Analysis of the spectrum of elastic vibrations of a rod in a nonuniform elastic medium

Abstract

The spectrum of transverse vibrations of a thin nonuniform rod in an elastic medium with a variable rigidity coefficient and arbitrary boundary conditions of elastic fastening is studied. It is proved that the external medium can lead to an anomalous effect of increasing eigenfrequencies of lower modes of oscilla� tions with elongation of the rod length. The very fine properties of this variation depending on the length, mode number, and boundary conditions are revealed. Vibrations with standard methods of fastening are studied separately, and examples are calculated. 1. The problem of eigenfrequencies ω and forms u(x) of flat transverse vibrations of a thin nonuniform rod (see [1–7], etc.) with variable rigidity p(x) > 0 and density r(x) > 0 in a medium with elasticity q(x) ≥ 0 is studied. Using the Fourier method (procedures of sep�