Where the tire is in contact with the road, different solutions are found in each ... WfM = 0. (3). Because the ring is assumed to be inextensible, the circumferential.
K. Yamagishi Research Engineer, Bridgestone Tire Company, Tokyo, Japan.
J. T. Jenkins Associate Professor, Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, N. Y. 14853 Mem. ASME
Singular Perturbation Solutions of -the Circumferential Contact Problem for the Belted Radial Truck and Bus Tire
Introduction Here, a singular perturbation technique is used to obtain asymptotic solutions to the differential equations, boundary conditions, and continuity conditions which govern the circumferential behavior of a model steel belted radial tire. The formulation of the model is discussed in detail in a previous paper [1]. The belt is assumed to be inextensible and the tread region of the tire, consisting of the hard rubber in which the belt is embedded and the rubber of the tread, is idealized as a circular ring with a rectangular cross section which resists changes in curvature. The side wall of the tire is assumed to be an inflated membrane, with simple geometry and material properties, which, as a consequence of its inflation and subsequent deformation, exerts transverse forces upon the ring. The transverse force due to the inflation of the circular side wall modifies the initial tension in the circular ring due to the inflation pressures. That part of the transverse force due to the deformation of the side wall enters the model as a distribution of linear springs with a stiffness which may be calculated from the inflation pressure and the side wall geometry. The effect of the compression of the tread rubber is included by girdling the ring with an elastic foundation with a stiffness based upon the modulus of the tread rubber. Because the tension in the ring is large compared to its bending stiffness, bending is important only in limited regions in the neighborhood of the point at which the model tire loses contact with the roadway. Mathematically, the large difference in the size of the tension and the bending stiffness renders the coefficient of the highest derivative of the governing differential equations extremely small. This small parameter is the source of the numerical difficulties which are encountered when attempting to obtain exact solutions [1]. One possible alternative to obtaining exact solutions is to consider ap-
Contributed by the Applied Mechanics Division for publication in the JOURNAL OF APPLIED MECHANICS.
Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N. Y. 10017, and will be accepted until December 1,1980. Readers who need more time to prepare a discussion should request an extension from the Editorial Department. Manuscript received by ASME Applied Mechanics Division, March, 1979; final revision, November, 1979.
Journal of Applied Mechanics
proximate solutions which are series expansions in the small parameter. In situations in which the small parameter multiplies the highest derivative, an expansion of this type is not uniformly valid, but must be modified in regions where the highest derivative is important. These regions are the boundary layers. The expansions appropriate to each of the regions are joined, or matched, in their common region of validity. Such a singular perturbation analysis of the differential equations of the model radial tire is undertaken here for three reasons: first, because such an approach illuminates the physical phenomena by delineating the regions in which various physical effects are important; second, because the approximate solution which results is simple in form and easy to interpret; third, because such an analysis appears to provide the only means of obtaining solutions to contact problems for the cylindrical shell model of the tire. The problem considered here is similar to the "beam-string" discussed by Cole [2] and the nonuniform prestressed beam treated by Hutter and Pao [3]. An important difference is that the present problem is a contact problem; and, for a given vertical deflection of the tire bottom, the extent of tread in contact with the roadway is not known at the outset, but must be determined as part of the solution. Additional complications result from the initial curvature of the ring and the presence in the problem of a second small parameter, other than the ratio of the bending stiffness to the tension, which appears as a coefficient of high derivatives in the differential equation. In principle, these two small parameters may be varied independently, generating model tires of various sizes, side wall geometries, tread stiffnesses, and inflation pressures. However, in the problem at hand, it is convenient to relate the second of these small parameters to the first, converting the problem from one with two small parameters to a problem involving only one. In a tire of fixed geometry and tread displacement, as this parameter approaches zero the inflation pressure and the stiffness of the tread spring both increase without bound, but in a fixed ratio. In this "distinguished limit" corrections are obtained to previous analyses of simpler tire models, reviewed by Clark [4], which necessitate that a concentrated moment and a concentrated transverse shear be applied at the edge of the contact region. Where the tire is in contact with the road, different solutions are found in each of three regions. In the boundary layer closest to the
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contact with the road, 8 is 8*, and the displacements, tangent, transverse shear, bending, and circumferential tension must all be continuous. These conditions are equivalent to the requirements that, at 0 = 8*,
iiiiimnm
Vf,
IU C ( " =
(7)
WfV\
where I = 1,2, 3, and 4. The differential equations (1) and (3) in the contact region and the free region may each be integrated once to give
P
toc + 2
m m -b— Fig. 1
Wc = Wf,
a2T0\ ,„, a4 a4, 5>c + 1 + — ki + — k2 EI ) \ EI EI
a2T0 \ _ ' " EI
a*k2 ,_ , , a*k2_ „ a4c = — ( a - d o ) - — acos0 + — ,
The model of the steel belted radial tire
,n^ (8)
and point at which contact is lost, bending and tension dominate. In an intermediate layer, the effects of tension and the tread spring are most important, while the deformation in the outer region, furthest from the contact point, is governed by the side wall and tread springs. In that part of the tire not in contact with the road, there are different solutions in each of two regions. In the boundary layer closest to the contact point, bending and tension again dominate, while the deformation in the outer region is governed by the side wall spring. T h e Singular Perturbation Solutions The differential equation governing the radial displacement wc for the portion of the model steel belted radial tire shown in Fig. 1 which is at rest and in contact with a flat, smooth rigid roadway was determined in [1] to be
respectively. The constant of integration c is the same in both equations by virtue of the continuity conditions (7). In each region the displacement w is nondimensionalized by a and the nondimensional displacement is denoted by w. Estimates based on a real tire [1] indicate that the dimensionless ratio f4 = EI/a2To is extremely small; while the dimensionless ratios «i, «2, 013, and a.4, defined by ai1
a 2 fti a 2 =1
"~To~~
a2fe2 T0
a as = « 2 ~ , a
M'
CX4 = « 2
(10)
are all of order one. The asymptotic solutions to the differential equations which are obtained here are strictly valid only in the limit ( - ^ as 1 goes to zero with a\, OL% a% and 2(2) + 012W0 — 014+ as cos 0o*)
+ e 3 (u)3 (4> - u>3(2) + a2wi
- «30i* sin 0o*)
+ e4 u) 4 ( 4 ) - u)4iai0i* 2 cosh [Sofa sinh y^Kit
(50)
- 0O*)]| - D i V ^ s i n h [y/a^ir - 0O*)]
Es = D o V ^ I sinh Wa\(ir (41)
The exponentional solutions with negative argument cannot match with the outer solution. The constants which appear in the expressions for the approximate solutions are determined by the continuity conditions (7) and by matching solutions in adjoining parts in each of the two regions. The constant coefficients in the expansion (19) for the contact angle are determined, for a given tread displacement do, from the contact condition which follows from equation (2) by requiring that \ = 0 at 0 = 0*. A solution is obtained to third order, for example, when the contants in the coefficients of the powers of e up to and including the second in the expansions in each layer and in the outer regions have been determined. In matching the inner solution to the inner-inner solution in the contact region the first four terms in the inner solution expansion are first expressed in terms of the inner-inner variable r) = £/t. With rj fixed, these terms are expanded in powers of e and the first five terms, up to and including the fourth power of c, are retained. The coeffi-
(51)
- 0o*)] - 2 V ^ " 0 2 *
+ D2 cosh [ V ^ ( T - 0O*] + — , «i
and ws = Eio + En7i + Ei2ei.
(47)
(37)
to fourth order. Equating the coefficients to zero and solving the resulting differential equations, yields WQ
,
£1 = D0 cosh [sfa^br - 0o*)] + — , «i E4, = - A ) V ^ 0 i * sinh [V^idi- - 0o*)]
used in (35) result in the approximate differential equation W 4 ) - ">o(2) + e ( V 4 ) - wi) + eHw2i4) - u>2m) + e3(w3
