Чисельний розв’язок плоских динамічних задач теорії підкріплених циліндричних оболонок еліптичного перерізу при нестаціонарних навантаженнях

Abstract

Розглянуто плоску задачу про вимушені коливання поздовжньо підкріплених циліндричних оболонок еліптичного перерізу під дією розподіленого нестаціонарного навантаження. Наведено постановку та розроблено чисельний алгоритм розв’язування задачі. Наведено приклади розрахунку плоских задач динамічної поведінки підкріплених оболонок та проведено аналіз отриманих чисельних результатів.

In this paper, in the framework of the theory of shells and rods of Timoshenko-type, the problem of forced vibrations of longitudinal – cross stiffened cylindrical shells of elliptical cross section is considered. Numerical algorithm for solving the problem is constructed. The results of analysis of nonstationary vibrations of longitudinal - cross stiffened shells are presented. To derive the vibration equations for stiffened cylindrical shell the variation principle of stationary Hamilton – Ostrogradsky is used. After standard transformations in the variation functional, taking into account the conditions of the contact shell – cross rib, we obtained two groups of equations: 1) the wave equations of smooth cylindrical shell of the elliptical cross – section; 2) the wave equations of longitudinal rib, which is located along the – axis. The initial wave equations are supplemented by the corresponding boundary and initial conditions. The numerical algorithm for solving the initial – boundary value problems is based on the integro-interpolation method of constructing difference relations along the spatial coordinate and the explicit approximation of the time coordinate. According to the initial formulation of the problem the solution is constructed in a smooth area and is pasted on the lines of discontinuity (lines of design center of gravity of the cross- ribs on the middle surface smooth shell). Results of theoretical study of the stability conditions of obtained difference equations are presented. As a numerical example, the problem of the dynamic behavior of longitudinal – cross stiffened cylindrical shell with elliptical cross section under action of a distributed pulse loading is presented. Comparison with similar results of the dynamic behavior of stiffened cylindrical shells of circular cross section is made.