel centres.
7 Discussion and Design Results
7.1 Track Width and Wheelbase
The load transfer is a linear function of the wheelbase. Different static axle load
distributions will only shift the result in Y-direction. The differences between having a
wheelbase of 1525mm, the smallest allowed by the rules, and a wheelbase of 1700mm
are very small as seen in Figure 3.1. A longer wheelbase results in less longitudinal load
transfer. Therefore the wheelbase of the car will be set by the packaging conditions but
are to be kept as small as possible to make the race car react quicker on steering. This
will result in a wheelbase around 1700mm. The drawbacks of having a short wheelbase
is that it may lead to an unstable car at high speeds. But since the speeds are fairly low,
average speeds are around 45 km/h, this is considered not to be a problem.
The lateral load transfer as function of track width is also a linear functions as seen in
Figure 3.2. The results from tests investigating the influences on the ackermann angle
when the trackwidth is changed showed that the effects were small as seen in Equation
7.1, which shows the effects for two different track widths, 1250mm and 1350mm, at a
hairpin with an outer diameter of 9m.
1700
= 0.439 rad = 25.14°
r − 2 ⋅ tw 4500 − 12 ⋅1250
l
1700
δ=
=
= 0.444 rad = 25.46°
1
r − 2 ⋅ tw 4500 − 12 ⋅1350
δ=
l
1
=
(7.1)
During a 1G turn the change in vertical force on the curve inner wheel is 275N with a
trackwidth of 1250 mm and 254N with a trackwidth of 1350mm, in kilograms 28kg and
26kg. The difference may not be big but since a tyres ability to produce lateral force is,
among other parameters, a function of the tyre load a few kilos can be enough to pass
beyond the peak of the tyre force curve, the tyre will be overloaded. Since the very low
weight of a formula student car there are no tyres avivible developt especially for this
kind of light weight vehicles, therefore the risk of overloading a tyre due to lateral load
transfer is very low. The problem for this kind of light weight vehicle is the other way
around, the load transfer may lead to loss in lateral force on the curve inner wheel due to
underload.
In Figure 3.3 it can be seen that an increase in tyre load from 125lbs to 250lbs, an
increase of 100%, results in an 115% increase of the lateral force at 10 slip angle for a
20x8-13” Goodyear racing slick. The reason to use a wider trackwidth will not be to
eliminate the risks for a tyre overload, it is to prevent a tyre underload to happen.
The trackwidth is choosen to 1250mm in the front and 1200mm at the rear. The main
reason for having a smaller rear trackwidth is that wider tyres are fitted in the rear than
the front, the front tyres will be 20x6.2-13” and the rear tyres 20x7.2-13”, a difference
of 1.0” or 25.4mm. Having the same trackwidth front and rear would cause the rear
tyre’s inner line to be closer to the inside of the curve than the front tyre’s. This could
lead to the driver knocking down the cones that marks the track with his rear tyres
trying to take the shortest way possible only looking on his front wheels. Having a
bigger front trackwidth will also have the advantage of letting the frontaxle taking up a
bigger part of the rolling moment. The effect of this is that softer springs can be used at
the rear optimizing the rear traction and allowing more and earlier throttle on corner
exits.
7.2
Front Suspension Geometry
The final design of the front uprights was set in cooperation with the MME-students
designing the uprights and the Vehicle Dynamics-students designing the brake system
due to the packaging issues. The locations of the joints at the uprights can be found in
Table 7.1 and joints on the frame in Table 7.2. Figure 7.4 shows a model with the joints
marked.
Name
Wheel Centre
LCA outer
UCA outer
Tie Rod outer
X-location
0.0
0.0
20.0
-30.0
Y-location
645.0
607.0
572.0
572.0
Z-location
254.0
159.0
374.0
374.0
Adjustability
Y: +20.0
[mm]
Table 7.1. Location of joints at the front uprights.
Name
LCA front
LCA rear
UCA front
UCA rear
Tie Rod inner
X-location
-10.0
260.0
0.0
230.0
-40.0
Y-location
150.0
150.0
200.0
200.0
230.0
Z-location
119.0
119.0
300.0
300.0
305.0
Adjustability
Z: +20.0
Z: +20.0, +20.0*
Z: +20.0
Z: +20.0
Z: +20.0
[mm]
Table 7.2. Location of front suspension joints at the frame. * for Anti Dive adjustment
UCA outer
Tie Rod outer
UCA rear
Wheel Centre
LCA outer
Z
X
LCA front
LCA rear
Y
UCA front
Tie Rod inner
Figure 7.4. Location of the different joints in the front suspension.
The corresponding geometries of the front suspension can be viewed in Table 7.3.
Parameter
Anti Dive
Kingpin Inclination
Scrub Radius
Caster Angle
Trail
Camber Gain in Roll
Roll Centre Height
Value
0.0
9.2
4.6
5.3
19.1
0.32
13.8
Unit
[%]
[degrees]
[mm]
[degrees]
[mm]
[˚Camber/1˚Roll]
[mm]
Table 7.3. Front suspension geometry at unchanged set up.
By using the adjustment levels some parameters can be changed inside the intervals
listed in Table 7.4.
Parameter
Anti Dive
Camber Gain in Roll
Roll Centre Height
Ride Height
Camber Angle
Toe Angle
Low Value
0.0
0.16
2.8
25.0
-4.0
-2.0
High value
52.4
0.32
45.7
55.0
0.0
2.0
Unit
[%]
[˚Camber/1˚Roll]
[mm]
[mm]
[degrees]
[degrees]
Table 7.4. Parameters that can be changed in steady state.
The final design of the front suspension is a trade of between performance and
manufacturability. The hardest and most time-consuming part has been the design of the
front uprights. Due to lack of space available in the front rims the final design is a little
bit off from the design wanted. This mostly affects the kingpin inclination angle and the
scrub radius. It was desired to keep the kingpin inclination angle below 8.0 degrees but
the final design has a kingpin inclination angle of 9.2 degrees and this leads to a scrub
radius of 4.6mm.
7.2.1 Camber Characteristics
The parameter study shows that all parameters used in the study affects the camber
characteristics of the front suspension. The parameter showing the biggest influence is
angle of the plane made up by the upper A-arms in the side view. Also the interaction of
this parameter together with the other two parameters used, angle of the lower A-arm
plane in the front and side views, shows big influences on the camber gain during roll.
If more camber gain is wanted the interaction of the angle of the upper A-arms and the
angle of the lower A-arms in the side view is the combination that gives the biggest
positive influence. The biggest negative influence is, if less camber gain is wanted, the
angle of upper A-arms in front view.
7.2.2 Anti Features
As wanted, by tilting the plane made up by the lower A-arms in the side view, anti dive
feature was added to the front suspension. The design allows the rearward joint of the
two lower at the frame to be raised 20mm. Doing this slightly more than 50% anti dive
is added. Adding anti dive to the front suspension also gives some other effects. Adding
around 50% anti dive will decrease the camber gain with around 0.05 degrees Camber
per degree Roll, lower the roll centre height with about 20mm, increase the lateral
movement of the roll centre with 80mm at one degree roll and will increase the roll steer
with 0.02 degrees at one degree roll.
7.2.3 Roll Centre Characteristics
The parameter having the biggest positive influence on the roll centre height, the
parameter which raises the roll centre most up from the ground, is the angle of the plane
made up by the lower A-arms in the front view. Adding anti dive will lower the roll
centre. Making the angle of the upper A-arms in front view smaller will also lower the
roll centre.
Increasing the angle of the lower A-arms in the front view, high value of parameter B,
will increase the lateral movement of the roll centre for the front suspension while
decreasing the angle of the upper A-arms in the front view will make the lateral
movement of the roll centre smaller. Adding anti dive will also decrease the lateral
movement of the roll centre.
7.3 Steering
The purpose of the steering geometry design was to minimize bump steer and have the
possibility to adjust toe angles and the Ackermann geometry. Adjustable outer tie rod
joints make the adjustability of Ackermann geometry possible. Another criterion that
the steering system had to fulfil was that the car had to manage a hairpin with an outer
diameter of 9m. Table 7.1 and Table 7.2 show the locations of the joints in the steering
system and the adjustability levels wanted. Table 7.5 shows the corresponding levels of
adjustability for Ackerman geometry and toe angles.
Parameter
Toe Angles
Percent
Ackermann
Low Value
-2.0
-9.4/-23.2
High Value Unit
2.0
[degrees]
55.8/78.1 [%] at 10˚/35˚
Ackermann angle
Table 7.5. Adjustability of steering geometry.
7.3.1 Bump Steer
The goal was to minimize the bump steer. By having the tie rods in the same plane as
the upper A-arms the bump steer was minimized to 0.03 degrees steering angle per
10mm vertical travel in jounce and -0.01 degrees in rebound. If the tie rods not are
located in the same plane as the upper A-arms this will affect the bump steer
characteristics a lot. The parameter study shows that moving the inner end of the tie
rods up by 10mm will increase the bump steer with 0.30 degrees steering angle per
10mm wheel travel in both jounce and rebound. This corresponds to a curve radius
about 425m.
7.3.2 Ackerman
By adjusting the outer ends of the tie rods in Y-direction the amount of Ackerman can
be changed. By moving the outer end of the tie rod 20mm further out from the
centreline of the car the Ackerman is raised from around 0% to 65% at a steering angle
corresponding to a curve with a radius of 11.5m. By moving the steering rack forward
the Ackerman geometry can be reduced. A movement of 20mm forward will result in
10% less Ackerman for the 11.5m radius curve. Moving the inner end of the tie rods up
from the plane made by the upper A-arms in the front view will not affect the
Ackerman. The parameter study also shows that with a steering rack movement of
30mm the outside turn diameter is 9180mm with the Ackerman set to the low value and
8180mm with the Ackerman set to the high value. At 35mm steering rack movement the
corresponding values are 8040mm and 6850mm. The reasons for the smaller outer turn
diameter with Ackerman set to high value is not the change in percent Ackerman, the
reason is that the change made to the steering geometry also changes the steer camber
characteristics as shown in Figure 5.12 and Figure 5.13. With Ackerman set to low the
change in camber of the curve outer wheel is -1.1 degrees at 35mm steering rack
movement and -1.3 degrees with Ackerman set to the higher value. For the curve inner
wheels the values are +5.8 degrees and +7.7 degrees, both curve inner and curve outer
wheels are leaning into the curve as shown in Figure 7.5.
Figure 7.5. At big steering angles both curve inner and curve outer wheel leans into the curve.
7.4 Rear Suspension Geometry
The designs of the rear uprights are not the same as the design of the front uprights,
since inboard brakes are used. The operation conditions for the rear suspension are
slightly different since no steering is applied here. The locations of the joints at the
uprights can be found in Table 7.6.
Name
Wheel Centre
LCA outer
UCA outer
Tie rod outer
X-location
1650.0
1590.0
1650.0
1650.0
Y-location
600.0
590.0
590.0
590.0
Z-location
250.0
140.0
370.0
140.0
Adjustability
[mm]
Table 7.6. Location of joints at the rear uprights.
The locations of the joints on the frame can be read in Table 7.7. The toe link
configuration at the rear suspension is a so-called ungrounded toe link; it is connected to
the lower control arm instead of connected to the frame.
Name
LCA front
LCA rear
UCA front
UCA rear
X-location
1550.0
1850.0
1550.0
1850.0
Y-location
150.0
150.0
300.0
300.0
Z-location
109.0
109.0
250.0
250.0
Adjustability
Z: +20.0
Z: +20.0
Z: +20.0
Z: +20.0
[mm]
Table 7.7. Location of rear suspension joints at the frame.
The corresponding geometries of the rear suspension can be viewed in Table 7.8, which
shows the parameters at the unchanged set up.
Parameter
Anti Squat
Anti Lift
Kingpin Inclination
Scrub Radius
Caster Angle
Cambergain positive Roll
Cambergain negative Roll
Roll Centre Height
Value
0.0
0.0
0.0
10.0
14.6
0.73
1.05
76.9
Unit
[%]
[%]
[degrees]
[mm]
[degrees]
[˚Camber/1˚Roll]
[˚Camber/1˚Roll]
[mm]
Table 7.8. Rear suspension geometry at unchanged set up.
“Camber Gain positive Roll” refers to the curve outer wheel measured at 0.5 degrees
roll angle at contact patches and “Camber Gain negative Roll” refers to the curve inner
wheel.
By using the adjustment levels in Table 7.7 some parameters can be changed inside the
intervals listed in Table 7.9, which also shows the adjustment levels for static camber,
toe angles and ride height.
Parameter
Anti Lift
Anti Squat
Camber Gain positive Roll
Camber Gain negative Roll
Roll Centre Height
Ride Height
Camber Angle
Toe Angle
Low Value
0
0
0.58
0.84
53.4
25.0
-4.0
-2.0
High value
29
24
0.84
1.18
107.5
55.0
0.0
2.0
Unit
[%]
[%]
[˚Camber/1˚Roll]
[˚Camber/1˚Roll]
[mm]
[mm]
[degrees]
[degrees]
Table 7.9. Parameters that can be changed in steady state.
Note that anti squat and anti lift can’t be changed independently of each other. The
parameters characteristics during operation of the car can be viewed in section 4.4.
7.4.1 Camber Characteristics
The parameter study shows that all parameters used in the study affects the camber
characteristics. The parameter showing the biggest influence is angle of the plane made
up by the upper A-arms in the side view, the same parameter as in the front suspension.
Setting this parameter to high value will decrease the amount of camber gain for the
curve outer wheel and increase the amount of camber gain for the curve inner wheel.
7.4.2 Anti Features
By tilting the plane made up by the lower A-arms in the side view anti features can be
added. The available anti features for the rear suspension are anti squat and anti lift.
Anti squat and anti lift are coupled to each other since they both are effects from the
same changes on the suspension. Since inboard brakes are mounted at the rear
suspension the amount of anti lift will always be larger than twice the amount of anti
squat present. The design allows the forward joint of the two lower at the frame to be
raised 20mm. Doing this slightly more than 24% anti squat is added. Adding this
amount of anti decreases the amount of cambergain with 0.08 degrees per degree roll, it
also raises the static roll centre height with more than 25mm.
7.4.3 Roll Centre Characteristics
The parameter having the biggest positive influence on the roll centre height, raises the
roll centre further up from the ground, is the angle of the plane made up by the lower Aarms in the front view. Adding anti effects will lower the roll centre. Making the angle
of the upper A-arms in front view smaller will instead raise the roll centre. The
variations in the roll centre height during roll are very small; at the unchanged set up the
vertical variation is about 0.5mm. The lateral movement of the roll centre for the same
setup is slightly smaller than 13mm.
8 Future Work
To be able to carry this work further the first thing to do is to take complete measures of
the Formula Student race car built. These measures can then be used to update the
computer models used. With models that correspond to the real car new simulations can
be made and further development is possible. This gives the opportunity to have two
development processes running simultaneously, one on the race car and another in the
computer. This also gives the opportunity to test how well the computer models agree
with the real race car.
One important parameter, not so much discussed in this thesis work, is the tires. As the
tires are the part of the race car having the biggest dynamic influence a lot of work can
be carried out investigating how different tires would affect the handling qualities of the
race car. Different sizes, brands and rubber compounds are available at the market.
Testing different tires against each other to find out which tires fits the race car best
could be of bigger importance than improving the design of the suspension. Especially
since the car is raced under such conditions that the wheel travel is very small and not
affecting the suspension geometry much.
If KTH decides to develop a new car for the 2005 Formula Student event the 2004 car is
a very useful tool in the engineering process. The car can be used in different tests and
evaluated to become base to start the new development process from. The computer
model of the car can be updated with drivetrain and brakes and used for evaluation of
the car’s dynamic behaviour. By using the computer model unwanted behaviours on the
2004 car can be prevented from appear on the 2005 car as well.
9 Nomenclature
Fzi
l
ax
Fz i
lat
hCG
twi
Vertical axle load, i=1 for front axle and i=2 for rear axle
Wheelbase
Length from front axle to centre of gravity
Acceleration in X-direction
Length from ground up to centre of gravity
Lateral load transfer, i=1 for front axle and i=2 for rear axle
Lateral acceleration
Height to centre of gravity
Track width, i=1 for front axle and i=2 for rear axle
10 References
10.1 Literature
[1]
2004 Formula SAE Rules, SAE Inc, USA
[2]
Milliken, William F. & Milliken, Douglas L. (1995), Race Car Vehicle
Dynamics, SAE Inc, USA
[3]
Reimpell, J. & Stoll, H. & Betzler, J.W. (2001), The Automotive Chassis:
Engineering Principles 2nd Edition, Butterworth-Heinemann, England
[4]
Wennerström, Erik (2001), Fordonsteknik, KTH, Stockholm
[5]
Beckman, M & Agebro M. (2002), A Study of Vehicle Related Parameters
Influencing the Initial Phase of Ramp Rollovers, Department of Vehicle
Engineering KTH, Stockholm
[6]
Race Tire Analysis and Plotting Toolbox Open Wheel Release version 7.2
(2001), The Goodyear Tire and Rubber Company
[7]
BOSCH Automotive Handbook 5th Edition (2000), Robert Bosch GmbH,
Germany
10.2 Oral References
[8]
Christer Lööw, Öhlins Racing, Stockholm
[9]
Erik Lycke, Endurance Race Car Driver, Stockholm