Tire Model With Temperature Effects For Formula SAE Vehicle
applied sciences Article Tire Model with Temperature Effects for Formula SAE Vehicle Diwakar Harsh 1 and Barys Shyrokau 2, * 1 2 * Rimac Automobili d.o.o., 10431 Sveta Nedelja, Croatia; diwakar.harsh@rimac-automobili.com Department of Cognitive Robotics, Delft University of Technology, 2628CD Delft, The Netherlands Correspondence: b.shyrokau@tudelft.nl 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 Figure 1. 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. 1. DUT17 Automotive Engineers the Formula Formula Student Student Figure DUT17 Formula Formula Society Society of of Automotive Engineers (FSAE) (FSAE) vehicle vehicle at at the Germany competition [2]. Germany competition [2]. achieve the the best best FSAE FSAE vehicle vehicle performance, performance, the team developed various tools to evaluate To achieve vehicle dynamics and to predict its behavior. However, predictions are based on estimated friction coefficient, neglecting the thermal effect coefficient, effect on the tire performance. Frequently, lap simulation is based on a point point mass mass or orsimplified simplifiedvehicle vehiclemodels modelsallowing allowingfor forthe thedevelopment development top-level concept ofof thethe top-level concept of of the new vehicle. It can quantify vehicle parameters such as the height of the center of gravity the new vehicle. It can quantify vehicle parameters such as the height of the center of gravity andand lift lift coefficient. However, to design various vehicle components specifically, control algorithms, coefficient. However, to design various vehicle components and, and, specifically, control algorithms, a full a full vehicle simulation including a comprehensive tire model is required. 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 The physical physical approach approach provides provides more insights regarding tire behavior and better represents the whole operation range; however, it lacks accuracy, specifically considering camber and other other phenomena phenomena [4,5]. [4,5]. The empirical approach provides a higher influence, conicity, plysteer, and modeling accuracy in the predefined tested range but According to accuracy in the predefined tested range butrequires requiresextensive extensivespecial specialtests. tests. According [6,7] thethe tire temperature, especially for to [6,7] tire temperature, especially forracing racingapplication application[8], [8],has hasaasignificant significanteffect effect on on the force capability of the tire. Several advanced tire models using the brush element approach incorporated thermaleffects effectshave havebeen beenproposed proposed [9–11]; however, their parametrization requires intensive with thermal [9–11]; however, their parametrization requires intensive test test sessions, commonly unfeasible for FSFor teams. Forvehicle, a FSAE complex models sessions, commonly unfeasible for FS teams. a FSAE lessvehicle, complexless models incorporating incorporatingeffect temperature effectusing are proposed usingapproach the physical approach basedmodel on the [12,13]. brush model temperature are proposed the physical based on the brush They [12,13]. Theya demonstrate a good with the experimental tire data collected the demonstrate good correlation withcorrelation the experimental tire data collected using the indoor flatusing track tire indoor flat track test the machine [14]. physical-based Also, the proposed physical-based models were compared to test machine [14].tire Also, proposed models were compared to the original magic the original magic formula demonstrating close accuracy; however, original magic formula does not formula demonstrating close accuracy; however, original magic formula does not include the effect of include the effect offorce temperature onofthe temperature on the capability theforce tire.capability of the tire. model incorporating incorporating Thus, the goal of the proposed study was to develop an empirical tire model effectsand andtotoevaluate evaluate performance compared to experimental the experimental measurements. The thermal effects itsits performance compared to the measurements. The main main contribution the study the improvement the accuracy ofempirical the empirical tire model using contribution of theofstudy is theisimprovement of theofaccuracy of the tire model using the the proposed thermal model and modification of the formula magic formula to incorporate temperature proposed thermal model and modification of the magic to incorporate temperature effects. effects. The paper is structured as follows. Section 2 describes the experimental setup, test program, The paper is structured as follows. setup, test and and experimental results. A thermal tireSection model 2isdescribes proposedthe in experimental Section 3 discussing the program, basic thermal 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 equations comparing them to well-established models. Section 4 presents the widely used magicand formula with the extension related to the thermal temperature effect. The complete vehicle model used magic formula with the extension related to temperature effect. complete including thermal and modified magic formula is the presented in Section 5; The it includes thevehicle whole model including thermal and modified magic formula is presented into Section 5; itand includes the whole model simulation for steady-state and transient maneuvers compared real tests simulation without the simulation for steady-state and transientwith maneuvers compared real and teststhe and simulation without temperature effect. The paper concludes a discussion of theto work recommendations for the temperature effect. Theinpaper concludes with a discussion of the work and the recommendations future research presented Section 6. 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, The experimental tests were performed at Dynamic Testuncombined Center AG [15] Vaufellin, Switzerland. The test program is shown in Table 1 covering sliplocated testinginusing sweep Switzerland. Theαtest is shown in Table 1 covering uncombined slip testing sweep tests of slip angle andprogram wheel slip κ under various normal loads Fz for three tires: using Apollo Single tests of slip angle α and wheel slip κ under various normal loads F z for three tires: Apollo Single Compound, Apollo Double Compound, and Hoosier. 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 cornering sweep Pure cornering sweep 0 0 600/1000 600/1000 Pure acceleration/braking 0 sweep Pure acceleration/braking 0 sweep The measurementswere wereperformed performedoutdoor outdoor using using aa test tire forces The measurements test truck truck(Figure (Figure2). 2).ToTomeasure measure tire forces and moments, a Kistler wheel force transducer [16] was used. The measurement accuracy of the and moments, a Kistler wheel force transducer [16] was used. The measurement accuracy of the wheel wheel force transducer is 1% of the full scale. For the investigated load range, it resulted in approx. force transducer is 1% of the full scale. For the investigated load range, it resulted in approx. 5% 5% absolute measurement error including crosstalk. The normal load during the test was controlled absolute measurement error including crosstalk. The normal load during the test was controlled using using hydraulic suspension. The target normal load was obtained by the adjustment of hydraulic hydraulic suspension. The target normal load was obtained by the adjustment of hydraulic pressure. pressure. The wheel was driven through the axle unit from the motor for acceleration tests and using The wheel was driven through the axle unit from the motor for acceleration tests and using a reverse a reverse gear for braking. Another actuator was mounted on the tie-rod in order to generate the tire gear for braking. Another actuator was mounted on the tie-rod in order to generate the tire slip angle. slip angle. Figure 2. Test setup. Figure 2. Test setup. Pure Cornering Results 2.2.2.2. Pure Cornering Results Tire lateral performancewas wasassessed assessed by by performing performing sweep Tire lateral performance sweep tests testsatataaconstant constantroad roadwheel wheel steering rate of 15 /s. The tests were conducted under various normal loads F z of 600 and 1000 N steering rate of 15 /s. The tests were conducted under various normal loads Fz of 600 and 1000 N correspondingly. During thetests, tests,the thevertical verticalforce force was was fluctuated and other correspondingly. During the fluctuateddue duetotoroad roadmodulation modulation and other irregularities, which as a result affected the lateral force F y measurement. To obtain the lateral force, irregularities, which as a result affected the lateral force Fy measurement. To obtain the lateral force, this influence was compensated using the normalized ratio between lateral and normal forces and this influence was compensated using the normalized ratio between lateral and normal forces and then then multiplying by mean vertical load. Figure 3a shows the effect of vertical load on the utilized multiplying by mean vertical load. Figure 3a shows the effect of vertical load on the utilized friction friction coefficient µ. The results are aligned to the load sensitivity phenomena that utilized friction coefficient µ. The results are aligned to the load sensitivity phenomena that utilized friction coefficient coefficient reduces as normal load increases. The test results of Hoosier and Apollo tires (double reduces as normal load increases. The test results of Hoosier and Apollo tires (double compound) are compound) are compared in Figure 3b. The peak utilized friction for the Hoosier tire was around 5% compared in Figure 3b. The peak utilized friction for the Hoosier tire was around 5% higher than the higher than the Apollo one; however, the average utilized friction was in the same range. Apollo one; however, the average utilized friction was in the same range.
Appl. Sci. 2019, 9, x FOR PEER REVIEW Appl. Sci. 2019, 9, x FOR PEER REVIEW Appl. Sci. 2019, 9, 5328 (a) (a) 4 of 21 4 of 21 4 of 21 (b) (b) Figure 3. Normalized lateral friction µ vs. slip angle α: (a) different normal loads Fz (Apollo); (b) Figure 3. Normalized lateralfriction frictionµµvs. vs. slip normal loads Fz (Apollo); (b) Figure 3. Normalized lateral slip angle angleα:α:(a)(a)different different normal loads Fz (Apollo); different tire types (Fz 600 N). different tire types (F z 600 N). (b) different tire types (Fz 600 N). Acceleration Braking Results 2.3. Pure Pure Acceleration and Braking Results 2.3. 2.3. Pure Acceleration andand Braking Results For the longitudinal force measurement, the wheel was lifted, accelerated, and pressed onto For longitudinal the longitudinal force measurement, wheelwas waslifted, lifted,accelerated, accelerated, and and then For the force measurement, thethe wheel then pressed pressedonto onto the The brake torque was applied realize sweep tests of wheel slip while forward the ground. ground. brake torque was appliedtoto torealize realizesweep sweeptests tests of of wheel wheel slip slip while while the the the ground. TheThe brake torque was applied the forward forward velocity of truck was kept constant. The effect normal load on utilized friction much less velocity of the the truck kept constant. Theeffect effectofof ofnormal normalload loadon on utilized utilized friction friction is is velocity of the truck waswas kept constant. The is much much less less compared to lateral friction (Figure 4a). The performance of the Apollo tire was similar compared to compared to lateral friction (Figure 4a). The performance of the Apollo tire was similar compared to compared to lateral friction (Figure 4a). The performance of the Apollo tire was similar compared to the Hoosier one (Figure 4b). The difference was found to be 3% which can be related to the ambient the Hoosier one (Figure 4b). The difference was found to be 3% which can be related to the ambient the Hoosier one (Figure 4b). The difference was found to be 3% which can be related to the ambient and track conditions. conditions. and and tracktrack conditions. (a) (a) (b) (b) Figure 4. Normalized longitudinal friction µ vs.wheel wheel slipκ:κ:(a) (a) different normal normal loads F z (Apollo); Figure 4. Normalized longitudinal friction µ vs. (Apollo); Figure 4. Normalized longitudinal friction µ vs. wheelslip slip κ: (a)different different normal loads loads F Fzz (Apollo); z 600 N). (b) different tire types (F (b) different tire types (Fz N).N). z 600 (b) different tire types (F600 Related to Effect 2.4. 2.4. Results Related to Temperature Effect 2.4. Results Results Related to Temperature Temperature Effect Both the force capability and lifetime depend on tire temperature. Since BothBoth the the force capability andand thethe tiretire lifetime depend the tire tirecarcass carcass force capability the tire lifetime dependon ontire tiretemperature. temperature. Since Since the the tire carcass an elastic element, the temperature change will result in the modulus of elasticity of the rubber and is anis elastic element, the temperature change will result in the modulus of elasticity of the rubber is an elastic element, the temperature change will result in the modulus of elasticity of the rubber and influence the cornering stiffness [17]. The effect of temperature on the normalized and therefore therefore influence the cornering stiffness [17]. The effect of temperature on the normalized therefore influence the cornering stiffness [17]. The effect of temperature on the normalized longitudinal lateral friction is Figure can be observed that the longitudinal andand lateral friction is shown inin Figure 5.5. temperature affects longitudinal and lateral friction is shown shown in Figure 5.ItIt Itcan canbe beobserved observed that that the the temperature temperature affects affects the stiffness (curve slope) and the peak friction. the stiffness (curve slope) andand thethe peak friction. the stiffness (curve slope) peak friction.
Appl. Sci. 2019, 9, x FOR PEER REVIEW Appl. Sci. 2019, 9, 5328 5 of 21 5 of 21 (a) (b) Figure Temperature effectsonontire tireforces forcesfor forApollo Apollotire tire (F (Fzz 1000 Figure 5. 5. Temperature effects 1000N): N):(a) (a)normalized normalizedlongitudinal longitudinal friction µ vs. wheel slip κ; (b) normalized lateral friction µ vs. slip angle α. friction µ vs. wheel slip κ; (b) normalized lateral friction µ vs. slip angle α. 3. Thermal Model 3. Thermal Model 3.1.3.1. Proposed Thermal Proposed ThermalModel Model A lumped parameter [18] and andthe thethermal thermalmodel modelconsists consists three A lumped parameterapproach approachisisused usedaccording according to [18] of of three bodies such as as thethe tread with temperature temperature TTcarcass , and inflation bodies such tread with temperatureTtread Ttread,, the the carcass carcass with temperature , and thethe inflation carcass with temperature Tgas Theroad road surface surface with ambient with gasgas with temperature Tgas . .The ambientair airprovides providesthe theboundary boundaryconditions conditions with fixed temperaturesTT road and and Tamb . .InIn contrast to the study [19],[19], a single valuevalue of tread temperature was fixed temperatures contrast to the study a single of tread temperature road amb without the need toto define a acomplex temperature distribution across thethe contact patch wascalculated calculated without the need define complex temperature distribution across contact patch area. The temperature change of tread, carcass, and gas is described as: area. The temperature change of tread, carcass, and gas is described as: . Ttread Ttread Qsliding Qtread road Qcarcass tread Qtread amb Qsliding Qtread road Qcarcass tread Qtread amb Mtread SStread treadM tread . Qdamp Qcarcass tread Qcarcass amb Qcarcass gas Tcarcass Qdamp Qcarcass Stread Q Qcarcass gas carcass Mcarcass carcass amb . T carcass Qcarcass gas Scarcass M carcass T gas S gas M gas (1) (1) Qcarcass gas T gasdue to sliding; Q where Qsliding is the heat flow tread road is the heat flow between tread and road; S gas M gas Qcarcass tread is the heat flow between carcass and tread; Qtread amb is the heat flow between tread and is the heatheat flowflow due due to sliding; heat flow between tread and road; where air; Qsliding Qtreaddeflection; ambient Qdamp is the to carcass road is the Q carcass amb is the heat flow between is the heat flow between carcass and tread; is the heat between Qcarcassand Q carcass ambient air; Q is the heat flow between carcass and flow inflation gas; tread Stread and is the tread tread amb carcass gas specific heat capacity of carcass; Scarcass the specific heat capacity the specific heat ambient air; Qdamp is the heat flow due toiscarcass deflection; the heat Sflow between carcass Qcarcass ambofiscarcass; gas is capacity of inflation gas; M is the tread mass; M is the carcass mass; M is the inflation and ambient air; Qcarcass gas tread is the heat flow between carcass the specific heat carcass and inflation gas; Stread isgas gascapacity mass. of carcass; Scarcass is the specific heat capacity of carcass; Sgas is the specific heat capacity of According totread [18], twotread heatmass; generation should inflation gas; M is the Mcarcass processes is the carcass mass;beMconsidered: gas is the inflation gas mass. According Due to carcass deflection to [18], two heat generation processes should be considered: Due to carcass deflection Qdamp Ex Fx E y F y Ez Fz Vx ( ) (2) Qdamp Ex Fx E y Fy Ez Fz Vx (2) where Ex , Ey , and Ez are the carcass longitudinal, lateral, and vertical force efficiency factors; Fx , Fy , andwhere Fz areEthe lateral, and normal forces; Vx and is thevertical longitudinal velocity. factors; Fx, Fy, x, Eylongitudinal, , and Ez are the carcass longitudinal, lateral, force efficiency FzDue to sliding frictionlateral, in theand contact patch and are the longitudinal, normal forces; Vx is the longitudinal velocity. Due to sliding friction in the contact patch Qsliding µd Fz vs (3) Qsliding μ d Fz vs (3) where µd is the dynamic friction coefficient; vs is the sliding velocity. To define the shift along the µ axis with compound temperature, the dynamic friction model [18] was modified. Based on the friction model [20], Equation (4) is proposed incorporating the shift along
Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 21 Appl. Sci. 2019, 9, 5328 6 of 21 where µd is the dynamic friction coefficient; vs is the sliding velocity. To define the shift along the µ axis with compound temperature, the dynamic friction model [18] was modified. Based on the friction [20], Equation is proposed incorporating the µ axis due to temperature using model the parameters µ and(4) h as temperature dependent.the Theshift ideaalong was the µ axis due to temperature using the parameters µ and h as temperature dependent. The idea was taken from research [21] where the parameters of the friction models from [20] and [22] were made taken from research [21] The where theequation parameters of the friction models fromis[20] and [22] were made temperature-dependent. final for the dynamic friction model described: temperature-dependent. The final equation for the dynamic friction model is described: h i 2 ( vs ) Kshi f t (Ttread Tre2f ))) µd (T ) µbase µpeak (T ) µbase e (h(T)(log (4) 10 Vmax v h (T ) log10 s K shift (Ttread Tref ) (4) Vmax μ d (T ) μbase μ peak (T ) μbase e The temperature-dependent peak friction coefficient µpeak is defined: The temperature-dependent peak friction coefficient µpeak is defined: µpeak (T ) a1 T2 a2 T a3 (5) μ peak (T ) a1T 2 a2T a3 (5) where a1 , a2 , and a3 are the tuning parameters for a second order polynomial function. where a1, a2, and a3 are the tuning parameters for a second order polynomial function. Dimensionless parameter h is related to the width of the speed range in which the friction Dimensionless parameter h is related to the width of the speed range in which the friction coefficient varies significantly [23]. The following temperature-dependent adjustment is made similar coefficient varies significantly [23]. The following temperature-dependent adjustment is made similar to the adaptation of the compound shear modulus in [18]: to the adaptation of the compound shear modulus in [18]: b T Ttread e 2 b2tread h(Th)(T ) b 1b1 be T b e 2e 2reTreff (6) (6) where b11 and b22 are are the the tuning tuning parameters. parameters. The proposed modifications cover both sliding and non-sliding friction components. components. Temperature-dependent peak friction friction coefficient coefficientµµpeak corresponds to the peak friction properties corresponds to the peak friction properties of peak of compound. The parameter h affectsthe thecurve curveslope slopeand andresults resultsin inthe the change change of of the shear thethe tiretire compound. The parameter h affects modulus the lower lower limit limit of of friction friction coefficient coefficient modulus of the tire tire compound. compound. The peak friction friction coefficient coefficientµµpeak peak,, the µ , and the parameter h are identified from measurement data. The simulation and experimental µbase base, and the parameter h experimental results for the obtained normalized normalized lateral lateral friction friction coefficient coefficient are are shown shown in in Figure Figure 6. 6. The model shows . similar qualitative behavior observed in the the experimental experimental data data at at the the slip slip angle angle above above 44 . Figure 6. Comparison modified friction friction model. model. Figure 6. Comparison between between measurement measurement data data and and modified heat flow between tread andand ambient air; ii) Four types types of of heat heatflows flowsare areconsidered: considered:i) (i) heat flow between tread ambient air;heat (ii) flow heat between carcass and ambient air; iii) heat flow between tread and carcass; iv) heat flow between flow between carcass and ambient air; (iii) heat flow between tread and carcass; (iv) heat flow gas. carcass and inflation gas. Heat flow with the ambient air Qtread amb and Qcarcass amb is defined as:
Appl. Sci. 2019, 9, 5328 7 of 21 Heat flow with the ambient air Qtread amb and Qcarcass amb is defined as: Qtread amb Htread amb (T tread Tamb ) Qcarcass amb Hcarcass amb Tcarcass T gas (7) The heat flow coefficient Hcarcass amb is constant and the heat flow coefficient Htread-amb is assumed to be a function of the longitudinal vehicle speed Vx and taken as a linear function [18]: Htread amb 2Vx 10 (8) Heat flow between the tread and carcass Qcarcass tread is calculated as: Qcarcass tread Hcarcass tread (Tcarcass Ttread ) (9) Heat flow between the carcass and inflation gas Qcarcass gas is defined as: Qcarcass gas Hcarcass gas Tcarcass T gas (10) Using heat flow coefficient Htread road , heat flow between tread and road Qtread road can be found as: Qtread road Htread road Acp (Ttread Troad ) (11) The contact patch area, Acp , is calculated assuming a constant width of the contact patch b and the variable contact patch length a which was adjusted compared to [18] to better match the data of the FSAE tires: Fz 0.7 Acp 2ab 0.12p 0.7 b (12) bar 3000 Since the volume of the gas is constant, the pressure is directly proportional to the Kelvin temperature and thus the inflation pressure pbar can be calculated as: pbar pcold T gas 273 Tamb 273 (13) where pcold is the pressure of the tire at ambient temperature. 3.2. Comparison with the Established Models The Sorniotti model [12] describes an empirical model to estimate tire temperature as the function of the actual working conditions of the component. To evaluate the temperature effect on tire forces, a combination of the estimated temperature with a tire brush model [24] was used. The model was empirically tuned using experimental data to demonstrate the variation of tire performance as temperature function. The thermal model considers distinct thermal capacities related to the tread and carcass. The tread thermal capacity is related to the heat flux caused by the tire–road forces and carcass thermal capacity is affected by rolling resistance and exchanges heat with the external ambient. Other heat fluxes corresponded to the ambient and exchange between the two capacities. The original model [12] did not consider the heat flow between the road and tread. Therefore, the heat flow term Ptread,road was introduced to the original model to improve the accuracy according to [25]. The model is described as: Ceq carcass dTcarcass Prolling resis tan ce Pconduction Pambient,carcass dt Ceq tread dTtread dt PFx,tire PFy,tire Pconduction Pambient,tread Ptread,road (14)
Appl. Sci. 2019, 9, 5328 8 of 21 Power fluxes corresponding to the cooling flux due to the temperature difference between carcass and ambient Pambient,carcass , and tire tread and ambient Pambient,tread are defined as: Pambient,carcass hcarcass (Tambient Tcarcass ) Pambient,tread htread (Tambient Ttread ) (15) Power fluxes related to conduction between tread and carcass Pconduction and between tread and road Ptread,road are calculated as: Pconduction hconduction (Ttread Tcarcass ) Ptread,road Htread road (Ttread Troad ) (16) Compared to the Sorniotti model, the proposed thermal model due to a higher differential order should capture more dynamics related to heat flow and potentially produce better results. The Kelly and Sharp model [18] states that in racing applications, the temperature of the tread significantly affects both the tire stiffness and the contact patch friction. It should be noted that the effect on the friction is higher. Since the rubber viscoelastic properties depend on temperature, the maximum performance on the racetrack is only available in a specific temperature range. The tire model is also based on the brush model. Using the bristle stiffness cp the adhesion part can be presented: wcp Gtread cp (17) htread where wcp is the contact patch width; htread is the tread height. The shear modulus of the tread Gtread is defined as: Gtread log(GTA Glimit ) log(GTB Glimit ) GTA Glimit KG Ttread Glimit , where KG e K T G GA TGB TGA e (18) where GTA is the reference shear modulus at temperature A; GTB is the reference shear modulus at temperature B; Glimit is the limit shear modulus value
applied sciences Article Tire Model with Temperature E ects for Formula SAE Vehicle Diwakar Harsh 1 and Barys Shyrokau 2,* 1 Rimac Automobili d.o.o., 10431 Sveta Nedelja, Croatia; diwakar.harsh@rimac-automobili.com 2 Department of Cognitive Robotics, Delft University of Technology, 2628CD Delft, The Netherlands * Correspondence: b.shyrokau@tudelft.nl Received: 22 October 2019; Accepted: 4 .
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