Численное моделирование колебаний жидкости в составных оболочках вращения при перезагрузках

Abstract

Исследованы свободные колебания жидкости в составных оболочках вращения, состоящих из цилиндрической и конической частей. Предполагается, что жидкость является идеальной и несжимаемой, а ее движение, вызванное колебаниями оболочки − потенциальное. Использован метод собственных форм. Задача сведена к решению системы сингулярных интегральных уравнений. Применен метод граничных элементов. Определены частоты и формы колебаний жидкости в составной цилиндрическо-конической оболочке. Сделан анализ влияния перегрузок на частоты колебания составных оболочек.

The free fluid vibrations in the shells of revolution having cylindrical and conical parts are considered. To simulation the region occupied by the fluid, a mathematical model is developed based on the following hypotheses: the fluid is incompressible and inviscid, and its motion, caused by shell oscillations-potential, the only small oscillations are considered. In the course of the work, a mixed boundary-value problem is formulated for the Laplace equation. The modes decomposition method is applied. The problem is reduced for the solving of the system of singular integral equations. In the course of the solution, it was revealed that the kernels of integral equations contain complete elliptic integrals of the first and second kind. An elliptic integral of the second kind is calculated using Gauss's standard quadrature formula. For an elliptic integral of the first kind, an approach based on the characteristic property of the arithmetic mean AGM is used. To solve outer integrals, special Gaussian quadrature formulas are applied. The developed method is applied further for the analysis of free vibrations of shell structures. The boundary element method is applied in a straightforward form. At the first stage, a necessary number of boundary elements is determined to find the eigenfrequencies with a given accuracy. The frequencies and forms of fluid oscillations in a compound cylindricalconical shell are determined. A comparison is made between the frequencies of axisymmetric fluid vibrations in a cylindrical shell obtained using by the method and the analytical formula developed in this work. Also, the values of the first seven vibration frequencies of composite shells are given for different lengths of the cylindrical part. The analysis is made of the effect of overloads on the vibration frequencies of composite shells.