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Dynamics Modeling of a Race Car Tiago Mota Garcia de Oliveira, Universidade de Lisboa - Instituto Superior Tecnico´

Abstract—This thesis focus on the development of a software focuses more on the possibility of altering components, while tool to simulate automotive vehicles dynamics. The tool is respecting the motorsport regulations. oriented to motorsport activities but can be applied to more Srisuchat, [1], makes a vehicle division, based on the forces generic vehicles. Besides contributing to the track performance, on a team management and economical perspective this tool that apply to it. In his line of work, the bloc division in not can also be used to reduce the track time thus increasing the explicit but when you analyse the pseudo code of the model, effectiveness of a racing team. you will find that the vehicle is studied based on physics The relevant components to simulate the dynamics of a expressions that characterize its behaviour. vehicle are first studied, similarly to works done in vehicle computer simulation area. The model developed benefits of the advantages of studied models, bridging their drawbacks. Finally II.SIMULATION ARCHITECTURE the model developed is tested in different situations: the turn 1 The architecture of the vehicle modeled in this work is of Autodromo do Estoril (AFPS), where it is possible to test the shown in figure 1. The purpose was that of reducing the model in a strong braking situation, the chicane of AFPS, where it is possible to test the model in a direction-changing situation number of blocks to the number of vehicle components to and the parabolic turn of AFPS where it is possible to test the be modeled and to calculate the impact of each block on model executing a turn in an acceleration situation. its adjacent ones in the same simulation cycle iteration. The The results obtained are consistent with real data obtained model calculates the response of the vehicle to the following under real racing conditions. variables: engine rpm, throttle position, brake pedal position, Keywords—Vehicle Dynamics, Computer Simulation steering angle and the flux of air generated by the vehicle’s motion. I.INTRODUCTION ACH season, motorsport becomes more complex and E competitive. Therefore, there is a constant need for performance increases. Two important phases that contribute to performance can be identified: the first one, regarding the development of the vehicle and the second one regarding tuning, as optimized as possible, of the vehicle components. This project refers to the second phase - the tuning of components of a vehicle - resorting to a computer simulation tool that studies the behavior of each component. The performance of the tool developed depends directly on the choice of the vehicle simulation model. In order to do so, several theoretic approaches were anal- ysed, all of them representing the vehicle’s behaviour in its whole and consistent with performance reality. Marchesin, [4], suggested that the vehicle is divided in four Fig. 1. Vehicle’s model blocs, and two other representing the external factors that influence the vehicle. This work isolates the most complex The model of each of the blocks in figure 1 are described components, in order to develop them gradually. in the following subsections. Fernandez,´ [5], divides the vehicle in six blocs, considering 1) Load Transfer: The load transfers that occurs at the an extra bloc that allows him to test different driving aid vehicle during acceleration and turning, can be calculated by mechanisms, such as ABS ou ESP. This structure of analysis [1], conveys the necessary flexibility to test several models in each bloc. Acceleration mass (Height of Center of Gravity) Longitudinal Load T ransfer = T rack W idth In a different approach, Siegles, [7], divides the vehicle in (1) eleven blocs. Some of the blocs represent parts of the vehicle, while others refer to the calculation of certain parameters. This aL mass (Height of Center of Gravity) approach simplifies the independent development of each bloc. Lateral Load T ransfer = T rack W idth In [6], the main purpose is to transform a daily use vehicle (2) into a completion vehicle, so its division is made according Where aL is the load acceleration and can be calculated by V elocity2 to the components that alter its original geometry. This work T urnRadius . 2

The calculated value described in the equations above is 8 added/subtracted to to the load present at each wheel as the X I = (I + m a 2) (13) following expressions, zz zzi i i i=1 5) Engine Torque: The engine torque is calculated by, Lateral Load T ransfer−Longitudinal Load T ransfer PF ront Left T yre = 2 (3) Engine T orque = (T hrottle P osition)T orque(RPM) (14) −Lateral Load T ransfer−Longitudinal Load T ransfer PF ront Right T yre = 2 Where Torque(RPM) is the torque produced by the engine (4) for a given RPM, Revolutions Per Minute. This can be obtained from the characteristic curve of an engine. The engine torque is transmited to the wheels by the P = Lateral Load T ransfer+Longitudinal Load T ransfer Rear Left T yre 2 transmission shafts through differentials, where it is multiplied (5) by the gear ratio and differential ratio (ig and id). So the torque present at the wheels can be calculated as, P = −Lateral Load T ransfer+Longitudinal Load T ransfer Rear Right T yre 2 T = T i i (15) (6) W e g d 2) Tyre Model: The Pacejka Magic Formula was used to For the seek of simplicity it is assumed that no mechanical model the tyres. The lateral force can be calculated by, [2], losses occur. 6) Brakes: The braking force, BT is modeled by the following expression, [3], FY = Dsin[Carctan{Bx − E(Bx − arctan(Bx))}] + α (7) B = nµpAR (16) Where α is the slip angle and can be calculated by [1]: T d Where n is the number of friction pads, µ is the coefficient ˙ VLat + aCM φ of friction between the pads and the disc,p is the brake αDianteira = arctg( ) − δ (8) VLong pressure, A the brake piston area and Rd is the radius to the ˙ center of the pad. VLat − bCM φ αT raseira = arctg( ) (9) 7) Aerodynamic Forces: The aerodynamic forces calculated VLong in this work are the downforce, which is the descedant force Where δ is the steering angle, VLog is the longitudinal produced by the flux of air generated by the vehicle motion,and velocity, VLat is the lateral velocity, aCM is the distance the Drag Force. They are calculated, respectevely as follows, between the front track and the centre of gravity, bCM is th distance between the centre of gravity and the rear track. F = 1 ρ C (W ing Area)(W ing Inclination Angle)V 2 3) Center of Gravity: To calculate the center of gravity, D 2 l (17) the vehicle was divided in eight blocks. Four of these blocks represent the vehicle’s sprung mass and the other four the 1 F = ρ C (F rontalArea)V 2 (18) vehicle’s unsprung mass. Knowing each block’s center of A 2 d gravity, the vehicle’s center of gravity can be obtained from, 8) Torque Delivered to Wheels: The force delivered to wheels is calculate from, Σximi Σyimi Σzimi CM T otal = ( , , ) (10) Σmi Σmi Σmi Fr = Fx − BT − FD (19) Where xi, yi and zi are the coordinates of the position of the centre of gravity for each block and mi is the mass of 9) Attitude: The attitude of the vehicle is obtained from the each block moments produced by forces and inertia, yielding the value of 4) Inertia Estimation: To calculate the moment of inertia angular acceleration. Using the Euler integration method it is the vehicle was also divided in eight blocks. The sprung possible to calculate the values of yaw, pitch and roll. mass blocks were considered perfect parallelepipeds and the unsprung mass blocks were considered hollow cylinders. The III.RESULTS value of inertia for each block is calculated according to its This work was developed in Matlab. The input data cor- geometrical form, in the end the vehicle’s moment of inertia respond to driver actions for the complete simulation period. is calculated through the parallel axes theorem, [3], The simulation runs a loop where it calculates the block’s in

8 figure 1 for each iteration. See figure 2. X 2 This work will test the developed tool through three distinct Ixx = (Ixxi + mi ai ) (11) i=1 situations: strong braking, direction-changing and executing a turn in acceleration. This situations will be testes in Turn 1, 8 X Chicane and Parabolic Ayrton Senna of AFPS. This turns can I = (I + m a 2) (12) yy yyi i i be observed in figure 3. i=1 3

Fig. 4. Values for gear, brake pedal, throttle and rpm variables

Fig. 2. Simulation Loop

Fig. 5. Values for steering wheel variable

Fig. 3. Fernanda Pires da Silva Race Track (From https://www.myracingcareer.com/en/track/est/)

A. Turn 1 of AFPS To simulate the execution of Turn 1 the driver profile corresponds to the data in figures 4 and 5. During the brake, there is a load transfer to the front of the vehicle. Still in the braking process, it is initiated the execution of the turn, which results in a load transfer - not only longitudinal, but also lateral, transferring the weight to the external side of the turn. The exit of the turn is made in acceleration, origination a longitudinal load transfer, now to the back of the vehicle. Fig. 6. Load transfer without downforce Below figure 6 represents the load transfer without account- ing for the downforce force, and figure 7 illustrates the load transfer accounting for the downforce. From figure 6 it is possible to verify the load transfer to 4

Fig. 7. Load transfer with downforce Fig. 9. Roll angle the front of the vehicle during the brake (starting at second 0), as well as the load transfer to the external side of the turn (between seconds 3 and 5) during its execution. By comparing this results, with the ones obtained on figure 7 , we can easily understand the influence of the downforce on the vehicle total weight, as well its variation with velocity. The lateral force generated by the tires during the simulation is illustrated in figure 8.

Fig. 10. Pitch angle

has reduced values. This results from that fact that the centre of gravity is very low, making the roll centre low too.The roll centre coupled with a considerable width creates a great moment of inertia around the x axe. Therefore, it will be needed a much higher lateral acceleration than the generated in this turn, in order to increase the value of the roll angle. In general, when braking, the pitch angle tends do be Fig. 8. Lateral force negative. However in figure 10 this never happens, possibly because of the low pitch axis, which makes for the low The lateral forceincreases when the steering wheel is turned. variations in the value of the pitch angle.The illustrated peak At this time, the lateral force has a negative value due to its in the graphic corresponds to the begging of acceleration. At orientation. From the 7th second on, the driver imposes a left this time, there is a longitudinal weight transfer, causing the angle to the steering wheel, in order to place the car in the back inclination of the vehicle. Also, the decompression of the right direction (to the straight line ahead). This situation makes front suspension at this phase influences the extra pitch in the the lateral force value positive, since it is oriented by the axes beginning of the acceleration. system used. The attitude of the vehicle is a result of the moments created around its center of gravity, and can be observed in the figures B. Chicane of AFPS 9 and 10: To simulate the execution of Chicane the driver profile In figure 9 it is possible to observe that the roll angle corresponds to the data in figures 11 and 12. 5

Fig. 11. Values for gear, brake pedal, throttle and rpm variables Fig. 13. Load transfer with downforce

Fig. 12. Values for steering wheel variable Fig. 14. Lateral force

The Load Transfer due to execution of Chicane is illustrated in figure 13: In this simulation the graphic that illustrates the load transfer with downforce is sufficient to understand how the load is transfered during Chicane’s execution. As can be seen in figure 13 the load is transfered from right side to left side at second 2.5, the moment when the driver change the steering wheel direction from left to right. The lateral force generated during the execution of Chicane is illustrated at figure 14: The value of lateral force starts to increase at the beginning of Chicane’s execution. At second 2,5 the driver change direction from left to right, this can be seen in the figure 14 since, from this point, the lateral force value starts to be negative. The attitude of the vehicle is a result of the moments created around its center of gravity, and can be observed in figures 15 Fig. 15. Roll angle and 16. From figure 15 the vehicle rolls around 0.0083 radians since the roll angles are bigger than that when turning left and smaller than that when turning right. 6

Fig. 16. Pitch angle Fig. 18. Values for steering wheel variable

In figure 16 it can be seen that the pitch angle decreases when braking to Chicane and starts to increase when executing it. Since chicane is executed with throttle pressed, the vehicle is accelerating and so this result is the expected one.

C. Ayrton Senna Parabolic of AFPS To simulate the execution of Ayrton Senna Parabolic he driver profile corresponds to the data in figures 17 and 18.

Fig. 19. Load transfer with downforce

Fig. 17. Values for gear, brake pedal, throttle and rpm variables

The Load Transfer due to execution of Ayrton Senna Parabolic is illustrated in figure 19: Since Ayrton Senna Parabolic is a right turn executed under acceleration it was expected that the rear left tyre was the one with more load on it. This is confirmed in figure 19. The lateral force generated during the execution of Ayrton Senna Parabolic is illustrated at figure 20. Fig. 20. Lateral Force Although observing figure 20 could induce that at some points the lateral force value is zero, this is not true. In this case the lateral force value has smaller order of magnitude. The attitude of the vehicle is a result of the moments created 7 around its center of gravity, and can be observed in the figures telemetry data observed under real racing conditions. Still, 21 and 22. further testing is necessary, including direct comparison with real telemetry. The tool can also be used to assess the performance of specific drivers and, eventually, suggesting specific actions to be taken in real driving. The modular structure allows a quick replacement of blocks. This is a useful feature in case of testing different models, for instance, for tyres (a key element in motorsports).

REFERENCES [1] Srisuchat , P., Development of a car physics engine for games, Bournemouth University, 2012. [2] Bakker, E., Nyborg, L. and Pacejka, H. B., Tyre modeling for use in vehicle dynamics studies,SAE paper 870421, Society of Automotive Enineers, 400 Commonwealth Drive, Warrendale, PA 15096, USA, 1986 [3] Harty D. and Blundell M,Multibody Systems Approach to Vehicle Dynamics,Butterworth-Heinemann,2007 [4] Marchesin, F.P., Analise´ de Desempenho de Ve´ıculos do Tipo Formula´ ,Universidade de Sao˜ Paulo, 2012 [5] Fernandez,´ J.G, A Vehicle Dynamics Model for Drivng Simula- tors,Chalmers University of Technology, 2012 Fig. 21. Roll angle [6] Santos Borges, Nuno M.,Estudo do Comportamento Dinamicoˆ do Punto Trofeu´ FEUP 2,Faculdade de Engenharia da Universidade do Porto,2010 [7] Siegles, B., Lap Time Simulation for Racing Car Design,School of Mechanical Engineering, The University of Leeds,2012

Fig. 22. Pitch angle

Analysing figure 21 it is possible to see that the roll angle is always smaller than 0.0083 radians which confirms the statement of the previous simulation. In 22 the pitch angle is alwaysbigger than 0.007 radians which confirms the statement of the previous simulation. Also, the pitch value reaches its maximum values when the value the throttle increases, when the driver presses it harder.

IV. CONCLUSION In this project a tool to simulate the dynamics behaviour of a vehicle was developed. This tool was tested under common racing conditionsat the Estoril Racing Track, such as strong braking (turn 1), change of direction (chicane) and turn executed under acceleration (parabolic turn). The block models presented here are a mathematical repre- sentation of the vehicle’s real components and contain uncer- tainties. Nevertheless, the results obtained are consistent with