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A problem of unilateral contact of two hanging chains

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Abstract

The unilateral contact of two closely hanging chains (heavy inextensible strings) under the action of gravity and small horizontal loading is considered. Each chain has one end fixed and the other free. The chains have different lengths; the linear densities of the chains may be variable. The horizontal loading slightly deflects the chains from the vertical line, so that the system is geometrically linear. The basic problem is to find the shapes of the chains. This problem is reduced to the problem of finding the density of the forces of interaction between the chains. The accurate formulation of the latter contact problem is propounded. The unknown density of the contact forces is assumed to be the sum of piecewise continuous (and one-sided continuous) function and delta functions describing the concentrated forces. The uniqueness of the solution of the problem is proved. The analytical solution is constructed in some special cases. The following contact patterns are found out: contact at one point, contact along the full length of the shorter chain, contact along a part of the shorter chain, contact at a point and along a part of the shorter chain.

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References

  1. Routh, E.J.: A Treatise on Analytical Statics: With Numerous Examples. Chapter X, vol. 1. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  2. Wang, C.Y.: Vibration of a vertically axially moving string or chain under the influence of gravity. Acta Mech. 228, 357–362 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  3. Schagerl, M., Steindl, A., Steiner, W., Troger, H.: On the paradox of the free falling folded chain. Acta Mech. 125, 155–168 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  5. Kim, J.H., Ahn, Y.J., Jang, Y.H., Barber, J.R.: Contact problems involving beams. Int. J. Solids Struct. 51, 4435–4439 (2014)

    Article  Google Scholar 

  6. Osipenko, M.A., Nyashin, YuI, Rudakov, R.N.: A contact problem in the theory of leaf spring bending. Int. J. Solids Struct. 40, 3129–3136 (2003)

    Article  MATH  Google Scholar 

  7. Feodosyev, V.I.: Selected Problems and Questions in Strength of Materials. Mir, Moscow (Translated from the Russian by M. Konyaeva). Problems 56, 57 (1977)

  8. Comez, I., El-Borgi, S., Kahya, V., Erdol, R.: Receding contact problem for two-layer functionally graded media indented by a rigid punch. Acta Mech. 227, 2493–2504 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, J.-S.: On the contact behavior of a buckled Timoshenko beam constrained laterally by a plane wall. Acta Mech. 222, 225–232 (2011)

    Article  MATH  Google Scholar 

  10. Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1962)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Mechanics and Biomechanics, National Research Polytechnic University of Perm, 29 Komsomolskii Prospect, Perm, Russia, 614990

    Michael Osipenko, Yuri Nyashin & Anton Kasatkin

Authors
  1. Michael Osipenko
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  2. Yuri Nyashin
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  3. Anton Kasatkin
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Correspondence to Michael Osipenko.

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Osipenko, M., Nyashin, Y. & Kasatkin, A. A problem of unilateral contact of two hanging chains. Acta Mech 228, 3923–3927 (2017). https://doi.org/10.1007/s00707-017-1904-6

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  • DOI: https://doi.org/10.1007/s00707-017-1904-6

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