Investigation of the semi-strip’s stress state in the case of steady-state oscillations

Abstract

The elastic semi-strip under the dynamic load concentrated at the centre of the semi-strip’s short edge is considered. The lateral sides of the semi-strip are fixed. The case of steady-state oscillations is considered. The initial problem is reduced to the one-dimensional problem with the help of the semi-infinite sin-, cos-Fourier’s transform. The one-dimensional problem is formulated in the vector form. Its solution is constructed as a superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is found with the help of the matrix differential calculations. The partial solution is expressed through Green’s matrixfunction, which is constructed as the bilinear expansion. The inverse Fourier’s transform is applied to the derived expressions for the displacements. The solving of the initial problem is reduced to the solving of the singular integral equation. Its solution is searched as the series of the orthogonal Chebyshev polynomials of the second kind. The orthogonalization method is used for the solving of the singular integral equation. The stress-deformable state of the semi-strip is investigated regarding both the frequency of the applied load, and the load segment’s length.