АНАЛІЗ СТІЙКОСТІ КОЛИВАНЬ РІДИНИ В ЖОРСТКИХ РЕЗЕРВУАРАХ ПРИ ПАРАМЕТРИЧНОМУ ЗБУДЖЕННІ
DOI:
https://doi.org/10.32782/2618-0340-2019-3-10Keywords:
parametric oscillations, shells of revolution, ideal incompressible liquid, boundary element method, Ince-Strutt diagramAbstract
Parametric oscillations of the shells of revolution partially filled with a liquid and subjected to perturbing longitudinal forces are investigated. It is assumed that the liquid in the shell is an ideal and incompressible one, and its motion caused by the applied load is irrotational. In these conditions, there exists a velocity potential of the fluid that satisfies the Laplace equation. To determine this potential in the fluid domain it is necessary to formulate the boundary conditions. On the wetted surface of the shell of revolution the non-penetration condition is fulfilled, and the kinematic and dynamic boundary conditions are applied on the free liquid surface. The dynamic boundary condition consists in equality of the liquid pressure on the free surface to atmospheric one. The kinematics boundary condition requires that liquid particles always remain on the free surface if they belong it initially. The spectral boundary value problem of determining the free liquid vibrations is formulated for an arbitrary fluid domain. For shells of revolution the spectral problem of determining the frequencies and modes of the liquid vibrations is reduced to solving the eigenvalue problem formulated for a discrete analogue of the system of singular integral equations. This system contains only one-dimensional integrals. The solution of this problem is carried out by the boundary element method. The effective method for evaluating one-dimensional singular integrals is developed. This approach is based on using the characteristic property of the arithmetic-geometric mean. Study of the liquid vibrations in the shell under the perturbing vertical force is reduced to the solution of the system of uncoupled Mathieu differential equations. The assessment of the motion stability is carried out using the Ince-Strutt diagram. A method is proposed that allows us to estimate the stability of parametric oscillations of the shell of revolution, partially filled with the liquid, and under the action of the periodic vertical driving force. To assess the stability of motion, it is necessary to know the spectrum of the natural frequencies of the liquid vibrations in the shell, the amplitude and frequency of the driving force.
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