International Conference on Innovations in Natural Science and Engineering (ICINSE 2018), Famagusta, Cyprus (Kktc), 03 January 2018, vol.1, no.1, pp.3-4
This paper presents an exact three-dimensional analysis of the free vibration and natural frequencies of an inhomogeneous pre-stressed hollow sphere filled with a compressible inviscid fluid. It is assumed that the initial stresses in the hollow sphere are caused with the uniform compressible radial forces acting on the outer surface as well as with the hydrostatic pressure acting on the inner surface of that. Under these radial forces, initial stresses in the sphere are to be inhomogeneous. These initial stresses in the sphere are determined analytically by using the concrete known expressions. The motion of the foregoing pre-stressed sphere is written within the scope of the three-dimensional linearized theory of elastic waves in initially stressed elastic bodies [1,2]. However, the motion of the fluid is written within the scope of the Navier-Stokes equations for compressible barotropic inviscid fluids . On interface surface between the sphere and fluid, the compatibility conditions are satisfied. For solution to the equations of motion related to the hollow sphere the discrete-analytical method developed in the works [3, 4] is employed. According to this method, the hollow sphere is divided into a certain number of sub-hollow spheres in each of them the initial stresses are taken as homogeneous one. Namely, this statement allows using the analytical solution method for the equations obtained for the potentials in the Helmholtz decomposition in the solution of the equations of elastodynamics. The solution to the corresponding equations for fluid motion is found analytically in the spherical coordinates through the spherical Bessel functions. Using the traction free condition outer surface of the hollow sphere, the contact conditions between the sublayers and compatibility conditions between the fluid and sphere on the inner surface of the sphere the frequency equation is obtained for determination of the natural frequencies for the hydro-elastic system under consideration. The aforementioned frequency equation is solved numerically, the results illustrated the influence of the existence of the fluid into the hollow cylinder, and the influence of the geometrical and mechanical parameters on the natural frequencies are presented and discussed.