# The vehicle performance envelope

By on

The FIA is currently looking at ways to curb the increases in vehicle speeds due to the tyre war between Bridgestone and Michelin. In addition further performance reductions such as smaller engines have been discussed as a way of returning some excitement and possibly overtaking to the sport.

Any changes will influence the size and shape of the performance envelope of the vehicle, and this is the subject of this article.

## The Envelope

The Performance Envelope can also be referred to as the g-g-V. This is because the envelope shows not only the well-known g-g diagram or ‘friction circle’ but how that varies with velocity.

## Logitudinal Acceleration

The key parameters affecting straight-line acceleration are engine torque, gear ratios, aerodynamic drag forces, and perhaps most importantly and relevant of all, the traction limit of the rear wheels.

Acceleration can be either traction limited or power limited. The approach used to define the longitudinal acceleration is first to consider the effect of the engine and gear ratios then to superimpose the effect of the traction limit and aerodynamic drag.

The tractive force at the rear wheels due to the engine is:

$F (N)= (Te*Ntf*eff)/r$

The corresponding vehicle speed is:

$V (m/S)= (n*r)/(60*Ntf)$

Te = Engine torque (Nm)
n = Engine speed corresponding to torque value (rpm)
Ntf = Transmission ratio (gearbox and final drive)
eff = Driveline efficiency
r = rolling radius of driven wheels (m)

The speed and force for each point on the engine torque curve are calculated for each gear and the resulting plot looks something like this.

Bear in mind that this is the maximum theoretical performance due to the engine. It will not be realised if the tyres can’t transmit it to the road.

The traction limit is defined for rear-wheel-drive cars as:

$Ft(N) = (Cf*Wr)/(1-Cf*(h/L))$

Cf = Peak tyre friction coefficient
Wr = Weight on the rear wheels (N)
h = Centre of Mass height (m)
L = Wheelbase (m)

This value will obviously increase slightly with speed as downforce adds to the weight on the rear tyres.

Until the engine produces less force than the traction limit, the traction limit defines the maximum acceleration Performance.

The other limiting factors are aerodynamic drag and rolling resistance. These can be modelled with a second order polynomial:

$Fd = K1 + K2*V + K3*V^2$

If we add these additional things to the graph we get something like this.

Where the drag curve crosses the tractive force curve is the maximum speed independent of gear ratio.

The resultant force at each speed is found be subtracting the drag force from the tractive force. If you then divide the force by the mass of the car the resulting plot is the maximum acceleration versus velocity.

## Braking

Longitudinal deceleration is easier to plot. There will be a theoretical value at zero velocity due to the static coefficient of friction and the static vehicle weight. The potential force will then increase parabolically due to aerodynamic drag adding to the friction force created by the tyres. This trend is plotted below.

## Longitudinal Envelope

Combining the acceleration and deceleration lines on one plot and following the g-g-V diagram convention of plotting velocity on the vertical axis we get the boundary of the performance envelope for straight-line operation.

The g-g diagram defines the maximum performance of a vehicle in any combination of longitudinal and lateral acceleration. These two parameters vary with speed. Therefore by definition any g-g diagram is specific to one velocity.

The first published reference to the g-g diagram was in 1970. In a CAL Labs report it was suggested that the four tyre friction circles could be collapsed into a single equivalent tyre/road interface or ‘vehicle friction circle’.

## The Friction Circle

All tyre forces arise as a result of slip within the contact patch. Slip can be either lateral or longitudinal.

Lateral slip is a result of the tyre being forced in a different direction to the vehicle it is attached to. A good way of visualising this is to consider a car travelling in a straight line and then to apply a steering angle. The result is shown below. As you can see the rubber is distorted and slips back as it leaves the contact patch. The elastic forces generated in this region have equal and opposite reactions with the road and this is the origin of lateral tyre force.

Longitudinal slip (driving) is caused by the torque from the engine trying to rotate the tyre faster than it would in a free-rolling condition. This compresses the tread as it enters the contact patch and this rubber must slip back as it leaves the contact patch. As with lateral force, this slip causes a resultant force. Braking is obviously the same process but in reverse.

The contact patch only feels slip, not how that slip was generated and as a result we can generalise lateral and longitudinal slip to a single resultant slip velocity and a resultant force.

If all the slip is in the longitudinal direction, all the force is in the longitudinal direction. If all the slip is in the lateral direction, all the force is in the lateral direction. Assuming that the resultant vector is a constant, the only way of connecting these two states is by plotting out a circle. This is the friction circle.

Because the resultant force vector on a vehicle is simply the sum of the four resultant tyre force vectors the g-g diagram will look much like the friction circle for a single tyre but with slight differences due to the longitudinal performance limits discussed earlier.

Seeing as this is an F1 site I have endeavoured to find some g-g plots for an F1 car. The only data available was the following diagram in Race Car Vehicle Dynamics of a 1984 F1 car. It shows the increase in cornering and braking due to the increase in downforce and the decrease in acceleration as power limitations kick in with increasing speed.

## The g-g-V Diagram

The final step is to combine the g-g diagrams and the longitudinal performance graphs to complete the performance envelope. The following diagram from a Racecar Engineering article is a perfect illustration of this:

It is important to note that the g-g-V diagram isn’t static, it will change in size and shape as things like track gradient and temperature vary. This article only explores the generic properties of the diagram.

## Pushing the Envelope

The performance envelope can graphically assist us in understanding various manoeuvres a driver might perform. I’m going to look at two, entering and exiting a corner.

When cornering you have to convert a longitudinal resultant force vector to a lateral resultant force vector. The driver can do this is two main ways; he can slam on the brakes, slow to the desired cornering speed then turn-in and negotiate the turn. Alternatively he can brake and gradually apply steering lock while gradually getting off the brakes.

These two approaches are plotted below. The first approach is shown, as are two attempts at approach two.

The g-g diagram graphically shows us that approach two gives a consistently larger resultant force vector and is thus quicker. The driver who uses approach two is ‘pushing the envelope’ to a greater degree than the driver who uses approach one.

Both entering and exiting a corner could be represented in this way but the full g-g-V diagram indicates a key difference. When entering a corner the reduction in speed means that as the g-g diagram is traversed the boundary is getting smaller so it is very easy to slip over the limit. Visualise a kind of screw-like path down the outside of the envelope.

On corner exit the envelope is getting larger due to the increase in speed, so as the limit is pushed the limit increases. It should be clear that this is a much more stable condition than corner entry.

I particularly like the late Mark Donohue’s take on the issue. He said that while balancing a car on the limit was like walking a tightrope, turning in was like jumping onto a tightrope blindfold.

## Conclusion

The cornering example illustrates that the shape of the g-g-V diagram influences the maximum performance and the stability of the vehicle. Any regulation changes will affect both the limit, and the vehicle’s stability near that limit.

When discussing a potential regulation change it is probably worth considering how that change will modify the performance envelope.

The explanations presented here of tyre dynamics are very cursory at best. For a fuller picture I would recommend Race Car Vehicle Dynamics. It is the best book on it’s subject, and is widely used by Formula One engineers at many of the teams.

Reference:

- Race Car Vehicle Dynamics (Milliken)
- Racecar Engineering and Mechanics (Van Valkenburgh)
- Fundamentals of Vehicle Dynamics (Gillespie)
- The Unfair Advantage (Donohue et al)
- Racecar Engineering Vol.5 No.3

By Ben Michell