TIRE FORCE AND MOMENT PROPERTIES FOR COMBINED SLIP CONDITIONS CONSIDERING CAMBER EFFECT Nan Xu, Konghui Guo ASCL State Key Lab, Jilin University Changchun Jilin, China April -1, 15
Outline Introduction Tire axis system and slip ratios Key factors for developing the analytical tire model Analytical tire forces and moments model Simulation analysis and experiment validation Conclusions
Introduction Tire mechanics characteristics under combined cornering/braking and camber situations have significant effects on the vehicle directional control. The analytical models, which usually include carcass model with elastic tread elements, establish the relationship between tire structure parameters and tire behaviors. Some key factors are considered in the model of this paper: arbitrary pressure distribution; translational, bending and twisting compliance of the carcass; effective carcass camber, dynamic friction coefficient; anisotropic stiffness properties and tire width. Combined longitudinal slip, lateral slip and camber. The analytical model in this paper is valuable for understanding tire properties and developing the semi-empirical models. 3
Tire axis system and slip ratios Tire revolution axis Wheel traveling direction V Unified definition for slip ratios: Sliding speed /updating speed Yt F x M y Zt M z F z Ot F y Xt M x Tire revolution direction S S x y Vsx V cos R V R r V V r sy e V sin R e e 4
Key factors for developing the analytical tire model Arbitrary pressure distribution Unified 3 parameter expression of CPD: for diff tire structures, loads, inflations & resistances ( u) A n14n1 n4n1 n n 1u 1 u Bu A 1 n n n 14 14 3 3 3 3 4 3 4 1 B n n n a 1.4 1.4 1. 1. 1. 1. η(u).8.6.4. =1 n=1 n= n=3. -1. -.5..5 1. Relative longitudinal coordinate u η(u).8.6.4. n= =.5 =1 =. -1. -.5..5 1. Relative longitudinal coordinate u 5
Key factors for developing the analytical tire model Carcass structure parameters y lateral translating deformation lateral bending deformation twisting deformation longitudinal translating deformation effective carcass camber y y (x) y y F K c y cy y ( x) F K ( u) cb y cb y( x) M z N x x F K c x cx c Fy Mx e c e F y M x a y c a x a (x) y cb a x a a x q z F z M x F z M x F y 6
Key factors for developing the analytical tire model Dynamic friction coefficient V s d s m s exp h log N exp V sm where, N(usually N=.8) is a factor to make the friction coefficient increase slightly around the origin. d V V s sm m h : describe the variation tendency s v sm v s 7
Shear force direction [deg] 8 6 4 Key factors for developing the analytical tire model Anisotropic stiffness properties measured force direction slip direction adhesion direction -.5 =1 =4 Slip direction =14 Adhesion direction 18 F ad 18 4 6 8 Slip direction [deg] X axis: Slip direction s arctan Sy Sx Y axis: Force direction F arctan Fy Fx Generally, contact area includes F ad both adhesion & sliding region Resultant direction varied from pure adhesion to totally slip In adhesion region: In sliding region: K S K S arctan F ad y y x x arctan S S F s y x 8
Analytical tire forces and moments model Deformation of carcass and tread element under combined slip condition wheel plane y c - y cr belt -a C D deflected carcass due to camber Δy B y o P t Y x c x O Δx P c V r t Vtcosα α A a θ V tread element X x contact line wheel spin axis 9
Analytical tire forces and moments model Tire forces and moments without sliding Fx a ktsx Fy K ys y K y sin M z Km S y Km sin K A A y 1 K y A3 A1 a K K K 3 1 K m 1 b K y b1k y m b y y a 3 A1 a 1 1 S 3a S 3 cf b x b x x A a 1 S K 3 K x y m A a 1 S b K 3 ab K S 3 F b r 3 x 1 y 1 y x x t K K y a kt y 3R ak 3 a kt e K b t cb K m c 3 a kt 3 1 1 K K cx cy 3 3 a kt N 1
Analytical tire forces and moments model Tire forces and moments in general case with sliding region Fx B7 KxSx B3 Fzsx PB K P B K S PB b K P P PB P B a F 1 7 5 9 1 8 1 5 3 sin 1 3 5 4 y m y y z sy F y P P5 PP 1 4 P B K P B K S P B b K P P P B P B a F 7 4 9 8 1 4 3 sin 3 4 4 y m y y z sy M z P P5 PP 1 4 1 P1 1 B9 B9 Sx B4Fz sxa N 1 P B B1S x BS x a B6Fzsx cfx Kcb 4 akt a P3 b1 B1 B11S x B3 B5 Fzsxb1 Fxb r t Re Re 3 P4 1 B1, P5 B7 a 1 1 1 B1 bd uc, B bd1 uc, B3 m uc 1 1 1 B4 m1 uc, B5 m uc, B6 md uc 1 uc 1 1 B7, B8 3 u u, B9 1 3u u 4 4 4 4 3 1 uc 1 uc 1 uc 1 uc B1, B11 1 uc 4 4 3 11 3 3 c c c c
Analytical tire forces and moments Tire width effect model vertical load, contact patch length and effective rolling radii have opposite variations in width. e z R y R y ew e e z y z y mw m e Re y a y R y z y w ew mw y aw Fzw y Fz a S xw R S e x e y R e y y y e a y a y 1 y
Simulation analysis and experiment validation Effect of carcass compliance on tire cornering stiffness A1 a 1 1 S 3a S 3 cf b x b x x A a 1 S K 3 K x y m Sx K y K y 1 1 1 b K A A y 1 8 x 14 bending characteristic ratio ε b twisting characteristic ratio ε θ The cornering stiffness will decrease with the increase of ε b or ε θ Cornering stiffness Ky [N/rad] 7 6 5 4 cornering stiffness 3..4.6.8 1 Bending characteristic ratio [-] b 13
Simulation analysis and experiment validation Cornering stiffness under combined slip condition K y 1 Sx1 b1 K 1 1 S 3 S 3 a cf ypure b x b x x 1.15 Normalized cornering stiffness [-] 1.1 1.5 1.95.9 -.5.5 Longitudinal slip ratio [-] cornering stiffness will increase when tire has a slight braking. 14
Simulation analysis and experiment validation Effect of carcass compliance on aligning moment under combined slip condition translating compliance coefficient: 5 c 1 1 K K cx cy Aligning moment Mz [Nm] c = = c = -5 1-6 c = -1 1-6 Fz=3N c = -15 1-6 -5-3 - -1 1 3 15 Longitudinal force Fx [N]
Simulation analysis and experiment Resultant force direction [deg] 36 34 3 3 8 6 4 validation Simulation results of tire forces and moments under combined slip condition 4 driving braking adhesion direction slip direction =1 resultant force direction 18 18 4 6 8 3 3 34 36 Slip direction [deg] Anisotropic stiffness properties. The anisotropy of tire slip stiffness will cause the variation of resultant force direction under different combined slip conditions. 4 =1 16
Simulation analysis and experiment Side slip with camber validation Lateral Force [N] 3 =-1 1-1 =1 - Fz = 3N -3 - -1 1 Slip Angle [deg] Aligning Moment [Nm] 6 =-1 4 = - -4 =1 Fz = 3N -6 - -1 1 Slip Angle [deg] The shift of lateral force The variation of curvature near the peak side force The severe asymmetry of aligning moment 17
Simulation analysis and experiment validation Longitudinal slip with camber 1 15 Lateral Force [N] 5-5 =-1 =1-5 - Fz = 3N -1-1 -.5.5 1 Longitudinal Slip Ratio [-] Aligning Moment [Nm] -1 =-1 Fz = 3N -15-1 -.5.5 1 Longitudinal Slip Ratio [-] Side force has an extreme variation when applying braking/driving force, and F y even changes its sign in the driving half of the diagram. The aligning torque, aroused by longitudinal force and shifted point of action due to camber, will generate an additional distortion of the carcass which results in an effective slip angle. 18 1 5-5 =1 =- =-5
Simulation analysis and experiment validation Combined slip with camber Longitudinal Force [N] 3 - =8 1-1 - =-1 Fz = 3N -3-1 -.5.5 1 Longitudinal Slip Ratio [-] Lateral Force [N] 3-8 -4 1 - -1 4 - =-1 Fz = 3N =8-3 -1 -.5.5 1 Longitudinal Slip Ratio [-] Aligning Moment [Nm] 15 1 5 =8-5 -1 =-1-15 - Fz = 3N -8 - -1 -.5.5 1 Longitudinal Slip Ratio [-] With the increase of slip angle a, the longitudinal force will decrease because of the limitation of friction coefficient the sliding velocity dependent friction coefficient can be observed the carcass compliance and width effect are reflected by the asymmetry of F y and M z 19
Simulation analysis and experiment validation Experiment validation--side slip with camber Lateral Force [N] 8 6 4 - -4-6 = 1 Test Data Analytical Tire Model = -1 Fz = 7N -8-3 - -1 1 3 Slip Angle [deg] Aligning Moment [Nm] 3 1-1 = -1 = 1 Fz = 7N - Test Data Analytical Tire Model -3-3 - -1 1 3 Slip Angle [deg]
Simulation analysis and experiment validation Experiment validation--longitudinal slip with camber Lateral Force [N] 8 6 4 - Fz = 4N =, = -1 Test Data Analytical Tire Model Aligning Moment [Nm] 3 1-1 Test Data Analytical Tire Model Fz = 4N =, = -1-4 - -6-1 -.5.5 1 Longitudinal slip ratio [-] -3-1 -.5.5 1 Longitudinal Slip Ratio [-] 1
Simulation analysis and experiment validation Experiment validation--combined slip with camber -5 Test Data Analytical Tire Model 15 Lateral Force [N] -1-15 - -5-3 Fz = 4N = 4, = -5-35 -4-4 Longitudinal Force [N] Aligning Moment [Nm] 1 5-5 -1 Fz = 4N = 4, = -5-15 Test Data Analytical Tire Model - -1 -.5.5 1 Longitudinal Slip Ratio [-]
Conclusions Firstly, arbitrary pressure distribution, translational, bending and twisting compliance of the carcass, effective carcass camber, dynamic friction coefficient anisotropic stiffness properties and tire width are the key factors for developing the analytical tire model. Secondly, the considerable and interesting effects on tire force and moment due to camber can be reflected well by the analytical model. It will be very helpful for researchers to understand the mechanism of tire force generation. Thirdly, for variety of cases with camber, the severe asymmetry and dramatic variations of lateral force and aligning moment are mainly due to the carcass compliance and tire width. Finally, considering all key factors, the analytical tire model is capable of describing all kinds of tire properties reasonably and accurately. The model parameters can also be identified from tire measurements and the computational results using the analytical model show good agreement with test data. 3
Acknowledgments The authors would like to thank the previous joint project between the Research and Development Center of General motors and the State Key Laboratory of Automotive Simulation and Control at Jilin University, from which the test data presented in this paper is produced. Special thanks are due to the National Natural Science Foundation of China (5145185) and the National Basic Research Program of China (973 Program) (11CB7111) for supporting authors research. 4
Any Questions? Thanks for your time and attention! Affiliation: ASCL State Key Lab, Jilin University Mailing address: No.5988 Renmin Avenue, Changchun, Jilin, 135, P.R.China Email: xu.nan61@gmail.com, guokonghui@gmail.com Phone: 431-85959-618