SIMULATION OF HYDROELASTIC OSCILLATIONS OF STRUCTURAL ELEMENTS USING THE HYPERSINGULAR EQUATION METHOD

Abstract

A method for determining the frequencies and modes of natural vibrations of structural elements in bilateral contact with a liquid has been developed. It is supposed that the fluid is perfect and incompressible one, and its motion induced by the structural element is vortex-free. Under these suppositions, there exists a velocity potential that satisfies the Laplace equation everywhere in the area occupied by the liquid. The non-penetration condition is set on the surfaces of the structural element. This condition is the equality of the normal components of fluid velocities and design. To evaluate the structure displacements, the equations of motion under the fluid pressure are used. The fluid pressure, in turn, is determined from the Laplace equation, where the boundary conditions contain an unknown velocity. So, a related problem for determining hydroelastic vibrations is obtained. To solve the formulated problem, the method of given forms is used. First, the frequencies and modes of the elastic element vibrations are determined without taking into account the pressure force from the fluid. According to the obtained forms, the representation of the structure displacements interacting with the liquid is received as corresponding series. Next, the Neumann boundary value problem for the Laplace equation is solved, and here the boundary conditions contain known functions, namely, the elastic element modes of vibrations obtained in the first stage. The solution of this problem is performed using the potential theory. The unknown function is represented as a double layer potential. The boundary conditions lead to a hypersingular integral equation with respect to an unknown density that is the fluid pressure. Further, this two-dimensional hypersungular equation is reduced to one-dimensional one. An effective method for numerical solution of this equation has been developed. The obtained numerical results are compared with known analytical solutions. A good agreement of the results is obtained that testifies reliability of the proposed method. Then the algorithm for evaluating the matrix of added masses was developed, that made it possible to find the natural frequencies of the circular elastic plate taking into account the liquid added masses.