Table 2: representative 2D tyre model for a V8 Supercar Parameter

Initial coefficient of friction Peak load

Value 2.2

850kg

Table 3 – predictive numbers for a rear roll centre change (all loads are shown in kg, the lateral forces are shown in N and V_pred is in km/h).

Figure 3: tyre schematic

RRC 240

RRC 250

FyR V_pred 674.96 133.82 694.58 54.47 9993.5 81.47 671.61 137.16 697.93 51.1 9938 81.27 Figure 4: FFact multiplier of traction circle radius vs tyre pressure

Table 4 – tests to run for set up referencing Test no Set up 1 2 3 4 5 6 7 8 9

base set up Figure 5: set-up reference dialogue from ChassisSim

The reason the change is so small is that the delta forces produced by the 2D model are miniscule. This is primarily due to the peak load value of 850kg. When we crunch the numbers, the magnitudes of the changes are so small because the gradients of the tyre force curve are tiny. This is due to the outside tyre loads approaching the optimum tyre load of the model. Don’t take my word for it, do the numbers yourself. A simple 2D tyre model is given by following equation 1. There are a number of things

you can do to mitigate this but, ultimately, it compromises

your model in other areas. For example, to go for set up sensitivity we can drop the peak load. It gives you sensitivity, but it compromises accuracy for high-speed corners and adding more downforce. Clearly, there is something else at play here. To figure a way out of this

mess, let’s consider what the tyre is actually doing. We’ll start with a tyre schematic, as in figure 3. What happens is, as we apply different set up conditions to the tyre, the internal temperature and the tyre pressure changes. There are a number of factors that come into play here. The first is the effect this has on

the contact patch variation (see equation 2). As we can see, as the pressure goes up, the contact patch length, and area, goes down. Consequently, there is less area for the forces to be applied and the tyre force goes down. Conversely, when the temperature and pressure is not optimal, there is more contact patch length and greater area. But the core of the tyre isn’t up to temperature yet. This is where the second effect comes into play. From the research that Michelin did in constructing the Michelin TaMe Tire model, the overall friction of the tyre is given by equation 3. This, combined with the

contact pressure, will have a significant effect on the tyre forces, and this is what we are approximating with set-up referencing. So, reviewing this, let me propose a modification to the traction circle radius equation, as shown by equation 4. What we have here is the 2D

model we discussed in equation (1) multiplied by a factor that is a function of hot tyre pressure. This function will look something like figure 4.

What to record tyre pressures + data

base set up + front TP -2psi tyre pressures + data base set up + front TP -1psi tyre pressures + data base set up + front TP +1psi tyre pressures + data base set up + front TP +2psi tyre pressures + data base set up + rear TP -2psi tyre pressures + data base set up + rear TP -1psi tyre pressures + data base set up + rear TP +1psi tyre pressures + data base set up + rear TP +2psi tyre pressures + data

Some of you might be a tad

annoyed at this point, but let’s take a moment to review things. From figure 1 we know we can achieve good correlation using a 2D model. However, when we investigate the nature of the numbers, the sensitivity needs to be improved. Furthermore, if we are not careful it can lead us up the garden path. And anybody that has spent more than five minutes with a real racecar knows how critical it is to get the tyre pressures right. So there must be something else coming into play, and all the results from the Michelin TaMe Tire model would indicate very strongly that tyre pressure plays a critical role. You also confirm this from running the car but, to quantify it, we need to run a series of tests, as shown in table 4. Remember, it is critical for

every run to record the hot tyre pressures. You’ll also note I’ve indicated a sweep of +/-1psi but, in reality, this is going to be dictated by your running experience and what the car wants. The critical thing to pay attention to is the lateral acceleration data. Due to the

January 2012 • www.racecar-engineering.com 61

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