The Effect of Altering Pacejka Coefficients on the Responsiveness of a Single Seat, Open Wheel, Racing Car 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 cornering force generated by a tyre and the vertical load carried by that tyre. This non-linearity enables weight transfer across the cornering car to reduce the cornering force that two tyres on an axle 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 axles 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 cars 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 slip angle 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