МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ РАСПРОСТРАНЕНИЯ ВИБРАЦИЙ В ТОНКОСТЕННЫХ КАРКАСИРОВАННЫХ КОНСТРУКЦИЯХ. 1. ОСНОВНЫЕ СООТНОШЕНИЯ И АНАЛИТИЧЕСКИЕ РЕШЕНИЯ ХАРАКТЕРНЫХ ЗАДАЧ

Abstract

The design features of the of thin-walled aerospace, shipbuilding, etc. structures in the form of a load-bearing frame sheathed with thin-walled panels, walls, bulkheads, etc. are discussed. Variants of constructive coupling of the specified thin-walled panels with supporting elements of the load-bearing frame and methods for their mathematical description in the classical mechanics of a deformable solid body are considered. It is proposed, without distorting the physical picture of the dynamic behavior of thin-walled panels, to present them in the form of multisupport thin bars resting on rigid elements of the load-bearing frame along part of their front surface. By the example of a plane dynamic problem of the mechanics of a bar with a fixed section of finite length on one of the front surfaces, it is shown that in the study of deformation processes, taking into account the compliance of the fixed section, it is necessary to introduce the concept of transformation of the stress-strain state parameters and the mathematical models used to describe them. Such a transformation takes place when crossing the border from an unfixed section to a fixed one (from a fixed to an unfixed one). Within the framework of the classical Kirchhoff–Love model, it is impossible to take into account the compliance of the fixed section of the bar, and when using the simplest refined shear model of S.P. Timoshenko, such accounting is possible when fixing the length only on one of the front surfaces. In particular, previously discovered and not described in the scientific literature phenomenon of the vibrations transmission through the support joints, regardless of their design, is carried out due to the transformation of the stress-strain state of the dynamically loaded section of the bar into longitudinal-shear vibration modes of the bar in the clamped area, followed by their retransformation into bending vibrations of the adjacent span. Within the framework of S.P. Timoshenko model, the main resolving equations are constructed, and the kinematic and force conditions for conjugation of fixed and non-fixed sections of the bar are formulated. On the basis of the developed mathematical model, exact analytical solutions of typical problems are constructed, confirming the transmission of vibration through the clamped sections of the bar due to the deformability of the marked sections. A significant increase of transverse shear stresses level in the clamped section of the bar in the vicinity of the junction of the unfixed section with the fixed one is revealed.