Chassis Stiffness

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_2000-01-3554

The Effect of Chassis Stiffness on Race

Car Handling Balance

Andrew Deakin, David Crolla, Juan Pablo Ramirez and Ray Hanley

School of Mech. Eng., The University of Leeds

Reprinted From: Proceedings of the 2000 SAE Motorsports

Engineering Conference & Exposition

(P-361)

Motorsports Engineering Conference & Exposition

Dearborn, Michigan

November 13-16, 2000

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2000-01-3554

The Effect of Chassis Stiffness on Race

Car Handling Balance

Andrew Deakin, David Crolla, Juan Pablo Ramirez and Ray Hanley

ABSTRACT

It is often quoted that to be able to make a race car handle ‘properly’ by tuning the handling balance, the chassis should have a torsional stiffness of ‘X times the suspension stiffness’ or ‘X times the difference between front and rear suspension stiffness’ [1].

This paper looks at the fundamental issues surrounding chassis stiffness. It discusses why a chassis should be stiff, what increasing the chassis stiffness does to the race engineer’s ability to change the handling balance of the car and how much chassis stiffness is required. All the arguments are backed up with a detailed quasi static analysis of the problem.

Furthermore, a dynamic analysis of the vehicle’s handling using ADAMS Car and ADAMS Flex is performed to verify the effect of chassis stiffness on a race car’s handling balance through the simulation of steady state handling manoeuvres.

INTRODUCTION

It is well known that to make a race car handle correctly, it must be possible to tune the handling balance. Tuning the handling balance means adjusting the level of grip available from either the front or the rear of the vehicle. When both the front and rear axles can produce a force to give the same lateral acceleration, the chassis can be said to be balanced.

Figure 1 illustrates the non-linear behaviour of a typical tyre used with Formula SAE racing cars. Figure 2 shows the Leeds University Formula SAE car. It can clearly be seen that if a pair of tyres on an axle had the same vertical load, then they could both produce the same maximum lateral force. If for example, the vehicle was cornering, then the lateral acceleration would cause a load transfer, equation 1. This lateral acceleration would increase the vertical load on the outside tyre and decrease the vertical load on the inside tyre by the same quantity. The result of this load transfer is that the two tyres combined can produce less lateral force.

t

h

Lat

m

LT

₌

a acc CG (1)

where LT is the lateral load transfer for an axle, LATacc is

the lateral acceleration, ma is the mass supported by that

relevant axle, hCG is the centre of gravity height and t is

the track width. This assumes a flexible chassis.

0 500 1000 1500 2000 2500 3000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Vertical load, N Maxim u m later al fo rce g en er ated , N

Figure 1 – Non-linear behaviour of a typical Formula SAE tyre, max. lateral force produced for a vertical load. Therefore a car understeers (a car that has too little grip at the front), the grip can be increased at the front by reducing the load transfer at the front and increasing the load transfer at the rear.

Figure 2 – Leeds University Formula 1999 SAE car Being able to control the load transfer distribution is therefore the key to being able to obtain a good handling balance. The lateral load transfer distribution can only be controlled however, if the chassis is stiff enough to transmit the torques.

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The question that is then raised is how stiff is stiff enough. The objective of this work was to go some way towards answering that question.

MODELLING

There are two sections of modelling within this paper. The first is a simple static analysis to determine the effects of chassis torsional stiffness on being able to maintain the desired lateral load transfer distribution. The second is a dynamic analysis of the effect of a flexible chassis using the ADAMS and ADAMS Flex. STATIC ANALYSIS OF CHASSIS STIFFNESS – A model calculating the static forces present in the chassis under steady state conditions has been developed. This considers the racing car to consist of two point masses, mf and mr for the front and rear respectively, connected by a torsional spring, Kch, and a suspension at each end of the vehicle represented by a roll stiffness, Krollf and Krollr, figure 3.

Krollf

Krollr

Kch

mf

mr

Figure 3 – Static model of the effect of chassis torsional stiffness on lateral load transfer distribution

From this model, equations 2, 3 and 4 were derived. φ1,

φ2 and φ3 are the front suspension roll angle, the rear

suspension roll angle and the chassis torsional twist respectively. Mf and Mr are the front and rear moments due to the lateral acceleration of the body masses.

3 1

φ

Kch

Krollf

Mf

=

−

(2) 3 2

φ

Kch

Krollr

Mr

=

+

(3) 2 3 1

φ

+

=

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These equations represents a very much idealised model of the vehicle as shown in figure 4.

Figure 4 – Idealised chassis model with two masses connected by a torsional spring

The real vehicle is much more equivalent to that shown in figure 5, where the mass is evenly distributed along the body. As long as the chassis is equally torsionally stiff at all points along the chassis then it can be shown that the idealised model still represents the actual chassis.

Figure 5 – Chassis model with uniformly distributed mass

The real vehicle however, does not have an evenly distributed mass with all mass having the same moment arm and each segment of the chassis having an equal torsional stiffness. In reality, figure 6 is something like an actual vehicle’s mass distribution. Heavier objects such as the engine, the driver safety cell and the driver are located close to the centre of gravity of the car. Also, the torsional springs may not be along the same axis as shown in figure 6. Therefore there are likely to be discrepancies between results from the idealised model and a real vehicle.

Figure 6 – Mass distribution of real vehicle

Additionally, there are compliances in the suspension, commonly referred to as the installation stiffness, which reduce the chassis torsional stiffness as seen at the wheels. These should also be considered as possible errors between the idealised model which has been proposed and the real vehicle.

MULTI-BODY HANDLING MODEL – A model of the Leeds University Formula SAE car has been developed in ADAMS to understand further the effect of a flexible chassis on handling. The basic model configuration with a rigid chassis is shown in figure 7. Two extensions to this model were created which included; a chassis separated into a front and rear section joined by a torsional spring along an axis at the wheel centre height, and a flexible chassis incorporated into the ADAMS model using ADAMS Flex.

In theory, the model containing a torsional spring could be used to validate the static model results. The results from the model incorporating an ADAMS Flex, flexible chassis, could be used to understand the effect on a real vehicle.

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The model with the torsional spring was developed to enable evaluation of multiple chassis torsional stiffnesses on the vehicle’s handling performance, as this just requires a single model parameter to be changed.

ADAMS Flex takes a modal neutral file format which is produced using the finite element method in a software package such as ANSYS, figure 8. This model is loaded such that a torsional force is put onto the chassis at the suspension rocker mounts. Ideally it should be loaded such that all the suspension wishbone and track rod forces load the ADAMS Flex model, however, this would increase the complexity significantly. As it was, there were 18 mode shapes represented in the model, eight of which were rigid body modes. The nominal torsional stiffness of the ADAMS Flex model was 1,300 Nm/deg.

Figure 7 – ADAMS model of the Leeds University Formula SAE Car.

Figure – 8 ADAMS Flex chassis template.

The overall vehicle parameter data used in the ADAMS model is shown in table 1.

Subsystem Value rear susp. 17.93 kg front susp. 16.70 kg rr. antiroll 1.99 kg frt. antiroll 1.99 kg steering 5.90 kg frt. wheels 21.00 kg rr. wheels 21.00 kg

chassis with driver 250.00 kg

Chassis Inertias

Ixx (roll) 7.33E+06 [kg*mm^2]

Iyy (pitch) 3.56E+07 [kg*mm^2]

Izz (yaw) 3.94E+07 [kg*mm^2]

Chassis C.G. Location frt. weight 46.5 % rr. weight 53.5 % height 300 mm Spring Rates frt. spring rate 61.5 [N/mm] rr. spring rate 87.9 [N/mm]

frt. antiroll bar rate 150 [Nm/deg] rr. antiroll bar rate 125 [Nm/deg]

Table 1 – Data for Formula SAE Car model

RESULTS

Results were produced to indicate how chassis stiffness effects, set up of the desired lateral load transfer. This was conducted both through static analysis and dynamic analysis.

STATIC ANALYSIS RESULTS – The static analysis results were performed for a range of vehicle, total suspension roll stiffnesses representing different vehicles. Dixon [2], gives a range of data values, table 2, for different types of racing vehicle. Total roll stiffnesses for typical Formula SAE cars are also included.

Car type Total roll stiffness, Nm/deg

Saloon 300 – 800

Sports car 2000

Sports prototype 18,000 Formula One 20,000 – 25,000 Formula SAE 500 – 1,500

Table 2 – Typical total vehicle roll stiffness, Nm/deg Figures 9, 10, 11 and 12 show the difference in front to rear lateral load transfer distribution for different roll stiffness distributions. This is calculated for a range of chassis stiffnesses and for total roll stiffnesses of 500, 1500, 5000 and 15000 Nm/deg respectively. All of these results assume that both the static load distribution is 50:50 and the front and rear centre of gravity heights are the same.

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The goal is to determine a chassis stiffness that ensures the vehicle’s handling is sufficiently sensitive to changes in the roll stiffness distribution. A large percentage of the difference in front to rear roll stiffness must therefore result in a difference in front to rear lateral load transfer, for example 80%.

Looking at figure 9, and the point where the roll stiffness distribution is 30:70, the lateral load transfer distribution can be anything from 30:70 to 40:60. If the difference between front and rear lateral load transfer is to be 80% of the difference between front and rear roll stiffness, then the lateral load transfer distribution must be at least 34:66.

It is clear from figure 9 that all but the least stiff of chassis shown, (100Nm/deg torsional stiffness), produces a load transfer distribution of 34:66 or greater at the point where the roll stiffness distribution is 30:70. Therefore if the criterion is that the difference between front and rear lateral load transfer is to be 80% of the difference between front and rear roll stiffness, then for softly sprung cars, (roll stiffness <500 Nm/deg), the torsional stiffness of the idealised chassis should be greater than 300 Nm/deg.

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

Front roll stiffness as % of total roll stiffness

F ron t load t ran sf er as % of t ot al load t ran sf er

Chassis stiffness 100 Nm/deg Chassis stiffness 300 Nm/deg Chassis stiffness 600 Nm/deg Chassis stiffness 1000 Nm/deg Chassis stiffness 2000 Nm/deg Chassis stiffness 4000 Nm/deg Chassis stiffness 8000 Nm/deg Chassis stiffness 16000 Nm/deg

Figure 9 – Lateral load transfer from a racing car with roll stiffness of 500 Nm/deg

With a total suspension roll stiffness of 1,500 Nm/deg, the modelled chassis stiffness required to produce a front to rear lateral load difference of 80%, of the roll stiffness distribution difference, is approximately 1000 – 2,000 Nm/deg, figure 10.

Similarly, when the roll stiffness is increased to 5,000 Nm/deg, a modelled chassis torsional stiffness greater than approximately 6000 Nm/deg is required, figure 11. For the vehicle with a roll stiffness of 15,000 Nm/deg, using the same 80% guideline, a modelled chassis stiffness of greater than 10,000 Nm/deg is required, figure 12. 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

Figures 13 and 14 explore what happens if the static load distribution is changed so more mass is supported by the rear suspension. The load distributions chosen were 45:55 and 40:60.

In both cases, the point at which chassis stiffness has no effect on the lateral load transfer distribution is where the ratio of front to rear roll stiffness is the same as front to rear weight distribution.

When the weight is moved more to the rear and the roll rate at the front is higher than the rear, less lateral load transfer difference between front and rear is achieved for

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the same roll stiffness distribution. Therefore a stiffer chassis is required. 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

Figure 13 – Lateral load transfer from a racing car with roll stiffness of 1500 Nm/deg with 45:55 load distribution

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

Figure 14 – Lateral load transfer from a racing car with roll stiffness of 1500 Nm/deg with 40:60 load distribution Figure 15 summarises the results from figures 9, 10, 11 and 12 for a suspension roll stiffness distribution of 60:40. From the graph it can be seen, for example, that if it is acceptable for the lateral load transfer difference to be only 80% of the roll stiffness difference, (lateral load transfer distribution difference is 20% less than the roll stiffness difference), then the ratio of total suspension roll stiffness to chassis torsional stiffness should be approximately 1.

If the ratio of front to rear roll stiffness is reduced so that it is closer to 50:50, then the chassis torsional stiffness is required to be slightly higher (up to 4% higher). Conversely, if the front to rear roll stiffness is increased above 60:40, a more flexible chassis can be used. If the engineer desires a loss of no more than X% of roll stiffness distribution ratio into lateral load transfer distribution, then as a rule of thumb, equation 5 can be used.

X/20 = Ratio (5)

where Ratio is the ratio of total chassis roll stiffness distribution to chassis torsional stiffness. If chassis torsional stiffness and suspension roll stiffness are

known, then this enables an approximate measure of how sensitive the vehicle’s handling balance will be to changes in roll stiffness distribution.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 5 10 15 20 25 30 35 40 45 50

Roll stiffness difference not turned into lateral load transfer difference, %

R ati o o f to ta l r ol l s ti ffne ss to c ha ss is to rs io na l s ti ffne ss

Figure 15 – Percent difference between lateral load transfer distribution and roll stiffness distribution for different ratios of roll to chassis stiffness

DYNAMIC ANALYSIS RESULTS – The dynamic analysis presented uses steady state analysis features contained within ADAMS. The vehicle was forced to follow a constant path radius and the required steering wheel angle and lateral acceleration generated for different speeds were determined.

Figure 16 shows the lateral acceleration vs. steering wheel angle for the vehicle configurations considered. These configurations include a stiff chassis, a chassis with torsional stiffness of 250, 1300 and 2500 Nm/deg and the chassis from the ADAMS Flex model. All these results were generated with the same suspension roll stiffness distribution.

For the chassis containing torsional springs to represent chassis stiffness, it is clear that the more flexible a chassis is, the more the vehicle tends to understeer. Subsequently, the roll stiffness distributions were tuned such that each of the vehicle models with a flexible chassis had the same total roll stiffness and achieved the same handling balance. The increase in the front to rear roll stiffness difference to give this same handling balance for each vehicle is shown in figure 17. This shows that the roll stiffness difference has to be increased for more flexible chassis in order to generate the same lateral load transfer distribution. An attempt was made to correlate these results with those obtained from the static analysis. This was not found to be possible, the likely cause was attributed to large castor angles on the front suspension which affect the height of the tyre contact patch centre when a significant steering angle is applied. The result of this kinematic effect was to change the front roll stiffness compared to that calculated from conventional theory. This will be the subject of future investigations.

The ADAMS Flex model shows an oversteering characteristic, figure 16. Looking closely at the torsional

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stiffness distribution of the chassis, figure 18, it is apparent that the front of the chassis is least stiff in this mode. This weakness at the front of the chassis effectively reduces the roll stiffness of the suspension at the front, thus producing less front load transfer and the oversteering characteristic. To achieve the same handling balance as the stiff chassis, the front suspension roll stiffness had to be increased, compensating for the weak area in the chassis, whilst the rear was reduced, thus maintaining the same total roll stiffness. -81 -80 -79 -78 -77 -76 -75 -74 -73 -72 -71 -70 -69 -68 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Lateral Acceleration, g steer in g wh eel an g le [d eg ] Rigid Chassis 250 N-m/deg 1300 N-m/deg 2500 N-m/deg ADAMS/Flex

Figure 16 – Lateral acceleration against steering angle for different ADAMS model configurations.

0.00 50.00 100.00 150.00 200.00 250.00 0 500 1000 1500 2000 2500

Chassis Torsional Stiffness, Nm/deg

C h ange in rol l ra te di st ri but ion f rom ba se lin e, %

Figure 17 – Difference in roll stiffness for flexible chassis to give the same handling balance compared to a stiff chassis.

DISCUSSION

It is clear from the set of results presented that there are discrepancies between the results obtained purely from static calculations to those that are obtained through dynamic analysis. With regard to the chassis with different torsional stiffnesses, these differences have been attributed to kinematic effects in the vehicle model, reducing the effective roll stiffness at one end of the vehicle. With regard to the ADAMS model containing the ADAMS flex representation of the chassis, the discrepancy is attributed to the distribution of chassis torsional stiffness along the vehicle length.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Longitudinal location, mm C h assis twist an g le, d eg Fr ont suspensi on m ount poi nt R ear suspensi on m ount poi nt

Figure 18 – Chassis torsional deformation along length of chassis model used with ADAMS Flex, front = 0mm. Equation 5, is very much a generalisation of the ratio of chassis torsional stiffness to total suspension roll stiffness, to produce a certain load transfer distribution from a certain roll stiffness distribution. The calculations suggest that if the vehicle weight distribution is approximately 50:50 then equation 5 can be used as a guide to determine how stiff the chassis should be. However, in order to do this, an understanding of what constitutes an acceptable loss of roll stiffness distribution into load transfer distribution is required.

It has also been shown that more subtle effects from changes in torsion stiffness along the chassis and kinematic effects in the vehicle will influence the results. Thus the chassis stiffness required will differ from vehicle to vehicle, however, this analysis gives an initial insight into the problem.

CONCLUSION

Two modelling strategies have been developed. A static analysis model can be used to calculate the effect of chassis torsional stiffness on achieving a desired handling balance. Also a dynamic handling model using ADAMS can be used to predict the effect of chassis torsional stiffness on dynamic handling manoeuvres. It has been shown that to translate a certain percentage of suspension roll stiffness distribution into a lateral load transfer distribution, the chassis torsional stiffness to total suspension roll stiffness must be a certain ratio. Therefore the chassis torsional stiffness must be a multiple of total suspension roll stiffness and not the difference between front and rear suspension stiffness as has previously been suggested. The chassis torsional stiffness referred to must include the installation stiffness of the suspension.

It has been shown that a Formula SAE car which has a total suspension roll stiffness of 500 – 1,500 Nm/deg requires a chassis stiffness between 300 and 1,000 Nm/deg to enable the handling to be tuned.

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A Formula One car and vehicles with a similar roll stiffness requires a chassis torsional stiffness in excess of 10,000 Nm/deg to enable the handling to be tuned. The dynamic results confirm that the stiffer the chassis, the less the difference in roll stiffness distribution has to be to achieve the same handling balance.

The ADAMS model demonstrates that the effective roll stiffness distribution can be affected by kinematic properties in the suspension which should be taken account of in any analysis.

The distribution of chassis stiffness along the length of a chassis also has an effect on the required roll stiffness distribution to achieve a good handling balance. Indeed a torsionally non-stiff region of a chassis close to the front or rear suspension can effectively reduce the roll stiffness of that suspension.

REFERENCES

1. Milliken, F.W.; Milliken, D.L.: ‘Race car vehicle dynamics’, SAE Int’l, 1995

2. Dixon, J.C.: ‘Tyres, Suspension and Handling’, Cambridge University Press, 1991.

CONTACT

Andrew Deakin

School of Mechanical Engineering The University of Leeds

Leeds, LS2 9JT, England, UK [email protected]