Modelling oscillations of a moving sheet with one rigid attachment condition

This work is dedicated to the study and analysis of a model initial boundary value problem on propagation of vibrations in a moving elastic sheet with one condition of rigid attachment at the ends of the sheet. The main problem is to find exact solutions for this problem. To construct the desired solution, the projection method is used, in which a set of basis functions from the auxiliary boundary value problem for vibrations of an I-beam with hinged ends is used. The mathematical model of vibrations is a linear partial differential equation of the fourth order with constant coefficients, which contains a mixed derivative in time and spatial variable. A method of constructing explicit exact solutions of this problem is given, which can be generalized to similar sets of problems. The article contains illustrations showing the deviations of the web from the equilibrium position and the dynamics of oscillations at different moments of time.