Dynamics of a rod undergoing a longitudinal impact by a body

A. K. Belyaev, C. C. Ma, N. F. Morozov, P. E. Tovstik, Tatiana Petrovna Tovstik, A. O. Shurpatov

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.

Original languageEnglish
Pages (from-to)310-317
Number of pages8
JournalVestnik St. Petersburg University: Mathematics
Issue number3
StatePublished - 1 Jul 2017

Scopus subject areas

  • Mathematics(all)


  • Hertz theory
  • impact
  • impact impulse
  • impact time
  • parametric resonance
  • rod
  • Saint-Venant theory
  • Sears theory

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