J. Cent. South Univ. Technol. (2008) 15(s1): 210-214
DOI: 10.1007/s11771-008-348-5
Adhesion coefficient of automobile tire and road surface
LIU Chang-sheng(刘长生)
(School of Traffic and Tansportation Engineering, Central South University of Forestry and Technology, Changsha 410004, China)
Abstract: The adhesion coefficient of automobile tire and road surface was analyzed and the formula about it was derived. Some suggestions about highway construction, driving safety of the drivers and the judgment of the traffic accidents were presented. The results show that the adhesion coefficient is a function with the extreme value. If there is atmospheric pressure in the tire, the load of the vehicle and the degree of the coarse on the road surface is not selected properly, it will reach the least and affect the safety of the running automobile.
Key words: constitutive model; microstructure; mechanics element; connecting type; adhesion coefficient; pavement profile; tribology analysis calculation; comprehension evaluation
Many accidents are caused by wheel-slippage when cars run on road, although vehicle status is good, speed is in the allowable range, driving has no mistake. The reason is that there are too skidding pavements. The problems were discussed here from the tribological aspect.
1 Adhesion coefficient
To sure the car travel safely on road, enough friction (adherent) force is needed between the tire and road, it is represented by adhesion coefficient. Adhesion coefficient φ, is the ratio between the maximum tread tangent stress τmax and road surface normal stress PG[1]:
(1)
2 Friction between tread and road surface
According to tribology, tread and road surface are a pair of objects with contact motion, also called friction pair. In the friction pair, because of tread’s manufacturing error, lots of micro rugged place exist in the contact face, the ridges part is called micro-convexs, and the valleys part is called micro-valleys.
As the height of road surface’s micro-convexs is inconsistent, when vehicle has normal loading, tread and road surface contacting and just in lots of small block regions (called contact spots), and the sum area of all spots is called profile contact area AG, the vertical(normal) loads on profile contact area are called profile normal contact stress PG.
2.1 Frictional force between tread and road surface
As there is friction between the tread and the road surface, running tires are sure to produce frictional force. The frictional force can be taken as relative motion resistance between tread and road surface, direction of frictional force is opposite to that of tire moving. Friction between tread and road surface belongs to solid external friction. Force magnitude of external frictional is decided by tangent direction displacement of tire. So, before calculating adhesion coefficient, the frictional force T between tread and road surface must be determined at first. Analysis indicates that the frictional force between tread and road surface includes two parts: One part is decided by road surface’s roughness, road surface sorts, pollution (oil stain), wetness, tire material and pattern etc, which are called adhesion component TM; The other part is that the road surface’s micro-convex indents into tread surface rubber producing deformation resistance as tire moving, called displacement component TR. It should be noted that the two component affects each other. When total frictional force is calculated, the sum of the two components may be adopted[2]:
T = TM+TR (2)
2.2 Influence elements of frictional force
In order to increase the frictional force between tread and road surface, avoid skidding on the road, the tread has all kinds of pattern. The test result indicates that tread pattern has a very important effect on tire’s working performance.
Tread patterns are used to clear moisture between tread and road surface’s actual contact area, increase the adhesion component of total frictional force. As the tread pattern protrudes out the tread, the actual contact stress is obviously bigger than that without tread pattern. In addition, structure of tire, shape of tread pattern, loading of vehicle, driving torque and brake torque of wheel and tire pressure affect the contact stress’s magnitude and distribution.
3 Force analysis on tire and road surface
3.1 Force analysis on tire
Fig.1 shows force diagram of driving wheel when mobile runs at speed v. As the tire is filled with compressed air, it has certain internal pressure. Under the action of internal force, outer force and torque, it will produce deformation. In Fig.1, G is radial load on axle, N; P is horizontal driving force of axle acting on tire, N; τ is tangent reaction that road surface acts to tire, N; r0 is tire radius, m; rd is tire tangent radius, m; L is contacting length between tire and road surface, m; h is tire pressure to flat value, m; N is normal reaction force of road surface, m; ω is rotating angular velocity of tire; Mt is driving torque, N?m; Mp is braking torque, N?m; a is length of arm of force, m.
Fig.1 Mechanical analysis of mobile tire: (a) Driving; (b) Braking
It can be seen from Fig.1 that, for hysteresis of tire, normal reaction force N acting position moves forward a short distance a. But a product is not big, when calculating frictional force between tire and road surface or profile normal contact stress, the product may be ignored. Load of tire makes the bottom of tire crash flat and produce contact area. Flat value h = r0-rd. When h is small, profile contacting area is almost rectangle. Flat value can be calculated by the following formula[2]:
(3)
where pw is air pressure of tire, MPa; 
and 
are constants, for truck tire, 
MPa-1, for car tire, 
MPa-1, 
; 
is tread’s transverse curvature radius, m.
To calculate adhesion coefficient, adherent force should be confirmed at first. Adherent’s magnitude is decided by profile normal contact stress. For the height of road surface’s micro-convex is inconsistent, the changing rule of profile contacting area’s normal stress will be very complicate. According to Ref.[2], the stress is decided by tire’s structure and air pressure, but has no large change. So in profile contacting area AG, there is compacting contact state between road surface and tread. Then the adhesion coefficient can be calculated by the average normal contacting stress of the contact area. In this way, profile normal contacting stress should be:
(4)
where G is radial load acting on axle.
To avoid skidding, automobile tire’s tread is designed with many patterns, for the existence of tread pattern, the contact area between tread and road surface is more similar to one rectangle[2]. The width of rectangle is tread width B, length is l (shown in Fig.1(a)), and l can be calculated by △OAB shown in Fig.1:
As h is very small(when tire air pressure and vehicle loading belong to normal range, h is very small), h2 is smaller, and it can be ignored. So
(5)
For existing pattern on tire’s surface, the tread pattern surface contacts with road surface. So profile contact area AG should multiply the tread pattern density coefficient k′ (road vehicle tire’s pattern density coefficient is 65%-80%[2]),
(6)
Then substituting Eqns.(3) and (5) into Eqn.(6), profile contact area is:
Let 
, put it into the above formula, so
(7)
Substituting Eqn.(7) into Eqn.(4), profile normal contact stress is:
(8)
For 
, according to Eqn.(3), ＜＜ 1, 
, 
＜ ＜ 
, and 
, 
, through calculating, 
.
To a concrete car, its load, tire structure parameter and tire air pressure are fixed values. In this case, adherent’s magnitude is decided by frictional force between tread and road surface, and frictional force is decided by road surface’s roughness, and roughness is decided by road surface’s profile.
3.2 Road surface’s profile
Road surface’s profile is the road geometry profile saved after road surface’s construction, usually it can be shown by micro-convexs’s number, and micro-convexs’s geometry profile and height distribution. H is height of micro-convexs, L is road surface’s sampling length. Fig.2 shows the section profile of concrete and pitch. It can be seen that road surface’s micro-convexs are similar to those of cut spheres, which lie on some base line and distribute along the height direction distribution. Fig.3 shows road surface micro-convexs vertical section[3-12]. Where Rmax is the max micro-convex profile peak-valley interval, Rp is the max height of micro-convex profile, Ra is the arithmetic mean value of micro-convex profile peak-valley interval, tp is the length of micro-convex profile supporting line (the wide black line above base line). Arithmetic mean value is the sum of at least five micro-convexs’s height divided by micro-convexs number.
Fig.2 Profile of concrete and pitch road surface: (a) New concrete road surface; (b) Concrete road surface used for a long time; (c) Pitch road surface
Fig.3 Road surface micro-convexs longitudinal section
According to Ref.[3], the road surface specifications propose the standard value of roughness by micro structure depth of road surface (TD), i.e. for highway, arterial highway and pitch road surface, TD≥0.55 mm; for concrete road surface, TD≥0.8 mm; for other grade concrete road surface, TD≥0.6 mm[3].
For micro-convex is similar to cut sphere, actually TD only expresses the arithmetic mean value of micro-convexs profile peak-valley interval Ra, it can’t indicate the basic profile of whole micro-convex. This problem can be explained by Fig.4.
Fig.4 Vertical section profile sketch map of road surface micro- convexs: (a) Arc-shaped; (b) Trapezium-shaped; (c) Sharp triangular-shaped
In Figs.4(a), (b) and (c), the three kinds micro-convexs have the same height, but because their peak top’s curvature radius r and support plane’s size tp etc are unlike, it makes the contact stress between micro-convexs and tread, and the depth within the tread rubber layer, it can produce great different frictional force. Without considering these factors, the large different adhesion coefficients will appear between the theory calculation and the measured value. Then road surface micro-convexs’s geometry parameters Rmax, r, v, b and 
are taken as the assessing parameters. Among the several assessing parameters, v, b and can be calculated by Eqn.(9), other micro-convexs parameter can be referred in Fig.3.
(9)
where tm is arithmetic mean value of supporting line’s length in micro-convexs vertical section; r is the average curvature radius of the top of micro-convexs, 
; 
and 
are the average curvature radii of the top of micro-convexs’ transverse section and vertical section, respectively.
4 Calculation of adhesion coefficient
From Eqn.(2), adherent (friction) force is the sum of adhesion part and deflective part, in which, friction adhesion part is decided by actual contacting area tread and road surface material molecules interaction. Frictional force deflective part is decided by road surface micro-convex that is pushed into road surface so as to produce hysteretic loss when deformation happens.
For single micro-convex, according to Eqn.(2), the adhesion can be written as:
(10)
where[2, 7]
TMi and TBi are single micro-convex frictional force’s adhesion part and deflective part, respectively; τ0 and β are friction parameters, which are decided by road surface condition (pollution, humidity), for rubber material, τ0=2.5 MPa, β=0.03 or β=0.05; pr is actual contacting stress, 
; 
is the depth of single micro-convex pushed into tread rubber, 
; aeff is tire rubber’s lag loss coefficient in complicated condition state, aeff=2.5a; a is lag loss coefficient in draging and pushing state, for rubber material, it is 0.09-0.13; 
is material’s elastic constant, 
.
The general friction (adhesion) of tread and road surface is[7]:
(11)
where 
is any section micro-convex quantity pushed into the equal depth.
To normal road surface, tangent stress (unit frictional force) is[2, 13]:
(12)
To typical road surface (concrete and pitch road surface, 
, 
) and automobile tread rubber material(elastic coefficient E=300-400 MPa, Poisson ratio
), tangent stress is:
(13)
When the radial loading and normal reaction force are equal, 
, adhesion coefficient equals the static friction coefficient.
Substituting Eqn.(13) into Eqn.(1), adhesion coefficient of tread and concrete and pitch road surface can be gotten:
(14)
According to Eqns.(8) and (14), there is
(15)
In Eqn.(15), the first and second parts of adhesion coefficient 
are adhesion part, the last part is deflective part. So, adhesion coefficient is decided mainly by road surface state (frictional parameter 
and 
), tread rubber property (elastic coefficient E, lag loss coefficient aeff), and it is also decided by tire geometry parameter (
, B, r0, ), tire air pressure Pw, wheel normal loading G and road surface’s roughness parameter integration value .
5 Ways to improve adhesion coefficient
5.1 Improving frictional force
In dry concrete and pitch road surface, adhesion coefficient’s deflective part is very small, which can be ignored[2]. From Ref.[4], tire is rolling on the wet road surface, the moisture on the road surface can be viewed as lubricant. As lubricant exists, the frictional force between tread and road surface should decrease, and the skidding and rolling proportion will increase distinctly at last, and the first and second parts’ value in Eqn.(15) will decrease evidently and tire skidding appears on the road surface. From Ref.[2], tire rolling on the wet surface has few influence on the last part in Eqn.(15). Then if it needs to keep enough adhesion coefficient on the wet road surface, the deflective part should be increased to compensate the decrease of adhesion part. To increase deflective part, the ways are: 1) increasing road surface roughness; 2) increasing tire air pressure; 3) increasing mobile loading, etc.
The first way can be done such as constructing high roughness road surface escarpment and crooked road, or laying gunny bags, grass shades in raining or snowy weather etc. The second and third ways would be completed by driver.
5.2 Avoiding the smallest value of adhesion coefficient
Eqn.(15) is a extremum function, , , , B, r0, , 
and E are variable, but to one type tire and road surface state(pollution, humidity), they are fixed. Pw, G and are controlled by road construction and driver, and have a wide changing range. Therefore, adhesion coefficient will change greatly along with the change of the three parameters. The extremum function can be used to prove that the adhesion coefficient will reach the smallest value.
Tire air pressure[2] is:
(16)
The loading of wheel is:
(17)
Road surface’s roughness is:
(18)
Eqns.(16), (17) and (18) show that, for the same type tire to different carrying capacity cars, for example 2.5 t, 5 t and 8 t, their tire air pressure should be different, carrying capacity i.e., the weight of goods and passenger, is also an important factor affecting friction cohesion of tire and road surface, if without good control. It can also decrease friction coefficient and accident may happen because of tire and road surface skidding.
6 Conclusions
1) When automobile runs on the road surface, as adhesion coefficient is not enough to produce skidding, accident will happen. The responsibility cannot be judged simply.
2) Adhesion coefficient appears in one of the three states, automobile may skid because adhesion coefficient is in the minimal value.
3) Concrete stiff road surface’s profile has no essential change after long time use, road profile has no great change.
References
[1] YU Zhi-sheng. Automobile theory[M]. Beijing: Machinery Industry Press, 2000: 16-18, 68. (in Chinese)
[2] KRAGELSKY I V, ALISIN V V. Friction, wear, lubrication handbook(Ⅱ)[M]. Moscow: Machine Manufacture Press, 1979: 50-55, 84-86.
[3] LU Ding-zhong, CHEN Jia-ju. Road base and road surface engineering (the second edition)[M]. Shanghai: Tongji University Press, 1999: 235-238. (in Chinese)
[4] NEALE M J. Tribology handbook[M]. London: Newnes- Butterworths, 1975.
[5] LI Jiang. Mechanical traffic accidents[M]. Beijing: Machinery Industry Press, 2000: 29-30. (in Chinese)
[6] WANG Jun-min, ALEXANDER L, RAJAMANI R. Friction estimation on highway vehicles using longitudinal measurements[J]. Journal of Dynamic Systems, Measurement and Control, 2004, 126(6): 265-275.
[7] TAEHYUN S, DONALD M. Model-based road friction estimation[J]. Vehicle System Dynamics, 2004, 41(4): 249-276.
[8] URBAKH M, KLAFTER J, GOURDON D, et al. The nonlinear nature of friction[J]. Nature, 2004, 430(6999): 525-528.
[9] STEFFEN M, MICHAEL U, KARL H. Estimation of the maximum tire-road friction coefficient[J]. Journal of Dynamic Systems, Measurement and Control, 2003, 125: 607-617.
[10] CIRACI S, BULDUM A. Atomic-scale study of friction and energy dissipation[J]. Wear, 2003, 254(9): 911-916.
[11] TAEHYUN S, DONALD M. Model-based road friction estimation[J]. Vehicle System Dynamics, 2004, 41(4): 249-276.
[12] YAMAZAKI S, FURUKAW A O, SUZUKI T, et al. Study on real time estimation of tire to road friction[J]. Vehicle System Dynamics Supplement, 1997, 27: 225-233.
[13] STEFFEN M, MICHAEL U, KARL H. Estimation of the maximum tire-road friction coefficient[J]. Journal of Dynamic Systems, Measurement and Control, 2003, 125: 607-617.
(Edited by CHEN Can-hua)
Received date: 2008-06-25; Accepted date: 2008-08-05
Corresponding author: LIU Chang-sheng, Professor; Tel: +86-731-5623728; E-mail: liuchangsheng0@163.com
