7.3 Stationary Tire Contact Forces
7.3.3 Tires Under Longitudinal (Circumferential) Forces
Adhesion coefficient Fundamentally two physical effects are responsible for the transmission of longitudinal forces in the contact area (Gillespie1992) (Fig.7.5):
• Traction (adhesion) through adhesion friction in the contact patch: surface adhesion occurs due to the intermolecular bonding forces between tire rubber and the material of the road surface. While the effect dominates the force generation between the tire and the road surface on dry roads, it significantly reduces on wet roads.
• Hysteresis friction: this leads to (positive) locking through the meshing effects between the tire contact patch (tread) and the road surface. This effect is due to the viscoelastic material characteristic of the tire rubber. The large damping of the rubber in the contact area results in a high hysteresis friction coefficient.
This effect is much less affected by water on the road surface.
Both effects depend on small relative motion between the contact partners tire tread and the road surface of the contact area. The ratio between the longitudinal force Fx and the normal force Fz is defined as the adhesion coefficient in the longitudinal direction
lẳFx Fz
: ð7:6ị
Fig. 7.4 Description of the rolling resistance
road surface
H A
hysteresis adhesion
contact patch Fig. 7.5 Description of the
forces between road and tires
The formation of the longitudinal forces can be described through the shear deformation of the tread in conjunction with the friction behavior between the tread and the road. This requires, however, a macroscopic description of this shear deformation, which can be incorporated into, for example, the multibody system formalism. The kinematics of the shear deformation will be described through the term longitudinal slip.
Circumferential slip (longitudinal slip)The circumferential slip is a kinematic quantity, which describes the state of motion of a driven, braked or non-driven rolling wheel. The wheel here is assumed to be a rigid body. This is why it is called also rigid body slip.
In a planar motion of a rigid wheel one distinguishes fundamentally between two states of motion
• pure kinematic rolling without sliding and
• combined rolling and sliding,
Table7.2. In vehicles the second case is relevant. First the motion of a planar wheel with the variables
• wheel radiusr,
• velocity vof the wheel center point,
• velocity vP of the (fictional) wheel contact point,
• angular velocity xof the wheel
will be investigated. In an ideal rolling wheel the velocityvP of the contact patch would be vanishing. In a real wheel (tire) this is not the case. The deviation, relative to the wheel speed is referred to as the slip. The acceleration slipsA and the brake slipsB are differentiated below. To this end the value of the relative velocity of the fictional tire contact point P is given relative to the larger of the two valuesvandxr. For a driven wheel the acceleration slip ðv\xrịis given by
sAẳvP
xrẳxrv
xr ð7:7ị
and for a decelerating wheel the brake slipðv[xrịis given by:
sBẳvP
v ẳvxr
v : ð7:8ị
Summarizing one can write:
sA;Bẳ j jvP
maxðv;xrịẳ jvxrj
maxðv;xrị: ð7:9ị
The slip is mostly given in percentage, i.e. for examplesAẳ90% instead of sAẳ0:9. The slip definition used here guarantees that the value of the slip in the extreme conditions for blocked wheel ðv6ẳ0;xẳ0ị and for a spinning wheel ðvẳ0;x6ẳ0ị takes the value of one. Thus the slip is normalized within the
interval [0, 1]. Special considerations require the cases of startup and motion at very low velocities. This is dealt with in detail in Schuster (1999). There can be found several other definitions of slip in the literature, Ref. for example (Rill1994) and again (Schuster1999). This is to be considered when using simulation soft- ware. This is especially true for the limiting cases of blocked and spinning wheels.
All definitions are similar however in that the slip of the rolling wheel is zero.
In the following these definitions of slip will be applied to radial tires. To this end the dynamic wheel radius determinable from the measurements of the rolling circumference will be first defined.
Rolling circumference and dynamic tire radiusThe rolling motion of a radial- ply tire is significantly different from that of a cross-ply tire, which is no longer relevant in modern passenger vehicles. The belt of a radial tire behaves as a non- elastic band along the outer tire structures due to the non-elastic nature of the steel wires, when compared with the surrounding rubber. Even under the most extreme loading conditions that can be encountered, the longitudinal elongation of the belt is under 1 % (Reimpell and Sponagel1988). It is therefore justified in assuming the rolling circumferenceU, in normal operating range, is a constant. The rolling circumferenceUis unwound on the road during one revolution. This quantity will be customarily measured on a tire towed with a velocity of 60 km/h, without drive or brake forces, and will be given in tire tables (DIN1986). The rolling circum- ferenceUdefines the dynamic tire radius
rdynẳ U
2p: ð7:10ị
Hence rdyn is the radius of an imaginary wheel-fixed circle, which on rolling over one revolution delivers the measured rolling circumference U. As the lon- gitudinally stiff belt lies, not on the surface, but below the carcass of the tire, the dynamic tire radiusrdyn is smaller than the manufacturing radiusr0. In belted tires the dynamic tire radius rdyn is closer to the construction radius r0 than to the statistical tire radiusrstat. The relationship between the radiir0,rstatandrdyncan be estimated using a simple geometric consideration. To this end, one compares the Table 7.2 Rigid body slip
vẳxr v\xr v[xr
Rolling wheel Driven wheel Spinning wheel Braked wheel Blocked wheel
No slip Drive slip Brake slip
sAẳ0 sBẳ0
sAẳvP
xr
ẳxrv xr
sAẳ1 sBẳvP
v
ẳvxr v
sBẳ1
unwinding length of an imaginary rigid disc, with a radius rdyn, with the half contact patch lengthL=2 (Fig.7.6):
Uẳrdynaẳr0sina!rdynẳr0
sina
a r0ð1a2
6ị: ð7:11ị If one now also considers the geometric relationship (see Fig.7.3)
rstatẳr0cosar0 1a2 2
!a2
2 ð1rstat
r0
ị ð7:12ị
and substitutes this relationship in Eq. (7.11), one obtains rdynr0 1a2
6
r0 11 3þrstat
3r0
ẳ2 3r0þ1
3rstat: ð7:13ị The wheel fixed circle with radiusrdynis hence the herpolehode of the motion.
The imaginary road-fixed straight line on which the tire unwinds is called the polehode. The instantaneous contact point M of the herpolehode and the polehode is the instantaneous center of rotation. One should note that the polehode lies under the road surface. In Table7.3 are a few examples for statistic and dynamic tire radii are given (Reimpell and Sponagel1988).
The difference between the statistic and dynamical radii is not, or rather is not primarily, due to the expansion of the tire belt due to the centrifugal forces, which are relatively small.
Circumferential slip of a wheel with belted tiresFrom now on, the rolling of the tire with a radiusrdynwill be assumed to be slip-less, i.e. it holdssA;Bẳ0. As such, a compromise will be accepted, that this normalization of the slip is not absolutely correct, as the dynamic tire radius is valid only for the specific velocity, at which the experiment was performed. The advantage is however, that the state slip-free is simple and comprehensible using a single standard experiment, without intro- ducing or considering contact mechanical processes in the contact patch. Fur- thermore it allows the definition of the slip of a rigid tire:
drive Slip v\xrdyn
ffl
: sA ẳxrdynv xrdyn
ð7:14ị
brake Slipfflv[xrdyn
: sBẳvxrdyn
v ð7:15ị
Summarizing one can write:
sA;Bẳ vxrdyn
maxðv;xrdynị: ð7:16ị
This kinematic analogous model of the slip-less rolling wheel is extended such that the rolling of the treads on the road is visible and describable. This helps to visualize and subsequently formalize the emergence of the circumferential forces on the braked, and the driven wheel respectively. The belt, with the tread, is considered here to be a circumferentially stiff closed band with a circumference Uẳ2prdyn;see (Ammon2013). Comparable to the tread of a tracked vehicle, this band is considered in the upper areas of the rigid wheel disc as a materialized herpolehode with a radiusrdynand in the patch area through an imaginary guidance plate parallel to the road surface (patch length L, entering point E, and exiting point A). Since the wheel disc, in the lower region of the guidance plate, would penetrate the circumferential belt and the road surface, it has to be dematerialized again in this region.
The treads carried by the belt are represented through elastic beam elements and profile stud elements respectively (Brush Model, Fig.7.7). The wheel center has a velocityv, and the wheel disc has an angular velocityx. Hence the belt runs with a longitudinal velocity of xrdyn. The belt section EA has, due to the assumed stiffness of the belt, in the patch the absolute velocity (sliding velocity):
vPẳvxrdyn: ð7:17ị This slip-less motion of the wheel, with a belted tire, is thus plausibly repre- sented using this model:
• Slip-less motion: vẳxrdyn!vPẳ0: The belt section EA is at rest. There occurs no shear or sliding motion of the profile elements.
herpolehode (circle)
polehode (straight line) Fig. 7.6 For the definition of the dynamic tire radius
Table 7.3 Examples of statistical and dynamical tire radii
Tires r0=mm rstat=mm U=mm rdyn=mm
185/65R15 311 284 1,895 302
195/65R15 318 290 1,935 308
205/65R15 324 294 1,975 315
• Driven wheel v\xrdyn!vP\0: There is an acceleration slip as per (7.14).
The profile elements in the contact patch are sheared to the front.
• Braked wheel: v[xrdyn!vP[0: There is a brake slip as per (7.15). The profile elements will be sheared to the back.
Force transmission in the circumferential directionTo explain the principle of the development of the circumferential forces, the motion of a single profile ele- ment through the patch area EA will first be investigated. The case of a driven wheel will be assumed exemplary:v\xrdyn. This view is also valid for a braked wheel (Fig.7.7).
Motion and shear deformation of a profile elementOne observes for the time being a single profile element on its path through the contact patch Fig.7.8. This profile element reaches at timet0ẳ0 the entry point Eðxẳ0ịof the contact area EA. Accordingly, it will be transported through the patch with the constant (acceleration-free motion of the wheel) circumferential velocityxrdyn[vof the belt. At timetthe profile will then be found at the position in the contact patch at:
xẳxrdynt: ð7:18ị A probable assumption is now, that the regarded profile element at the entry into the contact surface at time tẳ0, and also for a certain time after that, is adhered to the road surface. As the belt section EA has the absolute velocity vPẳxrdynvopposite to the driving directionv, this element will be deformed in the horizontal direction through a stretch^u tð ị:
^
u tð ị ẳvPtẳfflxrdynv
t: ð7:19ị
If one now represents the time t through x as in (7.18) and substitutes the driving slipsAfrom (7.14), one arrives at a linear relationship withx, with gradient sA, resulting in an increase of the deformation in the direction of the entry point E:
u xð ị ẳfflxrdynv x xrdyn
ẳsAx ð7:20ị
the element reaches the exit point AðxẳLịafter a time Tẳ L
xrdyn
ð7:21ị
under the patch and experiences the maximum deformation there
umaxẳu Lð ị ẳsAL: ð7:22ị Tangential stress on the profile elementsIf one assumes a linear-elastic behavior of the deformation of the profile element, then the profile element transfers a
shear stress in the length dx and breath dy, that is proportional to deformation uðxị:
sxðxị ẳkuðxị ð7:23ị with a corresponding constantk. The constant depends on
• the shear modulus of the elastic material,
• the height of the tread and
• the profile.
The assumption of a continuous stress distribution over the entire path surface does not consider any tire profile and is thus only an approximation.
The resulting circumferential forceFxtransferred by all the profile elements in the contact surface A is, under the assumption of a constant patch breadthb,
Fxẳ Z
A
sxð ịx dAẳ ZL
0
sxð ịb dxx ẳ1
2kbL2sA: ð7:24ị The forceFxis proportional to the triangular surface under the tangential stress distribution. The relationships (7.23) and (7.24) are only valid when the profile element is in adhesion over the entire length of the patchL. This is not generally applicable as through the friction behavior certain deviations have to be considered.
Adhesion through COULOMB’s frictionFor the adhesion between the profile elements and the road surface a coulomb friction will be assumed. Let the normal
P
unstreched belt
profile elements
Fig. 7.7 Mechanical analogous model for a belted tire per (Ammon2013)
pressure distribution pzð ịx be constant over the contact patch breath b and decreasing to the points E and A. The resulting vertical force (tire load) is then
FZẳ Z
A
pZð ịx dAẳ ZL
0
pZð ịb dx:x ð7:25ị
For the tangential stresssxð ịx the adhesion condition with the adhesion coef- ficientlH:
sxð ị x sxHð ịx with sxHð ị ẳx lHpZð ị:x ð7:26ị If the tangential stress surpasses the adhesion boundary at a point x, then the profile element begins to slide at this point. Therefore, instead of an adhesion stress, a sliding stress will occur
exit
bzw.
Entry
or
−
−
−
Fig. 7.8 Path of a profile element in a tread
sxGð ị ẳx lGpZðxị ð7:27ị with the sliding coefficientlG. For a friction pair rubber-asphalt the following rule applies
lG\lH!sxGð ị\sx xHð ị ẳx lHpZð ị:x ð7:28ị The contact area EA is thus divided into an adhesion zone EG and a sliding zone GA. With increasing slip sA the adhesion boundary G moves from the exit point A to the left. The resulting transferred circumferential force relates to the surface under the resulting tangential stress distribution, see Fig.7.9.
Slip-force curveThe transferred circumferential forceFxcan only be described in relation with the circumferential slipsA;B. As a general rule, this results in a similar curve, that is however different, dependent on the tires, with respect to the max- imum and the slope.
The typical curve force of this slip-force curve can be described, based on the preliminary work done in this chapter, as follows:
For small slip values the adhesion region extends almost over the entire patch length. The circumferential force Fx increases linearly thereafter, due to the assumptions made and according to the Eq. (7.24). It is therefore given as
Fxẳcss ð7:29ị with the circumferential stiffness of the tire
csẳ1
2kbL2: ð7:30ị
If the longitudinal slip increases further, the sliding region increases dispro- portionately and the resulting forceFxðsịbetween the road and the tire exceeds the maximum possible force Fx;max. This maximum value defines the maximum adhesion coefficient (Fig.7.10).
lmaxẳFx;max
FZ
: ð7:31ị
This maximum force adhesion coefficientlmax is significantly smaller than the adhesion coefficientlHof the material pair rubber and asphalt, that can be as high as 2 under the right conditions. It therefore always holds that:
lmax\lH: ð7:32ị The reason for this is that the maximum value of the adhesion friction in the patch can be achieved only in the transition zone between the adhesion and sliding friction. At very high slip values s the transferred circumferential force Fx
decreases to the value Fx;G, which occurs during pure sliding. That is the case when the wheel, either blocks during braking or spins during acceleration. The force adhesion coefficient corresponds to the sliding friction coefficient
lGẳFx;G FZ
: ð7:33ị
Characteristics of the circumferential curve are:
• the initial slopecsẳ dFdsx
sẳ0,
• the slip value smax and the corresponding maximum value of the slip force Fx;max,
• as well as the sliding forceFx;GẳlGFz. moderate slip
small slip high slip
sliding stress
adhesion limit
circumferential stress
adhesion sliding adhesion sliding adhesion sliding
Fig. 7.9 Circumferential stress in the slip area according to Ammon (2013)
sliding instable
region transition
adhesion/sliding linear
slope, adhesion
linear slope
Fig. 7.10 Typical curve of the circumferential
When no lateral forces need to be transmitted, then the circumferential slip curve will be influenced by the following variables:
• Constructive tire parameters (material, construction type, measurements).
A significant influence is the shear stiffness of the tread strip. With increasing stiffness, thesmaxmoves towards smaller values of slip, whereas, as a rule, the elevation oflmax=lGincreases, its breath however decreases. The characteristic form of the slip curve is mostly determined through the material, the con- struction and the measurements of the tire. The two other externally influencing parameters are the tire loadFzand the force adhesion coefficientslmaxandlG, which distort the fundamental form in different ways.
• Tire normal load Fz: increasing the tire load Fz leads to the most part to a change of scale of the ordinates of the slip curve. i.e. the position of the maximum is preserved. This can, on the one hand, be explained that with increasing tire load Fz the elastic deflection f and with it the patch length L increases, which in turn results in a bigger shear stiffness of the patch and hence leads to an increase in the initial slopecsofFxð ịs :On the other hand, the force adhesion coefficientslmax andlGremain principally the same. Through this the limits of the transferable circumferential forces increases proportion- ally to Fz: Up to a statistical operating tire load the forcesFx;max resp. Fx;G change proportionally to Fz. With increasing tire load the valuesFx;max resp.
Fx;G only increase slightly degressive. This is due to a deterioration of the friction coefficients resulting from the deformation of the tire structure. Due to this approximate proportionality betweenFxandFz, one often plots the ratio Fx=Fz over the slips.
• Maximum force adhesion coefficient lmax and the sliding friction coefficient lG: if lmax andlG were modified proportionally, then the abscissa and the ordinate will be scaled similar to stretching, with respect to the coordinate origin. The initial slope of Fxð ịs will be defined primarily through the shear stiffness of the tread and hence will remain the same. The limitFx;maxhowever, will move itself in the direction of the larger slip values, see Fig.7.11.
A fundamental problem in the modeling of the tire behavior is that the tire parameters vary within broad limits even in normal driving situations.