Tire models play a crucial role in racing simulations, dictating the interaction between the vehicle and the road surface. The Pacejka tire model, also known as the Magic Formula, is an empirical model developed by Dutch engineer Hans B. Pacejka. The model is based on fitting curves to experimental data and is renowned for its ability to accurately represent a wide variety of tire behaviors with a relatively simple equation. The Pacejka model takes into account factors such as load, slip angle, and camber angle, making it excellent for capturing the overall behavior of a tire under different conditions. However, one of its limitations is that it doesn’t provide a physical explanation for why the tire behaves as it does. Here is a very basic example of the Pacejka tire model:

F=D*sin[C*arctan([1-E]x+[E/B]arctan[Bx])]

In this equation, F is the force output, x is the slip ratio or slip angle, B is the stiffness factor, C is the shape factor, D is the peak value, and E is the curvature factor. The Pacejka tire model can be much more complex, taking into account factors such as temperature, wear, load sensitivity, and transient effects. These coefficients can vary based on a number of factors, including the type of tire, the load on the tire, and the inflation pressure. The Pacejka formula takes these coefficients, along with the slip ratio/angle as inputs, and then returns the calculated tire force. The limitations of the Pacejka tire model are revealed once the peak level of grip has been reached, which under high loads such as in oval racing or drifting can cause some serious issues due to inaccurate tire behavior.

Around 2010, developers began transitioning to physically based models like the Brush tire model. This model, which forms the basis of Assetto Corsa’s tire simulation physics, describes tire behavior based on the mechanics of the tire’s interaction with the road. It represents the tire’s contact patch with the road surface as a series of flexible bristles or brushes, which bend and generate forces as the tire rolls or slides over the road.

The Brush tire model operates on a set of core assumptions to simulate the interaction between the tire and the road surface. Firstly, it assumes that the normal load has a parabolic distribution along the contact patch, with the load tapering to zero at the edges. Secondly, the friction between the bristles, representing the tire rubber, and the ground is described by the simple Coulomb model: Fx,y = µFz if Fx,y > µFz. This model states that the force (Fx,y) is equal to the friction coefficient (µ) times the vertical load (Fz) if Fx,y exceeds µFz. Here, Fx and Fy represent the longitudinal and lateral forces generated by the stretching of the bristle. This assumption allows the contact patch to be divided into an adhesive region, where the bristles stick to the road, and a sliding region, where they slip. The third assumption is that the tire carcass is considered infinitely stiff, meaning it doesn’t deform under load. Lastly, each bristle is assumed to deform independently in the longitudinal and lateral directions, allowing for a detailed simulation of the tire’s response to different forces and conditions.

The Brush tire model provides a more intuitive understanding of tire behavior because it’s based on physical principles. However, it’s also more complex and computationally intensive than the Pacejka model. The Brush model needs to consider the distribution of pressure over the contact patch, the bending and deformation of the bristles, and the transient effects as the tire’s state changes.

With the advancement of game engines and home computers, the transition from the empirical, curve-fitting approach of the Pacejka tire model to the more physically-based, mechanistic approach of the Brush tire model was a logical progression for developers. This shift from Pacejka to Brush not only allowed for a more detailed and intuitive understanding of tire behavior but also enabled the more realistic and immersive driving simulation experiences we get to enjoy today.