The Optimum Weight Distribution for a Formula SAE Vehicle

Transcription

1 The Optimum Weight Distribution for a Formula SAE Vehicle Callum W. Watson 1 University of New South Wales at the Australian Defence Force Academy The aim of this report is to detail the investigation into the effects that the centre of gravity of a Formula SAE vehicle has on its ability to negotiate corners of varying radii at different velocities. The Formula SAE competition has varying static and dynamic components with a heavy weighting on the endurance event. This endurance event is conducted on a very technical course that encapsulates many corners of varying radii and limited opportunities for maximum speed to be accomplished. This demonstrates the need to know what influences of the vehicle handling can be optimised in order to gain the most advantage during the endurance event. There is little known about the effect that the centre of gravity has on the handling ability of a Formula SAE vehicle in certain situations and this thesis will investigate how to optimise the weight distribution so that the velocity the vehicle can negotiate the various turns is maximized. It also investigates the required tyre angle of each front wheel during the max velocity case and evaluates how the performance may be improved due to the findings. Contents I. Introduction 2 A. Motivation 2 B. Aims 2 C. Limitations 2 II. Background 3 A. Fundamental Physics 3 B. System Configuration 5 III. Review of Current Literature 6 IV. Project Work 6 A. Work Conducted 6 B. Results and Analysis 7 C. Significance of the Project 9 D. Problems Encountered 10 V. Conclusions 10 VI. Recommendations 10 Acknowledgements 11 References 11 1 LT, School of Engineering and Information Technology, ZEIT4501 Final Project Report, 2013 UNSW@ADFA

2 F FL F FR F RR F RL F CG R Nomenclature = Lateral Force on inside front wheel [N] = Lateral Force on outside front wheel [N] = Lateral Force on outside rear wheel [N] = Lateral Force on inside rear wheel [N] = Centripetal Force through vehicle Centre of Gravity (CoG) [N] = Radius [m] = Moments about the x-axis [Nm] = Moments about the y-axis [Nm] A. Motivation I. Introduction The Formula SAE competition gives students from Universities all around the world an opportunity to take part in a project from start to finish that entails both the design and manufacturing processes with regards to producing a racing vehicle. This provides a great opportunity to gain experience within the vehicle dynamics field for all students involved [1]. The University of New South Wales at the Australian Defence Force Academy is a participating University in the Australian competition and this thesis will focus on advancing their knowledge in relation to centre of gravity effects on vehicle handling. This competition includes both static and dynamic events where the vehicles are judged on everything from the design of the vehicle to the performance in three separate dynamic events [2]. The general track size and geometry ensures that the course is more technical with limited opportunity to reach maximum limited speed. This means that it essential to create a car that is able to outperform the opponents with their handling capability. This has created a demand on understanding all factors that influence the handling of the vehicles and how they may be optimised. B. Aims The aim of this thesis is to investigate the effects that the location, including both height and longitudinal position, of the centre of gravity of a Formula SAE vehicle has on its ability to negotiate corners of varying radii. This will help lead to the calculation of the maximum velocity a Formula SAE vehicle can have whilst negotiating turns of varying radii. This will then be calculated for a few different positions to see which centre of gravity location benefits the handling characteristics the most in order to find the most advantageous position. It will then look at the required angle of the front wheels to successfully negotiate the turn and the differences between the angles required with the centre of gravity position. This will be conducted by establishing a model that will be focused on the scenario when the inside rear wheel lifts of the ground whilst cornering. This allows for an simplified force balance to be performed and provides the opportunity to produce a program that will be able to determine the maximum velocity that the vehicle can attack the corner given different centres of gravity. Once this program can be tested and verified, it will provide the opportunity to look at the performance of a vehicle given a certain centre of gravity and figure out which configuration is most beneficial. This will lead into being able to design the car with the optimum centre of gravity already in mind. C. Limitations The limitations experienced throughout this thesis included such things as the availability and restrictions placed on required tyre data to determine key performance factors. This will not produce any detrimental effects as the final Final Project Report, 2013 UNSW@ADFA 2

3 program will require input by the user on the tyre properties specific to the tyre they plan on using for the competition. Another limitation on this thesis was the time frame available in order to develop the thesis further. This has meant that a single scenario model has been created for when the inside rear wheel lifts of the surface. This has created the ability to further develop the base model in follow on thesis projects. An example of this would be the effects of the centre of gravity during acceleration and braking and then comparing with steady state cornering in what would encapsulate a more overall view as normally there is a tradeoff between the two. This thesis will utilise data from the current UNSW@ADFA Formula SAE vehicle for most of the calculation parameters. These are simply used as a guide for the program as these values will require user input for other car setups. This will not affect the final outcome as it will only provide realistic values for input data to test the model. II. Background When dealing with vehicle dynamics there is a multitude of factors that influence the performance of a motor vehicle. These can normally be separated into two distinct categories which include the factors that affect handling and those that affect the power characteristics of the vehicle, more commonly referred to as the driveline. This thesis focusses on the handling ability of a Formula SAE vehicle during steady state cornering and looks specifically at how the handling changes when the centre of gravity is in different locations. For this to be successful a number of fundamental concepts have to be understood along with the effect that each of them have on each other. A. Fundamental Physics The major components comprise of a force and moment balance, tyre characteristics and a general understanding of suspension geometry. These three major components are very closely linked with tradeoffs and compromises often created between them to optimise performance. A brief description of these components is as follows. The overarching force that has to be opposed is the centripetal force acting through the centre of gravity of the vehicle. This requires the sum of the lateral forces created by the tyres to be equal and opposite too this force. It is easier to see visually that when the centre of gravity location is varied then the required angle of the front tyres is going to change due to the rear tyres being fixed. This is where the toe in/toe out angle of the rear tyres would be changed in order to try and create the most beneficial setup. This is typically easier to do when looking at either small or large radius turns but not a combination of both. A graphical representation of these forces can be seen in the following figure. F FL F FR Radius CoG Centre of turn F CG = F RL F RR Figure 1: Graphical Representation of lateral forces Final Project Report, 2013 UNSW@ADFA 3

4 This produces the following force balance equation: F CG + F FL + F FR + F RL + F RR = 0 In order to determine whether there is a feasible solution this force equation has to be satisfied. This is done by determining if the required angle of the front wheels to produce the most beneficial slip angle depending on the vertical load experienced by each tyre is within the steering limit. The equation may not be satisfied which would mean that the vehicle would not be able to negotiate a turn of that radius at the given velocity for that particular location of the centre of gravity due to tyre slip i.e., the traction in the tyres cannot produce enough lateral force to counter the centripetal force of the vehicle. The magnitude of the individual forces is largely dependent on the overall weight of the vehicle, but most importantly where the centre of that weight is located. This centre of gravity determines the distribution of the vertical forces placed on the front versus rear wheels and the height of the centre of gravity affects the weight transfer to the outside tyres whilst cornering [3]. This is due to the size of steering torque experienced from the side force created by cornering. This is depicted as the roll moment in figure 2 below. Figure 2: Diagram of Axis [4] Figure 2 aids in picturing the following equations to ensure feasibility of the system: and This shows how the centre of gravity height will affect the roll moment and the longitudinal location will affect the pitch. These moments will depict what equal and opposite forces experienced by each individual tyre, resulting in a final lateral force for each tyre mentioned previously and ultimately lead to determining the feasibility of the vehicle configuration. The physics involved when considering the contact between the tyre and the road is extremely complex and can easily become overwhelming. The characteristics and behavior of one tyre to another can differ from manufacturer to manufacturer and produce variable results. This thesis will concentrate on the Hoosier brand of tyre and specifically the 6-inch rim model [5]. This data has been sourced by the Formula SAE team from UNSW for the sole purpose of utilising realistic data for the modeling program. For any user of the program it is required that they place their own tyre data in to produce relevant results. These factors are relevant when discussing allowable slip angles for certain tyres. The slip angle on a tyre is the angle between the direction the tyre is facing compared to the actual direction the tyre is travelling and the optimum slip angles on different tyres vary with different vertical loads. This is due to a higher coefficient of friction for lower vertical loads at smaller slip angles [6]. This is illustrated in figure 3 depicting a tyre that is turning to the left. The tyre follows the red path due to the deformation in the tyre as it rotates around. Final Project Report, 2013 UNSW@ADFA 4

5 Due to restrictions on the use of the tyre data from the Tire Test Consortium for published work the graph could not be displayed, however it was used for the model to give a good approximation of the lateral forces available depending on the vertical load. This was also used to provide data on what slip angle produces the highest coefficient of friction depending on the vertical load applied. From this data it was clear that when the vertical load was less than the slip angle for maximum coefficient of friction was decreased to about five degrees. This compared to a higher vertical load requiring a slip angle of 10 degrees to create highest coefficient of friction. Figure 3: Demonstration of slip angle [7] Suspension geometry is another complicated system that is difficult to fully understand. It takes a lot of time and experience to discover what the consequences are for each little adjustment made as every adjustment can adversely affect the performance of the vehicle. It is for this reason that the thesis will be looking at a basic suspension setup which does not compare camber effects, or any change in caster angles [8]. It is based only on this case where the inside rear wheel lifts off of the ground with zero degree camber, which stiff suspension on the front and softer suspension on the rear will provide the conditions to produce this scenario. B. System Configuration As previously mentioned this thesis focuses on the simulation when the inside rear tyre lifts off of the surface leaving three tyres to produce the required lateral forces to balance out the centripetal force. The figure below depicts how a vehicle turning left would try to balance the forces. It can be seen that when a fixed slip angle is produced dependent on the vertical load of the inside rear wheel, a centre of turn can be calculated. Now depending on where that centre of turn and centre of gravity will determine the direction of the centripetal force. This is what will ultimately determine the required angle of the front tyres. Centre of turn Radius CoG F CG Figure 4: Force Direction change with change of Centre of turn Final Project Report, 2013 UNSW@ADFA 5

6 This is where we start to see the effects of all the major separate influences discussed previously creating complexity within the model. This model then has to be able to adjust values and methods with each changing variable. This is completely separate to previous study that I have undertaken on the handling characteristics of commercial vehicles and passenger cars. This is mainly due to the differences in the radii that the different vehicles are designed for. Commercial cars are designed for much larger radii due to the typical kinds of roads that they have to negotiate. Whenever there is a sharper corner they can slow down a lot nullifying some influences and tyre characteristics. In a Formula SAE vehicle however, it is aimed at getting around small radius turns at the highest possible velocity. The reason for looking at the case where the inside rear wheel lifts off of the ground was because it is a common occurrence during various forms of racing and more specifically during the endurance event at the Formula SAE competition. Even though this is the case there is no information available about how the handling of the vehicle changes when this is the case. This means that there could possibly be an advantage in purposely lifting the inside rear wheel if you have the right configuration. III. Review of Current Literature As mentioned, there was no information that would directly relate to this specific case, however, the following documentation was a key aspect in my research when trying to understand and learn about all the separate influences affecting the handling of a vehicle and more importantly, how all of those influences affect one another. The article written by Hiromichi Nozaki in 2006 [10] detailed closely related information in regards to some of the influences created the centre of gravity and proved to be good reference material that was able to point me in the right direction from the start. This journal was limited to only looking at longitudinal position of the centre of gravity and was simulator testing based which was not relevant for my research. The data obtained was using this simulator where three different test drivers where set the task of negotiating a turn with three different vehicle centre of gravity set ups. These results concluded that a 40:60 front to rear distribution was preferable due to the averaged quicker timings of the drivers. These findings provide some insight into what may be expected but deal with different system parameters which proves difficult to compare with as there is a larger focus on acceleration and braking effects compared to steady state cornering. Another couple of journals that helped build a more thorough understanding of the problem included such articles as Improvement of road ability of a formula SAE vehicle, by H. Oda et al, 2009 [11], which was limited by using the bicycle model which takes the front and rear axle to only have one tyre in the centre, and FTire The Tire simulation model, by M. Gipser 2007 [12], which was valuable in providing information on an exhaustive list of tyre characteristics. These articles both helped develop my understanding of tyre dynamics. A significant portion of my fundamental background knowledge was taken from a couple of textbooks on vehicle dynamics. These included Theory of Ground Vehicles, by J. Wong [13] and Car Suspension and Handling, by D. Bastow, G. Howard and J. Whitehead [14]. Both of these texts have provided a good foundation of understanding into the relevant principles of vehicle dynamics and all that was needed was to adapt to a different environment, as these texts focused more on commercial and military vehicles. A. Work Conducted IV. Project Work The start of this thesis required a lot of research into a topic that was very unfamiliar to me which meant that a large amount of self learning was involved. Once I had built a foundation into general vehicle dynamics I began looking further into the two major contributors to the Formula SAE vehicle handling systems. These included the Final Project Report, 2013 UNSW@ADFA 6

7 suspension and, what eventuated in being the most important for this thesis, tyre characteristics. Once a solid foundation had been set on the fundamentals of slip angles and suspension geometry the majority of the thesis had to begin. This included the formulation of the mathematical system that would be used to help provide the framework in the model program. I was Fortunate enough to be able to help with some testing on the previous UNSW Formula SAE vehicle that was able to provide me with some extremely relevant data. It provided me with the vertical loads experienced by the vehicle whilst conducting steady state turns at three different radii. This was able to provide me with some excellent values to compare with theoretical values being produced in my model. The model begins by defining the required slip angle of the rear outside tyre in order to find the centre of turn for various turn radii. This then leads on to finding all the different angles the front tyres need to be turned in order to counter the effect of the centripetal force direction. This centripetal force change in direction due to various centre of gravity positions is the determining factor when working out what angle the front tyres needed to turn and produced some results that were counterintuitive. These angles then had to be checked to ensure that they were within an acceptable range for a Formula SAE vehicle. This range was depicted as 25 degrees which is the common aiming mark for current Formula SAE vehicles. The next step was to incorporate a check to ensure that the vehicle didn t flip when subjected to the various centripetal forces. This was incorporated by making sure that when the run got to more than 1.7G then it would stop and move to the next iteration. The 1.7G value was taken from the 2013 Formula SAE competition rules that stipulate that all vehicles must be able to sustain the 1.7G mark in a tilt test. The last major check that is in place is to ensure that each of the three tyres did not experience too much of the load and begin to slip. From the tyre data received from the Tire Test Consortium it showed that under perfect conditions the tyres wouldn t slip until 2.5G was experienced. From the testing conducted on a rough surface it was found that the tyres where only able to produce approximately 1.6G. That meant that this model took the real data into account and made that the cut off force before tyre slip. The final model program that takes all of these things into account took a few different versions and updates to remove all the problems that were encountered along the way. This model program has been developed until it was able to produce the desired results where some were expected and others seeming counterintuitive. B. Results and Analysis The results will be split up into the three major findings from the thesis. The first one will look into the effects of the centre of gravity height. This was arguably the test that was most likely going to produce expected results and these became very clear from the adjacent graph. Figure 5 depicts two lines that show the maximum velocity at which the vehicle can negotiate turns of various radii. This is depicted with radius on the horizontal axis in (m) and velocity on the vertical axis in (m/s). The top red line is for the centre of gravity at 0.2m above the ground and the green line for height Figure 5: Variation in maximum velocity due to CoG height Final Project Report, 2013 UNSW@ADFA 7

8 of 0.3m above the ground. These limits were taken as the acceptable range of a Formula SAE vehicle however they can be manipulated by the user to find out the effect of anywhere in between. I chose the limits to show the upper and lower boundaries. The results produced from varying the centre of gravity height were expected due to any increase in height producing a larger roll moment of the car. This increases the vertical load on the outside front tyre which is beneficial, however, it does it at the cost of decreasing the vertical load on the inside front tyre. It actually decreases the vertical load so much that the inside front tyre is no longer able to contribute enough lateral force in order to balance the centripetal force. The next set of results looks at the comparison between changes in longitudinal position of the centre of gravity. The following figure has compared the centre of gravity at positions of 0.5m from the rear axle and 0.75m from the rear axle. These are approximately 35:65 and 50:50 front/rear ratio respectfully. These values are once again towards the realistic boundaries of possible centre of gravity positions due to certain specification requirements in the 2013 Formula SAE competition rules. The model program can be tailored by the user to test other variations of longitudinal position. Figure 6: Maximum velocity comparison for two different centre of gravity positions Figure 6 shows the maximum velocity with the centre of gravity 0.5m in front of the rear axle on the left and 0.75m in front of the rear axle on the right. From the values shown it is clear that as the centre of gravity is moved closer to the front the maximum velocity is increased. The range of radii varies due to the model ending the iteration when the velocity goes over 30.55m/s. This is the due to 30.55m/s being the performance limited speed of a Formula SAE vehicle, which means that if it has reached its maximum speed at 32m radius turns it can negotiate any larger turn. One of the reasons behind these results is determined by how the vehicle weight is distributed. Whilst cornering the front tyres normally contribute to a larger percentage of the lateral force. By placing more weight onto the from tyres it allows for more centripetal force before the tyres become overloaded and slip. Another reason why the vehicle is able to maintain a faster speed around corners when the centre of gravity is further forward is due the location of the centre of turn in relation to the centre of gravity. The further forward the centre of gravity is, the less the angle required on the front tyres to negotiate the turn and hence, the increased speed. Final Project Report, 2013 UNSW@ADFA 8

9 Angle (degrees) Angle (degrees) The third set of results produced from the model program investigates the required angles for both front tyres to achieve the optimum slip angle depending on the vertical load applied to each tyre. The following graphs compare the tyre angle for three centre of gravity positions including 0.5m, 0.75m and 1m from the rear axle. The relationships are as follows: Required angle of inside front wheel Radius of turn (m) x=0.5m x=0.75m x=1m Required angle of outside front wheel Radius of turn (m) x=0.5m x=0.75m x=1m Figure 7: Required tyre angle at various radius turns for both inside and outside tyres As seen from figure 7, the steering angle required for the centre of gravity position located 0.75m from the rear axle remains relatively constant for both inside and outside tyres. The next two positions for the centre gravity show the effects of moving closer to and further away from the rear axle. The trends that are shown depict that the closer the centre of gravity is to the rear axle will require a larger turning angle of both inside and outside front tyres. This compared to the smaller angle required at smaller radius turns for the centre of gravity further away from the rear axle. The required steering angle then starts to increase as the radius increases was counterintuitive when first coming across these results. The results from the comparison of tyre angle required of both front tyres were hard to interpret initially due to the different trends seen in figure 7. After further deliberation there were two main explanations for these results. Firstly, there is the difference between the angles required of each individual tyre depending on longitudinal position. These trends are all to do with the angle of centripetal force created by the location of the centre of turn in Final Project Report, 2013 UNSW@ADFA 9

10 relation to the centre of gravity. To balance these forces it shows that smaller angles are required at a smaller radius turn when the centre of gravity is further forward. This is beneficial as a smaller steering angle is required and therefore less tyre clearance. It was also interesting to see that on the case where the centre of gravity is 0.5m from rear axle the required angle of the inside wheel started out larger than the outside wheel which was expected due to the difference in radius due to the width of the vehicle. However, as the radius is increased the two tyres swap with the outside tyre needing a larger angle. This is due to the change in vertical load on the tyres when negotiating different radius turns. With the roll moment increasing as the turn radius and velocity increase, the optimum slip angle of the inside front wheel decreases quicker than the outside tyre. Interestingly, this is not the case with the most forward centre of gravity. C. Project Significance This project could provide a very beneficial model program that would help predict what the optimal centre of gravity location would be for use within the design stage of the project. This is a crucial part of the design process as the majority of the layout would try to create that centre of gravity position. It will provide information on the effects that the centre of gravity has on the handling characteristics of the vehicle and provide the necessary tyre angle limitations. It will also provide the team with the steering relationship between inside and outside tyres whilst turning. This Thesis is providing some information on a topic that has no other sources of information available to the Formula SAE team and with some further development may be able to give them an edge on understanding the vehicle handling characteristics when the inside rear tyre lifts off of the surface. The project will also be able to change the toe in or toe out angle so that the effects can be better understood. There is a multitude of sources that have gone into detail about the benefits of toe in when all four tyre remain on the ground, however, the effects of this set up are not well known when it is only the three wheeled case. Given the importance of handling for Formula SAE vehicles, every effort should be made to take advantage of any improvements that can be incorporated. V. Conclusions This thesis was aimed at investigating the effects that the centre of gravity has on the handling characteristics of a Formula SAE vehicle during steady state cornering when the inside rear wheel lifts off of the surface. This was conducted by looking into the effect that the height of the centre of gravity had on the maximum velocity it could negotiate corners and as expected the maximum velocity decreased as the height increased. The longitudinal position of the centre of gravity was investigated with a centre of gravity located further forward on the vehicle producing the highest maximum velocities. This was then compared to the required tyre angle for each case showing some counterintuitive results. This showed that the relationship between angles required for different centre of gravity positions at different radii can vary significantly with less angle required the further forward the centre of gravity is. This was all completed to discover what the advantages could be, if any, to purposely lifting the inside rear wheel in order to create a Formula SAE vehicle that was able to negotiate various turn radii at a faster speed. This is due to the change in forces that could see a toe out angle be more preferential than the commonly used toe in angle. VI. Recommendations This thesis has been able to establish a program that can be utilised to find the maximum velocity a Formula SAE vehicle can negotiate various radius turns with a number of suspension geometry remaining fixed. This has provide a good opportunity to develop the model so that it can incorporate more complicated suspension systems and discover what effects they have. This thesis was also limited to the case where only three wheels remained in Final Project Report, 2013 UNSW@ADFA 10

11 contact with the surface during the steady state cornering. This could possibly lead into the case when the tyre remains on the surface; however, this situation is more commonly researched with a plethora of information available. This thesis was limited to the steady state cornering effects and could be combined into the investigation of the acceleration and braking effects. After putting these two together the centre of gravity could then be optimised for a specific track by looking at how much time a vehicle spends at certain radius turns. Acknowledgements Throughout the conduct of this thesis I received valuable insight and guidance from various individuals that I would like to express my thanks to. I would like to thank Mr. Alan Fien who has provided me direction and guidance when required. I would also like to thank FLGOFF Michael Olsen for the ability to help during the conduct of his testing that gave me valuable insight into the parameters of my project and Mr. Lorin Coutts-Smith for the help and guidance with regard to tyre handling characteristics. Finally, I would like to thank Dr. Tapabrata Ray for his assistance when having issues with the programming code and PLTOFF Brodie Kilkenny as a contributing member of the Formula SAE team that provided me with the required information on the previous and current Formula SAE vehicles. References Formula SAE Rules, revised 11/22/12, 2013 SAE International, cited on 19/5/ ibid. 3. Wong, J.Y., 2008, Theory of Ground Vehicles, Fourth Edition, John Wiley and Sons, inc., Ottowa, Canada. 4. Figure 2 image sourced from Cited on 26 May Kasprzak, E.M.( FSAE TTC Co-Director), 2012, Round 5 test data, Calspan and FSAE Tire test Consortium. 6. Bastow, D., Howard, G., and Whitehead, J., 2004, Car Suspension and Handling, Fourth Edition, SAE International, Warrendale, USA 7. Figure 3 image sourced from 26 May Bastow, D., Howard, G., and Whitehead, J., 2004, Op cit. 9. Wong, J.Y., 2008, Op cit 10. Nozaki, H., 2006 Preferable Front and Rear Weight Distributions of a Formula Car, a. SAE International, USA. 11. Oda, H., 2009, Improvement of road ability of a formula SAE vehicle, SAE Japan, Japan 12. Gipser, M., 2007, FTire The Tire simulation model, Taylor and Francis, England 13. Wong, J.Y., 2008, Op cit 14. Bastow, D., Howard, G., and Whitehead, J., 2004, Op cit. Final Project Report, 2013 UNSW@ADFA 11