ii

Transcription

1

2 ii

3 ABS (Anti Brake-lock System) Direct Yaw Control 4 Brush Model FEM(Finite Elements Model) iii

4 Magic Formula (1) (2) (3) Pure Slip Condition iv

5 v

6 Magic Formula Magic Formula Magic Formula vi

7 SAT Combined Slip Neo-FIALA Combined Slip Combined Slip Neo-FIALA Neo-FIALA ( [ η < 1] ) ( [ η < 1] ) ( [ η 1] ) ( [ η 1] ) Neo-FIALA vii

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9 1 Symbol Description Unit A Contact area m 2 a Coefficient of dependence on velocity of sliding friction coefficient - a 0 a 17 Magic Formula lateral force model parameter - A s Self aligning torque stiffness of a rigid ring model Nm/rad A s f Stability Factor s 2 /m 2 a y Lateral acceleration m/s 2 B Stiffness factor of Magic Formula - C Shape factor of Magic Formula - C q Compliance of asymmetric coefficient of contact patch 1/Nm C tr Lateral shared stiffness of the tread in FIALA model N/m C x Longitudinal shared stiffness of tread N/m C xc Compliance of the relative position of wheel center 1/N C y Lateral shared stiffness of tread N/m D Peak factor of Magic Formula - E Curvature factor of Magic Formula - E Elastic modulus of the beam Pa F Tire force vector - F x Longitudinal force N F x1 Longitudinal force in adhesive zone N f x1 Stress of x direction in adhesive zone N/m 2 F x2 Longitudinal force in sliding zone N f x2 Stress of x direction in sliding zone N/m 2 F y Lateral force N

10 2 Symbol Description Unit F y1 Lateral force in adhesive zone N f y1 stress of y direction in adhesive zone N/m 2 F y2 Lateral force in sliding zone N f y2 stress of y direction in sliding zone N/m 2 F yα Lateral force caused by slip angle N F y f Lateral force of front tire N F yγ Lateral force caused by camber angle (= Camber thrust) N F yr Lateral force of rear tire N F z Vertical load N F z f Vertical load of front tire N F zr Vertical load of rear tire N G Weighting factor of Magic Formula combined slip model - G mz Torsional stiffness of tread base N/rad G xα G ys Weighting factor of longitudinal force of Magic Formula under combined slip condition Weighting factor of lateral force of Magic Formula under combined slip condition - - h Height of wheel center m I Moment of inertia kgm 2 I z Moment of inertia of the beam kgm 2 k Radius of inertia m K e Control gain of driver-steer system rad K x Braking stiffness of a rigid ring model N K y Cornering stiffness of a rigid ring model N/rad k y Equivalent lateral stiffness of side wall N/m K y f Cornering stiffness of front tire N/rad K yr Cornering stiffness of rear tire N/rad l Contact patch center length m

11 3 Symbol Description Unit l f Distance front axle to c.g. m l h l ll l lr Boundary coordinate between the static zone and the sliding zone of contact patch Distance between front left wheel to lower laser sensor of front left wheel Distance between front right wheel to lower laser sensor of front right wheel m m m l r Distance rear axle to c.g. m l s Distance between upper and lower laser sensor m l ul l ur Distance between front left wheel to upper laser sensor of front left wheel Distance between front right wheel to upper laser sensor of front right wheel m m L x Look-ahead distance of driver m m Total mass of vehicle kg M z Self aligning torque (SAT) Nm M z1 Self aligning torque in adhesive zone N/m 2 M zr Residual torque Nm n Sholder exponent in contact pressure and D gsp (x ; n, q) - P max Maximum pressure of contact area Pa q Compliance of front inclination in contact patch - q Heat flex W R Radius of curvature m r Yaw angler velocity rad/s r e Effective free rolling tire radius m r h Boundary coordinate ratio between the adhesive zone and the sliding zone of contact patch - S Slip ratio (Positive : Braking) - s Laplace variable

12 4 Symbol Description Unit S h Horizontal shift of Magic Formula - S H f Horizontal shift of pneumatic trail - S Ht Horizontal shift of residual torque - S Hxα S HyS Horizontal shift of longitudinal force of Magic Formula under combined slip condition Horizontal shift of lateral force of Magic Formula under combined slip condition - - s.m. Static margin - S v Vertical shift of Magic Formula - S VyS Vertical shift of lateral force of Magic Formula under combined slip condition - T Tire surface temperature C t Pneumatic trail m t time s T 0 Tire surface average temperature during the test measuring the tire force parameters C T h Time constant of phase lead as driver-steer system s T k Time constant of phase lag as driver-steer system s T R Road surface temperature C u Longitudinal velocity of vehicle c.g. m/s u 0 Longitudinal constant velocity of vehicle c.g. m/s u δ Additional steer angle using front active steering system rad V Velocity of vehicle c.g. m/s v Lateral velocity of vehicle c.g. m/s V b Velocity of tire belt m/s V c Critical speed m/s V r Velocity of road surface m/s V s Slip velocity vector -

13 5 Symbol Description Unit V sx Slip velocity of x direction m/s V sy Slip velocity of y direction m/s W Heat capacity J/K w Contact patch width m x 1 Longitudinal distance from front contact edge m x 1 Non-dimensional longitudinal distance from front contact edge - x b x coordinate of the point of belt m x r x coordinate of the point of road surface m y 0 Lateral bending stiffness of tread base N/m y 1 Lateral distance of contact patch from rim center m y b y coordinate of the point of belt m y r y coordinate of the point of road surface m α Slip angle of tire rad α e Effective slip angle rad α f Slip angle of front tire rad α f l Slip angle of front left tire rad α f r Slip angle of front right tire rad α r Slip angle of rear tire rad β Slip angle of vehicle c.g. rad δ f Steer angle of front tire rad δ f l Steer angle of front left tire rad δ f r Steer angle of front right tire rad X Y x component of the distance between road surface and belt y component of the distance between road surface and belt m m ϵ Deflection compliance of belt 1/Nm

14 6 Symbol Description Unit η Relative change ratio of contact patch length with change of camber angle - γ Camber angle rad γ e Effective camber angle rad γ f l Camber angle of front left tire rad γ f l Camber angle of front right tire rad λ Thermal conductivity W/m 2 K µ Friction coefficient of tread rubber - µ B Braking force coefficient [ ] = F x /F z - µ d Sliding friction coefficient of tread rubber - µ d0 Sliding friction coefficient of tread rubber at V = 0 - µ s Adhesive friction coefficient of tread rubber - ω rotational angler velocity rad/s ω n Natural frequency rad/s ϕ Non-dimensional slip angle - φ Roll angle of vehicle rad ψ Yaw angle rad τ s Dead time of driver-steer system s θ Direction of friction force rad ξ Compliance of contact patch shift m/n ζ n Damping ratio -

15 7 JASO Z Z (+) Camber Vertical Load F z Angle γ Self Aligning Torque M z Direction of Tire Center Slip Angle α Over Turning Moment M x Longitudinal Force X (+) Direction of Road Surface Wheel Plane Road Plane Tire Rotational Axis Rolling Resistance Moment M y Y Lateral Force F y (+) Fig. 1 Definition of motion of the wheel and the forces and moments acting from the road on the wheel

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17 ( ) 1888 John Boyd Dunlop 1895 A. & E. Michelin 115

18 10 1 ABS(Anti Brake-lock System) Direct Yaw Control 4

19 Fromm Julien Brush model Fiala FIALA model (17) ( ) FIALA SAT (18) Frank FIALA model (19) FIALA model FIALA model (20) 2.3 (21) 6 (22) (25) (26) (32)

20 12 1 Pacejka Pacejka Stretched-String model (33) (34) SAT Stretched-String model SAT Stretched-String model FIALA model (35) Radt Milliken (36) 1987 Bakker (37) Magic Formula Magic Formula (38) (39), (40) (41) Jagt (42) Magic Formula Magic Formula (43)

21 (17) Experimental data only Using similarity method Through simple physical model Through complex physical model Degree of fit Number of full scale tests Effort Complexity of formulations Insight in tire behavior Number of special experiments [Magic Formula] [FIALA model] [FEM] Empirical Approach more Theoretical Fig. 1.1 Four categories of possible types of approach to develop a tire model Magic Formula (38) Magic Formula 2.4 MF-Tire Magic Formula Full-set Tire Model λ 3 Fiala

22 14 1 (FIALA model (44) ) Segel (Brush model (45) ) Fiala 2.3 FEM ABS(Anti-lock Brake System) MSC/ADAMS 1998 Zegelaar (46) 1999 Maurice (47) 2000 TNO Automotive SWIFT (Short Wavelength Intermediate Frequency Tire) Model Esslingen University Prof. Gipser 1987 (48), (49) 1999 ADAMS FTire (50) 1999 ADAMS RMOD-K (51) Gim Hankook tire model (52) (56) Guo UniTire (57) (59) 2005 Tire Model Performance Test(TMPT) (60) 1. : :

23 : : 3. : ABS : FEM 1.2 (61) 1.3

24 16 1 Fig. 1.2 TNO tire test trailer (61) 1.2 1) 2) 3) Magic Formula 1)

25 Fig. 1.3 Indoor test machine for tire force and moment measurement (MTS Flat-Trac III)

26 18 1 Magic Formula 2) 3) FIALA Model Neo-FIALA Model (3) (5) (6) (12) ( ) ( SAT ) ( ) (13) (16)

27 Magic Formula FIALA Model Fig. 1.4 The outline for the thesis 2 (1), (2)

28 20 1 FIALA model (18), (44), (62) Magic Formula (37), (38), (63), (64) 3 Magic Formula (3) (5) 4 Magic Formula (6) (12) 3 4 Magic Formula CAE 5 (13) (16) ( )

29 (18) Milliken (65) Segel (66) Radt (67) (Bicycle model) ( 2.1 ) V mv ( ) dβ dt + r = 2F y f + 2F yr (2.1) I dr dt = 2l f F y f 2l r F yr (2.2)

30 22 2 2F yr l r α r l y l f β V 2F yf α f δ x Fig. 2.1 Bicycle model in a cornering manoeuvrer F y f = F y f (α f ) (2.3) F yr = F yr (α r ) (2.4) α f, α r α f = δ f ( β + l ) f r V (2.5) ( α r = β l ) rr V (2.6) F y f = K y f α f (2.7) F yr = K yr α r (2.8)

31 (2.5) (2.6) (2.1) (2.2) mv ( ) ( dβ dt + r = 2K f β + l f I dr dt = 2l f K f ( β + l f V r δ ) ( V r δ 2K r ) ( + 2l r K r β l r β l r V r ) V r ) (2.9) (2.10) mv dβ { dt + 2(K f + K r )β + mv + 2 ( ) } l f K f l r K r r = 2K f δ (2.11) V 2 ( ) dr l f K f l r K r β + I dt + 2 ( l 2 f K ) f + lr 2 K r r = 2l f K f δ (2.12) V ( ) (2.11) (2.12) dβ/dt = 0 dr/dt = 0 β r 1 m l f V 2 2l l β = r K r l r 1 m l f K f l r K r l δ (2.13) V 2l 2 K f K 2 r 1 V r = 1 m l f K f l r K r l δ (2.14) V 2l 2 K f K 2 r (= 1/R) 1 R = r V = 2K f K r l 2K f K r l 2 mv 2 (l f K f l r K r ) δ (2.15) δ = 1 [ l mv 2 l ] f K f l r K r R 2K f K r l (2.16)

32 24 2 (2.16) A s f = m 2l 2 l f K f l r K r K f K r (2.17) δ = l R ( 1 + As f V 2) (2.18) a y = V 2 /R (2.18) δ = l R + A s f l a y (2.19) 2.2 δ [rad] l R 0 Understeer A sf l 1 Neutral steer Oversteer a y Fig. 2.2 The steer angle vs. lateral acceleration at constant path curvature 2.2 V c = 1 A s f 0 (62) A s f (2.17) l f K f l r K R (2.10) β

33 I dr dt + 2 ( l 2 f K ) f lr 2 K r r = 2 ( ) l f K f l r K r β (2.20) V l f K f l r K r s.m. = l f K f l r K r l ( K f + K r ) (2.21) ( ) (2.9) (2.10) miv s2 + 2m ( l 2 f K ) ( ) f + lr 2 K r + 2I K f + K r s + 4K f K r l 2 2 ( ) l f K f l r K r miv miv 2 I = 0 (2.22) s 2 + 2Ds + P 2 = 0 (2.23) 2D = 2m ( l 2 f K ) ( ) f + lr 2 K r + 2I K f + K r (2.24) miv P 2 = 4K f K r l 2 2 ( ) l f K f l r K r (2.25) miv 2 I I I = mk 2 (2.26) k l f l r K f K r (2.24) (2.25) 2D = 2 ( ) 1 + k2 K f + K r l f l r mv k 2 l f l r (2.27)

34 26 2 P 2 = 4K f K r l 2 m 2 k 2 V 2 ( 1 m ) l f K f l r K r V 2 2l 2 K f K r (2.28) ω 2 n = P 2 (2.29) 2ζ n ω n = 2D (2.30) ζ n = K f + K r 2 k K f K r l ω n = 2 K f K r l 1 + As f V 2 mk V ( 1 + k 2 ) ( ) ( ) /l f l r + 1 l f l r l f K f l r K r k 2 /l f l r k 2 (2.31) K f + K r 1 + As f V 2 (2.32) ω n ζ n K f, K r V ω n ζ n (2.32) K f K r Radt Handling Curve (67) 2.3 F y f F z f = F yr = a y F zr g (2.33) δ ( α f α r ) = l R (2.34)

35 Fig. 2.3 Handling diagram resulting from normalized tire characteristics.

36 Handling curve 2.2 a y = Vr = V2 R (2.35) 2.3 (2.34) 2.3 Handling curve 2.4 Fig. 2.4 A number of handling curves arising from the pairs of normalized tire characteristics shown left(1:front, 2:rear)

37 Handling Curve (cornering coefficient) (load transfer coefficient) (68) Allen (69) 1.1

38 30 2 (1), (2) 2.2 (1) (2.9) (2.2) (2.7) (2.8) (2.5) (2.6) m v = 2F y f + 2F yr (2.36) Iṙ = 2l f F y f 2l r F yr (2.37) ( ) v u0 ψ + l f r F y f = K y f δ f u δ u 0 ( ) v u0 ψ l r r F yr = K yr u 0 (2.38) (2.39) ẏ = v (2.40) ÿ = a y (2.41) ψ = r (2.42)

39 ( 2.5 ) δ(s) = K e T h s + 1 T k s + 1 e τs {y(s) + L x ψ(s)} (2.43) v u δ δ f ψ L x ψ y r u 0 L x l r l f Fig. 2.5 Driver model Padé e τs 1 τ/2s 1 + τ/2s (2.44) ẋ = Ax + Bu δ (2.45) x = [y, ẏ, ψ, ψ, δ f, δ f ] T (2.46) J = 0 { g1 (y + L x ψ) 2 + g 2 ψ 2 + g 3 (k e δ ψ) 2} dt (2.47) (2.47)

40 32 2 f 1 Σ + - f 2 Gain scheduler f 3 f 4 f 5 δ f δ sw r y u 0 f 6 s µ-estimator filter filter filter filter filter Fig. 2.6 Control signal flow of active front steering control system (2) VSC(Vehicle Stability Control) H (70) Brush model (71) FIALA model (44), (62), (72)

41 FIALA model Fiala ( 2.7 ) Radial Tire Tread Belt Sidewall Z Rim r Y X x 1 Fig. 2.7 Tire structure of FIALA model ( ) x 1 x 1 y 1 x 1 x 1 x 1 y 1 0 x 1 l h

42 34 2 Fig. 2.8 Tire deformation of FIALA model l h < x 1 l α l w 2.9 F y EI z d 4 y dx 4 + k yy = 0 (2.48) x = 0 dy/dx = 0 y 1b = λ 2k y F y e λx 1 (cos λx + sin λx) (2.49) λ = 1 ( ) 1 ky 4 (2.50) 2 EI z

43 Fig. 2.9 Deformation of belt and sidewall during cornering, and their elastic beam approximation. The tire illustrations are in an extremely stretched view for clarity. x 1 - y 1 y 1b y 1b = λ3 l 2 F y 2k y x l ( 1 x ) l (2.51) α 0 x 1 l h y t y t = x 1 tan α (2.52) Y f y1 [ f y1 = C tr (y t y 1b ) = C tr x 1 tan α λ3 l 2 F y x ( 1 1 x )] 1 2k y l l (2.53) Y f y2 1.

44 Y 3. 2 p = 4p max x 1 l ( 1 x 1 l ) (2.54) p max p max = 3F z 2wl (2.55) f y2 µ f y2 = µwp = 4wµp max x 1 l ( 1 x ) 1 l (2.56) l h f y1 = f y2 x 1 [ C tr x 1 tan α λ3 l 2 F y x ( 1 1 x )] 1 x ( 1 = 4wµp max 1 x ) 1 (2.57) 2k y l l l l x 1 l h = l 1 C tr lw tan α 4wµp max + C trλ 3 l 2 2k y (2.58) dx 1 f y1 dx f y2 dx F y lh l F y = f y1 dx 1 + f y2 dx 1 0 l h lh [ = C tr x 1 tan α λ3 l 2 F y x ( 1 2k y l 0 1 x 1 l )] l x ( 1 dx 1 + 4wµp max l h l 1 x 1 l ) dx 1 (2.59) F y α = 0 F y = K y tan α K2 y K tan 2 y 3 α + 3µF z 27µ 2 Fz 2 tan3 α (2.60)

45 K y = df y dα = α=0 wc tr l 2 2 (1 + C ) (2.61) trλ 3 l 3 12k y K y α = 0 F y SATM z M z = lh 0 = C tr lh + o l l h ( x l ) l f y1 dx l h ( x l ) f y2 dx 1 2 ( ( x l ) [ x 1 tan α λ3 l 2 F y x 1 2 2k y l ( x l 2 ) x ( 1 4wµp max l 1 x 1 l 1 x 1 l )] dx 1 ) dx 1 (2.62) F y α = 0 M z = l K y 6 tan α K2 y K tan 2 y 3 K 4 α + tan 3 y α tan 4 α (2.63) 6µF z 18µ 2 Fz 2 162µ 3 Fz 3 (2.60) (2.63) SAT FIALA model SAT (2.60) (2.63) ϕ = K y F z tan α (2.64) F y µf z = ϕ ϕ2 3 + ϕ3 27 M z µf z l = ϕ 6 ϕ2 6 + ϕ3 18 ϕ4 162 (2.65) (2.66) (2.65) (2.66) SAT (2.65) (2.66) ϕ 3 F y /µf z 1 ϕ 3/4 SAT M z /µf z l 27/512

46 38 2 Fig Relationship between normalized F y and normalized α Fig Relationship between normalized M z and normalized α

47 FIALA model (20), (21), (73) (75) 2.12 C x ω Z V Tread Belt Sidewall Rim z 1 h r e 0 l x 1 X Fig Tire model while braking V ω r e V r (= V) V b (= r e ω) V r > V b 1 t x b = V b t (2.67) 1 t x r = V r t (2.68) t x r x b = (V r V b ) t (2.69)

48 40 2 S = V r V b V r (2.70) X 1 x 1 x 1 = (V r V b ) x 1 /V r = S x 1 (2.71) f x1 f x1 = C x S x 1 (2.72) w l p = 4p max x 1 l ( 1 x 1 l ) (2.73) F z F z = 2 3 p maxwl (2.74) p = 6F z wl x 1 l ( 1 x ) 1 l (2.75) µ s p l h f x Adhesive Sliding µ d p µ s p 0 C x Sx 1 l h l x 1 Fig Braking force profile in circumference direction

49 l h ( 2 3 p l h maxµ s 1 l ) h = C x S l h (2.76) l l ( l h = l 1 K ) xs 3µ s F z K x = C xwl 2 2 (2.77) (2.78) K x S = 0 F x l h µ s µ d f x2 f x2 = µ d p (x 1 ) (2.79) F x F x = = lh o lh 0 f x1 dx 1 + l wc x sx 1 dx 1 + l h f x2 dx 1 l l h 6F z l 3 µ dx 1 (l x 1 ) dx 1 (2.80) (2.80) l h 0 ( F x = K x S 1 K ) 2 ( xs Kx S + F z µ d 3µ s F z 3µ s F z µ B = K ( x S 1 K ) 2 ( xs Kx S + µ d F z 3µ s F z 3µ s F z ) 2 ( 3 2K xs 3µ s F z ) 2 ( 3 2K xs 3µ s F z ) ) (2.81) (2.82) l h 0 l h = 0 (2.80) F x = µ B F z (2.83) µ B F x F z S = V b V r V b (2.84)

50 n (18), (23) µ d (2.79) (2.83) V µ d = µ d0 av (2.85) S V (2.85) V = S V l (2.86) l l h µ d = µ d0 as V l l l h (2.87) n p = ( ) n {( ) n ( 2 l p max x 1 l ) n } l 2 2 (2.88) n n = 4 (2.88) p = n + 1 n 2 n F z l n+1 w [( ) n ( l x 1 l ) n ] 2 2 (2.89) l h (2.72) (2.89) n + 1 n 2 n F z l n+1 w [( ) n ( l l h l ) n ] = C x S l h (2.90) 2 2

51 F x1 (2.80) 1 2 F x2 F x2 = l n + 1 l h n = n + 1 n 2 n F z µ d l n+1 2n F z µ d l n+1 [( ) n l 2 ( l 2 ( x 1 l ) n ] 2 ) n (l l h ) 1 n + 1 dx 1 ( ) n ( l h l ) n+1 2 (2.91) n F x F x = C xs wl 2 h 2 + n + 1 ( 2n F z n l µ n+1 d0 as V l ) ( ) n l l l (l l h ) 1 ( ) n+1 1 h 2 n ( l h l ) n+1 2 (2.92) 2.14 Fig Braking force characteristics with vehicle velocity change on wet road

52 X Y C x C y C x = C y = C r e ω V α V r (= V) V b (= r e ω) 2.15 M z V r F x α V b F F y Fig Axis of forces and velocities of tire 2.16 ABD AC A B C AB BC t (x r, y r ) x r = V r t cos α (2.93)

53 y 1 Adhesive B Sliding D A V r α θ C x 1 V b y 0 X f y µ s p µ d psinθ µ d p 0 f x µ s p[=(f x 2 +f y2 ) 1/2 ] C x Sx 1 sinα µ d pcosθ x 1 µ d p 0 C x Sx 1 cosα l h l x 1 Fig Deformation and forces of tire while combined slip condition

54 46 2 y r = V r t sin α (2.94) t (x b, y b ) x b = V b t (2.95) y b = 0 (2.96) X Y X Y X = (V r cos α V b ) t (2.97) Y = V r t sin α (2.98) V r t = x 1 V b t = x 1 (2.70) (2.84) (S > 0) X = S x 1 (2.99) Y = x 1 sin α (2.100) (S < 0) X = S x 1 (2.101) Y = x 1 (1 + S ) tan α (2.102) f x1 = CS x 1 cos α (2.103) f y1 = Cx 1 sin α (2.104) f x1 = CS x 1 (2.105) f y1 = Cx 1 (1 + S ) tan α (2.106)

55 F x1 F y1 f x1 f y1 l h F x1 = lh 0 wcs x 1 cos αdx 1 (2.107) F y1 = lh 0 wcx 1 sin αdx 1 (2.108) F x1 = lh 0 wcs x 1 dx 1 (2.109) F y1 = lh 0 wcx 1 (1 + S ) tan αdx 1 (2.110) SAT M z1 (Z ) f x1 f y1 M z1 = lh 0 wc [ ( x 1 sin α x 1 l ) ( y 0 + x ) ] tan α S x 1 cos α dx 1 (2.111) M z1 = lh 0 wc [ ( x 1 tan α(1 + S ) x 1 l ) ( y 0 + x ) ] tan α S x 1 dx 1 (2.112) y G y y 0 = F y /G y (µ s p) 2 = f 2 x1 + f 2 y1 (2.113) l h α S ( l h = l 1 K ) tan 2 α + S 3µ s F 2 z (2.114) K = Cwl2 2 (2.115)

56 48 2 V V r V b 2.16 ( ) θ S α S tan θ = tan α (0 θ π) (2.116) F x2 F y2 X Y F x2 = F y2 = l l h µ d wp cos θdx 1 (2.117) l l h µ d wp sin θdx 1 (2.118) SAT M z2 BC M z2 = l l h [ ( ) ( x1 l µ d wp l h l l h tan α + y 0 cos θ + x 1 l ) ] sin θ dx 1 (2.119) 2 S α SAT F x = KS (l h /l) 2 cos α + µ d F z (1 l h /l) 2 (1 + 2l h /l) HS (2.120) F y = K (l h /l) 2 sin α + µ d F z (1 l h /l) 2 (1 + 2l h /l) H tan α (2.121) M z = K (l h /l) 2 [ (4l h 3l) sin α 6y 0 S cos α 2S l h sin α ] /6 µ d F z S H [ (1 l h /l) 2 (1 + 3l h /l) l h tan α/2 + y 0 (1 l h /l) 2 (1 + 2l h /l) ] µ d F z H (1 l h /l) 2 (l h /l) 2 l tan α (2.122)

57 F x = KS (l h /l) 2 + F z µ d (1 l h /l) 2 (1 + 2l h /l) HS (2.123) F y = K (1 + S ) (l h /l) 2 tan α + µ d F z (1 l h /l) 2 (1 + 2l h /l) H tan α (2.124) M z = K (l h /l) 2 [ (1 + S ) (4l h 3l) tan α 6y 0 S 2S l h tan α ] /6 µ d F z S H [ (1 l h /l) 2 (1 + 3l h /l) l h tan α/2 + y 0 (1 l h /l) 2 (1 + 2l h /l) ] µ d F z H (1 l h /l) 2 (l h /l) 2 l tan α (2.125) H = ( 1 tan2 α + S l 2 h = l 1 K ) tan 2 α + S 3µ d F 2 z Table 2.1 Values for calculation of Sakai model Symbol Value Unit Symbol Value Unit F z 4,000 N l m K 57,200 N/rad G y 250,000 N/m µ s µ d F y [kn] 3 Slip Angle [deg] Slip Ratio [-] Fig Examples of simulation results (slip ratio vs. lateral force)

58 50 2 F x [kn] 3 2 Slip Angle[deg] Slip Ratio[-] Fig Examples of simulation results (slip ratio vs. longitudinal force)

59 Slip Angle [deg] 1 2 Slip Ratio [-] M z [Nm] Fig Examples of simulation results (slip ratio vs. SAT) F y [kn] 3 2 Slip Angle [deg] F x [kn] Fig Examples of simulation results (longitudinal force vs. lateral force)

60 Slip Angle [deg] F x [kn] M z [Nm] Fig Examples of simulation results (longitudinal force vs. SAT) 2.4 Magic Formula Magic Formula Pacejka 1987 (37) (TNO) MF-Tire (64) ( ) Magic Formula ( SAT) Pure Slip ( ) 2.22 sine

61 Self Aligning Torque Braking Force Slip Angle Slip Ratio Lateral Force Fig Basic form of steady-state tire characteristics y = D sin (Bx) (2.126) y SAT x (α) (S ) x (2.126) 2.22 x (2.126) y = D sin [C arctan (Bx)] (2.127) Magic Formula (2.127) D B C D C x (2.127) [ π ] y = lim y = D sin x 2 C (2.128) C = 2 y = 0 (2.129) C SAT

62 54 2 Φ = (1 E)x + (E/B) arctan(bx) (2.127) x (2.127) y = D sin [C arctan {Bx E (Bx arctan (Bx))}] (2.130) x y y(x) = D sin [C arctan {Bx E (Bx arctan (Bx))}] (2.131) Y(X) = y(x) + S v x = X + S h (2.132) Magic Formula 2.23 y Y D y S v S h x m arctan(bcd) x X Fig A typical tire characteristic indicating the meaning of the coefficients of Eqs. (2.131) and (2.132)

63 B : C : Stiffness Factor BCD Stiffness Shape Factor C 1.30 C 1.65 SAT C 2.40 D : E : S h : Peak Factor Curvature Factor Horizontal Shift S v : Vertical Shift C Delft Tyre 96 SAT M z0 = t F y0 + M zr (2.133) t(α t ) = D r cos [ C t arctan { B t α t E y (B t α t arctan(b t α t )) }] (2.134) M zr = D r cos [arctan(b r α r )] (2.135) [ π ] y = D r cos 2 C t (2.136) α t = α + S Ht α r = α + S H f (2.137) SAT 2.24 (2.134) (2.136) 2.25 SAT SAT

64 56 2 Y y -S h D X, x 2 BC x 0 -y Fig A typical tire SAT characteristics indicating the meaning of the coefficients of Eqs.(2.134) and (2.136) M z F y α -t F y M zr α -S Hf t α -S Ht Fig The images of SAT characteristics. Upper: the product of lateral force and pneumatic trail, Middle: residual torque [Eq.(2.135)], Lower: pneumatic trail

65 Combined Slip Magic Formula Combined Slip( ) 1993 Michelin Bayle (76) Pure Slip cosine Combined Slip G = D cos [C arctan(bx)] (2.138) Combined Slip 2.26 (2.138) Pure Slip α = 0 F x S = 0 F y Pure Slip 1 Combined Slip F x F y Pure Slip F x0 F y0 G xα = F x = F x0 G xα (α, S, F z ) (2.139) cos [C xα arctan {B xα α s E xα (B xα α s arctan (B xα α s ))}] cos [C xα arctan {B xα S Hxα E xα (B xα S Hxα arctan (B xα S Hxα ))}] (2.140) α s = α + S Hxα (2.141)

66 58 2 Fig Dimensional graph of combined slip force characteristics

67 F y = F y0 G ys (α, S, γ, F z ) + S VyS (2.142) G ys = cos [ C ys arctan { B ys S s E ys ( ByS S s arctan ( B ys S s ))}] cos [ C ys arctan { B ys S HyS E ys ( ByS S HyS arctan ( B ys S HyS ))}] (2.143) S s = S S HyS (2.144) SAT M z = t ( ) ( ) ( α t,eq Fy + M zr αr,eq + s Fy, γ ) F x (2.145) t ( ) α t,eq = Dt cos [ C t arctan { ( B t α t,eq E t Bt α t,eq arctan ( ))}] B t α t,eq cos (α) (2.146) ( ) M zr αr,eq = Dr cos [ arctan ( )] B t α r,eq cos (α) (2.147) ( ) 2 Kx α t,eq = arctan tan 2 α t + S 2 sgn(α t ) (2.148) α r,eq = arctan tan 2 α r + K y ( Kx K y ) 2 S 2 sgn(α r ) (2.149)

68

69 61 3 (3) (5) Magic Formula ( ) ( ) ( ) (3), (4) Magic Formula SAT 1)

70 62 3 2) 3.2 (39)(40) ( ) Magic Formula 6 Magic Formula Magic Formula (41) Magic Formula Magic Formula 4 ( α S γ F z ) 4 ( F x F y SATM z [ OTM]M x )

71 (α, γ, F z ) ( ) 2 (F y, M z ) ( ) 6 6 ABS (77)(78) (79)(80)(81)(82) 6 (83)(84) (85) 6 (80) α γ 1. ( (86) ) ( 3.1 )

72 ( 500g ) 2. 6 CCD CCD 20g 100g 6 Fig. 3.1 The outline of measurement system for tire forces, torques and attitude angles CCD 3.2 CCD ( 1/60 )

73 Fig. 3.2 Sample of the picture analyzing the slip angle Fig. 3.3 The principle of the slip angle measurement

74 ( ) 0.2deg 3.1 (inverse tangent) ( 3.4 ) Fig. 3.4 The principle of the camber angle measurement 3.5

75 Fig. 3.5 Measurement system of tire forces, torques and attitude angles ( 3.7 ) /65R15 230kPa 1

76 68 3 Fig. 3.6 Example of relation between vertical load versus slip angle when vehicle running Slip Angle 0 :Light Weight :Middle Weight :Heavy Weight 0 Vertical Load Fig. 3.7 Example of relation between vertical load versus slip angle while vehicle running with dead weight

77 3.4. Magic Formula 69 2,214kg +200kg, +400kg 3.1 Table 3.1 Initial load condition of the experimental vehicle Vehicle + Driver Vehicle + Driver Vehicle + Driver + weight 200kg + weight 400kg Front Left Tire 5.45 kn 5.96 kn 6.42 kn Front Right Tire 5.66 kn 6.15 kn 6.70 kn Rear Left Tire 5.45 kn 5.96 kn 6.51 kn Rear Right Tire 5.15 kn 5.59 kn 6.06 kn kN 3.4 Magic Formula Magic Formula Magic Formula (3.1) (3.9) F y (x) = D sin [C arctan {Bx E (Bx arctan (Bx))}] (3.1) F y (α) = F y (x) + S v (3.2)

78 70 3 : Base : Base+200kg : Base+400kg Fig. 3.8 Test result (slip angle vs. lateral force) : :: Slip Angle(deg) Vertical Load(kN) Fig. 3.9 Test result (slip angle vs. vertical load)

79 3.4. Magic Formula 71 x = α + S h (3.3) C = a 0 (3.4) D = ( ) ( a 1 F 2 z + a 2 F z 1 a15 γ 2) (3.5) BCD = a 3 sin (2 arctan (F z /a 4 ) (1 a 5 γ )) (3.6) E = (a 6 F z + a 7 ) ( 1 (a 16 γ + a 17 ) sgn (α + S h ) ) (3.7) S h = (a 8 F z + a 9 + a 10 γ) F z (3.8) S v = a 11 F 2 z + a 12 F z + ( ) a 13 F 2 z + a 14 F z γ (3.9) α, γ, F z α γ F z F y B, C, D, E, S h, S v (factor) a 0 a 17 (parameter) (38) 1. 2 B, C a 0 a a 0 a (43) Magic Formula

80 72 3 (5) ( Levenberg-Marquardt ) Matlab TM (The Mathworks Inc. ) Coefficient Identification Measured Data MF Coefficients Pure Slip Identification (Newly developed) Pure Slip Parameter Identification MF Pure Slip Parameters Combined Slip Parameter Identification MF Combined Parameters Fig The flow of tire model identification Magic Formula Magic Formula Magic Formula SAT (38) Shape factor C Curvature factor E

81 3.4. Magic Formula Lateral Force(kN) Fz=3(kN) Fz=6(kN) Fz=9(kN) Slip Angle(deg) Fig The identified results without constraint conditions (γ = 0[deg]) 1.0 < C < 1.65 (3.10) ( ) C2 < E < 1 (3.11)

82 Lateral Force(kN) Fz=3(kN) Fz=6(kN) Fz=9(kN) Slip Angle(deg) Fig The identified results with constraint conditions of Eqs. (3.10) and (3.11) (γ = 0[deg]) a 17 = 0 where γ = 0 (3.12) S h = S v = 0 where γ = 0 (3.13) 0[deg] 0[N] 0[deg] 3.13 / µ max 3.14 S h S v S h /γ = 0.1(deg) (3.14)

83 3.4. Magic Formula 75 4 Lateral Force(kN) Fz=3(kN) Fz=6(kN) Fz=9(kN) Slip Angle(deg) Fig The identified results with constraint conditions of Eqs. (3.10), (3.11), (3.12), and (3.13) (γ = 0[deg]) 5 Lateral Force(kN) F z = 7 [kn] γ =-4(deg) γ = 0(deg) γ = 4(deg) Slip Angle(deg) Fig The identified results with constraint conditions of Eqs. (3.10), (3.11), (3.12), and (3.13) (Fz = 7[kN])

84 76 3 S v /γ < 25F z (N deg) (3.15) 0[deg] Peak factor D a 15 > 0 (3.16) (3.10) (3.16) Lateral Force(kN) γ = 0 [deg] Fz=3(kN) Fz=6(kN) Fz=9(kN) Slip Angle(deg) Fig The identified results with all constraint conditions (γ = 0[deg]) µ max 3.17 Magic Formula

85 3.4. Magic Formula 77 6 Lateral Force(kN) F z = 7 [kn] γ =-4(deg) γ = 0(deg) γ = 4(deg) Slip Angle(deg) Fig The identified results with all constraint conditions (F z = 7[kN]) Lateral Force(kN) Measured Data Identified Data Error = Sigma = Slip Angle(deg) Fig The comparison between the measured data and the identified results

86 ( 1.3 ) 3M # Table 3.2 Tire test machine specification Maximum tire size Maximum speed 910mm ±250km/h Maximum vertical force 25,000N Maximum longitudinal force Maximum lateral force Maximum over turning torque Maximum aligning torque Maximum spindle torque Maximum slip angle Inclination angle ±10,000N ±15,000N ±10,000Nm ±2,000Nm ±2,800Nm ±30deg -12deg to +45deg

87 Lateral Force(kN) γ = 0 [deg] Fz=3(kN) Fz=6(kN) Fz=9(kN) Slip Angle(deg) Fig The identified results from the data on the flat belt test machine(γ = 0[deg]) Lateral Force(kN) F z = 7 [kn] γ =-4(deg) γ = 0(deg) γ = 4(deg) Slip Angle(deg) Fig The identified results from the data on the flat belt test machine(f z = 7[kN])

88 µ max µ max kN 7.3kN µ max kN µ max [deg] [deg] 3.15 µ max 10[deg] [deg]

89 [deg] The Camber Angle(deg) Initial weight Initial + 200kg Initial + 400kg Tire Slip Angle(deg) Fig The relationship between the slip angle and the camber angle in this test (5)

90 Multi-Component Force Transducer Laser Displacement Sensors Fig Experimental setup for measuring the tire data (39), (40) 6 ( 3.22 ) α f l = δ f l α f r = δ f r ( β + l ) f r V x ( β + l ) f r V x (3.17) (3.18)

91 γ f l = φ tan l ul l ll l s (3.19) γ f r = φ + tan l ur l lr l s (3.20) Laser displacement Sensor ϕ Vehicle Center Line l ul l ll Fig Measurement method of camber angle /55R16 230kPa 5.77kN [588kgf], 5.47kN [558kgf] 3 3.3

92 84 3 Table 3.3 Initial camber angle conditions Set of lower arm Left[deg] Right[deg] Original arm Long arm Short arm Fig Measured results (slip angle vs. vertical load) -7 7[deg]

93 Fig Measured results (slip angle vs. camber angle) ( 3.26 ) ( ) 3.27 ( )

94 86 3 Fig The identified results of lateral force using the test data in this section Fig The identified results of lateral force using the data on the flat belt test machine

95 Fig Comparison with the identified model data and the test data ( 3.28 ) TIME 3.29

96 88 3 Fig The example of tire input pattern using tire test machine (87)(88) TIME Procedure Isolated Magic Formula 3. Magic Formula 4.

97 Fig The example of definition of TIME test procedure

98

99 91 4 (6) (12) (87) (6), (9) (7), (8), (10) (12) 4.2 (6), (9) 4.1

100 92 4 Fig. 4.1 The example of lateral acceleration and steering angle of the fish hook test ( 4.1 ) 4.2 SUV , 4.5

101 Fig. 4.2 Maximum lateral acceleration of 1st peak on fish hook test Fig. 4.3 Test sequence for measuring lateral force

102 94 4 Fig. 4.4 Test results of different skip angular velocity (slip angle versus tire surface temperature) Fig. 4.5 Test results of different skip angular velocity (slip angle versus lateral force)

103 Magic Formula (7), (8), (10) (12)

104 96 4 W dt dt = q λa(t T R ) (4.1) q = F V s (4.2) F = Fx 2 + Fy 2 (4.3) V s = V 2 2 sx + V sy (4.4) = (V cos α r e ω) 2 + V 2 sin 2 α (4.5) (4.2) α = 0 q = F x (V rω) = F x VS S = 0 q F y Vα (4.1) λ W λ (4.1) κ = λa/w dt dt = q W κ (T T R) (4.6) q = 0 0 (4.6) κ = dt dt 1 (4.7) T T R κ (4.2)

105 (4.6) Angle[deg] κ Heating ( :205/65R15) Slip Surface Tire Temperature Tire Measuring Surface Time[sec] ] Fig. 4.6 Slip angle measuring the parameter κ and W κ[sec 00 5 Time[sec] Fig. 4.7 Measured results of κ [Vertical load:4600n] κ ( ) κ

106 λ (4.6) κ W W (4.1) W = q dt dt + κ (T T R) (4.8) κ q (4.2) (4.5) F x, F y, V sx, V sy dt/dt, T W q 4.6 ( ) ( 1 Wheel Plane ) 4.8 W ( 4.6 ) W 4.8 W κ (4.8) 4.8 W W W W

107 4.4. W[Nm/K] Measured Result at -10deg +10deg Time[sec] Fig. 4.8 Measured results of W [Vertical load:4600n] Magic Formula 0 0 Pure Slip Condition 0 Combined Slip Condition Pure Slip Magic Formula (64) F i0 (p) = D i sin [ C i tan { ( 1 B i p E i Bi p tan 1 (B i p) )}] + S vi (4.9) α + S hx if i = x p = S + S hy if i = y (4.10) Magic Formula D i ( ) K i = B i C i D i (x = 0 ) (4.9) F i0 (p, T) = D i (T) sin [ C i tan 1 { B i (T)p E i ( Bi (T)p tan 1 (B i (T)p) )}] + S vi (4.11) K i (T) = B i (T) C i D i (T) (4.12) D i K i (4.11)

108 100 4 D i D i D i D i D x D y 4.9, D i 1 D i (T) = D i0 {1 + a i (T T 0 )} (4.13) T 0 Base Magic Formula Parameter K i K i 0.01 ±1[deg] K x, K y K y 1[deg] K i 1 K i (T) = K i0 {1 + b i (T T 0 )} (4.14)

109 Force[N] Measured Breaking Identified Resutls Data 20Tire Surface 40 Temperature[oC] Force[N] Fig. 4.9 Measured data and identified results of D x (T) [Vertical load:4600n] Tire Side Measured Identified Results Data[Plus] Data[Minus] 20 Tire Surface 40 Temperature[oC] Fig Measured data and identified results of D y (T) [Vertical load:4600n]

110 102 4 Stifnes[N] Braking Measured 40Tire 45 Surface 50Temperature[oC] 55 Identified Data 60Results65 Force/Tire Angle[N/deg] Tire Side S lip Measured Identified Results Data[+1deg] Data[-1deg] 20Tire Surface 30 40Temperature[oC] Fig Measured data and identified results of K x (T) [Vertical load:4600n] Fig Measured data and identified results of K y (T) [Vertical load:4600n]

111 K y D y 1 K y Combined Slip Magic Formula Combined Slip F x (α, S, F z ) = F x0 (S, F z ) G xα (α, S, F z ) (4.15) F y (α, γ, S, F z ) = F y0 (α, γ, F z ) G ys (α, γ, S, F z ) + S VyS (4.16) (4.11) F x (α, S, F z, T) = F x0 (S, F z, T) G xα (α, S, F z ) (4.17) F y (α, γ, S, F z, T) = F y0 (α, γ, F z, T) G ys (α, γ, S, F z ) + S VyS (4.18) Combined Slip Pure Slip (11) 4.5 MTS ( 1.3 ) 60km/h ( 1.3 ) Magic Formula

112 [m] (62) 4.13 Magic Formula 2sec 8sec Fig Measurement sequence for longitudinal force tire model and comparison between measured data and simulation results

113 Fig Simulation results of slip ratio vs. braking force using traditional model and developed model [Vertical load:4600n] -10 0[%] -10[%] 15[%]

114 106 4 Fig Measured data and simulation results of slip ratio vs. braking force [Vertical load:4600n] Fig Measured data and simulation results of slip ratio vs. tire surface temperature [Vertical load:4600n]

115 Angle[deg] sec sec Slip Time[sec] Fig Measurement sequence for lateral force tire model and comparison between measured data and simulation results sec 4sec Combined Condition Combined Condition / /

116 108 4 Force[N] Side Tire Measured Simulation Data Results Slip -10 Angle[deg] Fig Measured data and simulation results of slip angle vs. lateral force [Vertical load:4600n] Tire Surface oc] Temperature[ Measured 0 Simulation Data Results Slip -10 Angle[deg] Fig Measured data and simulation results of slip angle vs. tire surface temperature [Vertical load:4600n]

117 / Fig Test sequence for validating the development tire model under the combined condition (89)

118 110 4 Fig Measured data and simulation results of tire force and tire surface temperature under the combined condition

119 Magic Formula Magic Formula 4.22 ( ) Fig Steering angle of Fish Hook pattern test

120 112 4 Fig Lateral acceleration on Fish Hook pattern test comparing the different tire surface temperature Fig Simulation resutls of lateral acceleration on Fish Hook pattern test comparing the different tire surface temperature

121 Fig The second peak value of lateral acceleration comparing the test results and simulation results with the effect of tire surface temperature

122 114 4 [m/s 2 ] Magic Formula

123 115 5 (13) (16) Magic Formula (38) FIALA model (44) ( ) Fiala- Fiala 2 n (18)

124 116 5 (27) (30)(31)(32) n (90) Fiala SAT SAT (91)(92) Combined Slip (13)(14) (93) (15)(16) Magic Formula Fiala- (90) (2.73) p(x 1 ) F z (2.75) p(x 1 ) = 6F z wl x 1 l ( 1 x ) 1 l (5.1)

125 n (2.88) n = 4 n = 4 p(x 1 ) F z (27) p(x 1 ) = 5 ( ) F z l 4 l 5 w 2 ( x 1 l ) 4 2 (5.2) n F x1 = lh 0 C x ws x 1 dx 1 (5.3) l F x2 = µ d w p(x 1 )dx 1 (5.4) l h (2.87) p(x 1 ) p(x 1 ) = n + 1 n ( Fz wl D x1 ) gsp ; n, q l (5.5) D gsp (x ; n, q) = (1 2x 1 n ) [ 1 q(2x 1) ] (5.6) D gsp (x ; n, q) 5.1 n q p(x 1 ) l h C x S l h = µ s p(l h ) = µ s n + 1 n ( Fz wl D x1 ) gsp ; n, q l (5.7) p(x 1 ) µ s, µ d0, p(x 1 ), K x

126 118 5 Fig. 5.1 Generalised skewed parabola D gsp (t ; n, q) for description of contact pressure profile in circumference direction Fig. 5.2 Verification of new concept tire model (called Neo-FIALA tire model) comparing measured data (+) and model fitting results (solid line)

127 5.3. SAT µ S 5.3 Fig. 5.3 Analytical results of Neo-FIALA tire model parameters µ s µ d0 µ s µ d K x n q 5.3 SAT FIALA model

128 120 5 Fiala Fiala SAT (18) SAT (30) SAT ( ) SAT (91), (92) Direction of wheel handling α Moving direction of tire Contact patch O Share deformation Fig. 5.4 Contact patch of cornering tire when α (a) F y SAT M z F y = K y tan α = C ywl 2 tan α (5.8) 2 M z = A s tan α = ( Cy wl f ) xwl 2 tan α (5.9) 2

129 5.3. SAT 121 F y 0 Lateral force x 1 F x Longitudinal force 0 Contact patch α Tread base w εlf y 3 M z G mz l (a) Shear deformation of tread rubber (b) Tread base deformation by lateral force F y (c) Tread base rotation by SAT M z Fig. 5.5 Deformation of tread rubber and tread base during cornering at small slip angle α M z 1 SAT 2 SAT f x 2.7 Fiala F y = K y (tan α 1 3 ϵlf y ) (5.10) ( 5.5(b) ) SAT 5.5(c) α e α e = α M z G mz (5.11) α SAT FIALA model C y = C tr (5.12)

130 122 5 FIALA model approach Tread base deflection εlf - y 3 Slip angle α Slip angle correction α = α e - M z G mz (-) α e (-) Shear deformation of tread rubber (Calculation of lateral and longitudinal force profile at contact patch) F y M z Tread base rotation - M z G mz Fig. 5.6 Calculation flow of lateral force and SAT at small slip angle α ϵ = λ3 = (EIz ) k 4 y (5.13) 2k y G mz = k y πh 3 (5.14) α 5.7 V Contact patch (CP) α F x F y Fig. 5.7 The tire forces during cornering 5.2 n p(x 1 ) D gsp (t ; n, q) = (1 2t 1 n ) [ 1 q (2t 1) ] (5.15) p(x 1 ) = n + 1 ( Fz n wl D x1 ) gsp ; n, q l (5.16)

131 5.3. SAT 123 O p(x 1 ) SAT M z q x c /l q = C q M z (5.17) x c l = 1 2 ξm z l 2 (5.18) 2.3 FIALA model SAT l h F y lh F y (α) = w 0 f sy (α, l h ) = µ s p (l h ) (5.19) l f sy (α, x 1 )dx 1 + µ d w p(x 1 )dx 1 (5.20) l h 1 2 SAT M z lh M z (α) = w 0 + CP l f sy (α, x 1 ) (x 1 x c )dx 1 + µ d w p(x 1 )(x 1 x c )dx 1 l h f x (α, y 1 ) (y 1 y c )dx 1 y 1 (5.21) 1 2 SAT 3 SAT SAT α e = α M z G mz (5.22) q = C q M z (5.23) x c l = 1 2 ξm z l 2 (5.24)

132 r h = l h /l 2K y r h [ tan αe ϵlf y (1 r h ) ] = n + 1 n µ s F z D gsp (r h ; n, q) (5.25) 3. F y F y (α) = 2K y rh 4. SAT M z 0 [ t tan αe ϵlf y x 1 (1 x 1 ) ] dx 1 + n n µ df z D gsp (x 1 ; n, q)dx 1 r h (5.26) rh [ M z (α) = 12A s x1 tan α e ϵlf y x 1 (1 x 1 ) ] ( x 1 x ) c dx 1 0 l + n ( n µ df z l D gsp (x 1 ; n, q) x 1 x ) c dx 1 r h l +A x r h tan α e (5.27) A x = f x wl 2 /2 (5.28) Neo-FIALA SAT F y SAT M z F y µ s µ d M z M z 5.2 Combined Slip

133 5.3. SAT 125 Tread base deflection εlf - y 3 Slip angle α Slip angle correction α = α e - M z G mz (-) α e Shear deformation of tread rubber (Calculation of lateral and longitudinal force profile at contact patch) F y M z Change of contact patch pressure p(x 1 ) Tread base rotation - M z G mz Fig. 5.8 Calculation flow of lateral force F y and SAT M z

134 (a1) Tire A: Fy Fy(a) (kn) 4 2 Fya (Adhesive) Fys (Sliding) Fya +Fys a (deg) Mz(a) (kn m) 0.1 (a2) Tire A: Mz Mzy (Side force torque) Mzx (Longitudinal force Mzy +Mzx torque) a (deg) (a3) Tire A: p(x1) 400 (kpa) a = 0 a = 2 a = 6 a = x1 (cm) Fig. 5.9 Identified results of lateral force F y, SAT M z and circumferential contact pressure profile for Tire A

135 5.3. SAT (b1) Tire B: Fy Fy(a) (kn) 4 2 Fya (Adhesive) Fys (Sliding) Fya +Fys a (deg) Mz(a) (kn m) 0.1 (B2) Tire B: Mz Mzy (Side force torque) Mzx (Longitudinal force Mzy +Mzx torque) a (deg) (b3) Tire B: p(x1) 400 (kpa) a = 0 a = 2 a = 6 a = x1 (cm) Fig Identified results of lateral force F y, SAT M z and circumferential contact pressure profile for Tire B

136 (c1) Tire C: Fy Fy(a) (kn) 4 2 Fya (Adhesive) Fys (Sliding) Fya +Fys a (deg) Mz(a) (kn m) 0.1 (c2) Tire C: Mz Mzy (Side force torque) Mzx (Longitudinal force Mzy +Mzx torque) a (deg) (c3) Tire C: p(x1) 400 (kpa) a = 0 a = 2 a = 6 a = x1 (cm) Fig Identified results of lateral force F y, SAT M z and circumferential contact pressure profile for Tire C

137 5.4. Combined Slip Neo-FIALA Combined Slip Neo-FIALA Combined Slip (13), (14) 1. Pure Slip Combined Slip 2. Combined Slip Pure Slip SAT M z Combined Slip F x Combined Slip 1. n D gsp ( x 1 ; n, q) D gsp ( x 1 ; n, q) = (1 2 x 1 1 n ) [ 1 q (2 x 1 1) ] (5.29) 2. F x ( q x c ) x c l q = C q F x (5.30) = C xcf x (5.31)

138 α M z ( α e ) α e = α + M z G mz (5.32) 4. r h = l h /l [r h < 0 r h = 0 ] [ K x r h S 2 + { tan α e ϵlf y (1 (1 + S ) r h ) } ] 2 1/2 n + 1 = µ s F z D gsp (r h ; n, q) (5.33) n 5. θ ( θ = tan 1 tan α ) S (5.34) 6. µ d µ d (S, α, V) = µ d0 α v V { 1 + (S 2 1) cos 2 α } 1/2 7. F x (S, α, V) 1 r h (5.35) F x (S, α, V) = K x r 2 h S n n µ d(s, α, V)F z cos θ D gsp ( x 1 ; n, q)d x 1 (5.36) r h 8. F y (S, α, V) rh [ F y (S, α, V) = 2K y x1 tan α e ϵlf y (1 + S ) x 1 {1 (1 + S ) x 1 } ] d x n + 1 n µ d(s, α, V)F z sin θ 9. SAT M z (S, α, V) 1 r h D gsp ( x 1 ; n, q)d x 1 (5.37) rh [ M z (S, α, V) = 12A s x1 tan α e ϵlf y (1 + S ) x 1 {1 (1 + S ) x 1 } ] d x n + 1 ( n µ d(s, α, V)F z l sin θ D gsp ( x 1 ; n, q) r [ h +r h tan α e 4A s S (r h ) 2 n + 1 n µ d(s, α, V)F z l cos θ x 1 x c l 1 ) d x 1 r h D gsp ( x 1 ; n, q) 1 x 1 1 r h d x 1 ] (5.38)

139 ݽ µèùö ÓÖÒ Ö Ò Ú ËÐ Ò 5.4. Combined Slip Neo-FIALA 131 Û ÓÒØ ØÈ Ø Ë Ö ÓÖÑ Ø ÓÒ Ð Ð ÐØ Ø ÓÒܽ ݽ µ ÓÑ Ò ËÐ Ô ÓÒØ ØÈ Ø Ú ËÐ Ò Û Ë Ö ÓÖÑ Ø ÓÒ Ü½ м Ð ÐØ Ø ÓÒ Ý Ë Ö ØÖ ËØ Ø Ö Ø ÓÒ ÙÖÚ ËÐ Ò Ö Ø ÓÒ Ä Ø Ö Ð ÓÖ ¼ Ú Ð¼ Ð ËÐ Ò Ð Ü½ Fig Deformations of tread rubber and tread base during cornering at small angle α ϵ (a), during cornering and breaking simultaneously (b) and the lateral force profile

140 r h = l h /l [r h < 0 r h = 0 ] [ K x r h S 2 + { (1 S ) tan α e ϵlf y (1 r h ) } ] 2 1/2 n + 1 = µ s F z D gsp (r h ; n, q) (5.33 ) n 5. θ ( ) θ = tan 1 (1 S ) tan α S (5.34 ) 7. F x (S, α, V) F x (S, α, V) = K x r 2 h S + n n µ d(s, α, V)F z cos θ D gsp ( x 1 ; n, q)d x 1 (5.36 ) r h 8. F y (S, α, V) rh [ F y (S, α, V) = 2K y (1 S ) x1 tan α e ϵlf y t (1 x 1 ) ] d x n + 1 n µ d(s, α, V)F z sin θ 9. SAT M z (S, α, V) 1 rh [ M z (S, α, V) = 12A s (1 S ) tan αe ϵlf y x 1 (1 x 1 ) ] ( 0 n + 1 n µ d(s, α, V)F z l sin θ [ +r h tan α e 4A s S r 2 h n + 1 n 1 r h D gsp ( x 1 t; n, q)d x 1 (5.37 ) x 1 x ) c d x 1 l ( D gsp ( x 1 ; n, q) x 1 x c r h l µ d(s, α, V)F z l cos θ 1 ) d x 1 r h D gsp ( x 1 ; n, q) 1 x 1 1 r h d x 1 ] (5.38 ) F x F y SAT M z

141 5.4. Combined Slip Neo-FIALA 133 Õ Ü Ð ÓÒØ ØÔ Ø ÖÓÒع ÒÐ Ò Ø ÓÒÓ ÓÒØ Ø ÔÖ ÙÖ Ò Û Ö ¹ ØÓ ËÐ ÔÖ Ø Ë Î ÐÓ ØÝÎ ËÐ Ô Ò Ð ÓÖÖ Ø ÓÒ ÑÞ ÅÞ Ë Ö ÓÖÑ Ø ÓÒÓ ØÖ ÖÙ Ö Ç Ü ËÐ Ô Ò Ð ÐÙÐ Ø ÓÒÓ Ð Ø Ö Ð Ò ÐÓÒ ØÙ Ò Ð ÁÆÈÍÌ Ë Û ÐÐÖÓØ Ø ÓÒ ÐØ Ø ÓÒ ÅÞ Ð Ý ÓÖ ØÖ ÙØ ÓÒ Ò ÓÒØ ØÔ Ø ÑÞ Ä Ø Ö Ð ÓÖ Ý ÄÓÒ ØÙ Ò Ð ÓÖ Ü ÇÍÌÈÍÌ Ë Ð ¹ Ð Ò Ò ØÓÖÕÙ ÅÞ Fig Calculation of longitudinal force F x, lateral force F y and self-aligning torque M z under combined slip condition using Neo-FIALA model

142 Combined Slip Neo-FIALA Magic Formula Combined Slip SUV Pure Slip Combined Slip Combined Slip Pure Slip Neo-FIALA Magic Formula Pure Slip tire: 235/50R18 velocity: 80 km/h Longitudinal Force [N] Measured Fz = 2.9, 5.0, 7.1 kn 6000 Magic Formula Neo FIALA Slip Ratio [ ] Fig Results of slip ratio vs. longitudinal force under pure slip condition Neo-FIALA Magic Formula

143 5.4. Combined Slip Neo-FIALA tire: 235/50R18 velocity: 80 km/h 4000 Lateral Force [N] Measured Fz = 2.9, 5.0, 7.1 kn 6000 Magic Formula Neo FIALA Slip Angle [deg] Fig Results of slip angle vs. lateral force under pure slip condition ( ) 0.03 Neo-FIALA 2 Magic Formula Neo-FIALA n 5.16 SAT Magic Formula 15 SAT Neo-FIALA (2.62) (5.27) SAT 0

144 tire: 235/50R18 velocity: 80 km/h Self Aligning Torque [Nm] Measured Fz = 2.9, 5.0, 7.1 kn 150 Magic Formula Neo FIALA Slip Angle [deg] Fig Results of slip angle vs. self-aligning torque under pure slip condition SAT 0 SAT 5.4 Combined Slip Neo- FIALA Pure Slip Combined Slip Magic Formula Combined Model Pure Slip (Pure Slip Parameters) Combined Slip Combined Slip Parameters Combined Slip Magic Formula Tire Model Combined Slip Combined Slip

145 5.4. Combined Slip Neo-FIALA 137 Table 5.4 The structural parameters of Neo-FIALA tire model calculated by measured data under pure slip condition F z = F z = F z = 2.9[kN] 5.0[kN] 7.1[kN] K x (= K y )[kn] A s [knm/rad] µ s µ d [S = 0.0] µ d [S = 0.5] µ d [S = 1.0] n C q C xc [ C y kn/m 3 ] ϵ[1/knm] G mz [knm/rad] [ k y kn/m 2 ] [ EI z knm 2 ]

146 138 5 Combined Slip Combined Slip Magic Formula Pure Slip Neo-FIALA SAT Lateral Force [N] tire: 235/50R18 load: 2.9 kn slip angle: 2 deg velocity: 80 km/h Measured Magic Formula Neo FIALA Longitudinal Force [N] Fig Results of longitudinal force vs. lateral force under combined slip condition Neo-FIALA Magic Formula Neo-FIALA Pure Slip Combined Slip

147 5.4. Combined Slip Neo-FIALA tire: 235/50R18 load: 5.0 kn slip angle: 2 deg velocity: 80 km/h Measured Magic Formula Neo FIALA Lateral Force [N] Longitudinal Force [N] Fig Results of longitudinal force vs. lateral force under combined slip condition Lateral Force [N] tire: 235/50R18 load: 7.1 kn slip angle: 2 deg velocity: 80 km/h Measured Magic Formula Neo FIALA Longitudinal Force [N] Fig Results of longitudinal force vs. lateral force under combined slip condition

148 140 5 Self Aligning Torque [Nm] tire: 235/50R18 load: 2.9 kn slip angle: 2 deg velocity: 80 km/h Measured Magic Formula Neo FIALA Longitudinal Force [N] Fig Results of self-aligning torque vs. lateral force under combined slip condition Self Aligning Torque [Nm] tire: 235/50R18 load: 5.0 kn slip angle: 2 deg velocity: 80 km/h Measured Magic Formula Neo FIALA Longitudinal Force [N] Fig Results of self-aligning torque vs. lateral force under combined slip condition

149 5.4. Combined Slip Neo-FIALA 141 Self Aligning Torque [Nm] tire: 235/50R18 load: 7.1 kn slip angle: 2 deg velocity: 80 km/h Measured Magic Formula Neo FIALA Longitudinal Force [N] Fig Results of self-aligning torque vs. lateral force under combined slip condition

150 Neo-FIALA ( ) Neo-FIALA (15), (16) 5.23 Fz=2[kN], γ=0[deg] Fz=4[kN], γ=0[deg] Fz=8[kN], γ=0[deg] Fz=2[kN], γ=3[deg] Fz=4[kN], γ=3[deg] Fz=8[kN], γ=3[deg] Fig The shape of contact area and contact pressure without camber angle (upper) and with camber angle (lower), Tire size: 195/65R15 91H, Inflation pressure: 200kPa

151 5.5. Neo-FIALA n ( 5.24 ) 5. ( 5.25 ) 6. ( 5.26 ) 7. SAT γ γ e 2. η γ e = γ + ηf yγ G my (5.39) η = w l tan β = 2hw l 2 sin γ e (5.40)