МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ РАСПРОСТРАНЕНИЯ ВИБРАЦИЙ В ТОНКОСТЕННЫХ КАРКАСИРОВАННЫХ КОНСТРУКЦИЯХ. 2. КОНЕЧНО-ЭЛЕМЕНТНЫЕ МОДЕЛИ И ЧИСЛЕННЫЕ ЭКСПЕРИМЕНТЫ

Abstract

Based on the refined S.P. Timoshenko shear model, one-dimensional finite elements for modeling the dynamic response of flat rods with clamped section of finite length on one of the front faces have been constructed. To analyze their stationary dynamic response under harmonic external action, a system of resolving equations in a complex form is formed. Three models of kinematic conjugation of clamped and free sections of rods are developed using the equation of connection between the rotation angle of the cross section and axial displacement at the boundary between the marked parts of the rod, the transitional finite element, and the concept of a single finite element with nodes located on one of its front faces. It is noted that for practical implementation, the most convenient model is one that uses a single finite element to represent fixed and free sections of the rod. On the basis of the noted model, a finite element solution of the problem of transverse bending vibrations of a cantilevered flat rod under vibration loading conditions by a periodic axial force applied to the end section of a clamped section of finite length, as well as the problem of transverse bending vibrations of a flat rod with two free ends and clamped length section between them under vibration loading by a transverse force on one of the free ends of the rod was found. The results of the finite element solution of the noted two problems are in good agreement with the previously obtained exact analytical solutions found on the basis of the S.P. Timoshenko shear model. The presence of a significant transformation of the parameters of the stress-strain state of the considered rods during the transition through the boundary from free to the clamped length areas on one of the face surfaces is noted.