The Effect of Altering Pacejka Coefficients on the Responsiveness of a Single Seat, Open , Racing Simulator

John Olsen & John Page School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, N.S.W., 2052. [email protected] & [email protected]

Abstract. We alter the Pacejka coefficients of tyres fitted to a Formula Ford race car simulator to introduce a reasonable amount of non-linearity into the relationship between the generated by a tyre and the vertical load carried by that tyre. This non-linearity enables across the cornering car to reduce the cornering force that two tyres on an can generate. Adjustments to the weight transfer which occurs at one end of the car with respect to the other end of the car with anti-roll bars can the be used to enable the tyres on both of the car to run at similar slip angles thus enabling control of the tendency of the car to either understeer or overseer.

either the front and/or rear suspension systems can 1. INTRODUCTION influence the roll stiffness. The anti-roll bar is a This work documents recent work on the development suspension component that twists when the car rolls in of a simulator for a single seat, open wheel, racing car. cornering. The weight transfer across the car while The purpose of the simulator is to train students firstly cornering is ultimately determined by the height of the in how to recognise the basic handling characteristics of centre of gravity and the cars track, i.e. the distance a racing car and secondly, the effect of making simple between the centreline of the cars tyres across the car. adjustments to certain elements of the car, e.g. anti-roll The anti-roll bar may be thought of as being a device bars, springs and dampers, on the cars handling that adjusts the level of weight transfer that occurs at characteristics. This is especially useful for those one end of the car with respect to the other. students involved with Formula SAE - Australasia. In our previous work (Olsen et. al., 2008), we found that the ‘Racer’ simulation of a Swift SC93 Formula Ford racing car was superficially very good, but had some engineering shortcomings. In particular, the simulated cars handling characteristics were relatively unaffected by changes in the cars roll stiffness’s made by altering the stiffness’s of the cars anti-roll bars. The following paragraph offers an explanation of this statement. By handling characteristics, we mean the tendency of a car to exhibit understeer, oversteer or neutral steer characteristics during cornering. Understeer, oversteer and neutral steer are defined in terms of the slip angles a tyre operates at when a lateral force is applied to that tyre as during cornering. The is the difference between the direction in which a tyre points and the direction in which it travels. An understeering car is a Figure 1: Understeer and Oversteer. Figure adapted car whose front tyres operate at a higher average slip from similar figure in reference 3. angle than the rear tyres. An oversteering car is a car whose rear tyres operate at a higher average slip angle Returning to the virtual world, we found previously that than the front tyres. Figure 1 shows this idea more changing the height of the simulated cars front and rear clearly. Neutral steer is a condition in which the average roll centres was the only way to alter the cars handling front tyre slip angles equal the average rear tyre slip characteristics. In practice, this would involve changing angles. We consider understeer approaching neutral the suspension geometry of the front and rear steer to be the optimal car handling characteristic. On a suspension systems. This is impractical when one real car, it is usual to adjust the cars handling considers how simple it can be to change the stiffness characteristic by altering the roll stiffness of the front of an anti-roll bar. and/or the rear suspension systems. The cars suspension We believe that the failure of the simulated car to geometry, determined at the design stage controls this respond to changes in the anti-roll bar settings is due to to a certain extent but the addition of anti-roll bars to the tyre characteristics. In the Racer simulation, the

Pacejka ‘magic formula’ (Pacejka, 2006 and Beckman) Formula Ford Swift are listed in table 1. Note that the controls the tyre characteristics. Changing the Pacejka slip angle was defined earlier and that the camber angle coefficients can alter the relationship between the is set at zero degrees. coefficient of friction of a tyre in the longitudinal and Figures 2a) through to 2j) show the effect of lateral direction and the associated slip ratio or slip individually varying some of the Pacejka coefficients. angle for a given tyre loading. In this work, we present The figures show the coefficient of friction (= F/Fz) plots of the effect of these coefficients. It is especially plotted against . The figures are plotted in pairs so as to important to understand the effect of the coefficients as show the effect of changing the vertical load. The we vary the load on a tyre. Knowledge of the effect of vertical load in each case is varied from 1500N to each coefficient enables the selection of a better set of 4800N. Figures 2a) and 2b) show the effect of the shape coefficients to mimic the behavior of a real tyre. This factor a0. The shape factor varies the detailed shape of should enable changes in the anti-roll bar settings to the curves; however these shapes are not significantly alter the simulated car’s handling characteristics. affected by an increase in vertical load. Figures 2c) and st st To enable a car to react to the anti-roll bar settings, the 2d) show the effect of the 1 peak coefficient a1. The 1 relationship between the lateral or cornering force the peak coefficient has little effect on the detailed shape of tyre generate to the vertical load it supports should be the curves but increasing the vertical load while st non-linear. In our previous work, this relationship was changing the 1 peak value does change their shape. nd very close to being linear. This was the main reason Figures 2e) and 2f) show the effect of the 2 peak nd why the car could not react to its anti-roll bar settings. coefficient a2. The 2 peak coefficient has a significant Our intention in this work is to make adjustments to the effect on the detailed shape of the curves which does Pacejka tyre coefficients so as to introduce some non- not vary significantly as the vertical load is increased. st linearity into this relationship. Figures 2g) and 2h) show the effect of the 1 stiffness st coefficient value a3. The 1 stiffness coefficient has the effect of decreasing the steepness of the curves as its value is decreased. The effect is largely independent of 2. THE EFFECT OF ALTERING THE the vertical load. Figures 2i) and 2j) show the effect of PACEJKA COEFFICIENTS nd nd the 2 stiffness coefficient value a4. The 2 stiffness coefficient has the effect of increasing the steepness of 2.1 The Pacejka magic formula the curves as its value is decreased. The effect does depend on the vertical load. The Pacejka magic formula for the lateral (or cornering) We did not plot the effect of the 3rd stiffness coefficient force component is: because the initial camber angle was zero and so it

would have no effect. We do not show the effect of the F = Dsin ()Carctan{}SB − E[]arctan(SB)− SB + S v remaining Pacejka coefficients as they have no significant effect. In this work we are only interested in the lateral or cornering force generated by the tyres and so: 2.2 The selection of the appropriate coefficients

C = a 0 In our previous work, one figure showed the cornering

2 force generated plotted against the vertical load carried a1Fz a 2 Fz D = + by a tyre at a number of different slip angles. The 1,000,000 1,000 curves in this figure were essentially linear in the region were the vertical loads are typical of those experienced ⎛ ⎧ F ⎫⎞ by tyres fitted to our Formula Ford car. We concluded ⎜ z ⎟ B = a 3 sin 2 arctan ⎨ ⎬ ()1− a 5 γ C D that this linearity prevents adjustments made to the anti- ⎜ 1000 a ⎟ ⎝ ⎩ 4 ⎭⎠ roll bar from changing the handling characteristic of the car load. a 6 Fz E = + a 1,000 7 In this paper, figures 3a) and 3b) show cornering force plotted against the vertical load for a series of Sh = a 8 γ + a 9 Fz + a10 slip angles for the front and rear tyres respectively. Adjustments to the Pacejka coefficients a2 (increase of ~10%) a (decrease of ~80%) and a (decrease of Sv = (){}a11Fz + a12 γ + a13 Fz + a14 , 3 4 and ~80%) were found to alter the peak value and the stiffness value so as to produce a set of curves that gave S = α + S h an appropriate level of non-linearity without changing

the cornering force generated by the tyres significantly. where a0 through to a14 are the Pacejka coefficients, α is the slip angle and γ is the initial camber angle and Fz is the vertical load carried by the tyre. The lateral Pacejka coefficients for the original front and rear tyres of the

2 2 1.9 1.9 1.6 1.8 1.6 1.8 1.7 1.7 1.2 1.6 1.2 1.6 0.8 1.5 0.8 1.5 1.4 1.4 0.4 1.3 0.4 1.3 1.2 1.2 α 0 α 0 μ 1.1 μ 1.1 -0.4 1.0 -0.4 1.0

-0.8 -0.8

-1.2 -1.2 The effect of shape factor a0 The effect of shape factor a0 -1.6 Fz = 1500N -1.6 Fz = 4800N -2 -2 -20 -16 -12 -8 -4 0 4 8 12 16 20 -20 -16 -12 -8 -4 0 4 8 12 16 20 α α 2a) 2b)

2 2 -07 -07 1.6 -12 1.6 -12 -17 -17 1.2 -22 1.2 -22 0.8 -27 0.8 -27 -32 -32 0.4 -37 0.4 -37 -42 -42 α 0 α 0 μ -47 μ -47 -0.4 -52 -0.4 -52

-0.8 -0.8

-1.2 -1.2 The effect of peak value a1 The effect of peak value a1 -1.6 Fz = 1500N -1.6 Fz = 1500N -2 -2 -20 -16 -12 -8 -4 0 4 8 12 16 20 -20 -16 -12 -8 -4 0 4 8 12 16 20 α α 2c) 2d)

2 2 1375 1375 1.6 1425 1.6 1425 1475 1475 1.2 1525 1.2 1525 0.8 1575 0.8 1575 1625 1625 0.4 1675 0.4 1675 1725 1725 α 0 α 0 μ 1775 μ 1775 -0.4 1825 -0.4 1825

-0.8 -0.8

-1.2 -1.2 The effect of peak value a2 The effect of peak value a2 -1.6 Fz = 1500N -1.6 Fz = 4800N -2 -2 -20 -16 -12 -8 -4 0 4 8 12 16 20 -20 -16 -12 -8 -4 0 4 8 12 16 20 α α 2e) 2f)

2 2 1625 1625 1.6 1675 1.6 1675 1725 1725 1.2 1775 1.2 1775 0.8 1825 0.8 1825 1875 1875 0.4 1925 0.4 1925 1975 1975 α 0 α 0 μ 2025 μ 2025 -0.4 2075 -0.4 2075

-0.8 -0.8

-1.2 -1.2 The effect of stiffness value a3 The effect of stiffness value a3 -1.6 Fz = 1500N -1.6 Fz = 4800N -2 -2 -20 -16 -12 -8 -4 0 4 8 12 16 20 -20 -16 -12 -8 -4 0 4 8 12 16 20 α α 2g) 2h)

2 6 1.6 7 8 1.2 9 3500 1 degrees 2 degrees 10 3 degrees 4 degrees 0.8 5 degrees 6 degrees 11 3000 0.4 12 7 degrees 8 degrees 13 9 degrees 10 degrees α 0 2500 μ 14 -0.4 15 2000 -0.8

-1.2 1500 The effect of stiffness value a4

-1.6 Fz = 1500N Cornering Force 1000 -2 -20 -16 -12 -8 -4 0 4 8 12 16 20 500 α 0 2i) 0 500 1000 1500 2000 2500 Vertical Load 2 6 3a) 1.6 7 8 1.2 9 3500 1 degrees 2 degrees 0.8 10 3 degrees 4 degrees 11 5 degrees 6 degrees 3000 0.4 12 7 degrees 8 degrees 13 9 degrees 10 degrees α 0 μ 14 2500 -0.4 15 2000 -0.8

-1.2 1500 The effect of stiffness value a4 -1.6 Fz = 4800N Cornering Force 1000 -2 -20 -16 -12 -8 -4 0 4 8 12 16 20 500 α

2j) 0 0 500 1000 1500 2000 2500 Figures 2a) through to 2j): Vertical Load The coefficient of friction μα plotted against tyre slip 3b) angle α. Figures 3a) and 3b): Table 1: Cornering force generated by a tyre plotted against The initial and final Pacejka coefficients for the the vertical load carried by a tyre for a number of front and rear tyres. different slip angles. Figure 3a) represents the front tyres while figure 3b) represents the rear tyres. Pacejka initial initial final final coefficient front rear front rear 0 1.4 1.4 1.4 1.4 The effect of non-linearity like that shown in figure 3a) 1 -32.0 -32.0 -32.0 -32.0 does the following (also see Lauda, 1977). With 2 1625.0 1665.0 1825.0 1925.0 reference to figure 3a), if the two tyres on the front of 3 1875.0 1915.0 400.0 500.0 the Formula Ford each carry a vertical load of 1000N 4 11.0 11.0 2.0 2.0 and operate at a slip angle of 10 degrees, the maximum 5 0.013 0.013 0.013 0.013 cornering force both tyres could generate is (2×1720N 6 -0.14 -0.14 -0.14 -0.14 =) 3440N. This is assuming that no weight transfer 7 0.14 0.14 0.14 0.14 occurs across the car in cornering (which can only 8 0.019 0.019 0.019 0.019 occur if the centre of gravity of the car is at the road 9 -0.019 -0.019 -0.019 -0.019 surface or if the car is infinitely wide). If we allow for 10 -0.18 -0.18 -0.18 -0.18 some weight transfer to occur, so that the outside tyre of 11 -11.0 -11.0 -11.0 -11.0 the cornering car carries 1500N and the inside tyre 12 -0.48 -0.48 -0.48 -0.48 carries 500N, the corresponding cornering force 13 10.2 10.2 10.2 10.2 generated by both tyres will be (2410N + 900N =) 14 -2.4 -2.4 -2.4 -2.4 3310N, a reduction of 130N. The more the weight transference, the more the reduction in cornering force. Table 2: The effect of the non-linearity allows weight transfer to Anti-roll bar settings for testing the tyres. reduce the cornering force generated by the tyres on a particular axle. experiment front anti-roll rear anti-roll bar bar 3. RESULTS 1 80,000 0 2 40,000 20,000 We need to test the car to determine whether the non- 3 40,000 40,000 linearity in the cornering force vs. vertical load 4 40,000 80,000 relationship is sufficient. To test the car, we have

downloaded and used the Northampton Oval circuit. sa0 sa1 sa2 40 This circuit has two short straights connected by two sa3 speed throttle 180 degree corners. Table 2 shows how the anti-roll 35 bars settings used in the four experiments. If we 30 increase the stiffness of the roll resistance at one end of 25 the car with respect to the other, there will be more 20 weight transfer at that end of the car. The settings given 15 in table 2 clearly intended to change the cars handling 10 characteristic from understeer to oversteer. 5

Figures 4a) through to 4d) show the effect of the and Throttle Speed Slip Angle, 0 -5 changes. The figures show the raw slip angles for all 0 250 500 750 1000 1250 1500 1750 2000 four tyres, the speed of the car and the throttle position Distance (m) plotted against the distance the car has travelled. It will 4a) be noticed that the speed of the car fluctuates, going up on the short straights and down through the corners. sa0 sa1 40 sa2 sa3 The position of the throttle is included as it can speed throttle influence the cars handling characteristic. When the 35 throttle is fully open, it has a value of one and when it is 30 closed, it has a value of zero. 25 20

Figure 4a) shows that the car has a marked tendency to 15 understeer. This is because the slip angles of the two 10 front tyres are significantly higher than those of the two 5 rear tyres. Figure 4b) shows that the car has only a Slip Angle, Speed and Throttle and Speed Angle, Slip 0 small amount of understeer which is perhaps the -5 optimum setting. Figure 4c) shows that the cars now 0 250 500 750 1000 1250 1500 1750 2000 Distance (m) has a small tendency to oversteer. Finally Figure 4d) shows that the car has more of a tendency to oversteer. 4b) To summarize, the figures show that we can alter the understeering tendency of the car towards oversteer by 40 sa0 sa1 sa2 sa3 speed throttle increasing the roll stiffness of the rear axle of the car 35 with respect to the front axle. 30

25

3.1 Further driving observations and concurrent 20 work 15

Although the lateral properties of the tyres were 10 changed, the longitudinal properties were not. This may 5 have been responsible for the fact that accelerating the Throttle Slip Angle, Speed and 0 car failed to amplify the handling characteristic of the -5 car. Future work will focus on this point. 0 250 500 750 1000 1250 1500 1750 2000 Distance (m) 4c) Concurrently, work is underway on taking data generated by the simulation and dumping it in real time 40 sa0 sa1 sa2 to a second computer. Data fed to the second computer sa3 speed throttle 35 will control the motion of a six-axis motion platform. 30 The motion platform will compliment the force feedback through the wheel to give the driver 25 more feel. A future paper will report on this work. 20 15 4. CONCLUSIONS 10 5

We altered the Pacejka coefficients of the tyres fitted to Slip Angle, and Throttle Speed 0 a Swift Formula Ford simulator so as to introduce some -5 non-linearity into the relationship between the 0 250 500 750 1000 1250 1500 1750 2000 Distance (m) cornering force generated by a tyre and the vertical load applied to the tyre. Testing of the tyres modified in this 4d) way indicates that we could change the handling characteristic of the car from understeer to oversteer by Figures 4a) through to 4d): altering the roll resistance of suspension systems by Slip angle, throttle position and velocity plotted against means of anti-roll bars. the distance travelled in metres.

REFERENCES

1. Olsen, J., Page, J. & Vulovic, Z. (2008) “Initial investigations into simulating an open-wheeler racing car”, SimTecT 2008.

2. http://www.racer.nl/.

3. Daniels, J. (1988) “Handling and roadholding, at work,” Motor racing publications, Great Britain.

4. Pacejka, H. B. (2006) “ and ”, 2nd Edition,” SAE International, Warrendale, PA, USA.

5. Beckman, B. “Physics of Racing” http://phors.locost7. info/contents.htm.

6. Lauda, N. [in collaboration with Diplom-Ingeniuer Dr Fritz Indra]., (1977) “The art and technicalities of grand prix driving”, William Kimber & Co. Limited.