TIRE MODELING

Background

In today’s vehicle development, engineering an accurate representation of tire force and moment characteristics has never been more important. The tire is the link between the vehicle and the road and determines the driving characteristics that can be realized. They are critical factors for understanding and improving the performance of the vehicle. Today, there are many different tire models that can be used for tire analysis. The most widely used is the Magic Formula Tire Model by Dr. Hans Pacejka; however, empirical models, brush and ring models, and finite element models are also popular for studying tire performance.

FSAE Tire Testing Consortium

The FSAE Tire Test Consortium (FSAE TTC), located at Calspan TIRF, is a volunteer-managed organization of member schools who pool their financial resources to obtain high-quality tire force and moment data targeted for Formula SAE and Formula Student competitions. The FSAE TTC’s role is to gather funds from registering member schools, organize and conduct tire force and moment tests and make the data available to individuals at member schools. So far they have completed 8 rounds of testing with tires from Hoosier, Continental, Avon, and Goodyear. This data provides invaluable empirical knowledge about the tires which allow teams to precisely and purposefully design their car around the tires. As was mentioned previously, there have been 8 rounds of testing.

Each of these tests was conducted on the Calspan Tire Research Facility (TIRF) flat-belt tire testing machine using 120 grit paper and at a velocity of 25 mph, which is the average velocity of an FSAE car during a race event. There are static and dynamic (rolling) spring rate tests which are run in conjunction with the lateral force - moment tests. These tests were specifically designed to simulate the average driving patterns and velocity of an FSAE vehicle in the dynamic events. The specific details of each test are documented and revised for each round of testing, but the main theme has remained the same.

Spline Model

**Figure 1:** This plot shows one of several sets of polynomial splines fitted to the experimental tire data, which is used to look for outliers and capture the usable data. This specific set of data represents the tire with the vertical load FZ at 100 lbs, the pressure at 12 psi, and the camber angle at 0 deg.

The spline model was created using the TTC data and MATLAB, specifically the spline fitting toolbox. We used these splines/surfaces to cover the performance of the tire for a given set of conditions. This spline was a simple tool used to quickly evaluate the differences between tested tires. Also, this method allowed the team to employ a tire model into our lap time simulation software. To perform the spline fitting, the data was first to read into MATLAB. Then, a GUI (guided user interface) allows the user to filter the data using a selection window. The GUI is important because the TTC changes the way the data is organized with each round of testing, and with each individual test, there are warm-up runs, which do not want to be analyzed. The data was then demultiplexed into individual channels. Using the start and stop points from the user, a spline was fitted to the slip angle channel to locate the points of interest for the data analysis. To capture the usable data between the maximum and minimum values and look for outliers, a polynomial spline was fitted to the data. See figure 1. This filtered data is then sorted into data arrays by slip, camber, and load. The last part is using the cubic smoothing spline to fit the data into three-dimensional splines. See figure 2.

**Figure 2:** The cubic smoothed spline showing lateral force vs slip angle vs vertical load.

Spline Model Limitations

This model, albeit easily generated and easily implemented, is limited in the range in which it can be implemented. For instance, this model is only applicable to the discrete camber angles the TTC tests the tires, 0, 2, and 4 degrees. Additionally, the model is limited in the range of normal loads FZ where it can be applied (50lbs - 250 lbs).

Pacejka Magic Formula Tire Model

Enter the Pacejka Magic Formula Tire Model. Although there are several different models that could be used to model the tire’s performance, the Magic Formula Tire Model is known for its accurate approximation of steady-state tire characteristics and is very fast in parameter identification due to separate model parameters being used to describe force and moment characteristics.

I started the Pacejka Magic Formula modeling by simplifying the problem into its simplest form, which is the 2D non-dimensionalized model that has 6 coefficients. Since I already had the data processing nailed down from the spline technique, I implemented a brute force method to fit the Pacejka curves to the data. This presented initial coefficients which got me in the ballpark of the optimized coefficients. To determine the optimal coefficients I implemented a single objective fitting scheme in MATLAB. I attempted several of MATLAB’s optimization techniques, but the one I landed on solves nonlinear data-fitting problems in a least-squares sense. Figure 3 contains these 2D Pacejka fits.

**Figure 3:** This figure contains the 25, 2D simplified Magic Formula fits. This data represents the Hoosier R25B 18x7.5-10 on the 7” wheel.

Once I had the 2D fits taken care of, it was time to tackle the 3D fits. I took the lessons I learned from the 2D fitting and applied it to the 3D fitting. I made use of the same nonlinear data-fitting optimization technique. This yielded a surface that is very close to the cubic spline interpolated surface but can be extended past the discrete data ranges the TTC tests. Below are several figures that reflect this.

**Figure 4:** This figure represents a slightly extended version of the spline model to help show the differences between the cubic smoothed spline and the Magic Formula surface.

**Figure 5:** This figure represents an extremely extended version of the spline model to help show the advantages of the Magic Formula surface compared to the cubic smoothed spline.

Future Modifications

Moving forward, work will be done to improve the fits by reducing the error between the evaluated points and the fitted spline. Additionally, these splines will be fitted for each TTC tested 10” tire at each pressure for further evaluation and comparison. Another area of future development would be modeling tire temperature and pressure effects.

I’m very much looking forward to seeing what my teammates do with this work after it I graduate this May. Hopefully I have helped pave the way for innovation and insight in this area of the vehicle.