As shown inFigure 2.33, the tire is rotating with an angular velocity,u, while traveling in a di- rection that forms an angle ofbto the rotation plane. The velocity component in the rotation plane is taken asu. Three forces act upon this tire, namely the longitudinal force,Fx, lateral force,Fy, and vertical force,Fz.
2.4.2.1 Braking
Figure 2.34shows how the front endpoint of the tire contact surface centerline is taken as the origin of the coordinate axes, with thex-axis in the longitudinal direction, and they-axis in the lateral direction. The point on the tread base directly on top of point O is taken as point O0. After a fraction of timeDt, the contact surface point moves from O to P, and the point O0on the tread base moved to P0. The projected point P0on thex-axis is marked as P00.
DuringDt, the distance in thex-direction of the contact point from point O is thex-coordinate of point P:
xẳuDt (2.43)
Thex-coordinate of point P0from point O0is as follows:
x0ẳR0uDt (2.44)
FIGURE 2.34
Tire deformation in contact plane.
FIGURE 2.33
Tire forces in three directions.
Therefore, the relative displacement of point P and point P0, i.e., the deformation of the tread rubber, is as follows:
xx0ẳuR0u u uDt
ẳsx
ẳ s 1sx0
(2.45)
whereby,sis the tire slip ratio in the longitudinal direction:
sẳuR0u
u (2.46)
and, the distance in they-direction of the contact point, from point O, i.e., they-coordinate of point P, is as follows:
yẳxtanbẳtanb
1sx0 (2.47)
Since there is no displacement of point P0in they-direction, the previous is the deformation of tread rubber in they-direction.
Therefore, the forces per unit length and width, acting on point P, in thex-direction andy- direction, respectively, aresx,sy:
sxẳ Kxðxx0ị ẳ Kx s
1sx0 (2.48)
syẳ Kyyẳ Ky
tanb
1sx0 (2.49)
The sign of these forces is taken as opposite to the axes direction. Furthermore, the resultant force magnitude is as follows:
sẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2xþs2y q
ẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2xs2þK2ytan2b
q x0
1s
(2.50)
WherebyKxandKyare the longitudinal and lateral tread rubber stiffness per unit width and unit length. When the tire longitudinal slip ratio and side-slip angle are produced, tire defor- mation occurs. As a result, a distribution of the contact surface force, proportional to x0, is generated at the tire contact surface.
Assuming a tire pressure distribution that is the same as that described inSection 2.3.1gives the following:
pẳ6Fz
bl x0
l
1x0 l
(2.51) As shown inFigure 2.35, the tire contact surface force is given byEqn (2.50)in the adhesive region denoted by 0x0x0s, andxx0sin the slip region, the tire contact surface force is given bymp.
In the adhesive region, the forces acting at the contact surface in thexandydirections aresx
andsy. In the slip region, the forces arempcosqandmpsinq. Here,qdetermines the direction of the tire slip.
SubstitutingsẳmpintoEqns (2.50) and (2.51)to findx0sand assuming a dimensionless variable,xs, gives the following:
xsẳx0s
l ẳ1 Ks
3mFz
l
1s (2.52)
where, if 13mFKsz1sl <0 thenxsẳ0, and the following is obvious:
lẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2þ
Kb
Ks
2
tan2b s
(2.53) Ksẳbl2
2 Kx;Kbẳbl2
2 Ky (2.54)
From the previous, the overall forces acting on the whole tire contact surface in thexandy directions are expressed as follows:
Whenxs>0, for a contact surface composed of adhesive and slip regions, the following results:
Fxẳb 0 B@
Zx0s
0
sxdx0þ Zl
x0s
mpcosqdx0 1
CA (2.55)
Fyẳb 0 B@
Zx0s
o
sydx0þ Zl
x0s
mpsinqdx0 1
CA (2.56)
And whenxsẳ0, the contact surface consists only of a slip region:
Fxẳb Zl
0
mpcosqdx0 (2.55)0
Fyẳb Zl
0
mpsinqdx0 (2.56)0
FIGURE 2.35
Force distributions in contact plane.
SubstitutingEqns (2.50)–(2.52)intoEqns (2.55), (2.56), (2.55)0, and (2.56)0givesFx,Fyin the following forms:
Whenxs>0, then,
Fxẳ Kss
1sx2s6mFzcosq 1
61 2x2sþ1
3x3s
(2.57) Fyẳ Kbtanb
1s x2s6mFzsinq 1
61 2x2sþ1
3x3s
(2.58) And, whenxsẳ0, then,
Fxẳ mFzcosq (2.57)0
Fyẳ mFzsinq (2.58)0
The direction of the slip force,q, is approximated by the slip direction at the slip start point.
tanqẳKytanb 1sx0 Kx s
1sx0 ẳKbtanb
Kss (2.59)
Therefore, the following results:
cosqẳs
l (2.60)
sinqẳKbtanb
Ksl (2.61)
Ks, as defined byEqn (2.54), is equivalent to the total longitudinal force per unit longitudinal slip ratio whens/0 atbẳ0.Kbis equivalent to the total lateral force per unit side-slip angle whenb/0 atsẳ0. This can be derived from the definition ofKxandKyby integrating the forces at the contact surface when the both slips are very small andsorbẳ0.Equations (2.57) and (2.58)confirm thatðvFx=vsịsẳ0;bẳ0ẳKsandðvFy=vbịsẳo;bẳ0ẳKb. Hence, ifFxandFyare used numerically in simulations with the model described here, it is practical to determine experimentally the value ofKs,Kb, depending on the tire vertical load.
The friction coefficient,m, is a function ofFzand the slip velocity,Vs, so an experimental equation that reflects the dependence ofmon tire load and slip velocity is desired. Here,Vsis defined as follows:
Vsẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðuR0uị2ỵu2tan2b q
ẳu ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2þtan2b p
(2.62) Thus, the tire longitudinal and lateral forces can be obtained numerically as functions of the longitudinal slip ratio,s, slip angle,b, tire load,Fz, and tire traveling speed,u.
FxẳFxðs;b;Fz;uị
FyẳFyðs;b;Fz;uị (2.63)
2.4.2.2 Accelerating
As in the case of braking, the deformation of tread rubber to the tread base at the contact surface gives the following:
xx0ẳuR0u
R0u R0uDtẳsx0 (2.64)
yẳxtanbẳ ð1ỵsịtanbx0 (2.65) wheresis tire longitudinal slip ratio during acceleration:
sẳuR0u
R0u (2.66)
Then, the following results:
sxẳ Kxsx0 (2.67)
syẳ Kyð1ỵsịtanbx0 (2.68)
sẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2xs2ỵK2yð1ỵsị2tan2b q
x0 (2.69)
As with braking, the point where the contact surface changes from the adhesive region to the slip region is found from the following:
xsẳx0s
l ẳ1 Ks
3mFzl (2.70)
where, if 13mFKszl<0, thenxsẳ0, and as follows:
lẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2þ
Kb
Ks
2
ð1ỵsị2tan2b s
(2.71) The forces acting in thex-direction andy-direction on the tire contact surface during ac- celeration are derived as follows:
Whenxs>0, then,
Fxẳ Kssx2s6mFzcosq 1
61 2x2sþ1
3x3s
(2.72) Fyẳ Kbð1ỵsịtanbx2s6mFzsinq
1 61
2x2sþ1 3x3s
(2.73) And whenxsẳ0, then,
Fxẳ mFzcosq (2.72)0
Fyẳ mFzsinq (2.73)0
where,
tanqẳKyð1ỵsịtanbx0
Kxsx0 ẳKbtanbð1ỵsị
Kss (2.74)
cosqẳs
l (2.75)
sinqẳKbtanbð1ỵsị
Ksl (2.76)
and the slip velocity,Vs, is as follows:
Vsẳu
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2
ð1ỵsị2ỵtan2b s
(2.77)
The longitudinal and lateral forces that act on a tire with side slip, while under braking or acceleration, are shown in Figure 2.36, which is plotted by using Eqns (2.57)–(2.58)0 and (2.72)–(2.73)0. It is clear from the theoretical analysis here that braking and acceleration affects the tire cornering characteristics, as explained inSection 2.3.2.