Tire Modelling and Selection
Please enjoy the 2-minute summary gallery below, or scroll further to explore my work in greater detail
Tires are one of the most important components of any vehicle, as they represent the only point of contact with the ground. Any force accelerating the vehicle is transmitted through the tires, making them the critical element in maintaining safety, performance and stability during vehicle operation. Thus it is of the utmost importance to understand the behavior of the tire and how it changes with various operating conditions. In this page, I will cover the steps I took to model tire behavior, and the analysis in selecting a tire for the Clemson Formula SAE vehicle.
Tire Modelling Strategy
Maximizing the performance of the tire is crucial for unlocking the performance of the car
Clemson University is part of the Formula SAE Tire Testing Consortium. This is a service provided by Calspan that grants students access to tire force and moment testing data for a range of FSAE tires, at a significantly discounted rate. Obtaining this data is critical to be able to capture and predict tire behavior. Since this data is protected intellectual property, all figures in this page will have normalized, unlabeled units.
A tire in action being tested on Calspan's testing rig
To process the data, I pull it into MATLAB. Data sets can contain different procedures sweeping through various parameters. The first step is to filter out unnecessary data and isolate only the information of interest. The variables I am most often concerned with are inflation pressure, normal load, inclination angle, tire slip, and resulting force generated.
The next step is to normalize the data for the vertical load (Fz) on the tire. Seeing the plot below on the left, it is evident that there is a significant amount of noise in the Fz channel. A key characteristic of tires is a phenomenon known as "Normal Load Sensitivity", where effective coefficient of the tire drops with increased load. This is a significant effect, and if you are trying to evaluate a tire at a specific set of conditions it is important to filter out the variation.
Noise in the normal load channel can be clearly seen here
This demonstrates the importance of correcting for normal load variation
Once the full data set was properly treated, I isolated individual slip angle sweeps, such as the sample sweep above. Each sweep captures a unique combination of load, pressure, and inclination angle. Once every sweep was isolated, they could be smoothed to filter out the remaining noise, and stitched back together to create a tire performance surface, like the ones pictured below. This process was repeated for all outputs of interest, especially lateral force, longitudinal force, and self-returning moment.
These surfaces are captured with a 4D cubic spline interpolation using the MATLAB CSAPS function. This creates a virtual lookup table that can produce a load/moment generated as a function of a slip, load and camber input. These surfaces are used in the majority of my vehicle dynamics models featured on this site. When incorporating these models, it is important to remember the key assumptions they carry with them. Namely, this performance was captured in a steady state condition, meaning that any dynamic simulations will have to be evaluated as quasi-steady state. In addition, these tire surfaces assume a constant pressure and temperature, and omit the effects of tire relaxation length. In reality, these variables are always shifting, which introduces a degree of uncertainty in the accuracy of the model. Thus, it is best to treat results as an idealized value of theoretical performance at best.
Choosing the right tire model is an exercise in understanding the goals of what you are trying to achieve and selecting your priorities accordingly.
Source: Pacejka, Hans B. “Basic Tire Modelling Considerations.” Tyre and Vehicle Dynamics, Butterworth-Heinemann, 2012, p. 85.
All things considered, CSAPS models provided 98% accuracy at a sliver of the total cost, making them the practical choice to form the foundation of tire exploration. Meaningful analysis can still be carried out as long as the primary considerations are accounted for - namely, exercising caution around the edges of the provided data range.
Recognizing the value of the Magic Formula, we waited until we had selected a tire compound to design the suspension around before devoting the resources to develop a MF model to that tire specifically, rather than wasting resources on tires we weren't going to use. My good friend Derek Moore was the first to successfully create an accurate Magic Formula fit. He too has a website that you should check out! I have worked with him for many years and he is a talented and capable designer.
The larger question at hand is, why did I select this method of representing tire performance? Many different tire models exist, and the most well known is a semi-empirical function known as the Magic Formula. Many versions exist, with varying degrees of accuracy and detail. Implementing a Magic Formula model would allow me to better capture more variables, in a robust function that yields more insight into the behavior of the tire. In addition, interpolation fits can be very computationally expensive to evaluate in a simulation, and lose their reliability once you have to extrapolate outside of the provided testing data range.
When I began my foray into tire modelling, one goal was to justify a tire compound selection for the FSAE car. This meant processing dozens of runs of tire testing data, making efficiency and accuracy the two biggest considerations. The key benefit of the CSAPS method is how fast it is. In an afternoon, I was able to process all of the data of interest for all 7 tires I was considering, giving me more time to critically compare their performance. When exploring different versions of the Magic Formula, I concluded that the time cost of developing an accurate model that captured the variables of interest would be prohibitive.
An assortment of curve fits used to create an MF5.2 Model
Tire Selection
Yeet or be yeeten
The next portion of this page is to detail the process taken to select a tire compound to use on the Clemson Formula SAE Vehicle.
Tire selection is a crucial early design decision for a Formula SAE car. There are dozens of size and compound variations available, and the final selection should work in harmony with the design goals of the vehicle.
For the Clemson team, maximum performance is a high priority, but it absolutely must not come at the expense of drive-ability. Peak performance is meaningless if the driver is unable to confidently and consistently achieve that peak day in and day out. This is reflected in our design criteria breakdown. The highest priority characteristics have a direct relation to either maximum performance or its accessibility.
For the following figures, numbers will be reincorporated for the facilitation of comparison, but to retain responsible data privacy considerations, construction and compound will be omitted.
In the following few images you will see a few of the figures generated for each tire to evaluate and compare different compounds "at a glance".
The friction vs load curve demonstrates the phenomenon of tire load sensitivity firsthand. As a race car accelerates along the track, weight is constantly transferring from one wheel to another. A shallower slope here indicates that the variation in tire performance is minimized under these conditions.
This annotated lateral force vs slip angle graph displays not only the peak force generated by the tire, but also the slip angle required to reach it. A tire that peaks sooner will not need to slide as much to reach max performance, which will help long term tire wear. In addition, we are looking for minimal "fall-off" after the peak slip angle, to ensure the tire doesn't suddenly lose grip if the driver overdrives the car.
Fixing the vertical load while varying the inclination (camber) angle helps display the tire behavior as the inclination angle to the ground increases. Different points to pay attention to are how much peak force falls off, where the peak slip angle shifts to, and the curve offset at the origin. This is an effect known as "camber thrust", and shows how much lateral force the tire is generating in pure straight-ahead running.
It is equally important to consider these graphs at loads closer to what would be seen in a cornering situation. As a vehicle accelerates around a corner, it experiences an inertial response that transfers load off of the inside wheels onto the outside wheels, loading them up more heavily. It can be seen on the chart that the outside wheel is generating the majority of the cornering load as a result.
The above charts represent results from cornering test procedures. Similar plots were made from drive/brake tests to evaluate the longitudinal performance of the tires. For each tire, all of the plots were generated for each individual tire pressure tested. From there the best pressure was selected, and the tires were compared to each other at their optimal operating pressure. How "best" was defined will be covered below.
Combining all these results, a few key summary tables were generated to allow for a quick comparison:
A handful of macro tire characteristics
Lateral force characteristics summary
Longitudinal force characteristics summary. Drive/brake data was not available for all the tires considered. However, every compound was represented at least once, allowing us to at least make decisions about the relative performance of compounds, if not construction dimensions.
These results are very useful, and from these tables alone a few key contenders start to differentiate themselves from the field. However, this does not necessarily reflect the performance of the tire in an actual driving condition. To do this, a vehicle with our specifications was simulated using a 2 degree of freedom bicycle model, like the one pictured here. This is not a complete vehicle representation, but it does provide an estimation of the slip angles achieved in a cornering scenario, and allows for a reasonable side to side comparison.
Two important modifications to the model are replacing the linear tire approximations with the CSAPS models (requiring a numerical solver) and the inclusion of aerodynamic downforce.
Bicycle model results for the vehicle navigating a 25 ft radius skid pad.
By now, a clear favorite was starting to emerge. But there was still one last consideration to explore, and that was how the tire performance evolved with wear and usage. Part of the TTC testing schedule includes repeating certain runs at the beginning and end of a test sequence, to allow a back to back comparison of fresh versus worn performance. Similar plots were generated to compare results:
Exploring the effect of tire wear on friction capacity
Camber can be seen here to have a much more drastic effect on a worn tire
Final Selection
Ultimately, the decision was to use Tire A for the development of the suspension. Firstly, it simply had the best grip all around, in almost all conditions. Drive-ability is important, but the friction advantage was too high to ignore. On top of that, it also scored very highly on metrics concerning peak drop-off and load sensitivity, which will point to more consistent and accessible performance. Cornering stiffness was the highest of all the tires, as was vertical stiffness, which will improve the resolution of suspension spring adjustments on ride characteristics.
That being said, it was not a perfect tire. It was by far the largest, heaviest option. The difference between the lightest and heaviest tire represented a potential 11 pound weight reduction, essentially for free! That too is hard to ignore. In addition, performance fell off at high camber angles, narrowing the operating window in which the tire would be effective. Lastly, peak performance came at very high slip angles, meaning the tires need to be worked harder to achieve peak performance, which will increase tire wear as well as driver performance demands.
In summary, tire A was not necessarily the most accessible, but it was by far the highest performing. Balancing the two considerations together, it was the best combination of both to develop a vehicle platform on.
Future Considerations
One significant effect not captured in the analysis above is tire temperature. Each tire has a unique operating range where it will perform the highest. The harder the tires are worked, the more they will heat up. This can come from a heavier vehicle, more power, more sliding, etc. Tire A is the biggest tire, meaning it also has the highest thermal mass. Clemson FSAE's 2019 vehicle weighed 475 pounds - for a space frame chassis powered by a four cylinder engine it could be considered a middleweight, but in context of the full FSAE field it is a heavy car, which increases the temperature generation in the tires. In a back to back comparison between Tire A and Tire E, the vehicle performed quite similarly initially, but Tire A demonstrated more consistent temperatures and performance, while Tire E overheated and fell off.
As it stands right now, our car is heavy enough that it needs more tire. But as we continue to shed weight every year, there will eventually come a point where a smaller tire might be better matched to the vehicle. This will require some amount of thermal modelling to predict tire temperatures. The most straightforward way is to model the tires as a lumped thermal mass, and use a lap simulation to estimate the energy generated at the contact patch. This model could be calibrated to the rate of temperature buildup in the TTC warm-up sweeps. Higher order effects such as roll center heights and Ackerman induced scrub would be omitted, but results could still show some promising insights.
After temperature effects are considered, the nest big topic to tackle would be tire relaxation length. Calspan carried out transient testing sweeps that can be used to calculate those values, but for the time constraints in place it was decided to leave out transient analysis. In the future, relaxation length could be modeled as a function of spatial frequency and included in step steer or frequency response analyses to evaluate the effects on response time and stability.