METHOD OF INTEGRAL EQUATIONS IN STABILITY ANALYSIS OF FLUID VIBRATIONS IN SHELLS OF REVOLUTION
DOI:
https://doi.org/10.32782/mathematical-modelling/2024-7-1-14Keywords:
boundary finite element method, liquid sloshing, parametric resonance, dampingAbstract
The purpose of the study is to develop a method for evaluating the stability of fluid movement in rigid shells of revolution under vertical and horizontal loads. It is assumed that the liquid filling the shell is an ideal and incompressible one, and its movement due to the action of loads is vortex-free. These assumptions make it possible to formulate a mixed boundary value problem for the Laplace equation with respect to the velocity potential. In the absence of loads, we come to the spectral problem of eigenvalues. For the numerical solution of this problem, the method of integral equations is used. The third Green's formula is applied, which allows obtaining a twodimensional system of singular integral equations for determining the potential of velocities. In the case of shells of revolution, the two-dimensional system of singular integral equations is reduced to one-dimensional one. Effective methods of calculating the one-dimensional singular integrals according to the Gaussian procedure using the average arithmetic-geometric value are applied. The boundary element method with constant density approximation is used. After solving the spectral problem, the eigenvalues and eigenmode of vibrations are obtained. These modes are further used as basis in the study of forced fluid vibrations under harmonic loads. Fluid vibrations in the rigid cylindrical tank have been studied. Resonance phenomena are considered including parametric resonance. Horizontal loads and combined horizontal-vertical loads are also studied. The resonance frequencies of the forcing forces are studied. The effect of Rayleigh damping is considered. The Rayleigh matrix is introduced artificially. By means of a computer experiment, the smallest damping coefficient, at which the oscillations are mitigated, has been determined. The cases of horizontal and vertical periodic loads are considered separately. The developed technique can be used to tune out unwanted resonant frequencies in the design of fuel tanks.
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