# Greg Smith considers the strengths of empirical modeling

When one hears the words ‘tire modeling’, one typically thinks first of either finite element or empirical modeling. These are the two most widely used types of tire model – and with good reason; however, the basic foundations of these are fundamentally different.

The FE approach models a tire as millions of tiny cells. Each cell has been given material properties and is connected to its neighboring cells. Simulating how each of these cells interacts with each other provides a good approximation of how the overall tire behaves. In a similar way, particles of sand pouring out of a cup can provide a good approximation of how water would pour out of a cup. An empirical model is very different. An empirical model is essentially an advanced curve fitting approach whereby an equation is parameterized so that it represents the measured performance of a tire.

At its most basic, an empirical model could be a simple equation used to model the temperature of a coffee as it cools down. Whether it’s a complex tire or a simple cup of coffee, the empirical modeling process is fundamentally the same. To explain how this works, we make a cup of coffee and regularly measure its temperature over time to gather data as it cools. Then we plot the test data and observe the shape of the curve so we can select or derive an equation that we know can represent that curve, in this case a second order polynomial will work just fine.

We also need a cost function, which is a measurement of how well the equation matches the data, something like R2 or RMS (root mean square). We also need an initial guess of the equation’s coefficients. If we’ve done this before we might use numbers from a similar cup of coffee.

Finally, all this information gets fed into an optimization algorithm. Some of these algorithms can be very complex, but it is really just a form of logic as to how new coefficients for the model are selected. As the optimization algorithm runs it might try setting a coefficient to three and the cost function returns a high error; in the second iteration it sets the coefficient to four and gets a lower error, then five and gets a high error again. This tells us that the best value for that coefficient is somewhere close to four. The optimization will continue until it finds a good set of coefficients that satisfy an end criteria.

Now we have a working empirical model. This same empirical modeling process applies to tires as it does to coffee. We source some tires and test them on a rig, then load the test data into software that runs the fitting. The output is a set of coefficients that, when fed into the modeling equation, produces a curve that closely matches the tire test data. One added complexity with tires is that there are multiple input and output conditions. The coffee model’s only input was time and the only output was temperature, whereas the tire model has multiple inputs and multiple outputs. This means a more complex set of equations is required to model the more complex behavior. As a result, empirical tire models, such as the widely used Magic Formula, include dozens of coefficients required to solve their numerous equations.

Once the tire model is complete it can be evaluated at virtually any load case. However, empirical models can rarely be extrapolated beyond their tested range without losing accuracy or becoming unstable, meaning the input testing conditions determine the range at which the resulting tire model is valid. For this reason, selecting exactly which tire load cases to test is a science (and perhaps another column) in itself. The true beauty of an empirical model is the speed at which it can be evaluated. At its core, it is simply an equation being solved and computers are excellent at quick math.

An empirical tire model can typically be run and its results outputted almost instantly. This means that many load cases can be investigated quickly to build up a wide understanding of the tire’s behavior. It also means the tire model can be integrated into a full vehicle model and used to support vehicle performance assessments and driving simulator activity. A driving simulator includes a human-in-the-loop and must therefore run in real time. To support this a tire model must run with a real-time factor of less than 0.25 for four models to simultaneously run faster than real time. This is an easy feat for an empirical model and completely impossible for an FE model. 