Dynamic bend of beams with binary edge fixing

Abstract

The deformation of a beam under short-time force loading by a pulse distributed along its length is considered. It is assumed that the conditions for securing the ends of the beam depend on the direction of deflection, that is, on the angles of rotation of its ends. Under the action of force loading, as well as for some time and after it, the angles of rotation are zero, since the ends of the beam are rigidly clamped, and after changing the sign of displacement they become freely supported in the cylinder hinges. Therefore, the motion of the beam is divided into two stages. On the first of them, when the edges of the beam are rigidly constrained, the solution of the problem is expressed in the functions of A. N. Krylov. A compact formula has been derived for calculating the positive displacement of the beam and the bending moments on the supports and in the median section. It is shown that the maxima of these quantities do not exceed their doubled static values. Formulas are also derived for calculating the time when the maxima and the formula for calculating the duration of the first stage of motion are reached. At the second stage of the movement, the unloaded beam with hinged edges is freely oscillated. The negative displacement of the beam and the bending moment in it are represented in trigonometric series. Numerical calculations are carried out. It is established that for small duration of the power pulse, in comparison with the period of the fundamental tone of the oscillations, the amplitude of the deflections toward the force loading is less than the amplitude of the deflections of the beam in the opposite direction. This also applies to the amplitudes of the bending moments along the middle of the beam. As the duration of the action of the pulse increases, this dynamic effect, characteristic of vibrational systems with an asymmetric elasticity characteristic, disappears. Based on the results of calculations, graphs of the time variation of deflections and bending moments in the characteristic cross sections of the beam are plotted. The effect of the number of computed terms in partial sums of series on the accuracy (convergence) of numerical results is also investigated. Thus, the used method of fitting solutions proved to be an effective way of obtaining analytical results in the solved nonlinear problem, the simplification of which was achieved due to the choice of a separate force load distribution along the length of the beam. It was he who allowed us to obtain several compact computational formulas for the first stage of the movement.