Contact between curves and rigid surfaces: covariant formulation and verifications. Alexander Konyukhov, Karl Schweizerhof Karlsruhe Institute of Technology, Institute of Mechanics, Germany,
[email protected],
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Summary: The solution of the problem of frictional interaction between a rope and a rigid cylinder is known since many years as Euler-Eytelwein problem or belt friction problem and is solved as a 2D problem describing the interaction between a rope and a circle. Here we generalize the solution under a first assumption that the rope is formed as a spiral line on a rigid cylinder. Various frictional cases are considered. The classical 2D result can be directly recovered. The path for further solutions with different geometries is then open. In this work the finite element model is derived for the curve-to-rigid surface contact and is verified via the generalized 3D Euler-Eytelwein formula.
Introduction Historically, the solution of a problem of the definition of frictional rope forces was reported by Euler in his Remarks on the effect of friction on equilibrium published by the Berlin Academy of science, see [1]. Since the first time publishing the Euler solution by Eytelwein [2] in his Handbuch der Statik fester K¨orper in 1808 the problem is spread through practical applications and became known as Euler-Eytelwein problem in standard books of technical mechanics, see in Gross et.al. [3], or as a belt or coil friction formula see e.g. in Maurer and Roark [4]. In this work the finite element model is derived for the curve-to-rigid surface contact and is verified via the Euler-Eytelwein formula.
Kinematics of the curve-to-rigid surface interaction The kinematics of the Curve-To-Rigid-Surface contact interaction is based on the combination of the Surface-To-Analytical-Surface and the Curve-To-Curve contact kinematics. Thus all parameters are formulated dually in the surface coordinate system (via Gaussian coordinates ξ 1 , ξ 2 ) and in the Serret-Frenet curve coordinate system (via coordinate ζ). The curve rs (η), see Fig. 1 is assumed to be deformable, thus, all parameters for the curve are considered as developed for the Curve-To-Curve contact in [5]. A rigid surface ρ(ξ 1 , ξ 2 ) is assumed to have arbitrary analytical description, e.g. via NURBS surfaces. In order to describe contact between deformable curves and rigid surfaces a Segment-To-Analytical-Surface (STAS) algorithm is modified as follows: A set of contact points (integration points) ηi are set on the curve segment AB: all kinematical parameters are considered then in the Serret-Frenet curve coordinate system τ , ν, β; The contact point (integration points) is projected onto the rigid surface. At each point all kinematical parameters are considered in the surface coordinate system ρ1 , ρ2 , n via the convective coordinates ξ 2 , ξ 2 . The combination of both CTC and STAS strategies leads to the Curve-To-Rigid (analytical) Surface contact algorithm and to the following definition of the coordinate system on the master rigid surface: ρs (η) = ρ(ξ 1 , ξ 2 ) + ξ 3 n(ξ 1 , ξ 2 ), (1)
Figure 1: Kinematics of the Curve-To-Rigid Surface (CTRS) Contact – definition of both the Gaussian surface and the Serret-Frenet curve coordinate systems. where ρs (η) is defining an integration point positioned on the curve, or in the case of the beam contact on the mid-line of the curvilinear beam element, r(ξ 1 , ξ 2 ) is the parametrization of the rigid “master” surface. The integration point ρs (η) is found in the direction of the normal n(ξ 1 , ξ 2 ) to the rigid “master” surface. The shortest distance between integration points and the surface denoted as ξ 3 plays the role of a penetration. Now the Closest Point Projection (CPP) procedure is exploited to define the penetration ξ 3 between this surface and the selected integration point S. A set of analytical solutions is possible for some surfaces such as cylinder, sphere and plane. The kinematical parameters for the CTRS contact are defined in the surface coordinate system ρ1 , ρ2 , n as follows: The measure of the normal interaction in the case of a curvilinear beam ξ 3 = (rs − ρ) · n − R.
(2)
where R-is a radius of the curvilinear beam (rope, cable etc.) as shown e.g. in Fig. 1. The measure of the tangential interaction in the rate form as : ξ˙i = (vs − v) · ρj aij ,
(3)
and later is reprojected into the Serret-Frenet curve coordinate system. The Euler-Eytelwein formula generalized for 3D spiral line is used to verify the obtained finite element algorithm.
References [1] L. Euler. Remarques sur l’effet du frottement dans l’equilibre, Memoires de l’academie des sciences de Berlin, 18:265–278, 1769. [2] J. A. Eytelwein. Handbuch der Statik fester K¨orper. Mit vorz¨uglicher R¨ucksicht auf ihre Anwendung in der Architektur. Vol. 2, Berlin, 1808. [3] D. Gross, W. Hauger, W. Schnell and J. Schr¨oder. Technische Mechanik. Vol. 1:Statik, Springer-Verlag Berlin Heidelberg, 2004. [4] E. R. Maurer and R. J. Roark. Technical mechanics: statics, kinematics, kinetics. New York: Wiley, 1944. [5] A. Konyukhov and K. Schweizerhof. Geometrically exact covariant approach for contact between curves, Computer Methods in Applied Mechanics and Engineering, 199:2510– 2531, 2010.
