Abstract
Slender thread like bodies (like cables, ropes, textilethreads or belts) are often used in technical applications. Becauseof their dimensions the one-dimensional continuum is the appropriatemechanical model for bodies of this type. Making use of the basicrelations of three-dimensional continua as a starting point the paperdevelops the general kinematic and kinetic relations of one-dimensionalcontinua for the case that the cross-sections will remain plane (Bernoullihypothesis), that large deflections are possible but the strains remainsmall and that the material is homogeneous and isotropic and behaveslinearly elastic. This results in the equations of motion of shearableand extensible rods (Timoshenko-beams). By neglection of shear deformationand of the rotational inertia of the cross-sections (assumptions thatcan be done in most technical applications) the equations of motionof Euler–Bernoulli-beams are derived in standard and concentratedform. The Euler–Bernoulli-beam equations contain the equations ofmotion of threads with zero bending and torsional stiffness. It isshown that the neglection of bending and torsional stiffness is onlyvalid if the tension is always positive. The second part of this paper[1] selects and develops appropriate numerical solution methods.The derived algorithms are used to solve problems from space and marineengineering.
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References
Weiss, H., 'Dynamics of geometrically nonlinear rods: II. Numerical methods and computational examples', Nonlinear Dynamics 30, 2003, 383–415.
Plagge, F., 'Nichtlineares, inelastisches Verhalten von Spiralseilen', Dissertation, Technische Universität Braunschweig, 1997. Dynamics of Geometrically Nonlinear Rods: I381
Szabo, I., Geschichte der mechanischen Prinzipien, Birkhäuser, Basel, 1979.
Truesdell, C. and Noll, W., The Nonlinear Field Theories of Mechanics, Springer, New York, 1965.
Wang, C.-C. and Truesdell, C., Introduction to Rational Elasticity, Noordhoff, Leiden, 1973.
Ciarlet, P.G., Mathematical Elasticity. Volume I: Three-Dimensional Elasticity, Elsevier, Amsterdam, 1987.
Kuhn, A., Steiner, W., Zemann, J., Dinevski, D., and Troger, H., 'A Comparison of Various Mathematical Formulations and Numerical Solution Methods for the Large Amplitude Oscillations of a String Pendulum', Applied Mathematics and Computation 67, 1995, 227–264.
Pedrazzi, C., 'Dog on lead mine-hunting underwater cable simulation', in Proceedings to the 12th European ADAMS Users' Conference, Marburg, Germany, November 18-19, 1997.
Schiehlen, W., Technische Dynamik, Teubner, Stuttgart, 1986.
Spong, M. W. and Vidyasagar, M., Robot Dynamics and Control, Wiley, New York, 1989.
Bremer, H., 'On the dynamics of elastic multibody systems', Applied Mechanics Reviews 54(9), 1999, 275–303.
Whitham, G. B., Linear and Nonlinear Waves, Wiley, New York, 1974.
Schnell, W., Gross, D., and Hauger, W., Technische Mechanik, Band 2: Elastostatik, Springer, Berlin, 1992.
Goeldner, H., Lehrbuch höhere Festigkeitslehre. Band I: Grundlagen der Elastizitätstheorie, Fachbuchverlag, Leipzig, 1991.
Franke, W., Ein Beitrag zur dynamischen räumlichen Elastizitätstheorie II. Ordnung eines geraden Stabes unter dem besonderen Aspekt transienter axialer Einwirkung, Shaker, Aachen, 1997.
Trommer, G., 'Das Aufdrehmoment eines Garnes', Faserforschung und Textiltechnik 17(8), 1966, 355–362.
Buerger, W., 'Theorie einfacher Wellen und Anwendungen in der Kontinuumsmechanik', Habilitation, Technische Hochschule Darmstadt, 1971.
Weiss, H., 'Zur Dynamik geometrisch nichtlinearer Balken', Dissertation, Technische Universität Chemnitz, 1999.
Triantafyllou, M. S. and Howell, C. T., 'Dynamic response of cables under negative tension: An ill-posed problem', Journal of Sound and Vibration 173(4), 1994, 433–447.
Hamel, G., Theoretische Mechanik, Springer, Berlin, 1949.
Reissner, E., 'On one-dimensional large-displacement finite-strain beam theory', Studies in Applied Mathematics 52, 1973, 87–95.
Simo, J. C., 'A finite strain beam formulation. The three dimensional dynamic problem. Part I', Computer Methods in Applied Mechanics and Engineering 49, 1985, 55–70.
Simo, J. C. and Vu-Quoc, L., 'Three dimensional finite strain rod model. Part II: Computational aspects', Computer Methods in Applied Mechanics and Engineering 58, 1986, 79–116.
Svetlizkij, V. A., Mechanics of Rods, Vysshaya Shcola, Moscow, 1987 [in Russian].
Antman, S. S., Nonlinear Problems of Elasticity, Springer, New York, 1995.
Dichmann, D. J., Li, Y., and Maddocks, J. H., 'Hamiltonian formulations and symmetries in rod mechanics', in Mathematical Approaches to Biomolecular Structures, J. P. Mesirov, K. Shulten, and D. Sumners (eds.), IMA Volumes in Mathematics and Its Applications, Vol. 82, Springer, New York, 1996, pp. 71–113.
Berger, R., Instationäre Bewegung und Stabilitätsverhalten eindimensionaler Kontinua, Fortschritt-Berichte VDI Reihe, Vol. 18, Nr. 189, VDI Verlag, Düsseldorf, 1996.
Stelzle, W., Geometrisch-nichtlineare Modellierung elastischer Balken mit Helix-Elementen, Fortschritt-Berichte VDI Reihe, Vol. 18, Nr. 224, VDI Verlag, Düsseldorf, 1998.
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Weiss, H. Dynamics of Geometrically Nonlinear Rods: I. Mechanical Models and Equations of Motion. Nonlinear Dynamics 30, 357–381 (2002). https://doi.org/10.1023/A:1021268325425
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DOI: https://doi.org/10.1023/A:1021268325425