The theory of longitudinal vibrations of a conical elastic body in an elastic medium

  • V. Bulgakov -
  • V. Adamchuk -
  • I. Holovach -
  • D. Orszaghova -
Keywords: solid elastic body, elastic medium, functional of Ostrogradsky-Hamilton, longitudinal vibrations, eigenforms, eigenfrequencies, Ritz method

Abstract

Aim. To elaborate the theory of longitudinal vibrations of a solid elastic body with one fi xed end in the elastic medium. The example of such a body may be found in a sugar beet root in soil, the latter being elastic medium. Methods. The principle of stationary action of Ostrogradsky-Hamilton and the Ritz method were applied in the work. Results. The Ritz method was applied to obtain the Ritz frequency equation for the oscillating process under investigation. The analytic expressions were defi ned to determine the fi rst and second eigenfrequencies of vibration and the amplitude of constrained vibrations of any of its cross-sections. The values of the fi rst and second eigenfrequencies of the elastic body under investigation with specifi c geometric and physical pa- rameters were found. The dependency diagrams for the fi rst and second eigenfrequencies on the coeffi cient of elastic contraction of soil as the elastic medium, and the dependency diagrams for the amplitude of constrained oscillations of the mentioned body on the coeffi cient c of elastic deformation of soil and the distance of the cross-section of the body from the conditional point of fi xation were drawn. The dependency diagrams for the amplitude of constrained oscillations of the elastic body on the change in the amplitude and the frequency of perturbing force were obtained. Conclusions. The impossibility of resonance occurrence was substantiated as the frequency of the perturbing force cannot equal the frequency of eigenvibrations of the elastic body due to technological and technical reasons. It was proven that the breaking of the elastic body is impossible with lon- gitudinal deformations due to the shortness of the amplitude of longitudinal vibrations of the mentioned body.
Published
2016-04-15
How to Cite
Bulgakov, V., Adamchuk, V., Holovach, I., & Orszaghova, D. (2016). The theory of longitudinal vibrations of a conical elastic body in an elastic medium. Agricultural Science and Practice, 3(1), 27-35. https://doi.org/10.15407/agrisp3.01.027