2023
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Перегляд 2023 за Ключові слова "519.6"
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Документ Відкритий доступ Effective decoupling method for derivation of eigenfunctions for closed cylindrical shell(Igor Sikorsky Kyiv Polytechnic Institute, 2023) Yudin, H.; Orynyak, I.By expansion into Fourier series along the circumferential coordinate, the problem for elastic thin-walled closed cylindrical shell is reduced to 8th order differential equation with respect to axial coordinate. In spite that the general structure of eigenvalues for this equation was known starting from 60-s of last century, they were derived only to some simplified versions of the shell theory. So, the main goal of paper consists in development of the general procedure for determination of the eigenvalues. The idea is based on that the theory of shell is actually formed by two much simple problems: the plane task of elasticity and the plate problem, each of them is reduced to much easily treated biquadratic equation. So, we start from either of two problems (main problem) while not taking into account the impact from another (auxiliary) problem. After computing its eigenfunctions, we gradually introduce the influence of auxiliary problem by presenting its functions as linear combination of functions for main problem. The results of calculation show the perfect accuracy of the method for any desired number of significant digits in eigenvalues. The comparison with known results for concentrated radial force shows the perfect ability to solve any boundary problem with any desirable accuracy.Документ Відкритий доступ Semi-analytical implicit direct time integration method for 1-D gas dynamic problem(Igor Sikorsky Kyiv Polytechnic Institute, 2023) Orynyak, I.; Kostyushko, I.; Mazuryk, R.Sharp wave treatment for 1-D gas dynamic problem is still a challenge for modern numerical methods. They often require too many space and time steps, produce spurious oscillation of solution, exhibit a strong numerical dissipation or divergence of results. This paper is further extension of authors’ idea of employment the analytical solution for space coordinate, where time step is a parameter which used in the space solution. Its peculiarity consists in development of additional linearization procedure of dependence between the pressure and density. It is performed in premise that actual pressure for each space element is close to the basic pressure, attained at previous moment of time. The efficiency of method is tested on the very popular task of Sod, where two different ideal gases in a tube are separated by diaphragm, which is suddenly broken. The problem considered in Lagrangian coordinates formulation. The results obtained show the very good method efficiency, which requires the essentially lesser time and space steps, leads to no spurious oscillation and give consistent and predictable results with respect to meshing. The accuracy of method is mostly controlled by time step, which should be larger than clearly stated theoretical lower limit. Other advantage of method is that it can calculate the process to any desired time moment, and space meshing can be variable in time and space and can be easily adapted during the process of calculation.