Vibration of a periodically supported beam on an elastic half-space

TLDR: In this article, the steady-state vibration of a periodically supported beam on an elastic half-space under a uniformly moving harmonically varying load is investigated, and it is shown that the equivalent stiffness is equal to zero for some relationship between the frequency and the phase shift.

Abstract: The steady-state vibration of a periodically supported beam on an elastic half-space under a uniformly moving harmonically varying load is investigated. The concept of equivalent stiffness of a half-space is used for problem analysis. It is shown that the half-space can be replaced by a set of identical springs placed under each support of the beam. The equivalent stiffness of these springs is a function of the frequency of the beam vibrations and of the phase shift of vibrations of neighboring supports. It is found that the equivalent stiffness is equal to zero for some relationship between the frequency and the phase shift. The reason for this is that the surface waves generated by all supports can come to any support in phase, providing an infinite displacement. It is demonstrated that the equivalent stiffness has a real and an imaginary part. The imaginary part arises due to radiation of waves in the half-space. The expressions are derived for the steady-state response of the beam to the moving load. The limiting case of a constant load is considered, showing that the load moving with the Rayleigh wave velocity causes resonance in the system.