applied sciences Article Tire Model with Temperature Effects for Formula SAE Vehicle Diwakar Harsh 1 and Barys Shyrokau 2,* 1 Rimac Automobili d.o.o., 10431 Sveta Nedelja, Croatia; [email protected] 2 Department of Cognitive Robotics, Delft University of Technology, 2628CD Delft, The Netherlands * Correspondence: [email protected] Received: 22 October 2019; Accepted: 4 December 2019; Published: 6 December 2019 Abstract: Formula Society of Automotive Engineers (SAE) (FSAE) is a student design competition organized by SAE International (previously known as the Society of Automotive Engineers, SAE). Commonly, the student team performs a lap simulation as a point mass, bicycle or planar model of vehicle dynamics allow for the design of a top-level concept of the FSAE vehicle. However, to design different FSAE components, a full vehicle simulation is required including a comprehensive tire model. In the proposed study, the different tires of a FSAE vehicle were tested at a track to parametrize the tire based on the empirical approach commonly known as the magic formula. A thermal tire model was proposed to describe the tread, carcass, and inflation gas temperatures. The magic formula was modified to incorporate the temperature effect on the force capability of a FSAE tire to achieve higher accuracy in the simulation environment. Considering the model validation, the several maneuvers, typical for FSAE competitions, were performed. A skidpad and full lap maneuvers were chosen to simulate steady-state and transient behavior of the FSAE vehicle. The full vehicle simulation results demonstrated a high correlation to the measurement data for steady-state maneuvers and limited accuracy in highly dynamic driving. In addition, the results show that neglecting temperature in the tire model results in higher root mean square error (RMSE) of lateral acceleration and yaw rate. Keywords: tire model; tire temperature; FSAE 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 Figure1. Appl. Sci. 2019, 9, 5328; doi:10.3390/app9245328 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 5328 2 of 21 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 21 Figure 1. DUT17 Formula Society of Automotive Engin Engineerseers (FSAE) vehicle at the Formula Student Germany competition [[2].2]. To achieveachieve thethe bestbest FSAEFSAE vehiclevehicle performance,performance, the team developed various tools to evaluate vehicle dynamics and to predict its behavior. Howe However,ver, predictions are based on estimated friction coefficient, neglecting the thermal e ffect on the tire performance. Freq Frequently,uently, lap simulation is based on a point massmass oror simplifiedsimplified vehicle vehicle models models allowing allowing for for the the development development of of the the top-level top-level concept concept of theof the new new vehicle. vehicle. It canIt can quantify quantify vehicle vehicle parameters parameters such such as theas the height height of theof the center center of gravityof gravity and and lift coelift fficoecient.fficient. However, However, to design to design various various vehicle vehicle components components and, and, specifically, specifically, control control algorithms, algorithms, a full vehiclea full vehicle simulation simulation including including a comprehensive a comprehensive tire model tire model is required. is required. For an accurate modeling of tire forces and moments, a physical, semi- or empirical tire model can be used [[3].3]. The physical a approachpproach provides more insights re regardinggarding tire behavior and better represents the whole operation range; however, it lacks accuracy, specificallyspecifically consideringconsidering camber influence,influence, conicity, plysteer,plysteer, andand otherother phenomenaphenomena [[4,5].4,5]. The empirical approachapproach provides a higher modeling accuracy in in the the predefined predefined tested tested range range bu butt requires requires extensive extensive special special tests. tests. According According to to[6,7] [6 ,7the] the tire tire temperature, temperature, especially especially for for racing racing application application [8], [8], has has a a significant significant e effffectect on on the force capability of the tire. Several advanced tire modelsmodels using the brush element approach incorporated with thermalthermal eeffectsffects have have been been proposed proposed [9 –[9–11];11]; however, however, their their parametrization parametrization requires requires intensive intensive test sessions,test sessions, commonly commonly unfeasible unfeasible for FS teams.for FS For team a FSAEs. For vehicle, a FSAE less vehicle, complex less models complex incorporating models temperatureincorporating e fftemperatureect are proposed effectusing are proposed the physical using approach the physical based approach on the brushbased modelon the [brush12,13]. model They demonstrate[12,13]. They a demonstrate good correlation a good with correlation the experimental with the tire dataexperimental collected usingtire data the indoorcollected flat using track tirethe testindoor machine flat track [14 ].tire Also, test themachine proposed [14]. physical-basedAlso, the proposed models physical-based were compared models to thewere original compared magic to formulathe original demonstrating magic formula close demonstrating accuracy; however, close accuracy; original magic however, formula original does magic not include formula the does effect not of temperatureinclude the effect on the of forcetemperature capability on of the the force tire. capability of the tire. Thus, the goal of the proposed study was to develop an empirical tire modelmodel incorporatingincorporating thermal eeffectsffects and and to to evaluate evaluate its its performance performance compared compared to the to experimentalthe experimental measurements. measurements. The main The contributionmain contribution of the of study the study is the is improvement the improvement of the of accuracy the accuracy of the of empiricalthe empirical tire modeltire model using using the proposedthe proposed thermal thermal model model andmodification and modification of the magicof the formulamagic formula to incorporate to incorporate temperature temperature effects. effects.The paper is structured as follows. Section2 describes the experimental setup, test program, and experimentalThe paper is structured results. A thermalas follows. tire Section model 2 is desc proposedribes the in experimental Section3 discussing setup, thetest basicprogram, thermal and equationsexperimental and results. comparing A thermal them totire well-established model is proposed thermal in Section models. 3 Sectiondiscussing4 presents the basic the thermal widely usedequations magic and formula comparing with the them extension to well-established related to the thermal temperature models. effect. Section The complete 4 presents vehicle the widely model includingused magic thermal formula and with modified the extens magicion formularelated to is the presented temperature in Section effect.5; itThe includes complete the vehicle whole modelmodel simulationincluding thermal for steady-state and modified and transient magic formula maneuvers is presented compared in to Section real tests 5; it and includes simulation the whole without model the temperaturesimulation for eff steady-stateect. The paper and concludes transient withmaneuver a discussions compared of the to work real andtests the and recommendations simulation without for futurethe temperature research presented effect. The in paper Section concludes6. with a discussion of the work and the recommendations for future research presented in Section 6. 2. Experimental Setup and Results 2.1. Setup and Test Program Appl. Sci. 2019, 9, 5328 3 of 21 2. Experimental Setup and Results 2.1. Setup and Test Program Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 21 The experimental tests were performed at Dynamic Test Center AG [15] located in Vaufellin, Switzerland.The experimental The test program tests were is shownperformed in Table at Dynamic1 covering Test uncombined Center AG [15] slip located testing in using Vaufellin, sweep testsSwitzerland. of slip angle Theα testand program wheel slipis shownκ under in Table various 1 covering normal loadsuncombinedFz for threeslip testing tires: using Apollo sweep Single Compound,tests of slip Apollo angle Doubleα and wheel Compound, slip κ under and Hoosier. various normal loads Fz for three tires: Apollo Single Compound, Apollo Double Compound, and Hoosier. Table 1. Tire test program. Table 1. Tire test program. Measurement Type α (rad) κ (–) Fz (N) Measurement type α (rad) κ (–) Fz (N) Pure corneringPure cornering sweepsweep