FIRST PRINCIPLES Equation 1
MTMax = CM _ MT(δcamb F′ MAX = fn(FZ
,TT MT = fn(α,FZ ( ) ,TT) ⋅MTMax
Fy = fn(α,FZ ,TT) ⋅CFy _ MT(δcamb Fx = fn SR,FZ ,TT) ⋅ ′ µ TC(δcamb
Table 1: parameter Parameter
F’MAX &
fn(α,FZ (
CFy _ MT δcamb ,Fz ( € € € µ′ TC δcamb ,Fz (
Equation 2 Cf
Cr ) fn SR,FZ
,TT ,TT
) ) ) What it does
This is the traction circle radius, or the Pacejka D term. This dictates your grip
This is the normalised slip curve, or the other terms of the Pacejka term that dictates how the forces vary with slip angle and slip ratio. We’ll talk about how this effects handling shortly
This dictates your lateral camber sensitivity. This is the go to function to dial in lateral camber sensitivity and your traction ellipse properties
This dictates the amount of longitudinal grip you’re going to have to accelerate. This, combined with our camber function, is going to dictate how we accelerate
,Fz) ⋅ ′ F MAX ,Fz) ⋅ ′ F MAX ,Fz) ⋅ fn(FZ ,TT )
tyre model the things we need to pay attention to are summarised in table 1, below left. The next thing we need to
keep in mind is the stability index. The formulation for this is shown in equation 2. (I’ve omitted the yaw rate term here, simply for ease of illustration). What this stability index is telling us that our understeer / oversteer is dictated by the difference between the traction circle radius multiplied by the normalised slip curve at either end. So to increase understeer, we either need to increase the traction circle radius at the rear or the slope of the rear normalised slip curve. The most direct way you can do this is by reducing the peak rear slip angle. So without further ado,
let’s get started by looking at some actual simulated data. To
global grip factor and multiply it by 10 per cent. In ChassisSim you simply click on the tyre graphic and you’ll see an edit box labelled Tyre force grip factor. The effect of this is shown in figure 2.
The grip factor is black, the baseline is coloured. As you can see, the effects are pretty obvious, with corner speed increasing everywhere. This does exactly what it is designed to do, so consequently is your first go to when dialing in grip. But what happens if you
increase grip factor and some corner speeds shift yet others don’t? What’s happening here is the simulation is telling you the tyre load characteristic isn’t right. To diagnose this is easy. All you need to do is pull up a trace of the corner speeds with the tyre loads. You note
Don't get white line
fever on having the perfect throttle trace
= ∂Cf ∂αf
= ∂Cr ∂αr
CT = Cf α=αf ⋅(Fm1 α=αr ⋅(Fm3 +Cr
stbi ≈ a ⋅Cf − b ⋅Cr CT ⋅wb
€ + Fm2 + Fm4 ) )
illustrate this, consider figure 1 that shows an F3 car going around a rather bumpy circuit, We need to get some
housekeeping out of the way first. The first trace is speed, the second trace is throttle, the third and fourth trace is front and rear dampers respectively, the fifth trace is neutral steer and actual steer, and the final trace is lateral and longitudinal acceleration. So our first port of call is to dial in the global grip. Let’s just say, for the sake of argument, we wanted to increase the grip. To do this we simply go to the
the tyre loads and you simply increase or decrease the force at the appropriate tyre loads. A handy feature I’ve put into ChassisSim is a quick tyre force edit feature. You click on the Tyre quick start and you then click on the Click here to edit Traction circle radius. The dialogue will look like what is shown in figure 3. You simply edit the tyre force
at the appropriate load and away you go. If you really get stuck, the Tyre force estimation feature is a very handy tool to construct this curve.
STEERING TRACE Our next port of call is to dial in the steering trace. Referring back to equation 2, our two biggest levers for this are going to be the global grip factor and modifying the peak slip angles, since this will change the normalised slip angle curve. Let’s look at figure 1. The steering trace in this model is very neutral so, to increase the understeer, we are going to:
Figure 1: initial F3 run 78
www.racecar-engineering.com • September 2011
• Increase the rear grip factor by 10 per cent
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