1973-62 simplified theory of rolling contact

8/12/2019 1973-62 Simplified Theory of Rolling Contact

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M E C H A N I C A L AND A E R O N A U T I C A L E N G I N E E R I N G AND S H I P B U I L D I N G 1

Simplifiedtheory ofrollingcontactJ . J . K A L K E RSubdeparlment of Mathematics,Delft University of TechnologyJulianalaan 132Delft-2208, The NetherlandsDelft Progr. Rep., Series C: Mechanical and aeronautical engineering andshipbuilding, 1 1973)pp. 1-10.Inthe present paper an approximate theory of rolling contact of elastic bodies is developed whichis verysimple touse.All salient features of rolling contact phenomena, trilh theexception ofthe phenomena due toelastic asymmetry, are icell reproduced. As a consequence it is notdifficult to gire theparameters of thesimplified theory such values that areasonable quantitative agreement icith theexact theory of steady-staterolling isobtained. Finally the simplified theory is well suited to roughly investigate the mechanical influenceof the surface layers which maycover the bodies.IntroductionIn the present paper two dry bodies areconsidered which roll over each other. Infirst instance the bodies may be regarded asrigid. Then,according to Coulomb's law ofdry friction,twostatesare possible, viz.1. The bodies roll without slip, and thetangential force falls below a fixed multipleof the normal force by which the bodies arepressed together.2. The bodies slide androllwhile the tangential force attains the fixed multiple of thenormalforce and acts in the direction of theslip.However, it has been observed experimentally,that the bodies slip a little even when theforce transmitted is below the maximum. Insomeapplications, such as the investigationinto the stability of railway trains, theseeffects are significant and the crude modeldescribed above cannot be used.Fo r an explanation, the elasticity of thecontacting bodies must be taken into account.This has been done by several authors,- referto the bibliography at the end of thispaper-, and the theory becomes quiteformidable, owing to the complexity of therelationships even of classical elasticity.In this paper the model is simplified in thesense that these complicated relations arereplaced by a much simpler relationship,whichappears to conserve many of the typicalfeatures of the conventional contact theory.Thusit has il lustrative value. Also it appearsto be possible to utilise the simplified theoryas an approximation of the more realistic,complicated model by adapting certainconstants. A program implementing thesimplified theorydoesa job in approximately1/100 of the time needed for the same jobby a program implementing the realistic,complicated model. Thus the simplifiedtheory has a greatpracticalvalue also.

Formulationof the problemConsidertwo elastic bodies which are pressedtogetherso that a contactareaforms betweenthem, see Fig.1. Acartesiancoordinate system{0; .T,y, i ) is introduced of which the planeofxan d v is the plane of contactan d in whichthe r-axis points vertically downward intoNotationsTh e exact model: the realistic complicatedmodel. ' . 8_

' 8x ' 8t1,2: if adistinction must be made betweenquantities of body 1 or 2, the quantities in question carry a superscript1o r2.(x, y, z): Cartesian coordinate system with

originin centre of the contact area,x- direction coincides with rolling-direction, r points vertically downwardinto 2. (see Fig.1)

A(y), B(y) (14. H I ) u, u 1 , o 2 above (3)a,b (1) u x, u,, u. (7a)C (1) V (5)Cj (13) V,V,',V>(3)F.,F, (13a, b) V , 1 , \rZ above (3)(5),(6)f (2) V (2),(10)G (13) X, Y (2)H (30) Z (1)h (44), (47) Zo (14.1,n, ni)L(y) (12) (38)1 (35) 6 (25a)M. (13c) 6 0 (32)N (40) X (25b)ss, (9) and below "(') .( / ) = u s(s x, s,) (7b) 4 (6)t time a r. o"_ G- 1stresses


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2body2, seeFig.1.The originisthe centreofthe contactarea.W e assume thatthecontact areaCandthenormal pressure Z actingon it can becalculated bymeans of theHertz theory.Forthisit issufficient that:1. the small displacement, small displacementgradient theoryofelasticity is applicable;2. thelargest diameterofthe contact areaissmall with respect to acharacteristic lineardimension of thebodiesat and near the contactarea;3. nocloseconformity may existbetweenthebodiesatthe contact area;4. thebodies must behomogeneousin theparts thataresensibly affectedby theelasticdeformation;5. either:a. thebodiesare madeofidenticalmaterials,orthey are incompressibleon b. thelevelofthe surface shear tractions{X, Y)is at each point of thecontact areamuch lower than thatofthe normal pressure:||X, Y\\


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4 1 1973) D E L F T P R O G R E S S R E P O R Texact relation described'in Ref. 1p.243 andRef. 2 p. 17 sqq.(9) is the response to sheartraction of a very thin elastic layer, mountedon a rigid substrate.It should be noted that E q. (9) is only anapproximation of the true stateof affairs ifthe bodies are made of identical materials.So we will exclude condition 5. b (see thebeginning of this section) from our considerations.If a prime (') denotes differentiation withrespect to x and a dot ( ) differentiation withrespect to timet, we arrivefrom (7), (9) and(2) to the following statement of the problem:v = ("x,u,) ;vx = Vox-Vy+VSJiX' -A~/*0(a)v y = Vo y + Vx+VSy(Y'- r/10(b)s(sx,sy) = yjV; relative slip; (c) (10)^ = P , = 0 - | i ( A - ,Y) l l 2 ), constant

Thisis the traction bound in accordance withtheHertztheory. However, thex, yderivativesat theedgesof the contact area are infinitelylarge. The rate of increase of the tangentialtraction is also infinitely large at theedgesofthe contact area in the exact theory, but it isalways finite in the simplified model, seeE q .(10 a,b),(11). Now, the only way in whicha stateof complete sliding may occur is whenthe initial slope of the traction bound issmaller in absolute value than de absolutevalue of the adhesion slope, see Fig. 2. Sincethis is not possible with traction bound (14.1)we seek an alternative.

I I . fZ = / Z 0 { l - xV - > - 2 / b 2 } (14.11)This traction bound is simple, but leads tonumericallyinaccurate results. It is used in thediscussion of the theory of steadystaterollingwith pure creepage of the following section. Inthe discussion of steady state rolling withcombined creepage and spin thethirdpossibilityis used:I I I . fZ =fZ0A(y){1 - x 2 / a 2 - y 2 l b 2 }

if |.x|>0.90^/(1-y 2lb 2)= 0.9L(y)= f Z 0 U ( l - x 2 l a 2 - y 2 l b 2 )+

+B(y)}if\x\


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6 1 1973) D E L F T P R O G R E S S R E P O R T

(19)

seen from (15) andFig.2,sx = ox+S x X ' S ; uI +S x (- u x / S I ) = 0,

X>0. (u x >0)sx =vx+SxX'


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8 1 1973) D E L F T P R O G R E S S R E P O R TAlso'wc willassume complete adhesion as insec. 3. Under those circumstances the exactmodel has been treated byKalkerin Ref. 7.T he governing equation is0 = sx = v(t) + Sx{X'-XiV(t)},M a; X = 0, \x\>a (34)This is a partial differential equation of thefirstorder forX.The rolling velocity V(t)isindependent of the jr-coordinate and we canwrite

V dt ~~ 5/ ' V(q)dq == distance traversed (35)

Hence forward,wewill,replace the time / bythe distance traversed, and we againdenoteby () differentiation with respect to /. (34)becomes0 =([)+SX{X'-X-},Ns ;oxX = 0, |x| > a

1' 8 /

(36)Thisequation is readily solved:X(x,[) =X ( x + l - l e , Q +

J'o Sj

I -. sxi {x + l - l 0 < a

(37a)dq ifx+ o> Z0

(37b)It is seen that when 11(g)= 0is constant fromthe distance / 0onward, and l2a>l0, thenX(x, t) =o(a-x)/Sx by (37b), the steadystateof sec. 3, independent of /, and independentof the initial traction distribution X(x, /0).T he condition /2a> /0 signifies that transience is completed after a contact width2aha s been traversed, a conclusion which isapproximately valid in the exact theory ofR e f . 7.A n important traction distribution is that dueto a shift without rolling, parallel to thex-axis, of one body with respect to the other.I tis called theMindlinshift and it is describedin Ref. 8. The displacement and traction due

K n d l i ;hR J Z

/ . 2 a a n d I>2aFig. 4. Transient rolling phenomena.to it are given byu= 6 =S xX^X = SIS Xif |.x| a. (38)W e start rolling at the distance / 0= 0 with aconstant creepagev;according to (37) and (38)X(x, I)= ( 5 / S J + . O dq-

= (5lS x) + vllS xif x+l


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10 1 1973) D E L F T P R O G R E S S R E P O R T

5L I

0 0 1 I ^ 1 2 1 0

F i g . 6. Pure spin, o, =u, = 0. x- a spinparameter. Simplified theory and theory ofR e f . 2.Thenin the moment calculations one must useS x = {Sb[i5CCr}+Lx, (46)S = {-7za*>2IAbll2C32G} + Ly

moment calculation,an d in the force calculation:S x = {8al3CnG}+Lx,S , = { 8 a / 3 C 2 2 G }+ . 7 ,h ={L,+Sy0}l{LJ+SyQlh0},

(47)S , 0 = S7(43) = 8 a / 3 C 2 2 G ,ho =A(44) = ^ ^ ^3M C 2 2

force calculation.I t is to be expected that the agreementbetweensimplified theory and exact theory tends toimprove asLx andL,becomelarger.A t this point we would like toremarkthat thedrastic decrease of the creepage coefficientbelow the value predicted by the exact theorywhichis described by Hobbs', may be due toan elastic layer covering the wheel and therail which is weaker in x-direction than in^-direction(LX>L,> 0 .It is also possible to investigate layers withnon-elastic response to shear by means of the

simplifiedtheory, but wewill not investigatethatfurther.

ConclusionIt has been shown to be possibledrasticallytosimplify the equation of elasticity and stillreproduce al l salient features ofrollingcontactphenomena with the exception of those whichar e a consequence of the elastic asymmetry ofthe bodies. Also, the simplified theory may beused as a quantitative approximation of theexact theory. Calculations with the simplifiedtheory are about 100 times faster than withthe three-dimensional exact theory.Finally,the presence of an elastic layer on thebodies may be taken into account, and it isproposed that the discrepancies betweenrailwayexperiments and theory may be due tojustsuch a layer.

1. A . E . H . Love,A treatise on the mathematicaltheory of elasticity, (4th E d. Cambridge 1926).2 . J . J . Kalker, O n therollingcontact between twoelastic bodies in the presence of dry friction(ThesisDelft, 1967).3. A.D. de Pater, 'On the reciprocal pressurebetween two bodies', in : Proc.Symp. RollingContact Phenomena, E d . J . B . Bidwell. (Elsevier,1962)pp . 29-75.4. J . J . Kalker, The transmission of force andcouple between two elastically similar rollingspheres" Proc. K N A W e t . Amsterdam67 (1964)p. 135-177.5. J . J . Kalker,'The tangential force transmittedby two elastic bodies rolling over each other withpurecreepage', WearIt (1968) p. 421-430.6. D.J . Haines and E . Ollerton, 'Contact stressdistributions on elliptical contact surfaces subjected to radialand tangential forces,Proc. Inst.Mech. Engrs. 17 9 (1964-1965)part.3.7 . J . J . Kalker, Transient phenomena in twoelastic cylinders rolling over each other with dryfriction', J. Appl. Mech.37 (1970) p. 677-688.8. R.D.Mindlin, 'Compliance of elastic bodiesin contact',J.Appl. Mech.16(1949) 259sqq.9 A . E . W . Hobbs, A survey of creep (BritishRailways R e s . Dept.Rept.D yn 52, 1967).

1973-62 simplified theory of rolling contact

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