Impact on the beam of the final length of the moving mass

Abstract

There have been considered solving direct problem of deforming of isotropic, elastic and hingedly supported beam of finite length. A roller moving at a constant speed along the axis of the beam acts on the beam. The roller has a cylindrical shape of a certain radius and a length that is greater than or equal to the width of the beam. The differential equations of beam motion are analyzed from the point of view of the influence of their components and especially the right-hand parts of the equations on the dynamic behavior of the beam in the case of using certain common materials of the beam and roller, and an option to reduce the equations to a more simplified form is proposed. The unknown functions included in the equations are sought in the form of Fourier series. This allows us to reduce the original equations to ordinary differential equations, which are solved using the Laplace transform. Expressions for coefficients in Fourier series are found using operational calculus and the residue theory. The results of the first numerical experiment on the study of the influence of the roller speed on beam deflections are presented in the form of curves in the figure. For a specific calculated mechanical system in the form of a steel roller, which moves along a steel beam at a constant speed under zero initial conditions, the research results are presented in the form of graphs of beam deflections for different speeds of the roller. The second numerical experiment was carried out to study the propagation of vibrational waves of the beam in the case of motion of the roller at a sufficiently high speed. For this, the figure shows the combined shapes of the beam and the position of the roller at different moments of action of the moving mass. The behavior of the beam at a high speed of movement of the roller was analyzed and a comparison of the deflections of the beam with the deflections of the static model of the beam was made. Further directions for the development of the problem in applied fields of technology and in inverse problems of identifying unknown parameters by indirect manifestations are outlined.