Study on the Stability Control of Vehicle Tire Blowout Based on Run-Flat Tire

Article Study on the Stability Control of Vehicle Tire Blowout Based on Run-Flat Tire

Xingyu Wang 1 , Liguo Zang 1,*, Zhi Wang 1, Fen Lin 2 and Zhendong Zhao 1

1 Nanjing Institute of Technology, School of Automobile and Rail Transportation, Nanjing 211167, China; [email protected] (X.W.); [email protected] (Z.W.); [email protected] (Z.Z.) 2 College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; ﬂ[email protected] * Correspondence: [email protected]

Abstract: In order to study the stability of a vehicle with inserts supporting run-flat tires after blowout, a run-flat tire model suitable for the conditions of a blowout tire was established. The unsteady nonlinear tire foundation model was constructed using Simulink, and the model was modified according to the discrete curve of tire mechanical properties under steady conditions. The improved tire blowout model was imported into the Carsim vehicle model to complete the construction of the co-simulation platform. CarSim was used to simulate the tire blowout of front and rear wheels under straight driving condition, and the control strategy of differential braking was adopted. The results show that the improved run-flat tire model can be applied to tire blowout simulation, and the performance of inserts supporting run-flat tires is superior to that of normal tires after tire blowout. This study has reference significance for run-flat tire performance optimization.

Keywords: run-ﬂat tire; tire blowout; nonlinear; modiﬁed model

Citation: Wang, X.; Zang, L.; Wang, Z.; Lin, F.; Zhao, Z. Study on the Stability Control of Vehicle Tire Blowout Based on Run-Flat Tire. 1. Introduction World Electr. Veh. J. 2021, 12, 128. As an important part of the vehicle driving system, tires are the only direct contact https://doi.org/10.3390/ part between the vehicle and the road surface. The main functions are to support the wevj12030128 vehicle, mitigate the impact of the road surface, and generate braking force, which have an important impact on comfort and other aspects of performance of the vehicle [1]. According Academic Editor: Joeri Van Mierlo to statistics, 46% of traffic accidents on expressways are caused by tire failures, and a flat tire alone accounts for 70% of the total tire accidents [2]. Received: 28 July 2021 How to improve the stability of vehicles after tire blowout has always been the core Accepted: 20 August 2021 problem of domestic and foreign scholars. CHEN set up an estimated calculation model Published: 21 August 2021 of additional yaw torque leading by tire blowout based on Dugoff tire model [3]. LI proposed a modified linear time-varying model predictive control (LTV-MPC) method Publisher’s Note: MDPI stays neutral based on the Pacejka tire model, and the stability region of the vehicle with active front with regard to jurisdictional claims in steering was expanded [4]. LIU established an extended three-degree of freedom model published maps and institutional affil- considering longitudinal velocity to weaken the influence of the steering wheel angle under iations. extreme conditions [5]. CHEN proposed a comprehensive coordinated control scheme for longitudinal and lateral stability based on the brush tire model. Speed tracking under the limit speed is achieved by means of longitudinal acceleration feedforward and state feedback controller [6]. GUO established a tire model suitable for operating coach and Copyright: © 2021 by the authors. estimated the sideslip angle, as well as the yaw rate of the operating coach in real time Licensee MDPI, Basel, Switzerland. based on the extended Kalman filter state estimator [7]. This article is an open access article In addition, some scholars have done a lot of research on the stability control after tire distributed under the terms and blowout. LIU proposed a fuzzy sliding mode control algorithm based on the current tire conditions of the Creative Commons Attribution (CC BY) license (https:// blowout control algorithm, and the tire blowout model is established using the UniTire creativecommons.org/licenses/by/ model [8]. JING established a linear variable parameter vehicle model considering the 4.0/). time-varying speed and the uncertainty of tire characteristics [9]. WANG proposed a

World Electr. Veh. J. 2021, 12, 128. https://doi.org/10.3390/wevj12030128 https://www.mdpi.com/journal/wevj World Electr. Veh. J. 2021, 12, 128 2 of 13

coordinated control approach based on active front steering and differential braking. The model predictive control method was adopted to control the front wheel angle for tracking the trajectory, and the differential brake control was adopted to offer an inner-loop control input and improve the lateral stability [10]. CHEN established the kinematics equation of the vertical load of the vehicle after tire blowout according to the vehicle dynamics model and the displacement mutation in the vertical direction of the tire wheel center [11]. ERLIEN found that the mandatory implementation of stability constraints may conflict with the expected collision avoidance trajectory and proposed that the stability controller should allow the vehicle to run outside the stability constraints to achieve safe collision avoidance [12]. WANG proposed a novel linearized decoupling control procedure with three design steps for a class of second order multi-input-multi-output non-affine system [13]. YANG proposed a composite stability control strategy for tire blowout electric vehicle with explicit consideration of vertical load redistribution [14]. CHOI designed a layered lateral stability controller based on LTV-MPC. The nonlinear characteristics of the tire are reflected in the extended “bicycle” model by continuously linearizing the tire force [15]. WANG proposed a nonlinear coordinated motion controller in the framework of the triple-step method to solve the path-following and safety problem of road vehicles after a tire blowout [16]. However, the tire blowout condition of the vehicle equipped with run-flat tire is more complicated, and there is no model that can be directly applied to inserts supporting run-flat tire under tire blowout condition. In addition, the inserts supporting run-flat tire is a typical run-flat tire based on the pneumatic tire structure, which consists of an auxiliary support body on the rim and a tire pressure detection device [17]. Because this type of run-flat tire is mostly based on the common rim design, it has the advantages of simple structure, convenient disassembly, strong zero-pressure bearing capacity, and so on. It is a new type of run-flat tire with great development prospects. In order to study the stability of the vehicle with inserts supporting run-flat tire after blowout, a dynamic model was established and modified based on the UniTire model. Afterwards, the model was modified according to the discrete curve of tire mechanical properties under steady conditions. The results show that the improved run-flat tire model is consistent with the characteristics of vehicle tire blowout, and the optimal control strategy of inserts supporting run-flat tire after blowout in straight running condition was obtained by comparing the effects of braking, steering, and combined control.

2. Establishment of Run-Flat Tire Model 2.1. Run-Flat Tire Model before Blowout The performance of inserts supporting run-flat tire is the same as that of a normal tire before tire blowout. Therefore, the model of inserts supporting run-flat tire is established according to the UniTire [18] theory and the published test data, which mainly analyse the parameters of the mechanical properties in the process of tire blowout. The longitudinal slip ratio and lateral slip ratio are defined as the ratio of slip velocity to rolling velocity. The definition is as follows: ( S = −Vsx = ΩRe−Vx , S ∈ (−∞, +∞) x ΩRe ΩRe x −V −V (1) S = sy = y , S ∈ (−∞, +∞) y ΩRe ΩRe y

where Sx is the longitudinal slip ratio; Sy is the lateral slip ratio; Ω is the tire rolling angular velocity; Re is the effective rolling radius; Vx and Vy are the longitudinal and lateral components, respectively, of the wheel center velocity V. World Electr. Veh. J. 2021, 12, 128 3 of 13

The normalized longitudinal slip ratio φx, normalized lateral slip ratio φy, and total slip ratio φ are deﬁned as dimensionless physical quantities.

 Kx·Sx φx =  µx·Fz  Ky·Sy φy = µ ·F (2) qy z  2 2  φ = φx + φy

where Kx is the longitudinal slip stiffness of the tire; Ky is the cornering stiffness of the tire; ux is the longitudinal friction coefﬁcient between tire and ground; uy is the lateral friction coefﬁcient between tire and ground; and Fz is the tire vertical force. The total tangential force expression of the complete E-index semi-empirical model is as follows: 1 F = 1 − exp −φ − E · φ2 − E 2 + · φ3 (3) 1 1 12

where E1 is the curvature factor. The aligning arm Dx of the steady semi-empirical model can be expressed as follows:

2 Dx = (Dx0 + De) exp −D1φ − D2φ − De (4)

where Dx0 is the initial aligning arm; De is the ﬁnal value of the aligning arm; D1 is the ﬁrst order curvature factor; and D2 is the quadratic curvature factor. According to the above theoretical model, the tire aligning torque can be obtained as follows: Mz = Fy(−Dx + Xc) − Fx · Yc (5)

where Xc and Yc are the offset caused by longitudinal force and lateral force, respectively. The formula of tire rolling resistance moment My is as follows:

My = (Rr_c + Rr_v · Vr) · Fz · Rl (6)

where Rr_c is the rolling resistance coefﬁcient; Rr_v is the rolling resistance speed constant of the tire; Vr is the longitudinal velocity of wheel center (Vr = Ω · Re ); and Rl is the load radius of the tire.

2.2. Run-Flat Tire Model after Blowout During the process of tire blowout, the vehicle is extremely unstable and the tire burst time is short. In order to simplify the model, the parameters of the tire are linearized [19]. According to the experiment of the mechanical properties of the tire blowout, it can be seen that the longitudinal slip stiffness, lateral deflection stiffness, and effective rolling radius of the tire are reduced to 28%, 25%, and 80% of the normal working conditions, respectively, and the rolling resistance coefficient of the tire increases by 30 times [20]. However, the above data cannot be fully applied to the inserts supporting run-flat tire. In order to study the parameter changes of the inserts supporting run-flat tire after tire blowout, the corresponding curves under the working conditions were obtained using the test bench.

2.2.1. Cornering Stiffness and Longitudinal Stiffness after Tire Blowout In order to simplify the research, the static loading method was used to analyse the longitudinal and lateral mechanical characteristics of inserts supporting run-ﬂat tire at zero-pressure. The tire size is 37 × 12.5R16.5. The relationship curves of tire lateral force and sideslip angle, longitudinal force, and longitudinal displacement under different loads are obtained, as shown in Figure1. World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 4 of 13

World Electr. Veh. J. 2021, 12, 128 4 of 13

and sideslip angle, longitudinal force, and longitudinal displacement under different loads are obtained, as shown in Figure 1. 7000 7000 350kPa 350kPa 0kPa 0kPa 6000 6000

5000 5000

4000 4000

3000 3000 Lateral force (N) 2000 2000 Longitudinal force (N) force Longitudinal

1000 1000

0 0 0123456 0 20406080100 Side slip angle (deg) Longitudinal displacement (mm) (a) (b)

Figure 1. CurveCurve of of inserts inserts supporting supporting run run-ﬂat-flat tire tire test test results. results. ( (aa)) Cornering Cornering stiffness stiffness curve; curve; ( (bb)) lon- longi- gitudinaltudinal stiffness stiffness curve. curve.

ItIt can can be be seen seen from from Figure Figure 1a1a that that the the side sideslipslip angle angle of of inserts inserts supporting supporting run-flat run-flat tire tire under zero-pressure zero-pressure condition condition is is 1.8 1.8 deg. deg. At At th thisis time, time, the the lateral lateral force force is isabout about 4700 4700 N. N. The critical point point of of sideslip sideslip is is selected selected to to calculate calculate the the stiffness. stiffness. The The slope slope at this at thispoint point is approximatelyis approximately equivalent equivalent to cornering to cornering stiffnes stiffnesss and the and cornering the cornering stiffness stiffness is about is 1.48 about × 101.485 N/rad× 105 (4700N/rad N/1.8 (4700 deg), N/1.8 which deg), is about which 0.90 is abouttimes the 0.90 rated times tire the pressure rated tire condition. pressure Similarly,condition. it Similarly, can be seen it canfrom be Figure seen from 1b that Figure the 1longitudinalb that the longitudinal stiffness becomes stiffness about becomes 0.95 times.about 0.95 times. ItIt can can be be concluded concluded that that the the cornering cornering stif stiffnessfness will will be be reduced reduced to to about about 90% 90% of of the the normal tire tire pressure pressure condition condition and and the the longitudinal longitudinal stiffness stiffness will will be be reduced reduced to to about about 95% 95% after inserts supporting supporting run-flat run-flat tire tire blowout. blowout. Therefore, the changeschanges ofof longitudinallongitudinal and and cornering cornering stiffness stiffness during during tire tire blowout blowout can canbe expressed be expressed as follows: as follows:  ()0.95KK− =−+xx00() K x TTsx K0 (0.95T Kx0−Kx0)  Kx = d T (T − Ts) + Kx0  d (7) (7) ()(0.90−Ky0−Ky0)  0.90KKyy00 K K=−+y = T ()TT(T − T Ks) + Ky0  ysyT d 0  d K K where Kx0x 0is theis the longitudinal longitudinal stiffness stiffness before before tire tire blowout;blowout; Ky0 yis0 the is the cornering cornering stiffness stiff- T T nessbefore before tire blowout; tire blowout;Td is thed duration is the duration of tire of blowout; tire blowout;Ts is the s time is the of tiretime blowout; of tire blow- and T is the simulation time. out; and T is the simulation time. 2.2.2. Change of Rolling Resistance Coefficient after Tire Blowout 2.2.2. Change of Rolling Resistance Coefficient after Tire Blowout The rolling resistance coefficient is the ratio of the required thrust to the wheel load The rolling resistance coefficient is the ratio of the required thrust to the wheel load when the wheel is rolling under certain conditions. The increase in the tire contact area after when the wheel is rolling under certain conditions. The increase in the tire contact area tire blowout is the main factor of the rolling resistance change. The dimension parameters after tire blowout is the main factor of the rolling resistance change. The dimension pa- of the contact impression of the inserts supporting run-flat tire were extracted under rameters of the contact impression of the inserts supporting run-flat tire were extracted zero-pressure condition and rated load, as shown in Table1 and Figure2. under zero-pressure condition and rated load, as shown in Table 1 and Figure 2. Table 1. Tire contact area parameters. Table 1. Tire contact area parameters. 2 F/NF/N W/mmW/mm L/mmL/mm SS/mm/mm2 WholeWhole contact contact area area 12,250 12,250 235 235 580 580 136,300 136,300 InsertInsert contact contact area area 12,250 12,250 132 132 258 258 34,056 34,056 WorldWorld Electr. Electr. Veh. Veh. J. 2021J. 2021, 12, 12, 128, x FOR PEER REVIEW 55 ofof 13 13 World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 5 of 13

Figure 2. Contact patch under the zero-pressure condition. FigureFigure 2. 2.Contact Contact patch patch under under the the zero-pressure zero-pressure condition. condition. It can be seen from Figure 2 that the green frame is the shoulder part, and the red ItIt can can be be seen seen from from Figure Figure2 that2 that the the green green frame frame is is the the shoulder shoulder part, part, and and the the red red frame is the tire part with insert. Under the zero-pressure condition, the insert takes part frameframe is is the the tire tire part part with with insert. insert. Under Under the the zero-pressure zero-pressure condition, condition, the the insert insert takes takes part part in in the load-bearing, and the colour of the tire shoulder is lighter where it is affected by the thein load-bearing,the load-bearing, and and the the colour colour of theof the tire tire shoulder shoulder is is lighter lighter where where it it is is affected affected by by thethe insert. The increase of rolling resistance after tire blowout is mainly caused by the increase insert.insert. The The increase increase of of rolling rolling resistance resistance after after tire tire blowout blowout is is mainly mainly caused caused byby thethe increaseincrease of contact area between elastic rubber and ground. The rolling resistance of a normal tire ofof contact contact area area between between elastic elastic rubber rubber and and ground. ground. The The rolling rolling resistanceresistance ofof aa normalnormal tiretire increasesincreasesincreases 30 3030 times timestimes after afterafter tire tiretire blowout. blowout.blowout. Compared Compared with withwith inserts insertsinserts supporting supportingsupporting run-flatrun-flat run-flat tire,tire, tire, thethe the insertsinsertsinserts are areare in inin direct directdirect contact contactcontact with withwith the the tire tire and and the thethe rolling rollingrolling resistance resistanceresistance of ofof this thisthis contactcontact contact areaarea area isis is affectedaffectedaffected after after after tire tire tire blowout. blowout.blowout. Hence, Hence, the the rolling rolling resistance resistance affected affectedaffected by by by insertinsert insert isis is approximatelyapproximately approximately equivalentequivalentequivalent to toto the thethe rolling rollingrolling resistance resistanceresistance of of the the rated rated tire tiretire pressure pressurepressure and andand the the the contact contact contact area area area of of insert of insert insert is aboutisis about about 25% 25% 25% of the ofof the wholethe wholewhole contact contactcontact area area from from Table TableTable1. The 1.1. TheThe rolling rollingrolling resistance resistance resistance of the of of the restthe rest rest areas areas areas is 30isis times30 30 times times of the ofof the normalthe normalnormal tire tiretire pressure; pressure;pressure; therefore, therefore, the the rollingthe rollingrolling resistance resistanceresistance of insertsof of inserts inserts supporting support- support- run-flatinging run-flat run-flat tire under tiretire underunder the zero-pressure thethe zero-pressurezero-pressure condition condition is about isis aboutabout 22.8 (0.25 22.822.8 + (0.25(0.25 0.75 + ×+ 0.75 0.7530) × times× 30) 30) times times of that of of underthatthat under under the normal thethe normalnormal condition. condition.condition. Therefore,Therefore,Therefore, the thethe change changechange ofof rolling resistance resistance coefficientcoefficient coefficient duringduring during tiretire tire blowout blowout blowout can can can be be ex- beex- expressedpressedpressed asas as follows:follows: follows: ()− (22.8R()22.8RR−rc_0R− rc) _0 R = RTTR=−+r_c0 rc_0r_c0 rc _0(T()()− T ) + R (8) r_c RTTRrcrc__0s s s rc rcr_c0 (8)(8) Td Td d

RRrc_0 wherewherewhereR r_c0rc_0is the is is thethe rolling rollingrolling resistance resistanceresistance coefﬁcient coefficient before beforebefore tire tiretire blowout. blowout.blowout. 2.2.3. Change of Effective Rolling Radius after Tire Blowout 2.2.3.2.2.3. Change Change ofof EffectiveEffective Rolling Radius after Tire BlowoutBlowout Effective rolling radius refers to the vertical distance from the tire rolling center to the EffectiveEffective rollingrolling radiusradius refers to the vertical distancedistance fromfrom thethe tiretire rollingrolling center center to to ground under a certain load when the tire is rolling steadily. The load characteristic curve thethe groundground underunder aa certaincertain loadload when the tire isis rollingrolling steadily.steadily. TheThe loadload characteristic characteristic of inserts supporting run-ﬂat tire under zero pressure condition was extracted, as shown in curvecurve ofof insertsinserts supportingsupporting run-flat tire underr zerozero pressurepressure conditioncondition was was extracted, extracted, as as Figure3. shownshown in in FigureFigure 3.3.

100100

8080

6060

4040 Displacement (mm) Displacement Displacement (mm) Displacement

2020

00 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 0 2,000 4,000 6,000Load (N) 8,000 10,000 12,000 14,000 Load (N) Figure 3. Load characteristic curve under the zero-pressure condition. FigureFigure 3. 3.Load Load characteristic characteristic curve curve under under the the zero-pressure zero-pressure condition. condition.

As can be seen from Figure3, the tire displacement under the zero-pressure is 100.36 mm, and the tire rolling radius and load radius under the rated load are 455 mm and 467 mm, World Electr. Veh. J. 2021, 12, 128 6 of 13

respectively. Hence, the rolling radius of the inserts supporting run-flat tire is about 0.92 times the normal rolling radius. The change in effective rolling radius during tire blowout can be expressed as follows:

(0.92Re0 − Re0) Re = (T − Ts) + Re0 (9) Td

where Re0 is the effective rolling radius before tire blowout.

3. Two Degrees of Freedom Model and Control System 3.1. Two Degrees of Freedom Model When a tire blowout occurs, the control system needs to determine the additional yaw moment and the additional active angle according to the deviation between the actual driving state and the ideal driving state, so as to keep the vehicle in the driving state before the tire blowout. At the same time, in order to facilitate the analysis of the relationship between the vehicle motion state and stability, the vehicle model is simpliﬁed, and the vehicle two degrees of freedom control model is used as the ideal vehicle motion model. The ideal expression of the two degrees of freedom model is as follows:

 . (aC −bCr)  mV β + r = − C + C β − f r + C δ  x f r Vx f 2 2 (10) . (a C +b Cr)  I · r = bC − aC β − f r + aC δ + ∆M  z r f Vx f z

Considering the inﬂuence of the additional yaw moment acting on the vehicle, the above formula is transformed into the equation of state:

" . # β a11 a12 β b1 c1 . = · + · δ + · ∆Mz (11) r a21 a22 r b2 c2

  −(C +Cr) bCr−aC " C # f f − 1 f 0 a11 a12 mVx mVx b1 mVx c1 where =  2 2 ; = aC ; = 1 ; a a bCr−aCf b Cr+a Cf b f c 21 22 − 2 2 Iz Iz Iz·Vx Iz Vx is the velocity of the vehicle along the x-axis; m is the total mass of the vehicle; β is the sideslip angle; r is the yaw rate; δ is the front wheel angle; a and b are the distances from the mass center to the front and rear axles, respectively; Iz is the moment of inertia of the whole vehicle around the z-axis; and Cf and Cr are equivalent cornering stiffness of front axle and rear axle, respectively. The yaw rate under ideal stable state is as follows:

V = x rd 2 δ (12) L(1 + K · Vx )

where L is the axis between the front and rear wheels distance. The expression of vehicle stability coefﬁcient K is as follows: ! m a b = − K 2 (13) L Cr Cf

The modiﬁed expression of ideal yaw angle is as follows:

V µg ∗ = x · ( ) rd min 2 δ , sgn δ (14) L(1 + K · Vx ) Vx

where rd is the ideal yaw rate; rd∗ is the ideal yaw rate after correction; µ is the adhesion coefﬁcient of pavement; and g is the acceleration of gravity. World Electr. Veh. J. 2021, 12, 128 7 of 13

The ideal sideslip angle of the mass center is corrected as follows:

2  b− maVx Cr L  βd = 2 δ  L(1+KVx )  2  µg maVx βd = 2 b − max Vx Cr L (15) ( maV 2 )  b− x 2  β ∗ = min Cr L δ , µg b − maVx · sgn(δ)  d L(1+KV 2) V 2 Cr L x x

where βd∗ is the corrected ideal sideslip angle of mass center and βd, βdmax is an interme- diate variable.

3.2. Differential Braking Control System The fuzzy sliding mode control algorithm is used to control the yaw moment after tire blowout, and the yaw moment is distributed to each wheel through the principle of braking force distribution, so as to control the stability of the vehicle. According to the comparison between the ideal yaw rate and sideslip angle of mass center and the actual one, the switching function of the sliding surface is selected. The expression is as follows: s = (r − rd) + ε(β − βd) (16)

where r is the yaw rate of the vehicle and rd is the ideal yaw rate of the vehicle; β is the sideslip angle of mass center; and ε is the weighting coefﬁcient. The controller output yaw moment is as follows:

h . . s i ∆M = I r + εβ − (a + εa )β − (a + εa )r − (b + εb )δ − ksat (17) z1 z d d 21 11 21 11 2 1 τ

s where sat τ is the saturation function. In the process of tire blowout, the sideslip stiffness of tire blowout does not change linearly, so it is difﬁcult to determine the magnitude of the compensated yaw moment, and the fuzzy controller is selected to compensate the yaw moment. In the design of the Simulink fuzzy toolbox, the “Mamdain” method is selected as the fuzzy controller. The two-dimensional structure was adopted in the fuzzy controller, that is, the input value is the difference of the sideslip angle of the mass center ∆β and the yaw rate ∆r, and the output value of the fuzzy controller is set to U. The basic domain of ∆β is [–5,5]. The basic domain of ∆r is [–1.5,1.5] and that of U is [–20,20]. The membership function of the fuzzy controller is selected as the Gauss membership function. Fuzzy control rules are shown in Table2.

Table 2. Fuzzy control rule table.

∆β ∆r NB NM NS ZO PS PM PB NB PB PB PB PB PM PS ZO NM PB PB PM PM PM PS ZO NS PM PM PM PM PS ZO NS NO PM PS PS ZO NS NS NM ZO PM PM PS ZO NS NS NM PS PS PS ZO NM NM NM NM PM ZO ZO NM NM NB NB NB PB ZO NS NM NB NB NB NB

According to the fuzzy control algorithm, it can be known that the relationship of the compensated yaw moment obtained by maintaining the stability of the vehicle is as follows:

∆Mz2 = K1 · U (18) World Electr. Veh. J. 2021, 12, 128 8 of 13

where ∆Mz2 is the compensated yaw moment. Therefore, the yaw moment to maintain the stability of the vehicle in the process of tire blowout is as follows: ∆Mz = ∆Mz1 + ∆Mz2 (19) Considering the inﬂuence of tire blowout on vehicle stability, the tire without blowout is preferred in the braking force distribution strategy. After the master-slave braking tire is determined, the braking force is determined according to the yaw moment, so as to keep the vehicle stable.

4. Simulation Results and Test Analysis E-class SUV is selected as the basis of vehicle dynamics simulation model in CarSim, as shown in Table3. The vehicle two degrees of freedom model is the basis of stability control. It describes the effects of yaw rate and sideslip angle of the mass center on vehicle yaw and lateral motion. In order to simplify the model analysis, it is taken as the ideal vehicle motion model.

Table 3. Vehicle and tire parameters.

Parameter/Unit Parameter Symbol Parameter Value Sprung mass/kg Ms 2290 Height of center of mass/mm h 810 Front axle distance/mm a 1180 Rear axle distance/mm b 1170 Wheel base/mm l 2950 Effective rolling radius/mm Re 455 Static load radius/mm R0 467 Rim diameter/mm D 419 Width of tire section/mm B 317 Radial stiffness/N·mm−1 Kt 405

In order to study the characteristics of the inserts supporting run-ﬂat tire, the UniTire tire model was built in Simulink. Based on CarSim, the simulation of run-ﬂat tire blowout was carried out under the straight driving condition. The tire blowout parameters of the front and rear wheels on the same side were extracted and compared with the normal tire. The tire size is 37 × 12.5R16.5 and the outer diameter of the insert is 840 mm.

4.1. Tire Blowout Dynamic Response When the vehicle runs in the straight running condition, the setting speed is 100 km/h, and the road adhesion coefﬁcient is 0.85. In order to simplify the analysis, the left front wheel and the left rear wheel were selected to carry out the tire blowout test, respectively, and the difference between the inserts supporting run-ﬂat tire and the normal tire in the uncontrolled state were compared after the tire blowout. (1) Left front tire blowout When the vehicle is running in the straight running condition, the left front wheel suddenly blows out in the third second. The driver does not take any operation during the tire blowout, and the running state of the vehicle was obtained. The change chart of tire blowout parameters is shown in Figure4. World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 9 of 13

When the vehicle is running in the straight running condition, the left front wheel World Electr. Veh. J. 2021, 12, 128 9 of 13 suddenly blows out in the third second. The driver does not take any operation during the tire blowout, and the running state of the vehicle was obtained. The change chart of tire blowout parameters is shown in Figure 4. 0.4 80 Normal tire Normal tire Run-flat tire Run-flat tire

0.3 60

0.2 40

0.1 20 Lateral acceleration (g) Lateral displacement (m) displacement Lateral

0.0 0

0246810 0246810 Time (s) Time (s) (a) (b) Normal tire Normal tire 12 Run-flat tire 0.0 Run-flat tire

9 -0.4

6 -0.8

Yaw rate (deg/s) rate Yaw -1.2 3 (deg) angle Sideslip

-1.6 0

0246810 0246810 Time (s) Time (s) (c) (d)

FigureFigure 4. 4. TheThe simulation simulation curve curve of of left left front front tire blowout: ( (aa)) lateral lateral acceleration acceleration curve; curve; ( (bb)) lateral lateral displacementdisplacement curve; curve; ( (cc)) yaw yaw rate rate curve; curve; ( (dd)) sideslip sideslip angle angle curve. curve.

ItIt can can be be seen seen from from Figure Figure 44 that that the the char characteristicsacteristics of ofthe the inserts inserts supporting supporting run-flat run- tireflat blowout tire blowout are similar are similar to the to normal the normal tire under tire under the straight the straight driving driving condition. condition. From From Fig- ureFigure 4a,b,4a,b, it can it can be beseen seen that that the the lateral lateral acceleration acceleration and and lateral lateral displacement displacement of of the inserts supportingsupporting run-flat run-flat tiretire areare slightly slightly larger larger than than the the normal normal tire, tire, and and the the lateral lateral displacement displace- mentreaches reaches 75 m 75 in m 10 in s. 10 Figure s. Figure4c,d indicate4c,d indicate that thethat parameter the parameter change change is unstable is unstable in the in thefirst first two two seconds seconds of tire of tire blowout; blowout; the yawthe yaw rate rate and and the sideslipthe sideslip angle angle change change sharply, sharply, then thengradually gradually stabilize stabilize in about in about fifth fifth second. second. (2) (2)Left Left rear rear tire tire blowout blowout When the vehicle runs normally in a straight line, the left rear tire blows out in the third Whensecond, the and vehicle the driver runs normallystill does innot a ta straightke any line,measures. the left The rear simulation tire blows results out in are the shownthird second, in Figure and 5. the driver still does not take any measures. The simulation results are shown in Figure5. World Electr. Veh. J. 2021, 12, 128 10 of 13 World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 10 of 13

Normal tire 40 Normal tire 0.20 Run-flat tire Run-flat tire

32 0.15

24 0.10 16

0.05 8 Lateral acceleration (g) Lateral displacement (m)

0.00 0

0246810 0246810 Time (s) Time (s) (a) (b) Normal tire 40 Normal tire Run-flat tire 0.0 Run-flat tire

32 -0.2

24 -0.4

16 -0.6

8 Sideslip angle (deg) Lateral displacement (m) -0.8

0 -1.0

0246810 0246810 Time (s) Time (s) (c) (d)

FigureFigure 5. 5. TheThe simulation simulation curve curve of of left left rear rear tire tire blowout: blowout: (a ()a )lateral lateral acceleration acceleration curve; curve; (b (b) )lateral lateral displacementdisplacement curve; curve; (c ()c )yaw yaw rate rate curve; curve; (d (d) )sideslip sideslip angle angle curve. curve.

ItIt can can be be seen seen from from Figure Figure 55 thatthat the the rear rear tire tire blowout blowout yaw yaw of of the the inserts inserts supporting supporting run-flatrun-ﬂat tire tire is is smaller smaller than than that that of of the the normal normal tire, tire, and and the the curve curve is is obviously obviously better better than than thatthat of of the the front front tires. tires. This This is is because because the engineengine ofof thethe modelmodel selected selected is is front-mounted, front-mounted, so sothe the vertical vertical load load on on the the front front axleaxle isis largerlarger thanthan thatthat of the rear rear axle. axle. When When tire tire blowout blowout occurs,occurs, the the rolling rolling resistance resistance increases, increases, and and the the front front tire tire is is generally generally the the driving driving wheel, wheel, whichwhich makes makes the the force force on on both both sides sides of of the the wh wheelseels uneven, uneven, forcing forcing the the car car to to lean lean to to the the sideside of of the the tire tire blowout. blowout.

4.2.4.2. Stability Stability Control Control Analysis Analysis InIn order order to tostudy study the the change change of the of thestabi stabilitylity parameters parameters of the of inserts the inserts supporting supporting run- flatrun-ﬂat tire after tire the after tire the blowout, tire blowout, differential differential braking braking control control is applied is applied to the to left the front left fronttire andtire the and left the rear left tire, rear respectively. tire, respectively. The runni The runningng speed speed of the of vehicle the vehicle is 100 is km/h, 100 km/h, and the and roadthe roadadhesion adhesion coefficient coefﬁcient is 0.85. is 0.85.In the In thir thed thirdsecond, second, tire blowout tire blowout occurs. occurs. The state The stateof the of vehiclethe vehicle is shown is shown in Figures in Figures 6 and6 and7, respectively.7, respectively. (1) Left front tire control results World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 11 of 13

0.4 80 Uncontrolled Uncontrolled Controlled Controlled Normal 0.3 Normal 60

0.2 World Electr. Veh. J. 2021, 12, 128 40 11 of 13 World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 11 of 13 0.1 20 Lateral acceleration (g)

0.0 (m) displacement Lateral 0.4 80 Uncontrolled Uncontrolled 0 -0.1 Controlled Controlled 0246810 Normal 0.3 0246810 Normal Time (s) 60 Time (s) (a) (b) 0.2 Uncontrolled 12 Uncontrolled 401.2 Controlled Controlled Normal 0.1 Normal 9 0.6 20 Lateral acceleration (g) Lateral displacement (m) displacement Lateral 0.0 0.0 6 0 -0.1 -0.6 3 0246810 0246810 Sideslip angle (deg) Yaw rate (deg/s) Time (s) Time (s) (a) -1.2 (b) 0 Uncontrolled 12 Uncontrolled -1.81.2 Controlled Controlled Normal 0246810 Normal 0246810 0.6 9 Time (s) Time (s) (c) (d) 0.0 6 Figure 6. The controlled simulation curve of left front tire blowout: (a) lateral acceleration curve; (b) lateral displacement curve; (c) yaw rate curve; (d) sideslip-0.6 angle curve. 3 Sideslip angle (deg) Yaw rate (deg/s) As can be seen from Figure 6, the yaw state-1.2 of the inserts supporting run-flat tire is 0 obviously improved after applying the differential-1.8 braking control, and the deviation from the original track under the control effect of lateral displacement is less than 3 m; the 0246810 0246810 change of yaw rate andTime sideslip (s) angle is ob viously reduced underTime the (s) control effect. The maximum value of yaw(c) rate after correction is 1.8 deg/s, and the (maximumd) deviation of sideslip angle after correction is about 1.2 deg. However, the direction of the sideslip angle FigureFigure 6.6. The controlled controlled simulation simulation curve curve of of left left front front tire tire blowout: blowout: (a) lateral (a) lateral acceleration acceleration curve; curve; (b) of the inserts supporting run-flat tire is opposite to that before control. (lateralb) lateral displacement displacement curve; curve; (c) yaw (c) yaw rate rate curve; curve; (d) sideslip (d) sideslip angle angle curve. curve. (2) Left rear tire control results As can be seen from Figure 6, the yaw state of the inserts supporting run-flat tire is 0.20 40 obviously Uncontrolledimproved after applying the differential braking Uncontrolled control, and the deviation Controlled Controlled from the original Normal track under the control effect of lateral displacement is less than 3 m; the 0.15 Normal change of yaw rate and sideslip angle is obviously30 reduced under the control effect. The maximum value of yaw0.002 rate after correction is 1.8 deg/s, and the maximum deviation of 0.10 0.001 sideslip angle after correction is about 1.2 deg. However,0.4 the direction of the sideslip angle 0.000 20 0.3 -0.001 0.2 of the inserts supporting run-flat tire is opposite to that0.1 before control. 0.05 -0.002 (2) Left rear tire control results 0.0 -0.003 -0.1 5678910 -0.2 10 5678910 Lateral acceleration (g) acceleration Lateral

0.20 (m) displacement Lateral 40 0.00 Uncontrolled Uncontrolled Controlled Controlled Normal 0 0.15 Normal -0.05 30 0246810 0246810 Time0.002 (s) Time (s) 0.10 0.001 (a) 0.4 (b) 0.000 20 0.3 -0.001 0.2 Figure 7. Cont. 0.1 0.05 -0.002 0.0 -0.003 -0.1 5678910 -0.2 10 5678910 Lateral acceleration (g) acceleration Lateral 0.00 (m) displacement Lateral

0 -0.05 0246810 0246810 Time (s) Time (s) (a) (b) World Electr. Veh. J. 2021, 12, 128 12 of 13 World Electr. Veh. J. 2021, 12, x FOR PEER REVIEW 12 of 13

6 0.6 Uncontrolled Uncontrolled Controlled Controlled Normal Normal 0.3 4

0.4 0.0 0.3 2 0.2 0.1 0.0 -0.3

-0.1 Sideslip angle (deg) Yaw rate (deg/s) -0.2 5678910 0 -0.6

0246810 0246810 Time (s) Time (s) (c) (d) Figure 7. The controlled simulation curve of left rear tire blowout: (a) lateral acceleration curve; (b) Figure 7. The controlled simulation curve of left rear tire blowout: (a) lateral acceleration curve; (b) lateral displacement curve; (c) yaw rate curve; (d) sideslip angle curve. lateral displacement curve; (c) yaw rate curve; (d) sideslip angle curve.

(1) ItLeft can frontbe seen tire controlfrom Figure results 7 that the left rear tire can basically follow the normal driving track, and the lateral displacement is about 0.25 m under the differential braking control.As The can lateral be seen acceleration from Figure and6, theyaw yaw rate stateare close of the to the inserts norm, supporting and the fluctuation run-flat tire is small.is obviously At 4 s, improvedthey can basically after applying return theto th differentiale normal state. braking Therefor control,e, the and stability the deviation of the vehiclefrom the with original control track after under tire blowout the control is ob effectviously of lateral improved displacement compared is with less that than without 3 m; the control.change of yaw rate and sideslip angle is obviously reduced under the control effect. The maximum value of yaw rate after correction is 1.8 deg/s, and the maximum deviation of 5.sideslip Conclusions angle after correction is about 1.2 deg. However, the direction of the sideslip angle of the inserts supporting run-flat tire is opposite to that before control. The tire blowout model of inserts supporting run-flat tire is built based on the UniTire(2) Left theory. rear tire Combined control resultswith the whole vehicle model built by CarSim, the co-simulation isIt carried can be out seen to from obtain Figure the 7change that the in leftcharacteristic rear tire can parameters basically followunder the straight normal drivingdriving condition. track, and the lateral displacement is about 0.25 m under the differential braking (1)control. The difference The lateral between acceleration the left and front yaw tire rate and are left close rear totire the blowout norm, condition and the fluctuation of inserts is small.supporting At 4 s, run-flat they can tire basically and normal return tire to thewas normal compared. state. The Therefore, results theshow stability that the of the vehiclecharacteristic with controlparameters after of tire the blowout two tires is obviouslyare similar. improved When the compared front tire withblowout that withoutoccurs, control. the yaw of the inserts supporting run-flat tire is larger, and when the rear tire blowout occurs, the yaw of the normal tire is larger. 5. Conclusions (2) The stability of the inserts supporting run-flat tire after tire blowout is controlled ac- cordingThe tire to blowout the difference model of between inserts supportingthe ideal yaw run-flat rate tireand is the built sideslip based angle on the and UniTire the theory.actual Combined value. The with simulation the whole vehicle results model show builtthat bythe CarSim, differential the co-simulation braking control is carried can out tobetter obtain maintain the change the running in characteristic track of parametersthe vehicle, under significantly the straight improve driving the condition. stability of (1) theThe vehicle, difference and betweenwhether thethe lefttrack front adjustment tire and lefteffect rear of tire the blowoutrear tire conditionof inserts support- of inserts ingsupporting run-flat tire run-flat is better. tire and normal tire was compared. The results show that the characteristic parameters of the two tires are similar. When the front tire blowout Authoroccurs, Contributions: the yaw X.W. of the performed inserts supporting the data analyses run-flat an tired wrote is larger, the manuscript; and when L.Z. the rearcontrib- tire uted toblowout the conception occurs, of the the yaw study; of theZ.W. normal performed tire isthe larger. experiment; Fen Lin contributed significantly(2) Theto analysis stability and of manuscript the inserts preparation; supporting Z.Z. run-flat helped perform tire after the tire analysis blowout with is constructive controlled discussions.according All authors to the have difference read and between agreed theto the ideal published yaw rate version and theof the sideslip manuscript. angle and the Funding:actual This value.research The was simulation partly supported results byshow the National that the Natural differential Science brakingFoundation control of China can under bettergrant 51605215, maintain and the runningpartly by track the China of the Po vehicle,stdoctoral significantly Science Fo improveundation theunder stability grant 2019T120450,of the the vehicle, Qing Lan and Project whether (Teacher the trackSu 2019 adjustment [3]), Research effect Foundation of the rearof Nanjing tire of Institute inserts of Technologysupporting (grant run-flat number. tire CKJA201906). is better. Conflicts of Interest: The authors declare no conflicts of interest. Author Contributions: X.W. performed the data analyses and wrote the manuscript; L.Z. contributed References to the conception of the study; Z.W. performed the experiment; Fen Lin contributed significantly to analysis and manuscript preparation; Z.Z. helped perform the analysis with constructive discussions. 1. Huang, J.; Guo, K.; Song, X.; Liu, W. Vehicle stability control method after tire blow-out. China Mech. Eng. 2009, 20, 2006–2010. All authors have read and agreed to the published version of the manuscript. 2. Liu, Y.; Yu, S.Y.; Gu, X.L. Trajectory control of vehicles with blowout tire on expressway based on fuzzy PID algorithm. J. Jilin Univ. Inf. Sci. Ed. 2015, 33, 380–388. World Electr. Veh. J. 2021, 12, 128 13 of 13

Funding: This research was partly supported by the National Natural Science Foundation of China under grant 51605215, and partly by the China Postdoctoral Science Foundation under grant 2019T120450, the Qing Lan Project (Teacher Su 2019 [3]), Research Foundation of Nanjing Institute of Technology (grant number. CKJA201906). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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