A new class of rigorous solutions of the functional differential equation of motion for the mechanical impulses in the 1D nonhomogeneus granular chain A new rigorous solution of the functional differential equation for signal propagation through the vertical granular

Authors: Gerasymov O.I.

Year: 2011

Issue: 11

Pages: 198-202


A new rigorous solution of the functional differential equation for signal propagation through the vertical granular chain with a nonlinear contacts has been found under the approximation of the weak nonlinearity in form of the Bessel function of first order. The linear combinations of the solutions we found in principle can satisfy an appropriate initial conditions. This new solution also is a significant supplement to the dispersive wave modes which has been discovered for such a system earlier and could be considered as an envelope function for them. The relevant experiments directed to the experimental studying of the discovered modes are under the current proceeding.

Tags: dynamic systems; energy/impulse transport; granular materials; Hertzian contact; nonlinear chain


  1. Zabusky N.Z., Kruskal M.D. Inverse problem of scattering. //Phys. Rev. Lett.-1965-Vol.15.-P.240-245.
  2. Zakharov V.T., Shabat A.V. Nonlinear waves. //Zh.Trsper.Teor.Fiz.(JETP) -1971.-619.-P.118-124.
  3. Bhatnagar P.L. Nonlinear waves in one dimensional dispersive systems.-Oxford: Clarendon 1979.-250p.
  4. Jackson E.A. Perspectives of nonlinear dynamics.-Cambridge, 1990.-350p.
  5. Ford J. Solitary waves. //Phys. Rep.-1992.-N213.-P.271-296.
  6. Nesterenko V.F. Tapered granular chains sound propagation. //J. Appl. Mech. Tech. Phys.-1984.-N5. P.733-738.
  7. Nesterenko V.F. Solitary waves in 1D system of inelastic balls. //J. Phys. (France)-1994.-Vol.IV, N55.-P.729-736.
  8. Nesterenko V. Dynamics of Heterogeneous materials.-New York: Springer, 2001.-350p.
  9. Coste C., Falcon E., Fauve S. Stationary states in vertical cradle excited from the bottom. //Phys. Rev. E-1997.-Vol.56.- P.6104.
  10. Sen S., Hong J., Bang J., Avalos E., Doney R. Soliton-like waves in low-dimensional granular systems.//Phys. Rep.-2008.-N462, P.21-66.
  11. E.Somfai, J-N.Roux, T.Snoeijer, M.van Hecke, W.Van Saarlos, Dispersive waves in 1D granular chain.//Phys. Rev. E-2005.-Vol.72.-P.021301.
  12. Герасимов О.І., Вандевалле Н., Співак А.Я., Худинцев М.М., Люмє Г., Дорболло С., Клименков О.А. Стаціонарні стани у 1D системі непружних частинок //Укр.фіз.журн. -2008.- Т.53, № 11. – С.1129-1137.
  13. Falcon E., Laroche C., Fauve S., Coste C. Bouncing balls under the vibrating substrate. //Euro. Phys. J. B-1998.-Vol.5.-P.111-117.
  14. E.J.Hinch, S.Saint-Jean, Signal transmission through the cradle. //Proc.Roy.Soc.London A-1999.-N455.-P.3201-3210.
  15. E.Avalos, T.Krishna Mohan, S.Sen, Propagation of energy throughout granular chains. //Phys. Rev. E-2003. – Vol.67.-P.060301(R)
  16. Job S., Santibanez F., Tapia F., Melo F. Mechanical impulse passing the inhomogeneous systems.//Phys. Rev. E-2009.-Vol.80.-P.025602(R)
  17. U.Harbola, A.Rosas, A.Romero, M.Esposito, K.Lindenberg, Wave transport in granular chains. //Phys. Rev. E-2009.-Vol.80.-P.051302.
  18. J.Lima Dias Pinto, A.Rosas, A.Romero, K.Lindenberg, Rigorous solutions for a equations governing the energy distribution in discrete nonlinear systems. //Phys. Rev. E-2010.-Vol.82.-P.031308.
  19. Landau L.D. Theory of elasticity, 3rd ed.- Heinemann: Butterworth, 1986.-200p.
  20. Pinney E. Ordinary Difference-differential equations.-Berkley and Los Angeles: University of California Press, 1955.-500p.
  21. Vandewalle N., Lumay G., Dorbolo S. and Gerasymov O. Towards the problem of energy (impulse) transport in 1D inhomogeneous granular chains. //Phys.Rev.E.-2011. (in preparation)
Download full text (PDF)