Mechanics of Axially Moving Structures at Mixed Eulerian-Lagrangian Description

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Part of the Advanced Structured Materials book series (STRUCTMAT, volume 81)

Abstract

We discuss a series of methods of the mathematical modelling of large deformations of axially moving strings, beams and plates. Both uni-axial and looped trajectories of motion are considered, which allows the application of these methods to such practically important problems as rolling mills or belt drives. Based on the principles of Lagrangian mechanics, we transform the variational formulations of structural mechanics to problem-oriented exact kinematic descriptions with mixed spatial and material coordinates. The discretization of an intermediate domain results in a consistent non-material finite element formulation with particles of continuum flowing across the mesh. This allows avoiding numerically induced oscillations in the solution, while keeping the discretization fine where necessary, e.g. in the regions of contact.

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Notes

Acknowledgements

The support of the author from the Austrian Science foundation FWF in the framework of the collaboration with the research team of the project “Eulerian Mechanics of Belts” [grant number I2093-N25] is gratefully acknowledged.

References

  1. Banichuk N, Jeronen J, Neittaanmäki P, Tuovinen T (2010) On the instability of an axially moving elastic plate. International Journal of Solids and Structures 47:91–99Google Scholar
  2. Bechtel SE, Vohra S, Jacob KI, Carlson CD (2000) The stretching and slipping of belts and fibers on pulleys. ASME Journal of Applied Mechanics 67:197–206Google Scholar
  3. Belyaev AK, Eliseev VV, Irschik H, Oborin EA (2017) Contact of two equal rigid pulleys with a belt modelled as cosserat nonlinear elastic rod. Acta Mechanica in pressGoogle Scholar
  4. Chen LQ (2005) Analysis and Control of Transverse Vibrations of Axially Moving Strings. ASME Applied Mechanics Reviews 58:91–116Google Scholar
  5. Ciarlet P (2005) An introduction to differential geometry with applications to elasticity. Journal of Elasticity 1-3(78/79):1–215Google Scholar
  6. De Lorenzis L, Temizer I, Wriggers P, Zavarise G (2011) A large deformation frictional contact formulation using NURBS-based isogeometric analysis. International Journal for Numerical Methods in Engineering 87(13):1278–1300Google Scholar
  7. Donea J, Huerta A, Ponthot JP, Rodriguez-Ferran A (2004) Arbitrary Lagrangian-Eulerian Methods. In: Stein E, de Borst R, Hughes T (eds) Encyclopedia of Computational Mechanics, vol 1: Fundamentals, John Wiley & Sons, Ltd, chap 14Google Scholar
  8. Eliseev V, Vetyukov Y (2012) Effects of deformation in the dynamics of belt drive. Acta Mechanica 223:1657–1667Google Scholar
  9. Eliseev VV (2006) Mechanics of Deformable Solid Bodies (in Russian). St. Petersburg State Polytechnical University Publishing House, St. PetersburgGoogle Scholar
  10. Eliseev VV, Vetyukov Y (2010) Finite deformation of thin shells in the context of analytical mechanics of material surfaces. Acta Mechanica 209(1-2):43–57Google Scholar
  11. Ghayesh MH, Amabili M, Païdoussis MP (2013) Nonlinear dynamics of axially moving plates. Journal of Sound and Vibration 332(2):391–406Google Scholar
  12. Hong D, Ren G (2011) A modeling of sliding joint on one-dimensional flexible medium. Multibody System Dynamics 26:91–106Google Scholar
  13. van Horssen WT, Ponomareva SV (2005) On the construction of the solution of an equation describing an axially moving string. Journal of Sound and Vibration 287(1-2):359–366Google Scholar
  14. Humer A (2013) Dynamic modeling of beams with non-material, deformation-dependent boundary conditions. Journal of Sound and Vibration 332:622–641Google Scholar
  15. Humer A, Irschik H (2009) Onset of transient vibrations of axially moving beams with large displacements, finite deformations and an initially unknown length of the reference configuration. ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik 89(4):267–278Google Scholar
  16. Humer A, Irschik H (2011) Large deformation and stability of an extensible elastica with an unknown length. International Journal of Solids and Structures 48:1301–1310Google Scholar
  17. Kim D, Leamy MJ, Ferri AA (2011) Dynamic Modeling and Stability Analysis of Flat Belt Drives Using an Elastic/Perfectly Plastic Friction Law. ASME Journal of Dynamic Systems, Measurement, and Control 133:1–10Google Scholar
  18. Kong L, Parker RG (2005) Steady mechanics of belt-pulley systems. ASME Journal of Applied Mechanics 72:25–34Google Scholar
  19. Leamy MJ (2005) On a perturbation method for the analysis of unsteady belt-drive operation. ASME Journal of Applied Mechanics 72(4):570–580Google Scholar
  20. Marynowski K, Kapitaniak T (2014) Dynamics of axially moving continua. International Journal of Mechanical Sciences 81:26–41Google Scholar
  21. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46(1):1278–1300Google Scholar
  22. Morimoto T, Iizuka H (2012) Rolling contact between a rubber ring and rigid cylinders: Mechanics of rubber belts. International Journal of Mechanical Sciences 54:234–240Google Scholar
  23. Mote JCD (1966) On the nonlinear oscillation of an axially moving string. Journal of Applied Mechanics 33:463–464Google Scholar
  24. Pechstein A, Gerstmayr J (2013) A Lagrange-Eulerian formulation of an axially moving beam based on the absolute nodal coordinate formulation. Multibody System Dynamics 30:343–358Google Scholar
  25. Reynolds O (1874) On the efficiency of belts or straps as communicators of work. The Engineer 38:396Google Scholar
  26. Rubin M (2000) An exact solution for steady motion of an extensible belt in multipulley belt drive systems. Journal of Mechanical Design 122:311–316Google Scholar
  27. Stoker JJ (1989) Differential Geometry. Wiley Classics Library, WileyGoogle Scholar
  28. Vetyukov Y (2012) Hybrid asymptotic-direct approach to the problem of finite vibrations of a curved layered strip. Acta Mechanica 223(2):371–385Google Scholar
  29. Vetyukov Y (2014a) Finite element modeling of Kirchhoff-Love shells as smooth material surfaces. ZAMM 94(1-2):150–163Google Scholar
  30. Vetyukov Y (2014b) Nonlinear Mechanics of Thin-Walled Structures. Asymptotics, Direct Approach and Numerical Analysis. Foundations of Engineering Mechanics, Springer, ViennaGoogle Scholar
  31. Vetyukov Y (2017) Non-material finite element modelling of large vibrations of axially moving strings and beams. Journal of Sound and Vibration submittedGoogle Scholar
  32. Vetyukov Y, Gruber PG, Krommer M (2016) Nonlinear model of an axially moving plate in a mixed Eulerian-Largangian framework. Acta Mechanica 227:2831–2842Google Scholar
  33. Vetyukov Y, Gruber PG, Krommer M, Gerstmayr J, Gafur I, Winter G (2017a) Mixed Eulerian-Lagrangian description in materials processing: deformation of a metal sheet in a rolling mill. International Journal for Numerical Methods in Engineering 109:1371–1390Google Scholar
  34. Vetyukov Y, Oborin E, Krommer M, Eliseev V (2017b) Transient modelling of flexible belt drive dynamics using the equations of a deformable string with discontinuities. Mathematical and Computer Modelling of Dynamical Systems 23(1):40–54Google Scholar
  35. Wickert JA (1992) Nonlinear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics 27(3):503–517Google Scholar
  36. Yang S, Deng Z, Sun J, Zhao Y, Jiang S (in press) A variable-length beam element incorporating the effects of spinning. Latin American Journal of Solids and StructuresGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Mechanics and MechatronicsVienna University of TechnologyViennaAustria

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