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

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1 Theoretical and experimental analysis of the rolling contact between two cylinders coated with multilayered, viscoelastic rubber G.F.M. Braat^, J.J. Kalker" "Delft University of Technology, Department of Technical Mathematics and Informatics, The Netherlands **0ce Nederland B.V. Research & Development, Department of Analytical Measurements, Venlo, The Netherlands ABSTRACT The rolling contact of cylinders coated with a rubber layer is studied. The rubber layer consists of one or more sublayers with different elastic and geometrical properties. Each sublayer may be viscoelastic. The numerical results are verified by experiments. A contactless measurement technique, Laser-Doppler Anemometry, is applied to measure the velocity in and outside the contact region. INTRODUCTION The frictional contact between rolling cylinders coated with one or more rubber layers has many applications in industry. For example in copiers these mechanical elements (or nips) frequently occur and are used to transport paper or to transfer the image onto the paper. The half-space approximation, which is based on the frictionless theory of Hertz\ is one of the methods that can be used to describe the underlying contact problem. Johnson^ summarizes this theory with friction. However, the half-space approximation is only applicable under certain conditions, i.e. the thickness d of the layer is large with respect to the dimensions of the contact region a. Meijers^ gives a solution for the frictionless indentation problem of a rigid cylinder and an elastic layer, founded on a rigid base, in terms of an infinite series for two cases (a/2d < 1) and (a/2d > 1). For thin, single layers the simplified theory of Kalker* may be applied. Frictional rolling contact of viscoelastic cylinders has been studied by Hunter^ The material is modelled as a viscoelastic halfspace with one relaxation time. The two-

2 120 Contact Mechanics dimensional^ and three-dimensional viscoelastic rolling contact problem has been treated by Wang. Kalker gives a numerical method to solve the problem of rolling contact of viscoelastic, multilayered cylinders on which this paper is based. Here we will briefly discuss the numerical model and present the numerical results. The computations are compared with experimental results. The measurements are obtained with the Laser-Doppler technique, which is a contactless method to measure velocities. Roller 1 sublayer 3 sublayer sublayi D Roller 2 Figure 1. Two multilayered cylinders in rolling contact STATEMENT OF THE PROBLEM In figure 1 we see a two-dimensional cross-section of two multilayered cylinders in contact. A coordinate frame xz is attached. The upper and lower cylinder (roller 1 and 2 respectively) are pressed together by a normal force F^ which causes an approach 6 of the axles. A torque M^ applied to the axles of the rollers causes a tangential force F^ applied by the adjacent body (i.e. paper or rubber surface) to the surface of roller a. The friction between the contacting surfaces is modelled by Coulomb's Law of Friction. V^ is the circumferential velocity of roller a. The difference between Vj and V^ is called creepage. It is assumed that the cylinders are rolling steadily, i.e. there is no explicit time dependence. The deformations of the rubber material are assumed to be small relative to the dimensions of the body (radius /? ). The theory of linear elasticity and viscoelasticity is applied.

3 Contact Mechanics 121 Viscoelastic properties are modelled with a simple three-parameter spring-dashpot model, the Standard Linear Solid, which is discussed in many textbooks on viscoelasticity, e.g. Findley\ The constitutive equation for such a linear viscoelastic material is given by: 0 + T CJ = EQ + (EQ + k ) T U) where T is the relaxation time, derivative. Q and k are the elastic moduli and (') is the time THE SOLUTION METHOD The numerical algorithm, which solves this contact problem, consists of two main parts: The determination of the influence functions and the contact algorithm. Influence functions or Green's functions are characteristic functions for the rubber material. They are determined by a Fourier transformation of the governing equations of elasticity and the boundary conditions with respect to x. The boundary conditions are summarized. Firstly the normal and tangential contact stress distributions (i.e. a(x)ao;(x,0) and i(x)&ixz(x,0)) are assumed to be known in advance. They will be determined by the contact algorithm. Secondly the displacements w/x,z), w/x,z) and the stresses o/%,z), T^z) are continuous at an interface z=d$ between two sublayers. The index s indicates a sublayer number and d$ is the cumulative thickness of sublayer 1 to s. Finally at the boundary between rubber layer and rigid cylinder (z-d), the displacements vanish: u^(x,d) = u^x,d) = 0. If one of the sublayers is viscoelastic, the Correspondence Principle^ is applied: The governing equations for the elastic and the viscoelastic problem are similar in the Fourier domain. In the Fourier domain, the transformed stresses and displacements at an arbitrary position z are expressed in terms of the Fourier transformed contact stresses:, z) (2), Z) or briefly by:, z) (3) An analytical inverse Fourier transformation of equation (3) is rather difficult because the explicit expressions for S^co, z) are very complicated. Therefore a numerical inverse transformation is preferred. The interval of integration is divided into two parts. The first interval u)e[0,oj is divided into small non-equidistant elements on which the influence functions are approximated by a second order polynomial, see figure 2. If the curve has a high gradient, the element width is small. On the second interval coe[qc,^) the influence functions are replaced by a limit function^. In the spatial domain, the potential contact region is divided into elements Ax on which the stresses and displacements are assumed to be constant. The midpoints are

4 122 Contact Mechanics one quadratic interpolation element subinterval 2 subinterval 1 subinterval 0 A ( > - A c»>0 / 4 A o> = A G>0 / 2 Aco A nw ( ) = =nwg AAw Figure 2. Non-equidistant inverse Fourier Transformation. given by x^=kax. The contact stresses p/xj are determined by the contact algorithm, which computes the normal and tangential stress distributions, the approach and the creepage. The normal and tangential contact conditions are solved separately by NORM and TANG, two algorithms, designed by Kalker". The contact problem has a non-linear geometry. After the calculation of the deformed state, the deformed surfaces do not coincide. This is the geometryproblem, which has its origin in two phenomena: Firstly, when the influence numbers are calculated, the rubber layers are assumed to be horizontal, while in the real geometry they are curved, see figure 1. Secondly, in the expression for the distance between the deformed surfaces (normal contact conditions), the tangential displacements are not taken into account^: + U- (i)/(x)\ U^ (2)/ ' (X)\ O * v (4} / where h(x) is the distance in the undeformed state. Expression (4) is called the approximate distance. This problem is solved by the addition of a correction e(x) to the expression for the approximate distance. The correction is determined after each completed contact iteration and repeated until convergence of e(x) has been reached. In fact the numerical problem is solved now. The inverse Fourier transform of (3) and the contact stresses determined by the contact algorithm, give the relevant quantities /fo,z) for an arbitrary z-coordinate by: EXPERIMENTS WITH LASER DOPPLER ANEMOMETRY Numerical results become more valid if they are verified by experiment. The purpose is to measure velocities locally at an arbitrary surface point x in and outside the contact region. Laser-Doppler Anemometry (LDA) is a technique to measure velocities of moving particles. It is an optical method with the advantage that it is contactless. The principles of LDA are discussed in Durst^.

5 Contact Mechanics 123 The measured velocity at x is denoted by v^x) and can be related to the computed normal strain in the ^-direction, e/%), by: (6) dt = -V This relation holds only for steady state rolling, where V is the rolling velocity. The basic idea is to focus two coherent laser beams by a converging lens. The region of intersection is called the measuring volume. When a moving particle is crossing this measuring volume, it scatters light from both laser beams with different frequencies, according to the Doppler-effect. rubber layer black roller (A) (B) Figure 3. Laser-Doppler measurements: (A). Particle moves through the measuring volume, (B). Laser beams focussed into the contact region The difference between these frequencies is the Doppler-frequency / which is related to the velocity of the particle. A scheme of the experimental set-up is given in figure 3, where one of the rollers is completely transparent, while the other is black. The measuring volume is placed in the contact region by using two mirrors A and B of which the latter can be rotated about the y-axis (i.e. the axle of the glass roller). The rotation of the lower mirror causes the measuring volume to be displaced along the circumference of the outer surface of the glass roller. Hence we are able to scan the surface and to perform measurements at different positions. For detailed information on the experimental part of this research work, the reader is referred to NUMERICAL AND EXPERIMENTAL RESULTS In this section an impression is given of the capabilities of the numerical algorithm, which have been implemented in a computer program, MULTILAYER. We examine a configuration which has two sublayers on the upper roller. The intermediate layer is relatively soft and thick (E=2.0 Nlmm\ d-4.0 mm), while the

6 124 Contact Mechanics d = mm * d = 0.2 mm x [mm] Figure 4. Normal stress distribution for a configuration with two sublayers toplayer is relatively hard (E=5500 Nlmnf). The thickness is varied (d=0, 0.025, 0.1 and 0.2 mm). The other roller is rigid. In figure 4 we see the normal stress distribution in the contact region. Peak stresses arise at the edges of the contact region, which are due to the bending of the upper layer. The peak values are proportional to the thickness of the toplayer. A similar result is obtained by Johnson^ for the indentation of a plate on an elastic foundation by a rigid cylinder. elastic, F? = 1.0 N/mm viscoelastic, F = 0.0 N/mm normal stress (A) normal stress Figure 5. Contour-lines of o^(x,z) for an elastic and a viscoelastic configuration In figure 5 an elastic configuration is compared with a viscoelastic configuration. The numerical data are: E=2.0 N/mm*, d=4.0 mm. v = 0.5,,,=7.0 Nlmm\ k=3.0 N/mm, x=0.007 s and V=100.0 mm/s. A contour-plot is shown of the normal stress distribution or,. From the shape of the contour-lines we can see that the elastic situation with tangential force shows a similar behaviour as the viscoelastic case, without tangential force. The normal stress is asymmetrical and hence rolling is resisted by a moment about the axle of the upper roller. (B)

7 o.os r t0.13 Contact Mechanics 125 o.oo Figure 6. Comparison between numerical and experimental results. /. F^ = 0.74 TV/mm, 0) F^ = 0.0 TV/mm, %) F^= 0.05 TV/mm In figure 6 the comparison between numerical and experimental results for configuration I is shown. Configuration I means that only the glass roller is covered with one single transparent rubber layer. The black roller is rigid. In figure 6B a tangential force is applied: A slip region occurs at the trailing (left) edge of the contact region. The numerical data for this configuration are: E=0.91 Nlmnr, d-2.0 mm, v=0.5, R,=50.0 mm and R2=25.0 mm. The coefficient of friction is 2.7, which has been measured in a separate set-up. enlargement with optimization o.n 0.60 o.oo Figure 7. Comparison between numerical and experimental results. Configuration II: F\ = 7.78 N/mm, (A) Fj = 0.0 N/mm, (B) Fj= 0.53 N/mm In figure 7 the results for configuration II are shown. Here the black roller is covered with a black (carbon-filled) rubber layer (E-2.04 N/mm^, d-2.7 mm). The strain in the contact region is not constant in this case. The coefficient of friction is 2.8. In the numerical results of figure 7 we see also the influence of the geometrical optimization. The agreement is better if the geometry is optimized, especially when a tangential force is applied, as indicated in figure 7B. Finally the transparent rubber layer of configuration I is replaced by a two-layer rubber coating (configuration III). The toplayer is very thin and relatively hard (d=0.020 mm, E=2.04 Nlmnf), while the intermeadiate layer is thick and soft (d-2.3 mm, E=0.83 Nlmnf).

8 126 Contact Mechanics o.oo Figure 8. Comparison between numerical and experimental results. Configuration III: F» = 1.78 N/mm, (A) F? = 0.0 N/mm, (B) F^ 0.53 N/mm CONCLUSIONS In this paper a numerical and experimental analysis is given of the contact between multilayered, viscoelastic, rolling cylinders. It has been demonstrated that the agreement between for theory and experiment is very satisfactory for three different configurations, with and without tangential force. The linear model is adequate enough to model this contact problem. The geometrical optimization has improved the agreement between theory and experiment. The viscoelastic model is suitable to obtain qualitative results. REFERENCES [1] H. Hertz, Uber die Beriihrung fester elastischer Korper. Journal fur reine und angewandte Mathematik. 92, pp , 1882 [2] K.L. Johnson, Contact Mechanics. Cambridge University Press, Cambridge U.K., 1985 [3] P. Meijers, The contact problem of a rigid cylinder on an elastic layer. Applied Science Research, 18, pp , 1968 [4] J.J. Kalker, The simplified theory of rolling contact. Delft Progress Report, 1, pp. 1-10, 1973 [5] S.C. Hunter, The rolling contact of a rigid cylinder with a viscoelastic half-space. Journal of Applied Mechanics. 28, pp , 1961 [6] G. Wang, Rollkontakt zweier viskoelastischer Walzen mil Coulombscher Reibung. PhD Thesis, Berlin, 1991 [7] G. Wang, J.J. Kalker, Threedimensional Rolling contact of two viscoelastic bodies. Proceedings of the Contact Mechanics International Symposium (Ed. A. Curnier), Presse Polytechnique Universitaire Press Romandes, pp , 1992 [8] J.J. Kalker, Viscoelastic multilayered cylinders rolling with dry friction. Journal of Applied Mechanics, 58, pp , 1991 [9] W.N. Findley, J.S. Lai, K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials. Dover Publications Inc., New York, 1976 [10] G.F.M. Braat, Theory and experiments on multilayered, viscoelastic cylinders in rolling contact. PhD Thesis, Delft, to be published in June 1993 [11] J.J. Kalker, Three-dimensional Elastic Bodies in Rolling Contact. Kluwer Academic Publishers, Dordrecht, Boston, London, 1990 [12] F. Durst, A. Melling, J.H. Whitelaw, Principles and Practise of Laser-Doppler Anemometry. Academic Press, London, 1981 [13] G.F.M. Braat, H.G.J. Rutten, Application of Laser-Doppler Anemometry in nips (in Dutch). Mikroniek, 1, 1993