Study of motion stability of a viscoelastic rod




Kelvin-Voigt model, critical force, viscosity coefficient, stability, beam functions


Stability of non-conservatively loaded elastic and inelastic bodies - a classic section of deformable solid mechanics that has been of interest for many years. In this paper, we study the motion stability of a free rod subjected to a constant tracking force on one of its ends. The defining ratio of the rod material is the Kelvin-Voigt model. The solution is presented in the form of an expansion in terms of beam functions. The number of terms of this expansion is substantiated. The values of the critical load in the presence and absence of viscosity are determined. The given analytical results are confirmed by numerical calculations.


  1. V.I. Feodos’ev, Selected problems and questions in the strength of materials, Mir Publisher, 1977.
  2. M. Zhravkov, Y. Lui and E. Starovoitov, Mechanics of Solid Deformable Body, Springer, 2023, doi:
  3. A. Luongo, M. Ferretti and F. D’Annibale, Paradoxes in dynamic stability of mechanical systems: investigating the causes and detecting the nonlinear behaviors, Springerplus, 2016, doi:
  4. A.P. Seiranyan, “Paradoks destabilizatsii v zadachakh ustoichivosti nekonservativnykh system,” Uspekhi mekhaniki, Vol. 13, pp. 89–124, 1990.
  5. A. E. Baikov and P. S. Krasil’nikov, “The Zigler effect in a non-conservative mechanical system,” Journal of Applied Mathemat-ics and Mechanics, Vol. 74, Issue 1, pp. 51–60, 2010, doi:
  6. A. P. Filin, Prikladnaya mekhanika tverdogo deformiremogo tela, Moscow: Nauka, Vol. 3, 1981.
  7. S. A. Agafonov and D. V. Georgiyevskii, “The dependence of the jump in the critical follower force for a viscoelastic bar on the form of the non-linear internal viscosity,” Journal of Applied Mathematics and Mechanics, pp. 367–373, 2011, doi:
  8. G. Samolyk, “Investigation of the Cold Orbital Forging Process of an AlMgSi Alloy Bevel Gear”, Journal of Materials Pro-cessing Technology, Elsevier, No. 213, pp. 1692–1702, 2013, doi:
  9. A. Luongo and F. D’Annibale, “A paradigmatic minimal system to explain the Ziegler paradox,” Continuum Mechanics and Thermodynamics, vol. 27, pp. 211–222, 2015, doi:
  10. C. Franco and J. Collado, “Ziegler paradox and periodic coefficient differential equations,” in Proc.12th International Confer-ence on Electrical Engineering, Computing Science and Automatic Control (CCE), Mexico City, Mexico, 2015, pp. 1–5, doi:
  11. F. D’Annibale and M. Ferretti, “On the effects of linear damping on the nonlinear Ziegler’s column,” Nonlinear Dynamics, vol. 103, pp. 3149–3164, 2021, doi:




How to Cite

I. Kostyushko and H. Shapovalov, “Study of motion stability of a viscoelastic rod”, Mech. Adv. Technol., vol. 8, no. 1(100), pp. 80–86, Mar. 2024.