Hipersingular integral equation method in numerical simulating frequencies and modes of circular plate immersed into liquid

Abstract

To study the frequencies and modes of vibrations of a circular plate immersed in a liquid, a new approach has been developed. The technic is based on the use of hypersingular integral equations and the method of prescribed shapes. It is assumed that a round thin elastic plate is immersed in an ideal incompressible fluid, and its motion is considered to be irrotational. Under these conditions, there is a velocity potential that satisfies the Laplace equation everywhere outside the plate, and the no-flow condition is satisfied on the plate surface. The fluid pressure has been determined by using the linearized Cauchy-Lagrange integral. During solving the boundary value problem with regard to the velocity potential, an integral representation in the form of a double layer potential was used. In this case, the potential density is proportional to the fluid pressure drop. The method of given forms made it possible to reduce the problem of determining the added masses of a liquid to solving hypersingular equations on a circular domain. During the research reduction of two-dimensional hypersingular integral equations to one-dimensional ones has been carried out. On condition of this, the inner integrals take the form of elliptic integrals of the second kind, which have no singularities. To calculate the external integral, which exists only in the sense of Hadamard, it is proposed to use the boundary element method. A procedure for calculating the elements of the matrix of a system of linear algebraic equations for finding the unknown density of the double layer potential has been developed. A numerical solution of the hypersingular integral equation has been obtained, and a comparison of the numerical and analytical solutions has been carried out. The right-hand sides of hypersingular integral equations are the forms of vibrations of a rigidly fixed circular plate obtained analytically. A technique for calculating the matrix of added masses has been developed, which made it possible to reduce the problem under consideration to solving the problem of eigenvalues.