Чисельна реалізація екстраполяції Річардсона для динамічних задач багатошарових циліндричних оболонок

This study considers forced vibrations of a heterogeneous elastic structure in the form of a multilayer cylindrical shell consisting of rigidly connected layers and reinforced with discrete ring elements. A mathematical model of vibrations of an elastic heterogeneous structure under the action of a non-stationary load has been constructed. The stressed-strained state of a multilayer cylindrical shell with discrete ring ribs was investigated using the geometrically nonlinear theory of Timoshenko-type shells and rods. The presence of a complex right-hand side and discontinuous coefficients in the spatial coordinates in the hyperbolic equations of vibrations of a heterogeneous elastic cylindrical shell (at the locations of the reinforcing ribs) necessitated the use of numerical methods for solving them. A numerical algorithm using Richardson extrapolations has been proposed for studying the constructed model. For example, a three-layer reinforced cylindrical shell is considered, taking into account the discreteness of the ribs' placement under dynamic loading with rigidly clamped ribs. The proposed numerical algorithm has made it possible to investigate the stressed-strained state of a three-layer reinforced elastic structure of a cylindrical type at any given moment in time. A comparative analysis of the numerical results of the calculations revealed that, according to the standard approach, the discrepancy in the deflection values for n = 40 and n = 160 reached 31%, for n = 80 and n = 160 it was about 5%, according to Richardson's approach for n = 40 ÷ 80 and the standard approach for n = 160, this difference was about 1%. A distinctive feature of this study is the use of Richardson extrapolation to identify the stressed-strained state of a three-layer reinforced cylindrical shell, which made it possible to increase the accuracy of the solution to the dynamic problem without reducing the calculation step. The study's results reported in this work could be used for investigating unsteady vibrations of shell structures at research and engineering organizations