A general theory for the Curve-To-Curve contact is applied to develop a special contact algorithm between curves and rigid surfaces. In this case contact kinematics are formulated in the local coordinate system attached to the curve, however, contact is deﬁned at integration points of the curve line (Mortar type contact). The corresponding Closest Point Projection (CPP) procedure is used to deﬁne then a shortest distance between the integration point on a curve and the rigid surface. For some simple approximations of the rigid surface closed form solutions are possible. Within the ﬁnite element implementation the isogeometric approach is used to model curvilinear cables and the rigid surfaces can be deﬁned in general via NURB surface splines. Veriﬁcation of the ﬁnite element algorithm is given using the well-known analytical solution of the Euler-Eytelwein problem – a rope on a cylindrical surface. The original 2D formula is generalized into the 3D case considering an additional parameter H-pitch for the helix. Finally, applications to knot mechanics are shown

Konyukhov, Alexander

Metzger, Andreas

Schweizerhof, Karl

English

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147ON CONTACT BETWEEN CURVES AND RIGIDSURFACES – FROM VERIFICATION OF THEEULER-EYTELWEIN PROBLEM TO KNOTSALEXANDER KONYUKHOV∗, KARL SCHWEIZERHOF† ANDANDREAS METZGER†∗Institute of MechanicsKarlsruhe Institute of TechnologyKaiserstrasse 12, 76131 Karlsruhe, Germanye-mail: Alexander.Konyukhov@kit.edu, http://www.ifm.kit.edu/14 203.php/†Institute of MechanicsKarlsruhe Institute of TechnologyKaiserstrasse 12, 76131 Karlsruhe, Germanyhttp://www.ifm.kit.edu/Key words: Curve-To-Curve Contact, Curve-To-Surface contact, Euler-Eytelwein Prob-lem, Belt Friction Problem, KnotsAbstract. A general theory for the Curve-To-Curve contact is applied to develop aspecial contact algorithm between curves and rigid surfaces. In this case contact kine-matics are formulated in the local coordinate system attached to the curve, however,contact is defined at integration points of the curve line (Mortar type contact). The cor-responding Closest Point Projection (CPP) procedure is used to define then a shortestdistance between the integration point on a curve and the rigid surface. For some simpleapproximations of the rigid surface closed form solutions are possible. Within the finiteelement implementation the isogeometric approach is used to model curvilinear cablesand the rigid surfaces can be defined in general via NURB surface splines. Verificationof the finite element algorithm is given using the well-known analytical solution of theEuler-Eytelwein problem – a rope on a cylindrical surface. The original 2D formula isgeneralized into the 3D case considering an additional parameter H-pitch for the helix.Finally, applications to knot mechanics are shown.1 INTRODUCTIONIn many engineering devices such capstan and belt drives frictional interaction betweenstructure and ropes, cables or belts can be modeled as the interaction between rigidsurfaces and deformable curvilinear beams or ropes without loss of tolerance. Several1XI International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XI E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) 148Alexander Konyukhov, Karl Schweizerhof and Andreas Metzgerdevelopments are required for this problem: a robust cable finite element and a contactalgorithm for the Curve-To-Rigid Surface contact interaction. The isogeometric approach,see [1], is employed together with a special finite beam element formulation allowing bothfinite rotations, see in [2] in order to obtain cable finite elements. The theory for Curve-To-Curve contact interaction developed in [3] is applied to obtain a contact algorithmdescribing the contact between a deformable curve (representing either the center-line ofthe beam or an edge of a solid body) and a rigid surface.The classical Euler-Eytelwein solution of the belt friction problem, see e.g. in [4], is firstemployed to verify the developed algorithm. The solution of this problem was reported byEuler in his Remarks on the effect of friction on equilibrium and has been first publishedby the Berlin Academy of science [5] in 1769. Since the first time publishing the Eulersolution by Eytelwein [6] in his Handbuch der Statik fester Körper in 1808 the problem isspread through the practical applications and became known as Euler-Eytelwein problemin many books of technical mechanics. Here we are proposing a generalization of theclassical 2D solution into the 3D case under the assumption that the rope is forming aspiral line on a rigid cylinder. This solution in due course is used for verification.Finally, combination of both Curve-To-Curve and Curve-To-Rigid Surface contact al-gorithms allows to step into modeling of more complex see-man knots (beginning of studysee in [7]) such as the clove-hitch knot. Numerical examples are illustrating the tying-up-processes.2 CURVE-TO-RIGIDANALYTICAL SURFACE CONTACTALGORITHMThe theory for Curve-To-Curve contact developed in [3] is employed to obtain a contactalgorithm between a deformable curve and a rigid surface as follows. A rigid surface isassumed to have arbitrary analytical description, e.g. via NURBS surfaces. All contactparameters for the Curve-To-Rigid Suface algorithm are defined similar to the Curve-To-Curve contact algorithm in the Serret-Frenet frame, see [3], obtained by the tangentvector τ , the normal vector ν and the bi-normal β of the curve line, see Fig. 1. A vectore is defining the shortest distance.In order to describe contact between deformable curves and rigid surfaces a Segment-To-Analytical-Surface (STAS) algorithm, discussed in [9], is modified using the projectionof the integration points which is set up on the “slave” curve onto the rigid “master”surface. The combination of this strategy with the Curve-To-Rigid (analytical) Surfacecontact algorithm leads to the following definition of the coordinate system on the mastersurface:ρs(ξ) = r(α1, α2) + pn(α1, α2), (1)where ρs(ξ) is defining an integration point positioned on the mid-line of the curvilinearbeam element, r(α1, α2) is parameterization of the rigid “master” surface. The integrationpoint ρs(ξ) is found in the direction of the normal n(α1, α2) to the rigid “master” surface.The shortest distance between integration points and the surface denoted as p plays2149Alexander Konyukhov, Karl Schweizerhof and Andreas MetzgerFigure 1: Curve-To-Curve contact algorithm is defined in the Serret-Frenet frame for both curves.the role of a penetration. The Closest Point Projection (CPP) procedure, which is thestandard contact local searching algorithm now turns into the determination of the surfaceconvective coordinates α1, α2 defined by equation (1). In general, Newtons method isexploited to solve eqn. (1) defining then a point on the rigid surface and the penetrationp between this surface and the selected integration point S. An analytical solution ispossible for some surfaces – one of the important examples for further verification is thecontact of a curve with a rigid cylindrical surface. Here, the rigid cylindrical surface withthe radius R is given as r = RC + zez + Reϕ(ϕ) with the normal vector n = eϕ(ϕ), seeFig. 2.The distance p and thereby penetration into the cylinder is defined asp ={�RC − ρs −((RC − ρs) · ez)ez� − R – for an outward normalR − �RC − ρs −((RC − ρs) · ez)ez� – for an inward normal(2)It should be noticed that for the contact between the cylinder and a cable with a circularcross-section with radius Rcable, the penetration is, of course, computed as penetration =p− Rcable.A coordinate z is used as a measure of tangential interactions for the definition offrictional interaction.3150Alexander Konyukhov, Karl Schweizerhof and Andreas MetzgerXYZOϕSRCeezϕzC zCsρApτνβFigure 2: Contact with a rigid cylinder given by an analytical equation.The contact integral is representing then the virtual work of contact traction along thecurve L and is computed numerically using the integration formula of Lobatto or Gausstype depending on modeling purposes. Thus, e.g. the part responsible only for the normalcontact is represented byδW =∫LNδpds. (3)For the derivation of tangent matrices results from the Curve-To-Curve contact approach,see in [3], are employed directly. Thus, the tangent matrix derived from the linearizedpart of eqn. (3) is given asKN =∫Lεr AT [e⊗ e]Ads (4a)+∫LN AT{k cosϕ1(1− rk cosϕ)τ ⊗ τ −1rg ⊗ g}Ads. (4b)Here, N is a normal contact force, εr is a normal penalty parameter, e is a vector definingthe shortest distance, g = τ×e is an orthogonal vector, r is the shortest distance betweenthe integration point and the rigid surface, k is the curvature of the curve, ϕ is the anglebetween the normal curve vector ν and the unit vector of the shortest distance e (e = e1in Fig. 1 and e = −eϕ in Fig. 2). For all geometrical parameters see Fig. 1. A is anapproximation operator for the curve line. The frictional tangential force T is computed4151Alexander Konyukhov, Karl Schweizerhof and Andreas Metzgervia the return-mapping algorithm. For the derivation of the tangential force T and otherparameters including the tangent matrices and more details see [3].3 GENERALIZATION OF THE EULER-EYTELWEIN FORMULA INTOTHE 3D CASE CONSIDERING PITCH HqdsβντRR−( + d )RThe equilibrium equation of the elementary part of the rope with length ds, see Fig. 3,is given in the following vector formdRds= q (5)The force vector R can be expanded in the natural Serret-Frenet coordinate system usedin the differential geometry of curves, see e.g. in [8], built by the unit tangent τ , the unitnormal ν and the unit bi-normal β vectorsR = Tτ +Nν + Bβ, (6)with T – a tangential force, N – a normal force, and B – a bi-normal force. The fullderivative of the force vector includes also a part due to changing the basis vectors τ , ν,β and is obtained via the Serret-Frenet formula, see e.g. in [8] and the application forthis problem in [11]. Taking this formula into account, the projection of eqn. (5) onto theaxes τ , ν, β results then in the system of ordinary differential equations:dTds+ kN = qτ – equilibrium in τ directiondNds− kT + κB = qν – equilibrium in ν directiondBds− κN = qβ – equilibrium in β direction(7)where k is the curvature and κ is the torsion of the curve, which are in general functionsof the arc-length parameter s.5152Alexander Konyukhov, Karl Schweizerhof and Andreas Metzger3.1 Solution of the equilibrium equation for a spiral line (helix)Now we consider a special case with the following conditions:(i) the rope is positioned on the rigid surface.(ii) the rope is loaded only by tangential forces T0τ 0 and T1τ 1 at both ends.(iii) the rope is beginning to slide preserving its geometrical shape, i.e. only a motionalong τ is possible. Thus the form of the helix is conserved.(iv) sliding is subjected to the Coulomb friction law such that T (s) = µN(s), where µ isa coefficient of friction.Thus, due to condition (iii), the first equilibrium equation in (7) is transferred into theequation of motion according to D’Alemberts principle withqτ = −ρ∂2u∂t2, (8)where u is a tangential component of the displacement and ρ is the linear mass density.However, equilibrium along ν and β axis is fulfilled and we consider the absence of thedistributed forces qν = 0, qβ = 0. The problem is then described by the system of ordinarydifferential equations: dNds− kT + κB = 0dBds− κN = 0(9)with the following initial conditions at s = 0 resulting from condition (ii):T (0) = T0, N(0) = N0, B(0) = 0. (10)Now, we consider a case with a constant curvature k and a constant torsion κ ofthe curve. This is a case of the spiral line – a helix – on the cylinder. The system ofequations (9) is transformed into a single differential equation of second order by takingthe derivative of the first equation in (9):d2Nds2− kdTds+ κdBds= 0 (11)and then using the second equation in (9) we obtaind2Nds2− kdTds+ κ2N = 0. (12)6153Alexander Konyukhov, Karl Schweizerhof and Andreas MetzgerNow taking into account the sliding condition (iv) we obtain a second order differentialequation :d2Nds2− µkdNds+ κ2N = 0 (13)with the initial conditions:N(0) = N0,dN(0)ds= kT0. (14)The solution of the ordinary differential equation (13) is obtained by using the character-istic equationλ2 − µkλ+ κ2 = 0 (15)λ1, 2 =µk ±√µ2k2 − 4κ22(16)Depending on the determinant D = µ2k2 − 4κ2 three solutions are possible. We considerthese values for the spiral line (helix). The geometrical characteristics for this line aredefined as follows, see Fig. 3XYZDBAOHϕαFigure 3: Spiral line (helix) with a pitch H .7154Alexander Konyukhov, Karl Schweizerhof and Andreas MetzgerParametrization of the spiral line (helix):r =R cosϕR sinϕH2πϕ+ const. (17)Within 0 ≤ ϕ ≤ 2π the line is making a full turn around the OZ-axis raising by thepitch z = H with the following geometrical items: arc-length of the spiral line (helix)s =�R2 +�H2π�2ϕ, ϕ ∈ [0, ...); curvature of the helix k = RR2 +�H2π�2 ; torsion of thespiral line (helix) κ = H/2πR2+H2π2 . Then positivity of the determinant can be expressedvia the geometrical parameters of the helix asD ≥ 0 =⇒ = µk ≥ 2κ =⇒ µ ≥HπR. (18)Omitting details of transformations we discuss the expressions for the force ratioTT0for all three possible cases.3.1.1 Case 1. Positive determinant D > 0, µ >HπRSince with H = 0 the determinant is positive D > 0, then this case is representing aspiral line with a very small pitch H in comparison with the radius R. The solution isgiven by the exponential functions as:TT0=�µk2ωsinhωs+ coshωs�eµk2s, (19)with ωω =�µ2k2 − 4κ22. (20)If we consider the 2D case with H = 0 in eqn. (19) the classical Euler-Eytelweinfor the rope on the cylinder is recoveredTT0=�sinh�µϕ2�+ cosh�µϕ2��eµϕ2 = eµϕ. (21)8155Alexander Konyukhov, Karl Schweizerhof and Andreas Metzger3.1.2 Case 2. Negative determinant D < 0, µ <HπRThe case is representing either a case with a spiral line with large pitch H , or with asmall coefficient of friction. The solution is given by the trigonometrical and exponentialfunctions:TT0=[kµ2ω̃sin ω̃s+ cos ω̃s]eµk2s, (22)with ω̃ω̃ =√4κ2 − µ2k22. (23)3.1.3 Limit case 3. Determinant is zero D = 0, µ =HπRThis case can be obtained just by the limit process with ω → 0 with the solution ineqn. (19) or in eqn. (22) aslimω→0TT0=[kµ2s+ 1]eµk2s. (24)A numerical computation for the cases with D > 0, D = 0, D < 0 and the classicalEuler formula with H = 0 is presented in Fig. 4 for the angle ϕ ∈ [0, 2π]. The computationis given with a coefficient of friction µ = 0.3 and radius R = 1.0 for the following cases:• Classical Euler case H = 0;• Positive determinant D < 0 with H = 0.75• Zero determinant D = 0 with H = µπR = 0.9424• Negative determinant D < 0 with H = 1.2One can see, that if the pitch H is increasing then the ratio of forces T/T0 is decreasing,thus, the standard 2D Euler case is representing the upper limit of the forces ratio T/T0with regard to the pitch H .The derived formula is used further for the verification of the Curve-To-Rigid Sufacecontact algorithm.4 APPLICATION TO KNOTSBoth the Curve-To-Curve (CTC) contact algorithm, see in [3], [7], and the developedCurve-To-Rigid-Surface (CTRS) contact algorithm are used to model the Clove-Hitchknot, see Fig. 5, presented also in [10].9156Alexander Konyukhov, Karl Schweizerhof and Andreas Metzger 0 1 2 3 4 5 6 7 8 0  1  2  3  4  5  6  7  8  9T/ToϕH=0D>0D=0D<0Figure 4: Comparison of all cases with determinant D > 0, D = 0, D < 0 and the classical Euler formulawith H = 0.The developed model allows to describe the tying-up mechanics of the Clove Hitchknot: during the tying up both loops of the knot are approaching each other. The upperloop of the knot is enforcing this motion. The knot is tying up and is not sliping becauseof the large frictional forces appearing between the two approaching loops.5 CONCLUSIONS- A special Curve-To-Rigid-Surface (CTRS) contact algorithm is developed in thecurrent contribution based on the application of the Curve-To-Curve contact theoryand Segment-To-Analytical Surface (STAS) contact algorithm.- Rigid surfaces can be approximated with arbitrary NURBS surfaces – in this casean iterative solution of the corresponding Closest Point Projection (CPP) procedureis necessary to define the contact point as well as the penetration.- An analytical closed form solution for the CPP procedure is possible for some simplecases e.g. for a cylinder.- The Euler-Eytelwein problem is generalized into the 3D problem as a frictionalsliding of a spiral line (helix) on a cylinder. It is shown that consideration of thepitch H leads to the reduction of the force ratio T/T0. The formula is used as abasis for the verification of the Curve-To-Rigid-Surface (CTRS) contact algorithm.10157Alexander Konyukhov, Karl Schweizerhof and Andreas Metzger(a) Untied knot (b) Model of the untied knot(c) Tied knot (d) Model of the tied knotFigure 5: Illustration and finite element model of an untied and tied Clove-Hitch knot- The developed Curve-To-Rigid (analytical) Surface (CTRS) contact algorithm to-gether with the Curve-To-Curve (CTC) contact algorithm is applied to study thekinematics of the tying up of the Clove Hitch knot.REFERENCES[1] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Inte-gration of CAD and FEA, John Wiley & Sons, 2009.[2] A. Ibrahimbegovic, On finite element implementation of geometrically nonlinearReissner’s beam theory: three-dimensional curved beam elements. Computer Methodsin Applied Mechanics and Engineering (1995) 122:11–26.[3] Konyukhov A., Schweizerhof K. Geometrically exact covariant approach for contactbetween curves, Computer Methods in Applied Mechanics and Engineering (2010)199:2510–2531.[4] E. R. Maurer, R. J. Roark, Technical mechanics: statics, kinematics, kinetics. -5.ed. New York: Wiley, 1944.[5] L. Euler, Remarques sur l’effet du frottement dans l’equilibre, Memoires del’academie des sciences de Berlin, 18 (1769), pp. 265–278.[6] J. A. Eytelwein, Handbuch der Statik fester Körper. Mit vorzüglicher Rücksichtauf ihre Anwendung in der Architektur. Vol. 2, Berlin, 1808, pp. 21–23.[7] Konyukhov A., Schweizerhof K. Geometrically exact theory for contact interactionsof 1D manifolds. Algorithmic implementation with various finite element models.11158Alexander Konyukhov, Karl Schweizerhof and Andreas MetzgerComputer Methods in Applied Mechanics and Engineering available online 2 April2011, doi:10.1016/j.cma.2011.03.013[8] Kreyszig E. Differential geometry. New York: Dover Publications, 1991.[9] Konyukhov A. Geometrically Exact Theory for Contact Interactions, Habilitationss-chrift, Karlsruhe, KIT, 2010.[10] Metzger A., Konyukhov A., Schweizerhof K. Finite element implementation for Euler-Eytelwein-problem and further use in FEM-simulation of common nautical knots.GAMM, 82nd Annual meeting 18-21 April 2011, Graz, Austria.[11] Konyukhov A., Schweizerhof K. Frictional interaction of a spiral rope and a cylinder– 3D-generalization of the Euler-Eytelwein formula considering pitch. submitted.12

On contact between curves and rigid surfaces – from verification of the euler-eytelwein problem to knots

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On contact between curves and rigid surfaces – from verification of the euler-eytelwein problem to knots

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