Hi everybody!

Once again, here we are talking about the modeling steps that can be used to build up the physics side of a car in a driving simulation software, rFactor in my case.

The car under analysis is, as I said in my previous post, an LMP2 prototype.

Although being a slower and technically less sophisticated vehicle compared, for example, to Audi R18 (on which I worked for quite some time last year to build up an rFactor model), this project has presented some bigger challenges, linked mainly to vehicle configuration and to the availability of data: this latter point meant, in fact, the possibility to dig much deeper into the details but, at the same time, asked for a more complex and precise modeling, to replicate as best as possible all the real car features. The final purpose of the work was also quite different than before, asking even more for a detailed investigation.

An LMP2 vehicle is, at a very high level, identified by a series of characteristics that influence its overall performance and its behavior quite significantly:

- very high downforce
- not equal front-to-rear weight distribution
- not equal front-to-rear downforce distribution
- very high ride height and pitch sensitivity
- different tires at the front and at the rear axle (different dimensions, different load sensitivity, different Force vs Slip curves, etc),

The above listed points, together with the availability of a pretty good amount of detailed data (including some telemetry data), have asked for a different approach in defining the first and very rough “coefficients” to be used to hit the right performance window straight from the start.

As usual, I will not go here into the detail of what each and every line in rFactor car files means and which values I assigned to each set of parameters, but I will show at high level which procedure I used to define them upfront, producing pretty satisfying results straight ahead.

Starting from the easy things, the first step has been to double check Aerodynamic drag, rolling resistance, mechanical losses and gearbox efficiency to replicate how the car accelerates and its top speed in different conditions and setups, basing on its engine power curve.

In a second step, I had to verify if and how much tire data needed to be scaled to obtain realistic cornering performances, using a simple vehicle model based on:

- Mass
- Mass distribution
- Downforce
- Vehicle geometry (tracks width, wheelbase, etc)
- CG height
- A Total Lateral Load Transfer Distribution assumption (TLLTD)

Assuming that you have accurate figures for all of the above data and that you trust all the information you got (and you don’t have reasons to do differently, also because most of the times these information are either very easy to measure or you don’t have chances to double check them, above all when we talk about aerodynamics), very often and very soon during your design process you will have to face the hard truth: tire data, this holy grail about which you have dreamt for so long and you have awaited so anxiously, are in most cases completely out of scale and, if not reworked and double checked, would (nearly always) produce completely inaccurate cornering and acceleration/braking performances.

As I already said in other posts, the first thing to consider is that most of the times you get the result of a data fit process, normally done to describe tire behavior through one of the many available versions of Pacejka Magic Formula (so no “real measured” or “raw” data).

Moreover, intrinsic tire data inaccuracy is easy to understand if you think for a second to how certain “real world” conditions cannot be properly replicated during lab testing (at least as far as I know), see for example (just to list a very few of them):

- Tire temperatures effect
- Tire wear effect
- Surface grip and its evolution with temperature and rubbing condition

But, like it or not, this is what you typically have, if you are lucky. And this is, as a consequence, what you have to work with trying, on one side, to reproduce in the most accurate way the results of the Pacejka fitting you got and, on the other side, to verify at least the magnitude of the forces through a comparison between simulated/calculated and on track logged data to avoid getting totally unrealistic cornering forces. Other finer things, like front-to-rear balance and tire curves shape will then be checked later on during the modeling and validating process.

Actually, if you have accurate enough vehicle and logged data (possibly from a session about which you which setup have been used) but no tire information, you can still create a tire model that could still be usable and that behaves closely to the real tire, but we will come to this more deeply in another post.

In a similar way to what described for Audi R18, during this preliminary tire data check I created an excel files that simulates a pure cornering condition for an LMP2 like vehicle. I built up a car model based on the available data regarding:

- Mass
- Mass distribution
- Downforce
- Vehicle geometry (tracks width, wheelbase, etc)
- CG height
- Total Lateral Load Transfer Distribution assumption (TLLTD)

What I assumed from the start here is that the car is in a steady state cornering condition where no brake and no throttle is applied. I have for this first investigation ignored tire self aligning torque and longitudinal forces.

Given a certain lateral acceleration and a certain speed (i.e. a certain Downforce), I was then able to define the vertical load acting on each tire (ignoring of course some aspects, like steering geometry, see caster).

*Tire Final Vertical Load = Static Load + Downforce +/- Weight Transfer*

Moreover, since I assumed a steady state cornering condition and I ignored for the time being the effect of tires self aligning torque, I could identify the Cornering Force that has to be produced by each axle, using the typical bicycle model approach (but here with four wheels):

*Axle Lateral Force = Mass% on Axle * Overall Lateral Force = Mass% on Axle * Overall Car Mass*Lateral Acceleration = Inner Tire Lateral Force + Outer Tire Lateral Force*

So, if a certain lateral acceleration Ay is fixed, each tire vertical load can then be calculated. With this in our hand (and assuming 0° camber on each wheel, for the time being), we can go inside each tire Magic Formula and calculate how much Cornering Force each tire could produce at a certain slip angle. At the end, Pacejka Magic Formula basically states (for pure cornering):

*Fy = f(Slip Angle, Vertical Load, Camber)*

Iterating on front and rear Slip Angle (assuming both tire of the same axle have the same Slip Angle, for the time being) we can finally find the ones that make the two axle tires to produce an overall cornering force equal to the one needed to have a steady state condition at that Ay.

We can then iterate on Ay until an equilibrium condition can still be found: at the end of this process, we will have then the maximum lateral acceleration at which the car could travel around a corner at a certain speed (or, if you prefer, the minimum corner radius the car can follow traveling at a certain speed).

Repeating this procedure at various speeds, we can finally produce a G-Speed Map, telling us how many gs the car can pull at a certain speed (or, if you want, under the action of a certain downforce).

A similar plot can be easily compared to the data you get out of a data logging file, possibly from a track where there are both low and high speed corners. Of course, when performing such a study with the aim to compare simulated vs logged data, it is important to try to reproduce with simulation real car settings as much as possible. In this case, this means mainly TLLTD, overall Downforce and its distribution front-to-rear.

The key of the process, is to then scale tire forces (Pacejka Magic Formula uses some scaling factors to do so) to finally have a match between simulated data and measured ones. I normally also take into account that I am ignoring camber effects in this early calculation, trying to stay a bit more on the lower side at the beginning to prevent the increase in cornering forces that normally camber brings.

This investigation is actually very similar to what described in Milliken and Milliken “Race Car Vehicle Dynamics” chapter 8, where the authors show their Force-Moment Method.

The idea behind their approach is to investigate car performance and reactions by keeping β (car body slip angle) constant while changing δ (front wheels steering angle) in step, then doing the opposite (δ kept constant and β changed in steps), finally building up a complete map of vehicle cornering capabilities in terms of a N (yaw moment) vs Ay plot.

The only difference is that, while the authors are investigating a complete map of vehicle performance and behavior also when the car is theoretically not in steady state conditions (so when the net Yaw moment is not equal to zero and although this “dynamic conditions” are investigated through a static simulation), we are just picturing the car in a very specific condition, more precisely when N=0 (so the points lying on the x axis of the plot above). Of course, for our purposes (an early tire data scaling) we are then interested (and, as a consequence, we are registering) only in the outer point lying on the x-Axis, where the maximum Ay in steady state conditions is reached.

On the procedure side, another small difference is that we are iterating on front and rear slip angles while Force-Moments Method (being a process that was originally developed as a constrained test of a real car) is iterating on β and δ. Referring to “Race Car Vehicle Dynamics” chapter 5 (and, in general, to the bicycle model theory) it is easy to link β and δ to front and rear Slip Angle.

From a pure computing perspective, excel is an amazing tool for this kind of studies, above all if you combine the use of normal cell formulas with some basic macros (to cover the iterative processes described above) and Excel Solver as a goal finder, to find the Slip Angles needed to have equilibrium at each and every Lateral acceleration step.

With slight modifications, such a tool could be used for a complete Force and Moments investigation or for more “vehicle dynamics related” studies.

Another great read!

By:

chris1603on August 21, 2014at 9:31 am