Acta Mechanica

, Volume 230, Issue 5, pp 1717–1723 | Cite as

Solving the paradox of the folded falling chain by considering the link transition and link geometry

  • Hong-Hsi LeeEmail author
  • Chih-Fan Chen
  • I.-Shing Hu
Original Paper


A folded chain, with one end fixed at the ceiling and the other end released from the same elevation, is commonly modeled as an energy-conserving system in one dimension. However, the analytical paradigm in the existing literature is unsatisfying: The theoretical prediction of the tension at the fixed end becomes infinitely large when the free end reaches the bottom, contradicting the experimental observations. Furthermore, the dependency of the total falling time on the link number demonstrated in numerical simulations is still unexplained. Here, considering the link transition between the two sub-chains and the geometry of each link, we introduce an additional term for the relation of balance of kinetic energy to account for the jump of link velocity at the fold. We derive analytical solutions of the maximal tension as well as the total falling time, in agreement with simulation results and experimental data reported in previous studies. This theoretical perspective extends the classical standard treatment, shows a simple representation of the complicated two-dimensional falling chain system and, in particular, specifies the signature of the chain properties.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Calkin, M.G., March, R.H.: The dynamics of a falling chain 1. Am. J. Phys. 57(2), 154–157 (1989). CrossRefGoogle Scholar
  2. 2.
    Tomaszewski, W., Pieranski, P.: Dynamics of ropes and chains: I. the fall of the folded chain. N. J. Phys. 7 (2005).
  3. 3.
    Schagerl, M., Steindl, A., Steiner, W., Troger, H.: On the paradox of the free falling folded chain. Acta Mech. 125(1–4), 155–168 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Orsino, R.M.M., Pesce, C.P.: Readdressing the classic falling U-chain problem by a modular nonlinear FEM approach. Acta Mech. 229(7), 3107–3122 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Irschik, H.: On the necessity of surface growth terms for the consistency of jump relations at a singular surface. Acta Mech. 162(1–4), 195–211 (2003). CrossRefzbMATHGoogle Scholar
  6. 6.
    Irschik, H.: The Cayley variational principle for continuous-impact problems: a continuum mechanics based version in the presence of a singular surface. J. Theor Appl. Mech-Pol. 50(3), 717–727 (2012)Google Scholar
  7. 7.
    Irschik, H., Belyaev, A.K.: Dynamics of Mechanical Systems with Variable Mass, vol. 557. Springer, Berlin (2014)zbMATHGoogle Scholar
  8. 8.
    Wong, C.W., Youn, S.H., Yasui, K.: The falling chain of Hopkins, Tait, Steele and Cayley. Eur. J. Phys. 28(3), 385–400 (2007). CrossRefGoogle Scholar
  9. 9.
    Geminard, J.C., Vanel, L.: The motion of a freely falling chain tip: force measurements. Am. J. Phys. 76(6), 541–545 (2008). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Biomedical ImagingNew York University School of MedicineNew YorkUSA
  2. 2.Department of PhysicsUniversity of CaliforniaDavisUSA
  3. 3.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan

Personalised recommendations