Forced oscillations of a multimodular beam on a viscous elastic base

Abstract

The aims of the research are to obtain and to solve equations of forced oscillations of beams made of different modular materials and located on a viscous elastic base. It is assumed that the beam, which has different resistance to expansion and compression and which is continuous and heterogeneous by thickness and length, performs forced oscillations under the action of a force that varies according to the cross-harmonic law. When solving the problem, the resistance of the environment is taken into account. Since the equation of motion is a complicated differential equation with partial derivatives with respect to bending, it is solved by approximate analytical methods. At the first stage, decomposition into variables is used, and at the second stage, the Bubnov - Galerkin orthogonalization method is used. Equations of dependence between the circular frequency and parameters characterizing the resistance of the external environment and heterogeneity are obtained. Calculations were carried out for the specific values of characteristic functions. Results are represented in the form of tables and curves of the corresponding dependencies. It is clear from the obtained equations that serious errors are made in solving problems of oscillating motion without taking into account the resistance of the environment and different modularity. In addition to this, as the values of parameters that determine the heterogeneity of the density increase, the value of the frequency difference changes significantly. The results can be used in reports on solidity, stability and gain-frequency characteristic of different modular beams, boards and cylindrical coatings, taking into account the resistance of the environment.