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Aerodynamic Influences on the Performance of the Grand Prix Racing Car J A Dominy BSc and R G Dominy Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 1984 198: 87 DOI: 10.1243/PIME_PROC_1984_198_134_02 The online version of this article can be found at: http://pid.sagepub.com/content/198/2/87

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On behalf of:

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81

Aerodynamic influences on the performance of the Grand Prix racing car J A Dominy,* BSc, PhD Transmission Research Group, Rolls-Royce Limited, Aero Division, Derby

R G Dominy, BSc Whittle Laboratory,Department of Engineering, University of Cambridge

The remarkable cornering power of the modern Formula 1 Grand Prix car is almost entirely due to its aerodynamic characteristics. As aerodynamic research generates much more data than can be track tested, it has been found necessary to develop an analytical model to investigate the effect of downforce, drag and centre of pressure on lap times round a sample circuit thus sijting the experimental results. This paper presents such an analysis and applies it to an investigation of the performance of a Formula 1 car round a typical circuit. NOTATION

aerodynamic downforce linear acceleration front braking force rear braking force total drag acting on the car outward force acting at the centre of gravity cornering force generated by the tyres outward force acting on front axle outward force acting on rear axle height of centre of gravity above the road front lateral weight transfer rear lateral weight transfer chassis mass front wheel/tyre mass rear wheel/tyre mass total mass = m, + 2(m, + mr) overturning moment proportion of downforce on rear axle proportion of weight on rear axle proportion of roll stiffness provided by rear axle load on front axle under braking load on rear axle under braking tyre radius corner radius front tyre radius rear tyre radius front wheel/tyre radius of gyration rear wheel/tyre radius of gyration torque torque to give angular acceleration of wheels and tyres torque to give linear acceleration to a mass m, front track rear track linear velocity static load on front axle static load on rear axle slip angle friction force coefficient This paper was presented at an Ordinary Meeting held in Birmingham on 7 February 1984. The M S was received on 29 January 1983 and was accepted f o r publication on 16 August 1983. * Formerly Arrows Racing Team Limited, Bletchley, Buckinghamshire. 12/84 (9 IMechE 1984

front tyre friction force coeficient

pf

pmax peak friction coefficient with no applied load

rear tyre friction force coefficient

pr

1 INTRODUCTION

To the racing car engineer the science of ground effect is that of harnessing the flow of air between the car and the road such as to generate an aerodynamic force pushing the car down on to the road surface. In order to generate this force, the underside of the ‘side wings’ either side of the driver are formed into a profile similar to a diffuser (Fig. la). If the low pressure under the car is to be maintained then air at ambient pressure must be prevented from passing under the side of the car. This sealing is achieved to a greater or lesser extent by flexible skirts which bridge the gap between the bodywork and the road, running between the wheels on the outside edge of the chassis (Fig. lb). Since the Lotus team introduced ground effect cars during the 1978 and 1979 seasons (1) most teams in Formula 1 have carried out a great deal of aerodynamic research. Although these results are not published in detail it is known that at 75 m/s an aerodynamic downforce of 22 kN can be developed. This is of the order of

(a)

‘

Flexible skirt

& (b)

Underwing profile

Flexible skirt

Fig. 1 Diagram of a modern Formula 1 racing car illustrating the flexible skirts and the side wing profile

0265-1904/84 $2.00

+ .05

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Proc Instn Mech Engrs Vol 198D No 7

J A DOMINY AND R G DOMINY

a8

three times the qualifying weight of the car and can result in lateral accelerations greater than 39. Wind tunnel testing of scale models enables a very large quantity of data to be generated; this is far more than could reasonably be tested on the full size car. It is, therefore, desirable to use an analytical model to sift through these data and to predict the performance of the car in each configuration. It has been shown that downforce, drag and centre of pressure can be applied to a simple chassis and tyre model to give a good general description of the steady state handling of a modern racing car (2). However, unlike chassis tuning, aerodynamic changes cannot be described in terms of cornering power alone since the drag component has a great effect on the straight line speed and hence lap times. If aerodynamic influences are to be evaluated then the analysis must consider the effect of downforce, centre of pressure and drag while allowing the model to be driven both round the corners and along the straights thereby giving an estimate of lap time which is, of course, of prime importance to the race engineer. 2 THE ANALYSIS

Previous methods of evaluating the performance of a racing car round a lap have been presented but have been unable to deal with the subtleties of ground effect (3). Furthermore the very stiff suspensions combined with limited movement (350 kN/m wheel rates and 25 mm wheel movements are not unknown) combined with the relatively smooth nature of the race track enables suspension movement as such and the resultant changes in camber and roll centres to be neglected for a first approximation. The main objective is to produce comparative rather than absolute results. The analysis is divided into two parts: ( i ) the modelling of the car and (ii) the circuit. In this case separate models are introduced for the chassis, power train, tyres and aerodynamics while the circuit algorithm must act as the driver to feel the limit of the car round the corners and to extract the utmost straight line performance. As with the real car, the race engineer is left to tune the chassis, balance the aerodynamics and optimize the gear ratios. It is also possible to evaluate the effect of weight and power, both of which have recently been controversial in Formula 1.

2.1 The chassis model

For the purpose of this analysis the chassis serves two functions while cornering. Firstly, by virtue of its mass and centre of gravity it dictates the cornering force trying to throw the car off the road and overturn it. Secondly, it must distribute the overturning moment between the front and rear thereby controlling the load distribution between the four tyres (Fig. 2). For a car travelling round a corner of radius r,: F , = M1u2/r, where F, is the outward force acting through the centre of gravity. If the weight distribution is such that a proportion, p , , acts on the rear axle then the outward force on the front and rear tyres is given by:

Ff = F,(1 - P,)

(la)

Fr = F o p ,

(lb)

wfi

wro

I

Front rolli

1

I

Fig. 2 Action of forces on the chassis

The static wheel loading front and rear is:

If the centre of gravity is at a height h above the road surface then the overturning moment is expressed as M , = F,h. Now, if the front and rear roll stiffness are such that a proportion, pr, of the overturning moment is resisted by the rear suspension then the front and rear weight transfer is:

where tr and tr are the front and rear tracks. The model as it stands does not evaluate the front and rear roll stiffness but is capable of showing the effect of stiffness variations. It is left to the race engineer to decide how this should be done, for instance by varying the spring rates, anti-roll bars or ride heights.

2.2 The engine and transmission model On the straights the model must reproduce the engine power-speed characteristic, the gear ratios and the final drive ratio. The rotational inertia of the wheels and tyres will also be considered since this can vary considerably from tyre to tyre. In all but first gear the free response of the engine and transmission system is very much greater than the acceleration of the car and in first gear the acceleration is limited by wheel spin and not the engine. Consequently the rotating inertia of the drive line can be neglected. At a velocity, u, the torque required to accelerate the total mass of the car against a drag is given by: TI = rr{[m, + 2(mf rnr)]a d }

+

+

and the torque required to accelerate a wheel of radius, r, radius of gyration, R , and mass, m, is:

T. = mR2a/r where a is the linear acceleration. Thus, for a four wheel car:

T. = 2aC(mfRi?/rf) + (mr R:/rr)I QIMechE 1984

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AERODYNAMIC INFLUENCES ON

For the system as a whole T T

= I,{

[m0

=

T,

rm

PERFORMANCE OF THE GRAND PRIX RACING CAR

+ TI,therefore:

+ 2(mf + m,)la + d}

+ 2a(mfR;/rf + m,R;/r,)

89

be given in terms of the measured downforce, Aref, at the reference velocity, uref: Af

=

Aref(U/Uref12(1 - ~

(54

c )

which can be rearranged to yield acceleration thus:

Ar = Aref(U/Uref)2Pc (5b) At both ends of the car the downforce is equally supported by each tyre.

where T can be determined from the instantaneous power and speed using the engine and transmission characteristics. The estimation of the total drag is discussed below.

2.5 Total tyre loads

2.3 The braking model Since it is the objective of this model to generate comparative rather than absolute results it has been possible to assume a constant braking force. Clearly, in reality the braking force will depend upon the downforce and therefore speed. A detailed model of the braking performance is given in Appendix 1 where it is assumed that the driver keeps the car just at the point of locking the wheels. It has not been used to generate the results given below due to the greater computing time involved. Even a model such as this becomes unreliable at very high speeds where the downforce is such that the driver can no longer lock the wheels. The braking force of 39 used for a Formula 1 car in this analysis has been gleaned from the brake manufacturers. 2.4 The aerodynamic model

Conventional racing car wind tunnel tests present data in terms of drag, downforce and centre of pressure. Since the regulations prohibit movable aerodynamic devices, these three parameters uniquely describe the aerodynamic performance of a particular configuration (Fig. 3). As the change in Reynolds number is small within the relevant speed range, the centre of pressure can be assumed to remain constant with speed, (at 75 m/s Re = lo'). Since, in the tunnel used, the test speed is limited by the fixed speed of the rolling road it has not been possible to verify this experimentally. For a centre of pressure such that a proportion, pc , of the downforce acts on the rear tyres then the front and rear downforce may be expressed as A, = A(1 - p,) and A, = Ap,. Tunnel tests often present the data at some reference speed, thus, for a given velocity, u, the downforce may Centre of pressure

\

As the car corners at a steady velocity the tyre loads can be obtained from equations (1, 2, 3, and 5). Thus: Front outer tyre load = +mil - P g ) + iAreXu/uref)2(l - P J + ~ o ( 1 pr)/t, Front inner tyre load = h t ( 1 - Pg) + tAreXU/Uref)2(1 - P C ) - Mo( 1 - Pr)/tf Rear outer tyre load = 3mt Pg + 3 ~ Aref(U/Uref12 c + Mo P J t r Rear inner tyre load = h t Pg + t ~ Aref(U/Uref)2 c - Mopr/tr 2.6 The tyre model Over the years a number of tyre models have been presented (e.g. 4, 5) but since there is no recent information available on current racing tyres it is not usually possible to provide the input data that the models require. Since it is not the purpose of this analysis to investigate tyres in detail, a highly simplified model has been used that considers variations in cornering force coefficient with only slip angle and normal force. It will be shown later that the relevant parameters may be deduced from direct observation of the car. It is not essential that the tyre model be exact. The objective is to optimize the car to balance the front and rear slip angles. If the car is unbalanced it is not important to know exactly how far out it is providing that the model gives some indication of the extent of the handling problem. This behaviour is defined by the shape of the friction coefficient-slip angle curve close to the peak slip angle. If the car is always driven at the limit of its adhesion then the cornering stiffness at low slip angles in unimportant. If the overall cornering force is to be realistic then the maximum friction coefficient must be broadly correct. Nevertheless, even if precise laboratory measurements of the tyre characteristics were available it has been shown that quite wide variations may occur under real conditions (6). The estimate described below is adequate for a comparative analysis. Two basic tyre models have been employed: that presented in (2) and the traction/cornering ellipse (7). The latter has been further simplified into a circle. For relatively small slip angles both methods give similar results. Consider a tyre acting under a normal load, W , and at a slip angle, a,(Fig. 4), then the limiting force at the tyre contact is p W - +[ W ] where W ] is a correction factor for p due to the load W . From Fig. 4 it is apparent that the torque required to drive the car a t a constant speed in the x direction is T = r,d/cos a and for a the cornering traction circle of radius pW force can be expressed as : Fc = {(pW - I , ~ [ W ]-) ~(rd/cos a)2 } 112

+[

+[v

t

Down force

Fig. 3 Aerodynamic forces acting on the chassis

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J A W M l N Y A N D R G DOMINY

-U

Lateral force J

I

Fig. 4 Forces acting on a slipping tyre

d = 0 for an undriven wheel and the force resisting the outward force due to cornering is:

F = F, cos u

+ F,

tan u

where F , is the force acting in a tangential direction; F, tan o! can be neglected for small slip angles and high downforces. At each end of the car the two wheels can be combined to give a hypothetical single tyre which has the properties of the real pair. The total cornering force at each end of the car is: F,

= [{(PWO -

$CKI) + (PW - $[WN2

- (rd/cos c ( ) ~ ] ’ / ~

where Woand are the total inner and outer tyre loads given above. The values of p used below are for a real tyre under a typical load. The effect of tyre width on force Coefficient is considered to be proportional to the load per unit width and the rear tyre is taken as the datum. 2.7 Circuit-driver algorithm

A satisfactory algorithm to simulate a racing car driving round a circuit must be capable of establishing the limiting speed and attitude of the car on curves of each radius encountered on that circuit and to accelerate along the straights, changing gear at the appropriate speeds and to brake into the following corner. The main difficulty is to obtain an adequate model of the car cornering. It is the authors’ experience that two top racing drivers may describe quite different techniques for driving the same car through a particular turn. It is, therefore, extremely difficult to form a general impression of the optimum behaviour of the car through a corner let alone to define it mathematically. However, it has been found that a steady state cornering analysis can give a good account of the fundamental handling characteristics of a particular car and direct observation confirms that modern Grand Prix cars corner in an orderly and controlled fashion, with little steering or throttle correction. Consequently, the steady

state model (2) has been used in this case. Each curve is considered to be an arc of a circle round which the car is driven at the maximum possible speed under a steady power setting. The technique for solving the cornering speed is iterative. For an initial low speed the lateral force on the rear of the car is calculated. Using a value of force coefficient corresponding to the peak slip angle the cornering force can be calculated. This is then resolved in a direction parallel to the lateral force. If the cornering force generated by the tyres is greater than the outward radial force then the speed is increased in steps. A bisection method is used to find the speed at which the forces are in equilibrium. This is considered to be the maximum speed at which the rear of the car can corner. The process is then repeated for the front. The lower limiting speed is then used to establish the slip angle at which the opposite end must run, giving the steady state cornering attitude. Straight line performance is analysed by a time step method. The car is considered to enter the straight at the exit speed of the previous corner. The correct gear ratio is chosen according to that speed. From the car speed and gear ratios the engine speed can be calculated and hence the engine power and wheel torque. The acceleration of the car can then be determined for the next time step. If, at the end of the time step, the engine speed is higher than the gear change point the car will change up a gear. In top gear the car may ‘over rev’ or a ‘rev limiter’ may be used to prevent engine damage. At the end of each time step the distance required to brake from the total speed reached to the limiting speed for the following corner is found and added to the distance travelled on the straight. If this is greater than the overall straight length then the calculation is complete and represents the total time needed to travel between two corners. For suitably small time steps this gives an adequate degree of accuracy. However, if larger time steps are used then there may be an unacceptable ‘overlap’ error at the end of the straight.

2.8 Setting up the model

Accurate measurements of the performance of Grand Prix cars are not generally made and those that have been attempted are not published. Similarly, data on current racing tyres are almost unobtainable. Consequently empirical techniques must be employed to provide estimates of the necessary parameters. Values of the lateral force may be obtained by timing the cars through a curve of known radius. It is then possible to estimate the maximum force coefficient. If this is repeated for several radii it is possible to evaluate the relationship between force coefficient and normal load. Typical characteristics are shown in Fig. 5. Information on the basic balance of the chassis can be obtained from the drivers. The model can then be adjusted to match the known characteristics of the car (Appendix 2). Total drag can be calculated from the terminal velocity and engine power. The aerodynamic component will be known from wind tunnel tests, the residual being rolling resistance and transmission losses. The total drag is considered to act as the square of the speed. Aerodynamic drag follows this law while mechanical

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AERODYNAMIC INFLUENCES ON THE PERFORMANCE OF THE GRAND PRIX RACING CAR

Normal force

91

75 m radius

n

1.5

\ 25 m radius

100 m radius\

loo0 m

Fig. 6 Layout of the test track: total length

=

3204 m

1.0

P

0.5

I 2

I

I

4

6

1 8

1

10

Slip angle, degrees

Fig. 5 Typical tyre friction characteristic: rear tyre

drag is unlikely to differ significantly and is, anyway, only a small proportion of the total. 3 APPLICATION OF THE MODEL

The objective of the model was to enable the effect of aerodynamic changes to be evaluated analytically. The most useful technique has been found to be to generate a carpet plot to examine the influence of aerodynamic drag and downforce on lap times round a sample circuit. Figure 6 illustrates the test track, and the car’s particulars are given in Appendix 3.

59-1

Figure 7 shows such a plot for a typical Cosworth powered Formula 1 car. Values of drag and downforce are quoted in Newtons at 67 m/s. This system is chosen as being least ambiguous. Coefficients become awkward as both plan and frontal areas can be radically altered by simple changes, for instance removing the front wings. In this case the plot also includes lines showing the maximum speed achieved on the loo0 m straight. This is a useful factor as, for a given lap time it is helpful for a car to have a greater top speed in the race than in practice to help overtaking. Typical values for drag and downforce at 67 m/s are 3800 N and 13 OOO N respectively. The curves show that for a given drag the slope reduces as downforce increases. This is a property of the particular car, which takes on a more severe understeering attitude that detracts from its maximum possible cornering speed. If the car were re-optimized aerodynamically for each value of downforce the curves would tend to straighten out. Although the lap time can be seen to be more sensitive to drag than downforce it must be realized that a reduction in drag of 100 N is extremely difficult to achieve whereas a change in downforce of 1OOO N is relatively easily obtained. For a Formula 1 car on a medium-speed circuit downforce is usually the dominant variable. The sample lap distance and speed are almost identical to those at Donington Park (although the track shape is quite different). This allows an interesting com-

2.8

.-

I

76 54.

53-

Fig. 7 Variation of lap time with aerodynamic downforce and drag for a Formula 1 racing car QIMechE 1984

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J A DOMINY AND R G DOMINY

92

67-

Downforce, N

65 -

64‘

Fig. 8 Variation of lap time with aerodynamic downforce and drag for a class ‘A’Clubmans car

parison with cars of different classes in an attempt to validate the model. Lap times of 57 s have been achieved by Formula 1 cars at Donington. Figure 8 shows a dragdownforce chart for a Class A Clubmans car and a point is shown for one particular car, a Mallock Mark 24. The predicted lap time of 71 s is within one second of the time achieved by this car. It is interesting to note that in this class, where little or no ground effect technology has crept in, quite modest increases in downforce give a useful improvement in lap time and that relatively large drag penalties can be tolerated. Figure 9 shows the effect of centre of pressure on lap time and cornering attitude, once again for a Formula 1 car. It has been shown (2) that a car’s balance is critical to its maximum cornering speed round a particular turn and this is again observed for the track as a whole. The cornering attitude (defined as af - a,) is plotted for the fastest (100 m radius) and slowest (25 m radius) bends. For this particular car and circuit it is more important to achieve a balance in fast corners than in slow. Because downforce rises as the square of speed it is almost impossible to achieve a perfect balance in both

fast and slow turns. In practice, since there is often an inconsistency of balance between the tyres used in qualifying and the race, achieving a reasonable balance is further complicated. The model has been used to investigate the effect of weight and power on performance. Both of these have been the cause of controversy over the past two seasons, since the introduction of turbocharged engines and ‘disposable ballast’. Figure 10 shows a carpet plot relating power and weight to lap time. In this case the lines of maximum speed refer to the 240 m straight since terminal velocity is almost independent of weight and this condition is approached on the long straight. The figure shows the lap time expected of a good Cosworth powered car at its qualifying weight. It is reasonable to believe that, in qualifying, the better turbocharged cars will have about 480 kW available but will weigh of the order of 50 kg more than the corresponding non-turbo. They can be seen to enjoy an advantage of about one second per lap which is typical of the performance observed during the 1982 season. 580 5857-

Attitude LOO m radius

62

Chassis weight, kg

56. 61-

-g VI

.g 60.

.-

P

0

3

59.

5554550

53-

58. I

52.

l-

30

40

50

60

70

80

Percentage aerodynamic load on rear Fig. 9 Effect of aerodynamic balance on lap time and cornering attitude for a Formula 1 racing car

Fig. 10 Effect of engine power and chassis weight on lap time for a Formula 1 racing car

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AERODYNAMIC INFLUENCES ON THE PERFORMANCE OF THE GRAND PRIX RACING CAR

93

Braking force at rear:

4 CONCLUSIONS

The analysis has demonstrated that a simplified model of a Formula 1 racing car can be used to evaluate aerodynamic and chassis changes in terms of lap times, straight line speed and cornering attitude. The main difficulty in any analytical work concerned with racing cars is the lack of measured data, published or otherwise, with which to verify the predicted values. However, in this case a degree of verification has been achieved, firstly by comparing the times of Formula 1 and Class A Clubmans cars round the same circuit and by comparing the effect of weight with the Formula 1 ‘folklore’ value of 1-1.5 s per lap per 50 kg (unfortunately the folklore does not specify a lap length or speed). Of greater importance is that the recognized interactions between chassis and aerodynamic configurations have been successfully modelled. Thus, the race engineer cannot only evaluate a great deal of wind tunnel data prior to track testing but can investigate those characteristics that give the greatest benefits on a particular track. Although both the car model and the driver-track algorithm are highly simplified they would require a considerable increase in computing power and a more detailed knowledge of the behaviour of the car (and driver) through corners if they are to be significantly improved. Since such data are unlikely to become available in the forseeable future and as this analysis is capable of giving realistic results as it stands, the extra complexity has not been considered to be justified at this stage.

thus the acceleration becomes: a1 = (bf

+ br)/mt

hence Su for a small time step at. The equations are solved to give a braking distance and time between two limiting velocities by means of time step integration. APPENDIX 2 Estimation of tyre data Consider a car rounding a corner of radius, rc, at velocity, u. For a car with no downforce m, u2/rc = m, p, therefore p = u2/rc. With downforce:

therefore p= Cmt

m, u2 + Aref(~/~ref)~Irc

For the front of the car:

REFERENCES 1 Wright, P. G. The influence of aerodynamics on the design of

Formula 1 racing cars. Int. J . of Vehicle Design, 1982, 3 (4), 383397. 2 Dominy, J. Frictional aspects of Formula 1 racing car performance. Tribology International,June 1981, 14 (3), 167-170. 3 Scherenberg, H. Mercedes-Benz racing car design and experience. S A E Trans, 1958,66,414420. 4 Moore, D. F. Friction and wear in rubbers and tyres. Wear, 1980, 61 (2), 273-282. 5 Bernard, J. E. et al. Tyre shear force generation during combined steering and braking maneuvers. SAE, paper No. 770852, 1977. 6 McCarty, J. L. Wear, friction and temperature characteristics of an aircraft tyre undergoing braking and cornering. NASA technical paper No. 1569,1979. 7 Ellis, J. Vehicle dynamics, London Business Books Ltd., 1969.

APPENDIX 1 Improved braking model

Test car data

Assuming braking is in a straight line and at the point of locking the wheels, then for an initial acceleration a,, and velocity, u,: a, mc h longitudinal load transfer = wb

Therefore load on front axle = static load + downforce + load transfer: At the rear: qr = mt

Pg

+ Aref(Uo/Uref)2Pc-

Braking force at front: bf = P4r - ll/C4fl

mc h/wb

This gives a mean value of p. By repeating this procedure for corners of different radii (hence speed and downforce) it is possible to estimate the effect of load, W, on p. For this investigation, the following relationx W). ship has been used: p = pma,.(l - 3.8 x Care is required in interpreting these data since the effective friction coefficient will depend upon a number of factors, notably the nature and condition of the road surface, ambient temperature and the particular set-up of the car. APPENDIX 3

Formula 1 Chassis weight, kg Front wheel weight, kg Rear wheel weight, kg Front wheel diameter, m Rear wheel diameter, m CG height, m CG position: YOfrom front axle C , position: % from front axle Roll stiffness: % on rear Front track, m Rear track, m Front tyre tread width, m Rear tyre tread width, m Max. friction coefficient Peak slip angle, degrees Max power, kW

600 18 20 0.6 0.66 0.35 60 50 60 1.78 1.63 0.18 0.36 1.8 8 380

Clubmans 380 15

‘A

15 0.508 0.558

0.35 60 55 50 1.46 1.41 0.18 0.18 1.5 8 135

Proc Instn Mech Engrs Vol 198D No 7

@IMechE 1984

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Aerodynamic Influences on the Performance of the Grand Prix Racing Car J A Dominy BSc and R G Dominy Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 1984 198: 87 DOI: 10.1243/PIME_PROC_1984_198_134_02 The online version of this article can be found at: http://pid.sagepub.com/content/198/2/87

Published by: http://www.sagepublications.com

On behalf of:

Institution of Mechanical Engineers

Additional services and information for Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering can be found at: Email Alerts: http://pid.sagepub.com/cgi/alerts Subscriptions: http://pid.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://pid.sagepub.com/content/198/2/87.refs.html

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81

Aerodynamic influences on the performance of the Grand Prix racing car J A Dominy,* BSc, PhD Transmission Research Group, Rolls-Royce Limited, Aero Division, Derby

R G Dominy, BSc Whittle Laboratory,Department of Engineering, University of Cambridge

The remarkable cornering power of the modern Formula 1 Grand Prix car is almost entirely due to its aerodynamic characteristics. As aerodynamic research generates much more data than can be track tested, it has been found necessary to develop an analytical model to investigate the effect of downforce, drag and centre of pressure on lap times round a sample circuit thus sijting the experimental results. This paper presents such an analysis and applies it to an investigation of the performance of a Formula 1 car round a typical circuit. NOTATION

aerodynamic downforce linear acceleration front braking force rear braking force total drag acting on the car outward force acting at the centre of gravity cornering force generated by the tyres outward force acting on front axle outward force acting on rear axle height of centre of gravity above the road front lateral weight transfer rear lateral weight transfer chassis mass front wheel/tyre mass rear wheel/tyre mass total mass = m, + 2(m, + mr) overturning moment proportion of downforce on rear axle proportion of weight on rear axle proportion of roll stiffness provided by rear axle load on front axle under braking load on rear axle under braking tyre radius corner radius front tyre radius rear tyre radius front wheel/tyre radius of gyration rear wheel/tyre radius of gyration torque torque to give angular acceleration of wheels and tyres torque to give linear acceleration to a mass m, front track rear track linear velocity static load on front axle static load on rear axle slip angle friction force coefficient This paper was presented at an Ordinary Meeting held in Birmingham on 7 February 1984. The M S was received on 29 January 1983 and was accepted f o r publication on 16 August 1983. * Formerly Arrows Racing Team Limited, Bletchley, Buckinghamshire. 12/84 (9 IMechE 1984

front tyre friction force coeficient

pf

pmax peak friction coefficient with no applied load

rear tyre friction force coefficient

pr

1 INTRODUCTION

To the racing car engineer the science of ground effect is that of harnessing the flow of air between the car and the road such as to generate an aerodynamic force pushing the car down on to the road surface. In order to generate this force, the underside of the ‘side wings’ either side of the driver are formed into a profile similar to a diffuser (Fig. la). If the low pressure under the car is to be maintained then air at ambient pressure must be prevented from passing under the side of the car. This sealing is achieved to a greater or lesser extent by flexible skirts which bridge the gap between the bodywork and the road, running between the wheels on the outside edge of the chassis (Fig. lb). Since the Lotus team introduced ground effect cars during the 1978 and 1979 seasons (1) most teams in Formula 1 have carried out a great deal of aerodynamic research. Although these results are not published in detail it is known that at 75 m/s an aerodynamic downforce of 22 kN can be developed. This is of the order of

(a)

‘

Flexible skirt

& (b)

Underwing profile

Flexible skirt

Fig. 1 Diagram of a modern Formula 1 racing car illustrating the flexible skirts and the side wing profile

0265-1904/84 $2.00

+ .05

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Proc Instn Mech Engrs Vol 198D No 7

J A DOMINY AND R G DOMINY

a8

three times the qualifying weight of the car and can result in lateral accelerations greater than 39. Wind tunnel testing of scale models enables a very large quantity of data to be generated; this is far more than could reasonably be tested on the full size car. It is, therefore, desirable to use an analytical model to sift through these data and to predict the performance of the car in each configuration. It has been shown that downforce, drag and centre of pressure can be applied to a simple chassis and tyre model to give a good general description of the steady state handling of a modern racing car (2). However, unlike chassis tuning, aerodynamic changes cannot be described in terms of cornering power alone since the drag component has a great effect on the straight line speed and hence lap times. If aerodynamic influences are to be evaluated then the analysis must consider the effect of downforce, centre of pressure and drag while allowing the model to be driven both round the corners and along the straights thereby giving an estimate of lap time which is, of course, of prime importance to the race engineer. 2 THE ANALYSIS

Previous methods of evaluating the performance of a racing car round a lap have been presented but have been unable to deal with the subtleties of ground effect (3). Furthermore the very stiff suspensions combined with limited movement (350 kN/m wheel rates and 25 mm wheel movements are not unknown) combined with the relatively smooth nature of the race track enables suspension movement as such and the resultant changes in camber and roll centres to be neglected for a first approximation. The main objective is to produce comparative rather than absolute results. The analysis is divided into two parts: ( i ) the modelling of the car and (ii) the circuit. In this case separate models are introduced for the chassis, power train, tyres and aerodynamics while the circuit algorithm must act as the driver to feel the limit of the car round the corners and to extract the utmost straight line performance. As with the real car, the race engineer is left to tune the chassis, balance the aerodynamics and optimize the gear ratios. It is also possible to evaluate the effect of weight and power, both of which have recently been controversial in Formula 1.

2.1 The chassis model

For the purpose of this analysis the chassis serves two functions while cornering. Firstly, by virtue of its mass and centre of gravity it dictates the cornering force trying to throw the car off the road and overturn it. Secondly, it must distribute the overturning moment between the front and rear thereby controlling the load distribution between the four tyres (Fig. 2). For a car travelling round a corner of radius r,: F , = M1u2/r, where F, is the outward force acting through the centre of gravity. If the weight distribution is such that a proportion, p , , acts on the rear axle then the outward force on the front and rear tyres is given by:

Ff = F,(1 - P,)

(la)

Fr = F o p ,

(lb)

wfi

wro

I

Front rolli

1

I

Fig. 2 Action of forces on the chassis

The static wheel loading front and rear is:

If the centre of gravity is at a height h above the road surface then the overturning moment is expressed as M , = F,h. Now, if the front and rear roll stiffness are such that a proportion, pr, of the overturning moment is resisted by the rear suspension then the front and rear weight transfer is:

where tr and tr are the front and rear tracks. The model as it stands does not evaluate the front and rear roll stiffness but is capable of showing the effect of stiffness variations. It is left to the race engineer to decide how this should be done, for instance by varying the spring rates, anti-roll bars or ride heights.

2.2 The engine and transmission model On the straights the model must reproduce the engine power-speed characteristic, the gear ratios and the final drive ratio. The rotational inertia of the wheels and tyres will also be considered since this can vary considerably from tyre to tyre. In all but first gear the free response of the engine and transmission system is very much greater than the acceleration of the car and in first gear the acceleration is limited by wheel spin and not the engine. Consequently the rotating inertia of the drive line can be neglected. At a velocity, u, the torque required to accelerate the total mass of the car against a drag is given by: TI = rr{[m, + 2(mf rnr)]a d }

+

+

and the torque required to accelerate a wheel of radius, r, radius of gyration, R , and mass, m, is:

T. = mR2a/r where a is the linear acceleration. Thus, for a four wheel car:

T. = 2aC(mfRi?/rf) + (mr R:/rr)I QIMechE 1984

Proc Instn Mech Engrs Vol 198D No 7

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AERODYNAMIC INFLUENCES ON

For the system as a whole T T

= I,{

[m0

=

T,

rm

PERFORMANCE OF THE GRAND PRIX RACING CAR

+ TI,therefore:

+ 2(mf + m,)la + d}

+ 2a(mfR;/rf + m,R;/r,)

89

be given in terms of the measured downforce, Aref, at the reference velocity, uref: Af

=

Aref(U/Uref12(1 - ~

(54

c )

which can be rearranged to yield acceleration thus:

Ar = Aref(U/Uref)2Pc (5b) At both ends of the car the downforce is equally supported by each tyre.

where T can be determined from the instantaneous power and speed using the engine and transmission characteristics. The estimation of the total drag is discussed below.

2.5 Total tyre loads

2.3 The braking model Since it is the objective of this model to generate comparative rather than absolute results it has been possible to assume a constant braking force. Clearly, in reality the braking force will depend upon the downforce and therefore speed. A detailed model of the braking performance is given in Appendix 1 where it is assumed that the driver keeps the car just at the point of locking the wheels. It has not been used to generate the results given below due to the greater computing time involved. Even a model such as this becomes unreliable at very high speeds where the downforce is such that the driver can no longer lock the wheels. The braking force of 39 used for a Formula 1 car in this analysis has been gleaned from the brake manufacturers. 2.4 The aerodynamic model

Conventional racing car wind tunnel tests present data in terms of drag, downforce and centre of pressure. Since the regulations prohibit movable aerodynamic devices, these three parameters uniquely describe the aerodynamic performance of a particular configuration (Fig. 3). As the change in Reynolds number is small within the relevant speed range, the centre of pressure can be assumed to remain constant with speed, (at 75 m/s Re = lo'). Since, in the tunnel used, the test speed is limited by the fixed speed of the rolling road it has not been possible to verify this experimentally. For a centre of pressure such that a proportion, pc , of the downforce acts on the rear tyres then the front and rear downforce may be expressed as A, = A(1 - p,) and A, = Ap,. Tunnel tests often present the data at some reference speed, thus, for a given velocity, u, the downforce may Centre of pressure

\

As the car corners at a steady velocity the tyre loads can be obtained from equations (1, 2, 3, and 5). Thus: Front outer tyre load = +mil - P g ) + iAreXu/uref)2(l - P J + ~ o ( 1 pr)/t, Front inner tyre load = h t ( 1 - Pg) + tAreXU/Uref)2(1 - P C ) - Mo( 1 - Pr)/tf Rear outer tyre load = 3mt Pg + 3 ~ Aref(U/Uref12 c + Mo P J t r Rear inner tyre load = h t Pg + t ~ Aref(U/Uref)2 c - Mopr/tr 2.6 The tyre model Over the years a number of tyre models have been presented (e.g. 4, 5) but since there is no recent information available on current racing tyres it is not usually possible to provide the input data that the models require. Since it is not the purpose of this analysis to investigate tyres in detail, a highly simplified model has been used that considers variations in cornering force coefficient with only slip angle and normal force. It will be shown later that the relevant parameters may be deduced from direct observation of the car. It is not essential that the tyre model be exact. The objective is to optimize the car to balance the front and rear slip angles. If the car is unbalanced it is not important to know exactly how far out it is providing that the model gives some indication of the extent of the handling problem. This behaviour is defined by the shape of the friction coefficient-slip angle curve close to the peak slip angle. If the car is always driven at the limit of its adhesion then the cornering stiffness at low slip angles in unimportant. If the overall cornering force is to be realistic then the maximum friction coefficient must be broadly correct. Nevertheless, even if precise laboratory measurements of the tyre characteristics were available it has been shown that quite wide variations may occur under real conditions (6). The estimate described below is adequate for a comparative analysis. Two basic tyre models have been employed: that presented in (2) and the traction/cornering ellipse (7). The latter has been further simplified into a circle. For relatively small slip angles both methods give similar results. Consider a tyre acting under a normal load, W , and at a slip angle, a,(Fig. 4), then the limiting force at the tyre contact is p W - +[ W ] where W ] is a correction factor for p due to the load W . From Fig. 4 it is apparent that the torque required to drive the car a t a constant speed in the x direction is T = r,d/cos a and for a the cornering traction circle of radius pW force can be expressed as : Fc = {(pW - I , ~ [ W ]-) ~(rd/cos a)2 } 112

+[

+[v

t

Down force

Fig. 3 Aerodynamic forces acting on the chassis

Proc Instn Mech Engrs Vol 198D No 7

QIMechE 1984

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J A W M l N Y A N D R G DOMINY

-U

Lateral force J

I

Fig. 4 Forces acting on a slipping tyre

d = 0 for an undriven wheel and the force resisting the outward force due to cornering is:

F = F, cos u

+ F,

tan u

where F , is the force acting in a tangential direction; F, tan o! can be neglected for small slip angles and high downforces. At each end of the car the two wheels can be combined to give a hypothetical single tyre which has the properties of the real pair. The total cornering force at each end of the car is: F,

= [{(PWO -

$CKI) + (PW - $[WN2

- (rd/cos c ( ) ~ ] ’ / ~

where Woand are the total inner and outer tyre loads given above. The values of p used below are for a real tyre under a typical load. The effect of tyre width on force Coefficient is considered to be proportional to the load per unit width and the rear tyre is taken as the datum. 2.7 Circuit-driver algorithm

A satisfactory algorithm to simulate a racing car driving round a circuit must be capable of establishing the limiting speed and attitude of the car on curves of each radius encountered on that circuit and to accelerate along the straights, changing gear at the appropriate speeds and to brake into the following corner. The main difficulty is to obtain an adequate model of the car cornering. It is the authors’ experience that two top racing drivers may describe quite different techniques for driving the same car through a particular turn. It is, therefore, extremely difficult to form a general impression of the optimum behaviour of the car through a corner let alone to define it mathematically. However, it has been found that a steady state cornering analysis can give a good account of the fundamental handling characteristics of a particular car and direct observation confirms that modern Grand Prix cars corner in an orderly and controlled fashion, with little steering or throttle correction. Consequently, the steady

state model (2) has been used in this case. Each curve is considered to be an arc of a circle round which the car is driven at the maximum possible speed under a steady power setting. The technique for solving the cornering speed is iterative. For an initial low speed the lateral force on the rear of the car is calculated. Using a value of force coefficient corresponding to the peak slip angle the cornering force can be calculated. This is then resolved in a direction parallel to the lateral force. If the cornering force generated by the tyres is greater than the outward radial force then the speed is increased in steps. A bisection method is used to find the speed at which the forces are in equilibrium. This is considered to be the maximum speed at which the rear of the car can corner. The process is then repeated for the front. The lower limiting speed is then used to establish the slip angle at which the opposite end must run, giving the steady state cornering attitude. Straight line performance is analysed by a time step method. The car is considered to enter the straight at the exit speed of the previous corner. The correct gear ratio is chosen according to that speed. From the car speed and gear ratios the engine speed can be calculated and hence the engine power and wheel torque. The acceleration of the car can then be determined for the next time step. If, at the end of the time step, the engine speed is higher than the gear change point the car will change up a gear. In top gear the car may ‘over rev’ or a ‘rev limiter’ may be used to prevent engine damage. At the end of each time step the distance required to brake from the total speed reached to the limiting speed for the following corner is found and added to the distance travelled on the straight. If this is greater than the overall straight length then the calculation is complete and represents the total time needed to travel between two corners. For suitably small time steps this gives an adequate degree of accuracy. However, if larger time steps are used then there may be an unacceptable ‘overlap’ error at the end of the straight.

2.8 Setting up the model

Accurate measurements of the performance of Grand Prix cars are not generally made and those that have been attempted are not published. Similarly, data on current racing tyres are almost unobtainable. Consequently empirical techniques must be employed to provide estimates of the necessary parameters. Values of the lateral force may be obtained by timing the cars through a curve of known radius. It is then possible to estimate the maximum force coefficient. If this is repeated for several radii it is possible to evaluate the relationship between force coefficient and normal load. Typical characteristics are shown in Fig. 5. Information on the basic balance of the chassis can be obtained from the drivers. The model can then be adjusted to match the known characteristics of the car (Appendix 2). Total drag can be calculated from the terminal velocity and engine power. The aerodynamic component will be known from wind tunnel tests, the residual being rolling resistance and transmission losses. The total drag is considered to act as the square of the speed. Aerodynamic drag follows this law while mechanical

Proc lnstn Mech Engrs Vol 198D No 7

QIMechE 1984

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AERODYNAMIC INFLUENCES ON THE PERFORMANCE OF THE GRAND PRIX RACING CAR

Normal force

91

75 m radius

n

1.5

\ 25 m radius

100 m radius\

loo0 m

Fig. 6 Layout of the test track: total length

=

3204 m

1.0

P

0.5

I 2

I

I

4

6

1 8

1

10

Slip angle, degrees

Fig. 5 Typical tyre friction characteristic: rear tyre

drag is unlikely to differ significantly and is, anyway, only a small proportion of the total. 3 APPLICATION OF THE MODEL

The objective of the model was to enable the effect of aerodynamic changes to be evaluated analytically. The most useful technique has been found to be to generate a carpet plot to examine the influence of aerodynamic drag and downforce on lap times round a sample circuit. Figure 6 illustrates the test track, and the car’s particulars are given in Appendix 3.

59-1

Figure 7 shows such a plot for a typical Cosworth powered Formula 1 car. Values of drag and downforce are quoted in Newtons at 67 m/s. This system is chosen as being least ambiguous. Coefficients become awkward as both plan and frontal areas can be radically altered by simple changes, for instance removing the front wings. In this case the plot also includes lines showing the maximum speed achieved on the loo0 m straight. This is a useful factor as, for a given lap time it is helpful for a car to have a greater top speed in the race than in practice to help overtaking. Typical values for drag and downforce at 67 m/s are 3800 N and 13 OOO N respectively. The curves show that for a given drag the slope reduces as downforce increases. This is a property of the particular car, which takes on a more severe understeering attitude that detracts from its maximum possible cornering speed. If the car were re-optimized aerodynamically for each value of downforce the curves would tend to straighten out. Although the lap time can be seen to be more sensitive to drag than downforce it must be realized that a reduction in drag of 100 N is extremely difficult to achieve whereas a change in downforce of 1OOO N is relatively easily obtained. For a Formula 1 car on a medium-speed circuit downforce is usually the dominant variable. The sample lap distance and speed are almost identical to those at Donington Park (although the track shape is quite different). This allows an interesting com-

2.8

.-

I

76 54.

53-

Fig. 7 Variation of lap time with aerodynamic downforce and drag for a Formula 1 racing car QIMechE 1984

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J A DOMINY AND R G DOMINY

92

67-

Downforce, N

65 -

64‘

Fig. 8 Variation of lap time with aerodynamic downforce and drag for a class ‘A’Clubmans car

parison with cars of different classes in an attempt to validate the model. Lap times of 57 s have been achieved by Formula 1 cars at Donington. Figure 8 shows a dragdownforce chart for a Class A Clubmans car and a point is shown for one particular car, a Mallock Mark 24. The predicted lap time of 71 s is within one second of the time achieved by this car. It is interesting to note that in this class, where little or no ground effect technology has crept in, quite modest increases in downforce give a useful improvement in lap time and that relatively large drag penalties can be tolerated. Figure 9 shows the effect of centre of pressure on lap time and cornering attitude, once again for a Formula 1 car. It has been shown (2) that a car’s balance is critical to its maximum cornering speed round a particular turn and this is again observed for the track as a whole. The cornering attitude (defined as af - a,) is plotted for the fastest (100 m radius) and slowest (25 m radius) bends. For this particular car and circuit it is more important to achieve a balance in fast corners than in slow. Because downforce rises as the square of speed it is almost impossible to achieve a perfect balance in both

fast and slow turns. In practice, since there is often an inconsistency of balance between the tyres used in qualifying and the race, achieving a reasonable balance is further complicated. The model has been used to investigate the effect of weight and power on performance. Both of these have been the cause of controversy over the past two seasons, since the introduction of turbocharged engines and ‘disposable ballast’. Figure 10 shows a carpet plot relating power and weight to lap time. In this case the lines of maximum speed refer to the 240 m straight since terminal velocity is almost independent of weight and this condition is approached on the long straight. The figure shows the lap time expected of a good Cosworth powered car at its qualifying weight. It is reasonable to believe that, in qualifying, the better turbocharged cars will have about 480 kW available but will weigh of the order of 50 kg more than the corresponding non-turbo. They can be seen to enjoy an advantage of about one second per lap which is typical of the performance observed during the 1982 season. 580 5857-

Attitude LOO m radius

62

Chassis weight, kg

56. 61-

-g VI

.g 60.

.-

P

0

3

59.

5554550

53-

58. I

52.

l-

30

40

50

60

70

80

Percentage aerodynamic load on rear Fig. 9 Effect of aerodynamic balance on lap time and cornering attitude for a Formula 1 racing car

Fig. 10 Effect of engine power and chassis weight on lap time for a Formula 1 racing car

Proc Instn Mech Engrs Vol 198D No 7

@IMechE 1984

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AERODYNAMIC INFLUENCES ON THE PERFORMANCE OF THE GRAND PRIX RACING CAR

93

Braking force at rear:

4 CONCLUSIONS

The analysis has demonstrated that a simplified model of a Formula 1 racing car can be used to evaluate aerodynamic and chassis changes in terms of lap times, straight line speed and cornering attitude. The main difficulty in any analytical work concerned with racing cars is the lack of measured data, published or otherwise, with which to verify the predicted values. However, in this case a degree of verification has been achieved, firstly by comparing the times of Formula 1 and Class A Clubmans cars round the same circuit and by comparing the effect of weight with the Formula 1 ‘folklore’ value of 1-1.5 s per lap per 50 kg (unfortunately the folklore does not specify a lap length or speed). Of greater importance is that the recognized interactions between chassis and aerodynamic configurations have been successfully modelled. Thus, the race engineer cannot only evaluate a great deal of wind tunnel data prior to track testing but can investigate those characteristics that give the greatest benefits on a particular track. Although both the car model and the driver-track algorithm are highly simplified they would require a considerable increase in computing power and a more detailed knowledge of the behaviour of the car (and driver) through corners if they are to be significantly improved. Since such data are unlikely to become available in the forseeable future and as this analysis is capable of giving realistic results as it stands, the extra complexity has not been considered to be justified at this stage.

thus the acceleration becomes: a1 = (bf

+ br)/mt

hence Su for a small time step at. The equations are solved to give a braking distance and time between two limiting velocities by means of time step integration. APPENDIX 2 Estimation of tyre data Consider a car rounding a corner of radius, rc, at velocity, u. For a car with no downforce m, u2/rc = m, p, therefore p = u2/rc. With downforce:

therefore p= Cmt

m, u2 + Aref(~/~ref)~Irc

For the front of the car:

REFERENCES 1 Wright, P. G. The influence of aerodynamics on the design of

Formula 1 racing cars. Int. J . of Vehicle Design, 1982, 3 (4), 383397. 2 Dominy, J. Frictional aspects of Formula 1 racing car performance. Tribology International,June 1981, 14 (3), 167-170. 3 Scherenberg, H. Mercedes-Benz racing car design and experience. S A E Trans, 1958,66,414420. 4 Moore, D. F. Friction and wear in rubbers and tyres. Wear, 1980, 61 (2), 273-282. 5 Bernard, J. E. et al. Tyre shear force generation during combined steering and braking maneuvers. SAE, paper No. 770852, 1977. 6 McCarty, J. L. Wear, friction and temperature characteristics of an aircraft tyre undergoing braking and cornering. NASA technical paper No. 1569,1979. 7 Ellis, J. Vehicle dynamics, London Business Books Ltd., 1969.

APPENDIX 1 Improved braking model

Test car data

Assuming braking is in a straight line and at the point of locking the wheels, then for an initial acceleration a,, and velocity, u,: a, mc h longitudinal load transfer = wb

Therefore load on front axle = static load + downforce + load transfer: At the rear: qr = mt

Pg

+ Aref(Uo/Uref)2Pc-

Braking force at front: bf = P4r - ll/C4fl

mc h/wb

This gives a mean value of p. By repeating this procedure for corners of different radii (hence speed and downforce) it is possible to estimate the effect of load, W, on p. For this investigation, the following relationx W). ship has been used: p = pma,.(l - 3.8 x Care is required in interpreting these data since the effective friction coefficient will depend upon a number of factors, notably the nature and condition of the road surface, ambient temperature and the particular set-up of the car. APPENDIX 3

Formula 1 Chassis weight, kg Front wheel weight, kg Rear wheel weight, kg Front wheel diameter, m Rear wheel diameter, m CG height, m CG position: YOfrom front axle C , position: % from front axle Roll stiffness: % on rear Front track, m Rear track, m Front tyre tread width, m Rear tyre tread width, m Max. friction coefficient Peak slip angle, degrees Max power, kW

600 18 20 0.6 0.66 0.35 60 50 60 1.78 1.63 0.18 0.36 1.8 8 380

Clubmans 380 15

‘A

15 0.508 0.558

0.35 60 55 50 1.46 1.41 0.18 0.18 1.5 8 135

Proc Instn Mech Engrs Vol 198D No 7

@IMechE 1984

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