Tire Modeling Membrane Method

8/8/2019 Tire Modeling Membrane Method

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Pneumatic Tire Modeling by Membrane MethodM.H.R. Ghoreishy'

Rubber Industries Engineering & Research Com pany, Tehran, Iran

Key Words:tire, design, mathematical modeling, thin shell theory, membrane method

ABSTRACTTires play important roles In cars performance.Slnce the early days of their production, many investigators have tried tofind the relationship between tires structural parameters and their performance .t t has been established that mathem aticalmethods a re the most effective techniques, and are divided into different me thods.This paper de als with the thin shell theory from the mem brane viewpoint and i ts related problems of stress analysisare discussed.In further steps, the calculated results of a m odel for a four ply bias t ire (7 .00-16) are presented and compa red with

experimental data.It is observed however that although this method is an approximate model and consists of many simplifyingassum ptions, the calculated results are in good agreem ent with experimental data.

INTRODUCTION

.It is comprised of parts or.However in general it can be said that

consists of three main parts :tread, carcassTread is that part of a tire which must have

arcass is the heart of the tire and consists of

.The bead, isinextensible hoops functioning as

.Figure I shows a section.This type of tire is well known as the

ntional tire.The other types are radial:Pulymer Rescarrh Cen[cr of Iran, P .O .tina

Tehran, Iran .

Figure 1 :A schematic view of tire section [1],

and bias-belted tires which are out of the scope ofthis paper.Before taking the mathematical modeling of

the tire into consideration, it is necessary to discussbriefly the tire constructing method .Bias tires arefabricated by the so called tire expansion process.Schematically illustrated in Figure 2, the carcass is

of Poi},ncr Server & Teci ,iologv . Vul. 2 , Nu .J, January, 1913 3 7


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Figure 2:T1re construction and exp ansion [3] .built on a cylindrical drum by applying pieces ofthe layers or plies, whose lengths equal thecircumference of the drum .The protective sidewalland tread rubber is then applied over thecarcass.At this stage, the green tire has beenmade.To bring the tire from its present cylindricalshape into its final toroidal shape, the expansionprocess is utilized in which, by applying pressureinto the green tire, the final shape will beproduced.During the expansion process, cord anglea of the green (or cylindrical) tire changes to theangle/3 as illustrated in Figure 2 .While the cordangle a is constant, the angle Q of the toroidal orexpanded tire is not constant but changes frompoint to point .The illustrative equation whichrelates the cord angle ft to any points of the tire toits green angle a, called "cosine law" will bediscussed in the next sections.The simplest stress analysis approach from atheoretical standpoint is the calculation of theshape taken by the inflation pressure and the stressdeveloped in an inflated but otherwise unloadedbias tire .The first rigorous solution to this problemwas obtained by Purdy in the U .S - . in 1928 [1].However it was not until 1956, when Hofferberthpublished his results in Germany, that such workbecame readily accessible to tire engineers.Biderman ' s investigations dealing with the samesubject appeared in 1957 in the Soviet Union [1] .

All the above mentioned investigations arebased on this assumption that the structure of tireobeys membrane laws.

The membrane method for tire stressanalysis is based on a number of simplifyingassumptions which are discussed in the nextsection .These assumptions can affect, to a greatextent the reality of tire structure, but as it will beseen later the results of analysis by this method arein very good agreement with experimental data.Therefore, we accept this type of analysis, despitethe fact that today, complicated techniques such asfinite element methods are extensivelybeing usedin tire design.Mathematical ModelingThe tire structure must be idealized before theconstruction of the mathematical model is initiated.This idealization is based on the following fiveassumptions [2].

a)The membrane is assumed to consist only ofthe cords, that is, the binding material andprotective rubber are neglected.

b)The carcass is thin-walled which means thethickness of the carcass wall is small in comparisonto its overall dimensions.This allows the use of themembrane theory of elasticity, since bending isneglected.

c)The plies of cords are coplanar.d)The cords act as a net that is, they are flexibly

linked at their intersection points .There is norelative motion between the cords at these points,but the deformation between the intersection isunrestrained.

e)The tire under consideration is symmetric,inflated but not loaded.

This pneumatic tire carcass with theneglected rubber may be visualized as a deformedmembrane in the form of a surface of revolution,obtained by rotation of a plane curve about an axisin the plane of the curve .This curve is known asthe meridian curve or the meridian shape of thetire (Figure 3).

From the theory of thin shells, themembrane equations can be deduced as follows [2]:r + + nos, +P,Y=0 (1 )

a = Green crown angle= Cured crown angle

3 8 !roman Journal of P o l y m e r S c i e n c e & T e d a n a l o g e V oL 2Nod, Jwwwy, 1993


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1 8(Nryy + di - Nocoso 4 Po- =0 (2)rrN

a~ o0(3 )=Pre rp

Where N,=force in the direction of the tangent tothe parallels per unit length, No=force in thedirection of the tangent to the meridians perunitlength, rr and r, are principle radii of curvature,T=N,,=Nr,=shearing force, P,= load componentin the direction of the parallel circle tangent,P,=load component in the direction of themeridian tangent, P= toad component normal tothe surface (Figure 3) [2].

Figure 3 :TIre meridian section curve with co-ordinates.

Principle Stresses and Cord TensionsThe stress generated in the membrane and thetensions in the cords can be calculated from theequilibrium equations (1), (2) and (3) utilizing themeridian equation .We assume that the meridiancurve has the following equation (Figure 3):Z =frydy(4where y, z represent the radial and lateraldirections, respectively.The equation of f(y) will bederived later.Now we concentrate on solving theequilibrium equations .First of all, the followingparameters should he defined .They are known asreduced stresses [31.

_ (yly cos.) N . (5 )N. _ ( y cosrly) N o (6 )

Iranian ]Dorval of Pohrmr Scicocc h la'haululr. Vol :, No. I, Ja, ,an, 1993

Rewriting the equilibrium equations in termsof these reduced stresses and further assumptions,will give:3N = o (7)aB

-yylv (8)No-No-11+(Y1P=0 ( A )These additional assumptions are:

- Because of symmetry the shearing forces mustbe vanished, i .e.T=N{,=N f =0

- It was assumed that the only external load isinflation pressure, so since the resultant force ofinternal pressure is normal to surface, it can bededuced, P,=Pr=0 and P= P, where P is theinflation pressure.In the above equations, )(and y ''denote dylda andd2yldz 2 , respectively; y'also is equal to cots* as canhe seen in Figure 3.With substituting the following equation [3]:

ax ay de dy [1)into equation (8) and solving the new differentialequation, one can obtain:

where the C is constant of integral.From Figure 3,it may be shown that, when y = y B , then f(yB ) = 0,so from equation (6), it can be deduced:Y = 3 ' t + N=012Now, substituting equation (12) into equation (II),and the resultant relation into equation (6), we canwrite:Nr = 'i-'a) ~+ f ( y ) (13 )2y f(y)We can derive the same general relation for Nd .Ifthis is done, it can be deduced that:No = pyl/lfl+(1- 2y Q2 -2)f ( f~) 1 I I (la)

These forces must be taken up by the carcass

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cords of the tire .For determining the cord tensionin the tire, consider the point A in Figure 4.The = 2 f r, [L + ~(i)l 1a dy (20)

Figure 4 :Diagram of farces acting at point A in Figure3 .position of this point is the saute as the point A inFigure 3.From the equilibrium condition, we canwrite:rM sinp = Nany (15)Where, r = cord tension, M = total number ofcords in tire an d # = angle between the cords andcircumferential direction. There are some relationsfor calculating fl from y or radius of tire but themost common formula which is most generally usedis the cosine Iaw.This formula is [2, 3]:

rnsp= (16)Where, a = a constant angle which is known asgreen angle, r, = bead point radius, (Figure 3),C,= average cord extension ratio.

It is common to express M, the total numberof cords in terms of no, or number of cord ends perunit length, which is a constant value in the greenstate.This expression is:M = 2 IInor,sna(17Therefore, we can write the cord tension in thefollowing form:

NyT = nor, siuasl lWith reference to Figure 3, one can derive thefollowing expression for cord length and meridianprofile length:L = 2 r, [1 + f 2 (Y)r dy ( 19)r, sin/3

Where L = cord length, L. = profile length andr,,naximumradius of tire profile at the crown.

There are also two parameters which can bederived from the membrane theory.They are beadtension force and shear stress applied from rubbermatrix to the cords .These formulas are [4]:T = (112)Py B 2-?)f(r,)P = ~ ) (y8 - y2 ) p p lc1+ (2)Car,slivWhere T = bead tension force, yB = radius of tireat maximum section width and U = shear stress.The value of shear stress at y = r, or at the crownis equal to zero.Determination of fT)It was seen that for computing the all parameters,one should know the f(y) or meridian equation .Thegeneral expression of meridian contour equationcan be obtained by solving of an ordinarydifferential equation which is known as the shapeequil brium differential equation [2, 3]:Cage, d # + mtgiq = dy ( 23 )

Y 2 - FaThis differential equation, is a special Bernoullitype [$] .This is a non-linear ordinary differentialequation, tractable by two successive iterations.After solving this equation, we will have [2, 31:

(y2 - yn 2) [G 2 - y2I ra (24)cosr(y )

[( r,2 -y E2)2 (C,2 a o Y e 2 r r r.2 )-(y2 -ya) 2 (G2 - y2)l'a

By integration of this equation, one can find thedesired formula for plotting the tire meridiancontour, thus:Z = ff(y) dy ( 25)

Now with the help of the above expressionalong with equation (24) one can compute all ofthe required parameters from the k nown values.

(18)

(21)

4 0 Iranian formal of 'Omer Science dr Tecbrsdogi& VolZ No.). /anima; 1 9 9 3


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RESULTS AND DISCUSSIONumerical Results

obtained relations will be

The size of the chosen tire is 7.00-16.Thebias tire's initial cord angle (or green angle) is

and the dimensions of the tire meridian curve.992mm for radius at maximum section width

.992mm for radius at the crown or.The average cord

.02 and the inflation. 4 1 3 7 M P a.

The calculations were performed at 21 points

Figures 5, 6 and 7 show the principle forces

the cord, respectively.50

,40 ,30

g 20 --10-0220 240 260 280 300 320 340 360 380Radius (mm )

M eridianircumferential:Principalfarces

For determining the whole tire contour or.The complicated shape of this

i+l = z; + ~ y,+l f(y)dy (26)of Polywnrr Science &Tcchaolr V ol.Ma Jamary, 1 9 9 1

Figure 6 :Card force

In computing f(y), it can be seen that thevalue of function will be infinite at the crown or aty = r so it is not possible to calculate the integralfrom ordinary methods of numerical computationsuch as trapozoidal or Simpsons.

Figure 7 :Shear stress

All investigators (for example see reference [3]),who have worked on the solution of this integral,used the mentioned methods with a slightdifference .They used r,-Ay instead of r, for theupper limit of the integral, where Ay is a smallquantity .The value of Ay can affect the results to agreat extent, therefore the conventional methodsof numerical integration are not suitable .To

0 .3,a.. 0 .2 s

i 0.20 .15 -0 . 1 -i005 -

240 260 380 300 320 340 360Radius (mm)

0,220 300

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overcome this difficulty we have computed theintegral from a well known numerical methodwhich is called 'Gaussian Quadrature" Technique[6J .

The five points Gaussian integration methodwas used and the coordinates of tire meridianprofile have beenobtained.Figure (8) shows thetire meridian profile which was obtained byGaussian numerical integration method .

equilibrium shape of tire under inflatiion .Utilizingthe later parameter, makes it possible to draw theinflated shape of a tire from the standard data, aoit can be used in designing of new tires.Also it isshown the calculated equilibrium shape is in verygood agreement with experimental data, thereforethe users of this method can be certain that it canbe applied to the tire design without any problem.

Figure & Inflated tire meridian pro fileFigure 9Comp arison between calculated and measuredinflated profile

Comparison with Experimental ResultsIn order to determine the ability of thismathematical model for simulation of the tire at itsinflated shape, the internal inflated profile of theprevious mentioned tire has been determined bysubtracting the tire thickness from the externalinflated profile.Thelatter shape has been measuredby a profilemeter .Figure 9 shows the tire meridianprofile which was obtained by numerical simulationand compares it with experimental data.It can beseen that the experimental results are in goodagreement with calculated ones.

CONCLUSIONSIt is shown that by using the thin shell theory, theequilibrium equation for a bias tire under inflationcan be deduced .Having solved these equations, onecan compute the structural parameters of tires,such as, principle forces per unit length, cord force,shear stress acting from rubber matrix to cords,cord length, meridian length, bead tension and the

N O T A T I O Nc c green angle

crown cured angle3 cord tension force

average cord extension ratioNo force in direction of tangent to the parallel per unit

lengthNo force In direction of tangent to the meridian per unit

lengthNor, Noe shear forceP inflation pressurePe load component in direction of parallel circle tangentP~ load component in direction of meridian circle tangentP, load component normal to the surfaceM total number of cordsrte cord ends per unit lengthL cord lengthL . profile lengthr, maximum or crown radiusr, drum or bead point radiusra, r* principle radi i of curvatureT shear force

4 2 Iranian ]owned of Polymer Science & Technafagv VeL2, Nal, January, 1993


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T bead tension forceU sheer stressy tire radiusY e radia at me>amum section widthz tire width

Date receiw&31 December 1991

REFERENCESfI jClarkS . ; 'Mechanics of Pne um atic T i res" , W ashingtonAC. : USDepartment of Transportation, {1971).

f 2] Walston, W.H . ; W .F.Ames, "Design and A nalysis ofInflated M emb ranes Reinforced with Extensib le Corns.Part I " Textile Research lawma4 pp I078-1087, (Dec.1978).f3JLauterbach, H.C ; W.F.Ames, 'Cord Stresses inInflated T ires" Textile Research Jour nal, pp 890-990,(Nov . 1959).

f4jDay, R .B . ; S .D.Gehm an, "The ory for t he M er id ianSect ion ofInflated Cord Tires" R ubb er Chemistry &Technology ; Vol.345, No 1, pp .11- 27, (1963).[5jMarr ig ian , G . "M athem at ical Notes Concerning theTire Contour Integral" Rubber Chemistry d rTechnology, V ol44, pp.122-I26 (1971).

f 6]Carnahan, Brice . "Ap plied Numerical Methods, NewYork : W iley, (1969).

Journal o f Polyorrcr Science & Technologr . VoL?, No .1, January, 1993 43

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Tire Modeling Membrane Method