Transformation Model of the Dynamic Deformation of an Elongated Cantilever Plate Mounted on an Elastic Support Element

Abstract

A transformation model of the dynamic deformation of an elongated orthotropic composite rod-type plate, consisting of two sections (fastened and free) along its length, is proposed. In the free section, the orthotropic axes of the material do not coincide with the axes of the Cartesian coordinate system chosen for the plate, and in the fastened section, the displacements of points of the contact’s boundary surface (clamping) with the elastic support element are considered to be known. The constructed model is based on the use for the free section of the relations of the refined Timoshenko shear model, compiled for rods in a geometrically nonlinear approximation without taking into account lateral contraction. For the section fastened on the elastic support element, a one-dimensional shear deformation model is built taking into account lateral contraction, which is transformed into another model by satisfying the conditions of kinematic coupling with the elastic support element with given displacements of the interface points with the plate. The conditions for the kinematic coupling of the free and fastened sections of the plate are formulated. Based on the Hamilton–Ostrogradsky variational principle, the corresponding equations of motion and boundary conditions, as well as the static conditions for the matching of sections, are derived. The constructed model is intended to simulate natural processes and structures when solving applied engineering problems aimed at developing innovative oscillatory biomimetic propulsors.