Transactions on Engineering Sciences vol 1, 1993 WIT Press, ISSN

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1 A stress analysis for contact problems involving rolling and sliding 0. Koca*^, F.-G. BuchhoW* "Institute of Strength of Materials, University of Tirana, Tirana, Albania ^Institute of Applied Mechanics, University of Paderborn, D-4790, Paderborn, Germany ABSTRACT In this paper a generalized model of the roll-slide contact problem is presented. By the analytical solution of this model the exact surface pressure distribution along the line of contact is obtained, including the effects of driving moments and/or tangential forces. By the aid of this model the well known problem of gears in contact is analysed, taking roll-and slide effects into account. The obtained analytical results are confirmed by experimental findings concerning the location where damage occures at gear surfaces. INTRODUCTION Most analytical models for the anlysis of contact problems are based on the classical HERTZ-solution, although in many applications rolling and sliding are involved respectively. In order to take these effects into account not anly a normal force P but also a driving moment M and/or a tangential force T have to be considered in a generalized roll-slide contact model. In this case the problem will loose its symmetry and the contact geometry should cover straight and curved contours of the elastic/rigid or elastic/elastic bodies in contact (Fig. 1). GENERALIZED CONTACT MODEL With reference to [3] and the notations given in Fig.l the generalized unsymmetric contact problem can be formulated as follows. By expressing the parameters x = a + (3 cos(40 und t = a + (3 cos(9) through ' Visiting Scientist at University of Paderborn.

2 128 Contact Mechanics Fig.l Notations and parameters for an unsymmetric contact problem. special coordinates and 6 the pressure distribution p(6) and the contour f(( )) are related by 7t Jp(6)-In cos(( ))-cos(9) sin (6) = (1) 1-cos(0) K where K denotes the effective modulus. Furthermore the conditions p(0) = p(7i) = 0 must hold and the normal force P is found through n -P = p-jp(e)-sin(9)-d0 (2) By expressing p(9) = <: sin (n-0) and f(< >) = B.-cos(n-< >) (3) n=0 through series of trigonometric functions and by substituting these expressions into eqs.(l) and (2) some relations between the unknown coefficients Cn and Bn can be established. But unfortunatelly the solution given by HERTZ is no more valid because obviously the problem will loose its symmetry due to the effect of the driving Moment M. This is confirmed by the FE-results presented in Fig. 2 for a corresponding contact problem between an elastic cylinder and a rigid half plane. If in the generalized case the contourf((()) is expressed as f(x) = A-x^+B-x^-H..= Bo+B^cos(<t)) + B2-cos( the pressure distribution can be found as sin(2-e) with (4) (5)

3 Contact Mechanics B-, C =0 fur (n>3). (6) unloaded contour loaded contour normalized pressure 0.0 -r,,, normalized coordinate x//s Fig.2 FE-results of contour and pressure distribution for an unsymmetric contact problem. By considering the equilibrium- and the contact boundary conditions we finally get 2 P -" 7C-P ' 4-(±M + P-a) and P-K A-f-3-a-B (7) With the assumption of B=const., which means that 6 is only a function of P but not influenced by M, and the following notations a 4-X " «-! x 2 ' M. =-? &,, A^^Ji.ViTs^, (10) P' j P the most relevant contact parameters ao, 5o and Mo can finally be expressed as functions of the loading parameter X alone. Their variations with X are given in Fig.3. (8) (9)

4 130 Contact Mechanics SOLUTION OF THE MODEL In order to solve the generalized contact model for criteria may be considered. three different Criterium I states that in the rear part of the line of contact no resultant force should act with reference to GRIWUTH [1] and analoguous to SOMMERFELD's criterium in hydrodynamics [10]. Based on this criterium Solution I of the generalized contact problem gives the following results (Sol.I, Fig.3). * *'* (%o=-0.526, 80 =0.062, Mo = P, 8 A = P K = P K (11) From Fig.3 it is to be seen that Soil leads to an unstable case of contact which also has been observed experimentally [1,2] when a corresponding test with a rubber cylinder on a rigid plane has been performed. With Arel =6% the comparison of experimentally measured and analytically calculated coefficients of rolling friction (Sol.I) show a rather good agreement coefficient A Fig.3 Variations of the contact parameters oco, So, and Mo with loading parameter X. Criterium II states that in the rear part of the line of contact the pressure distribution and its first derivative with respect to 6 should vanish at the same location. This corresponds with REINOLD's criterium in

5 Contact Mechanics 131 hydrodynamics [10] and leads to Solution II (Fig.3) of this contact problem with _ 1 2' =-0.366, 8. =-0.116, ML =0.116-P, A = P-K = P K (12) For Sol.II also a rather good agreement between the calculated and the measured coefficient of rolling friction are found (Arel = 5%) concerning experiments with a steel cylinder in contact with a steel plane [6,7]. Criterium III states that a limit value should exist for the transmissable driving moment M. Solution III (Fig.3) results in A, = 0.424, a, = , 8^ = , M^ = P, = P-K = P-K (13) and Arel = 3% for the calculated coefficient of rolling friction with respect to experimental findings referenced above [6,7]. X Q. t_ Z3 O) 0) CD normalized coordinate x/0 Fig.4 Effects of a driving moment M on normal pressure distribution (k = 0, HERTZ solution, A, * 0 new solution) In Fig.4 the effects of an increasing driving moment 0 < M < M^ on the normal pressure distribution p(6) are presented. Clearly the symmetry of p(9) for the HERTZ-solution and the rising unsymmetry of p(8) with increasing values for M or X is to be seen. Also for p^ it is found

6 132 Contact Mechanics Once the normal pressure distribution for the generalized contact problem is known we can calculate the stresses in the material of the bodies adjacent to the line of contact, which is a most important task for practical applications. With reference to [8] we may use the biharmonic funtion,-2'c (14) and the elliptical coordinates ' = p-sh(5).sin(n). (15) o p-normal pressure T tangential pressure a1-principal stress i.o -d.5 0:0 o:s i!o i!s normalized coordinate x/fi Fig.5 Principal stress and normal pressure distribution at the surface of the driven body For the stresses due to the normal pressure p we find ch(2- )-cos(2-t ) and similar expressions for ch(2-^)-cos(2-r ) (16) and (17)

7 Contact Mechanics 133 For the stresses due to the tangential pressure distribution T(x) = i-p(x) we get 0% (T) = ±(2 ' Y(T) - C\y (T)), C^yy (%) = ±G\y (l), G**xy (%) = ±C*xx (%).(! 8 On the basis of these results we finally are in the position to find the locations where the highest effective stresses will arise in the material and particulary important for fatigue where the material is subjected to tension stresses locally. With respect to Fig.5 this location is found at the end of the line of contact at the free surface where o^ = 2 i C^ (1 + X) > 0 arises in the driven body. APPLICATION OF THE MODEL TO GEARS IN CONTACT The generalized contact model can readily be applied to gears in contact as sketched in Fig.6 which is an important problem in engineering practice. With respect to the fatigue problem involved in this application we report the following findings T5 min ~ 1.23-C, (19) 0 2-t Here o, instead of.(1 + X)-p \=o. = a = p,.=ci<[a]. A A,=U 1 L Jm is relevant for the fatigue strength., <[o] (20) (21) last point of contact first point of contact Line of contact 02 Fig.6 Geometry and kinematics of gears in contact In agreement with the analysis of the generalized contact model the damage due to fatigue is found between the root of the tooth and the pitsch (Fig.7), but in particular adjacent to the pitsch where the largest tension stresses arise at the surfaces of both teeth.

8 134 Contact Mechanics Fig.7 Fatigue damage at gears, a) location, b) type of damage CONCLUSIONS In the presented generalized model of the roll-slide contact problem the effect of a driving moment has been investigated. When taking the rolland slide effects into account the analytical results agree well with experimental findings concerning coefficients of rolling friction and locations of fatigue damage at gears in contact. REFERENCES 1. Bowden, P.P. and Tabor, D., The Friction and Lubrication of Solids, Oxford University Press Vols.l and 2, London, 1954 and Yamaguchi, Y., Tribology of Plastic Materials, Elsevier, Amsterdam- Oxford-New York and Tokyo, Boicu, N., Contactul elastic linear, Editura tehnica, Bucuresti, Johnson, K.L. and Gray, C.G., Development of corrugations on surfaces in rolling contact, Proc.Inst.Mech.Eng.(London), pp , Kragelskii, I.V., Friction and Wear, Butterworths, London, Polzer, G., Meissner, P., Grundlagen zu Reibung und Verschleiss, VEB Fachbuchverlag, Leipzig, 1983, Moskva, Kaufman, H.N., Sliding Bearing Damage, CRC HANDBOOK of Lubrication (Theory and Practice of Tribology), Vol.2 pp , CRC Press.Inc.Boca Raton, Florida, Teodoresku, P.P., Probleme Plane in Teoria Elasticitati, Vol.2, Editura tehnica, Bucuresti, Tallian, T.E.., Failure Atlas for Hertz Contact Machine Elements, ASME Press, New York, Szeri, A.Z., Tribology, Friction, Lubrication, and Wear, Hemisphere Publishing Corporation, Washington-New York and London, 1980.

Figure 43. Some common mechanical systems involving contact.

Figure 43. Some common mechanical systems involving contact. 33 Demonstration: experimental surface measurement ADE PhaseShift Whitelight Interferometer Surface measurement Surface characterization - Probability density function - Statistical analyses - Autocorrelation