SAE - Design of a Suspension for a 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