# Tire Model with Temperature Effects for Formula SAE Vehicle
## 1 Introduction  
Formula SAE (FSAE) also known as formula student (FS) is a competition in which students design a single seat formula race car to compete against other FS teams from all over the world [1]. The competition motivates the students for innovative solutions to demonstrate their engineering talent and obtain new skills. It also allows the students to apply theoretical knowledge into practice in a dynamic and competitive environment. Since 2001 students have joined Formula Student Team Delft [2] on an annual basis to participate in the FSAE competition, earlier in the combustion class and electric class from 2011. Aiming to improve the FSAE vehicle performance, the team collaborated with Apollo Vredestein B.V. (Enschede, The Netherlands) to develop original tires. The FSAE vehicle (2017 version) from FS Team Delft equipped with such tires is shown in Figure 1.  
To achieve the best FSAE vehicle performance, the team developed various tools to evaluate vehicle dynamics and to predict its behavior. However, predictions are based on estimated friction coefficient, neglecting the thermal effect on the tire performance. Frequently, lap simulation is based on a point mass or simplified vehicle models allowing for the development of the top-level concept of the new vehicle. It can quantify vehicle parameters such as the height of the center of gravity and lift coefficient. However, to design various vehicle components and, specifically, control algorithms, a full vehicle simulation including a comprehensive tire model is required. For an accurate modeling of tire forces and moments, a physical, semi- or empirical tire model can be used [3]. The physical approach provides more insights regarding tire behavior and better represents the whole operation range; however, it lacks accuracy, specifically considering camber influence, conicity, physteer, and other phenomena [4; 5]. The empirical approach provides a higher modeling accuracy in the predefined tested range but requires extensive special tests. According to [6; 7] the tire temperature, especially for racing application [8], has a significant effect on the force capability of the tire. Several advanced tire models using the brush element approach incorporated with thermal effects have been proposed [9; 10; 11]; however, their parametrization requires intensive test sessions, commonly unfeasible for FS teams. For a FSAE vehicle, less complex models incorporating temperature effect are proposed using the physical approach based on the brush model [12; 13]. They demonstrate a good correlation with the experimental tire data collected using the indoor flat track tire test machine [14]. Also, the proposed physical-based models were compared to the original magic formula demonstrating close accuracy; however, original magic formula does not include the effect of temperature on the force capability of the tire. Thus, the goal of the proposed study was to develop an empirical tire model incorporating thermal effects and to evaluate its performance compared to the experimental measurements. The main contribution of the study is the improvement of the accuracy of the empirical tire model using the proposed thermal model and modification of the magic formula to incorporate temperature effects. The paper is structured as follows. Section 2 describes the experimental setup, test program, and experimental results. A thermal tire model is proposed in Section 3 discussing the basic thermal equations and comparing them to well-established thermal models. Section 4 presents the widely used magic formula with the extension related to the temperature effect. The complete vehicle model including thermal and modified magic formula is presented in Section 5; it includes the whole model simulation for steady-state and transient maneuvers compared to real tests and simulation without the temperature effect. The paper concludes with a discussion of the work and the recommendations for future research presented in Section 6.