Skip to main content
SpringerLink
Log in

Dynamics of Geometrically Nonlinear Rods: II. Numerical Methods and Computational Examples

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The paper selects and develops appropriate numerical solutionmethods for initial boundary value problems of the equations of motionof geometrically nonlinear extensible Euler–Bernoulli-beams. A finiteelement method that uses first-order Hermitian polynomials as interpolationfunctions for the rod axis position vector is used as discretizationtechnique. An averaging method for the calculation of net forces andmoments is developed that achieves a better approximation than thedirect calculation from the strains. Time integration is done usingan energy and momentum conserving algorithm that is proposed in thispaper and Newmark type methods. The derived algorithms are used tosolve problems from space and marine engineering. The obtained simulationresults are compared with results which have been already publishedin the literature or were calculated by different methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Weiss, H., 'Dynamics of geometrically nonlinear rods: I. Mechanical models and equations of motion', Nonlinear Dynamics 30, 2002, 357–381.

    Article  Google Scholar 

  2. Timoshenko, S., Young, D. H., and Weaver, W., Vibration Problems in Engineering, Wiley, New York, 1974.

    Google Scholar 

  3. Nayfeh, A. H. and Mook, D.T., Nonlinear Oscillations, Wiley, New York, 1979.

    Google Scholar 

  4. Shabana, A. A., Vibration of Discrete and Continuous Systems, Springer, New York, 1997.

    Google Scholar 

  5. Simo, J. C., Tarnow, N., and Doblaré, M., 'Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms', International Journal for Numerical Methods in Engineering 38, 1995, 1431–1473.

    Google Scholar 

  6. Bauchau, O. A. and Theron, N. J., 'Energy decaying scheme for non-linear beam models', Computer Methods in Applied Mechanics and Engineering 134, 1996, 37–56.

    Article  Google Scholar 

  7. Crisfield, M. A., Galvanetto, U., and Jelenic, G., 'Dynamics of 3-D co-rotational beams', Computational Mechanics 20, 1997, 507–519.

    Article  Google Scholar 

  8. Ibrahimbegovic, A. and Al Mikdad, M., 'Finite rotations in dynamics of beams and implicit time-stepping schemes', International Journal for Numerical Methods in Engineering 41, 1998, 781–814.

    Article  Google Scholar 

  9. Burgess, J. J., 'Bending stiffness in a simulation of undersea cable deployment', International Journal of Offshore and Polar Engineering 3, 1993, 197–204.

    Google Scholar 

  10. Dichmann, D. J. and Maddocks, J. H., 'An impetus-striction simulation of the dynamics of an elastica', Journal of Nonlinear Science 6, 1996, 271–292.

    Article  Google Scholar 

  11. Stelzle, W., Geometrisch-nichtlineare Modellierung elastischer Balken mit Helix-Elementen, Fortschritt-Berichte VDI Reihe, Vol. 18, Nr. 224, VDI Verlag, Düsseldorf, 1998.

    Google Scholar 

  12. 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.

    Article  Google Scholar 

  13. Berger, R., Instationaere Bewegung und Stabilitaetsverhalten eindimensionaler Kontinua, Fortschritt-Berichte VDI Reihe, Vol. 18 Nr. 189, VDI Verlag, Düsseldorf, 1996.

    Google Scholar 

  14. 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.

  15. Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill, New York, 1977.

    Google Scholar 

  16. Huebner, K. H. and Thornton, E. A., The Finite Element Method for Engineers, Wiley, New York, 1982.

    Google Scholar 

  17. Kuessner, M., 'Gemischte Finite-Elemente-Formulierungen für nichtlineare Probleme der Festkörpermechanik', Dissertation, Technische Hochschule Darmstadt, 1995.

  18. Shabana, A. A., 'Definition of the slopes and the finite element absolute nodal coordinate formulation', Multibody System Dynamics 1, 1997, 339–348.

    Article  Google Scholar 

  19. von Dombrowski, S. and Schwertassek, R., 'Analysis of large deformation problems in multibody systems using absolute coordinates', in Proceedings of the EUROMECH Colloquium 404 on Advances in Computational Multibody Dynamics, Lisbon, Portugal, September 20-23, 1999.

  20. Berneck, P., 'Die Anwendung direkter Zeitintegrationsverfahren auf Probleme der Strukturmechanik', Dissertation, Technische Hochschule Darmstadt, 1996.

  21. Belytschko, T. and Schoeberle, D. F., 'On the unconditional stability of an implicit algorithm for nonlinear structural dynamics', Journal of Applied Mechanics 42, 1975, 865–869.

    Google Scholar 

  22. Hughes, T. J. R., Caughy, T. K., and Liu, W. K., 'Finite-element-methods for nonlinear elastodynamics which conserve energy', Journal of Applied Mechanics 45, 1978, 366–370.

    Google Scholar 

  23. Kuhl, D. and Ramm, E., 'Constraint energy momentum algorithm and its application to nonlinear dynamics of shells', Computer Methods in Applied Mechanics and Engineering 136, 1996, 293–315.

    Article  Google Scholar 

  24. Simo, J. C., Tarnow, N., and Wong, K. K., 'Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics', Computer Methods in Applied Mechanics and Engineering 100, 1992, 63–116.

    Article  Google Scholar 

  25. Weiss, H., 'An exact energy and momentum conserving integration scheme for the nonlinear dynamics of geometrically exact threads', Zeitschrift fuer angewandte Mathematik und Mechanik 78, 1998, S805–S806.

    Google Scholar 

  26. La Budde, R. A. and Greenspan, D., 'Discrete mechanics-A general treatment', Journal of Computational Physics 15, 1974, 134–167.

    Article  Google Scholar 

  27. Tarnow, N. and Simo, J. C., 'How to render second accurate time-stepping algorithms fourth order accurate while retaining the stability and conservation properties', Computer Methods in Applied Mechanics and Engineering 115, 1994, 233–252.

    Google Scholar 

  28. Schagerl, M., Steindl, A., and Troger, H., 'Dynamic analysis of the deployment process of tethered satellite systems', in IUTAM-IASS Symposium on Deployable Structures: Theory and Applications, S. Pellegrino and S.D. Guest (eds.), Solid Mechanics and Its Applications, Vol. 80, Kluwer, Dordrecht, 2000, pp. 345–354.

    Google Scholar 

  29. Krupa, M., Kuhn, A., Poth, W., Schagerl, M., Steiner, W., Steindl, A., Troger, H., and Wiedermann, G., 'Tethered satellite systems: A new concept of space flight', European Journal of Mechanics A/Solids 19, 2000, S149–S164.

    Google Scholar 

  30. Blevins, R. D., Applied Fluid Dynamics Handbook, Wiley, New York, 1980.

    Google Scholar 

  31. Nordell, W. J. and Meggit, D. J., 'Undersea suspended cable structures', ASCE Journal Structures Division 107, 1981, 1025–1040.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Robotics and Mechatronics, German Aerospace Center (DLR), D-82234, Wessling, Germany

    H. Weiss

Authors
  1. H. Weiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Weiss, H. Dynamics of Geometrically Nonlinear Rods: II. Numerical Methods and Computational Examples. Nonlinear Dynamics 30, 383–415 (2002). https://doi.org/10.1023/A:1021257410404

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021257410404

Search

Navigation