The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.