On solutions of the equation of small transverse vibrations of a moving canvas

Abstract

The paper considers a model problem of one-dimensional small transverse vibrations of a canvas moving at a constant speed, which is fixed in a hinged manner. The oscillatory process is described by a linear differential equation of the 4th order with constant coefficients. In the model under consideration, the Coriolis force is taken into account, which leads to the appearance of a term with a mixed derivative in the differential equation. This effect makes it impossible to use the classical method of separating variables. However, families of exact solutions of the oscillation equation in the form of a traveling wave have been constructed. For the initial-boundary value problem, it was established that the solution can be constructed in the form of a Fourier series according to the system of eigenfunctions of the auxiliary problem on beam vibrations. For the considered oscillatory process, the law of conservation of energy is established and the uniqueness of the solution to the initialboundary value problem is proved.