Чисельний розрахунок частот вільних коливань некругової циліндричної оболонки з жорстко закріпленими торцями

Abstract

Визначаються частоти та форми вільних коливань тонкої циліндричної оболонки еліптичного поперечного перерізу сталої товщини з жорстко закріпленими торцями. Дослідження проводились методом скінченних елементів, який реалізовано на ліцензійному програмному засобі FEMAP з розв’язувачем NX Nastran. Достовірність отриманих результатів забезпечується використанням обґрунтованої математичної моделі, коректністю постановки задачі, розв’язком тестових задач та практичною збіжністю розрахованих частот при застосуванні методу скінченних елементів. У результаті дослідження розв’язано дві тестові задачі для кругової та не кругової циліндричної оболонки сталої товщини з жорстко закріпленими торцями. Результати тестових задач добре збігаються з результатами, отриманими іншими авторами експериментальним та чисельним методами. Проведено порівняльний аналіз частот і форм вільних коливань розглянутої оболонки з частотами і формами вільних коливань цієї ж оболонки з жорстко закріпленим одним торцем, інший торець вільний.

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In many cases, cylindrical shell constructions have a noncircular cross-sectional contour, variable thickness and different boundary conditions, and other features. In the study of the strength and stability of shell structures possession of information about the distribution of dynamic characteristics, in particular, the distribution of frequencies and forms of free oscillations is needed. Theoretical research of vibrations associated with considerable difficulties realization of the mathematical model and necessity solving difficult system of differential equations in partial derivatives with variable coefficients. The instability of solutions and the loss of the calculation accuracy required order occur in solving this system. There are some analytical solutions boundary problems describing vibrations noncircular cylindrical shells that can be used only in rare cases. Numerical methods for theory shell problem solving got wide applications in comparison with analytical methods. Based on the finite element method many modern programs for engineering calculations created. One such program is the solver FEMAP with NX Nastran. The proposed software tool used in several studies to determine the free vibration frequencies and forms of thin cylindrical shells elliptical cross section constant [1] and variable [2] thickness for different values of eccentricity [3] and different boundary conditions on one end face, the other an end face free [4 ]. Apart ready software developed by individual authors programs are often used. Thus, in two finiteelement implementations thin-walled structures natural oscillations were studied [5]. Finite element method becomes more popular in engineering calculations, but remember that the basic check any calculation results is only physical experiment. One of the most effective methods of investigation vibration is a method holographic interferometry, which allows real-time observe interference patterns which makes it possible real-time observe interference patterns of vibrations shell surfaces, accurately set moment of occurrence of resonance, select the desired mode vibrational excitation, explore the spectrum of free oscillation frequencies and forms with a sufficient degree of reliability. This experimental method used to study the console fixed cylindrical shells with circular cross-section [6] and noncircular cross-section constant [7] and variable thickness. The experimental results are in good agreement with numerical calculations. In [8] on the basis of holographic interferometry circular cylindrical shell with both rigid fixed end faces investigated, the experiment results made it possible to get empirical formula to calculate the free oscillation frequencies. In this paper estimates of two test problems are given, the results of which showed a good correspondence to the results obtained numerically [5] and experimentally [8] by other authors. The discrepancy between the numerical calculations does not exceed 4%, the difference between the numerical calculation, which is realized in FEMAP, and results obtained experimentally does not exceed 16%, which is acceptable and demonstrates the correctness of the proposed method for solving this class problems. The new calculation of free oscillation frequencies and forms of cylindrical shells elliptical cross section with rigid fixed both end faces carried, and conducted comparison of obtained the frequencies and frequencies received for the same shell with rigid fixed one end face, the other an end face free. A result of research we found that the rigid fixing of the second end face makes it possible to increase the first resonant frequency is twice, and the vibration mode does not match the first resonance frequency form of the console fixed shell. The obtained results make it possible to control the free vibration frequency spectrum of shell structures for its exclusion from the resonant regime by varying the boundary conditions, which is one on the actual problems of the research strength of shell structures.