Beam vibrations with gradient of properties along the beam thickness.

The paper analyzes the dynamic behavior of a beam, the mechanical properties of which vary along the thickness of the beam. Two cases are considered: the first is a layered beam (properties are piecewise continuous functions) and the second - material properties are a continuous function of the thickness of the beam coordinate. Due to its structure, such a beam does not have an axis of symmetry. For it, we find a neutral axis, relative to which the one-dimensional problem is solved. It is shown that in statics the problem splits into a plane problem and a beam bending problem, as in the case of homogeneous beams. In dynamics, there is no such disintegration: when a tangential load is applied to a beam, in addition to longitudinal vibrations, the beam simultaneously performs bending vibrations, and, conversely, when bending vibrations of the beam, a longitudinal stress-dynamic state simultaneously takes place. Here the complete problem is analyzed by mathematical methods. As a result, simple applied theories are obtained. It is shown that the complete problem is divided into two simpler problems. This is the first (main) problem and the second (auxiliary) problem. If a beam under the action of a tangential dynamic load performs quasi-tangential vibrations, then the main problem coincides with the problem of longitudinal vibrations of a homogeneous beam. The auxiliary problem is described by inhomogeneous equations, which include the tangential values found in the first problem. A similar pattern occurs for a bean performing forced bending vibrations under the action of an applied cross loading. [ABSTRACT FROM AUTHOR]