Liquid oscillation in a cylindrical-conical shell under the action of vertical and horizontal excitation

Vibrations of an ideal incompressible fluid in shells of revolution have been considered. The shells of revolution under consideration include cylindrical and conical parts. It is assumed that the shell is subjected to vertical and horizontal excitations. The liquid in the shells is supposed to be an ideal and incompressible one. The fluid flow is the irrotational. Therefore the velocity potential that satisfies the Laplace equation exists. The non-penetration conditions are applied to the wetted surfaces of the shell and the kinematic and dynamic conditions on the free surface have been considered. The liquid pressure as the function of the velocity potential is defined using the Bernoulli equation. The problem of determining the fluid pressure is reduced to solving a singular integral equation. The numerical solution of the equation has been obtained by the method of discrete singularities. The method of simulating the free and forced oscillations of the fluid in the shells of revolution has been developed. Розглядаються


Introduction
Shell structures for storage and transportation of fluid are the subject of numerous scientific researches. These shell structures are important for power plants, pumps, columns of oil evaporators, and other industrial constructions. There are a large number of various strength and vibration problems for shell structures that have been examined in recent decades.
The overview of scientific works has shown that many articles are focused on the problem of intense liquid sloshing in containers and fuel tanks. The overview on the problem of fluid splashing in cylindrical, conical and spherical reservoirs is presented in the works of R. A. Ibrahim [1,2]. On the other hand, shells consisting of cylindrical and conical parts have received less attention in the scientific literature although such thin-walled structures are widely used in various installations. A review of researches on this topic [2,3] shows that the dynamic characteristics of the reservoirs are significantly influenced by fluid sloshing and oscillations of the elastic walls. Free and forced oscillations of cylindrical reservoirs with a liquid are examined in [4][5][6][7]. It should be noted that the majority of studies do not take into account the impact of the force of gravity causing sloshing.

Formulation of the problem.
The problem of oscillation of a fluid in a compound rotation shell having cylindrical and conical parts has been considered. The moistened part of the surface of the shell is denoted as S 1 and the free surface as S 0 . Furthermore, it is suggested that the Cartesian coordinate system Oxyz is bound to the shell, the free surface coincided with the plane z = H = H 1 + H 2 at state of rest (Fig. 1).
We assume that H 1 is the length of the cylindrical part, and H 2 is the height of the conical part, R 1 and R 2 are the radii of the lower and upper parts of the shell, Fig. 2.1.
The mathematical model for the fluid-occupied region is based on the following hypotheses: fluid is incompressible and non-dissipative, fluid motion is non-vortex, and only small fluctuations are considered (linear theory). Therefore, we could enter the scalar potential of velocities Φ (x, y, z, t) which satisfy the Laplace equation. The fluid pressure p = p (x, y, z, t) which acts on the wetted surface is obtained from the linearized Bernoulli equation for the potential flux where g is gravitational acceleration, z is vertical coordinate of the point in the fluid, is a density of fluid, p 0 is an atmospheric pressure, ( ) = 0 is a vertical acceleration of the excitatory force, ( ) = 1 is a horizontal acceleration of the excitatory force, 0 is an amplitude of the forced influence.

Figure2.1. Shell rotation with fluid and the forms of oscillations of free surface
It should be noted that the assumption of flow potential is quite natural since, according to Kelvin's theorem, if the motion of an ideal incompressible fluid is potential at an initial moment, it will remain potential throughout the next period of motion.
First of all let us consider the problem of the shell free oscillations. We have formulated the boundary conditions for solving the Laplace equation. On the wetted surfaces of the shell, the boundary impermeability condition have been set [8]. The following boundary conditions have to be satisfied on the free surface of the liquid: where is a unit outer normal to the surface . Function ζ describes the shape and position of the free surface. The first of these conditions is dynamic one and states that the pressure on the liquid surface equals atmospheric pressure. The second condition is kinematic and means that points on the free surface at the initial moment of motion remain on it thereafter.
Having made the transformations at the specified boundary conditions, similar to [8] to find the velocity potential in the problem of free oscillations of a liquid, we have the following boundary value problem: Here n is an outer unit normal to the surface under consideration. In order to fulfill the conditions of solvability of the boundary-value problem, the Neumann condition must be satisfied [8]: Thus, a mixed boundary value problem for the Laplace equation is formulated.

The main research material
Consider the potential Ф in representation of the series. Φ = ∑= 1 ( ) . The modes φ k are solutions of boundary-value problem (2.2) which, according to [3,8], becomes the problem of eigenvalues. To define functions φ k , we use the direct formulation of the boundary element method. The main integral relation is presented in the form [9] 2 Φ(P 0 ) = ∬ where S = S 1 ∪S 0 ; points P and P 0 belong to the surface S. | − 0 | denote the Cartesian distance between points P і P 0 . Functions Φ, which is defined on the surface S 1 , is the pressure on the wetted surface, and the function = Φ , that is defined on S 0 , is a flow . Using boundary conditions and assuming that Φ( , , , ) = exp( ) ( , , , ) , we obtain a system of singular integral equations in the form [7,10] where φ 0 is value of the potential at the surface S 0 , φ is value of the potential at the surface S 1 . The numerical solution of the obtained system of singular integral equations is carried out by the method of discrete singularities. In [6] it is shown that the kernels of integral equations contain complete elliptic integrals of the 1st and 2nd kinds. Since the 2nd-order elliptic integral is a regular function, the standard Gaussian quadratic formula is used for the calculation. An approach base on the characteristic properties of the mean arithmetic geometric value of AGM (a, b), [11] is applied to the elliptic integral of the first kind. External integrals have a logarithmic feature. For their calculating the special Gaussian quadrature formulas [9] are used. The developed method of discrete singularities [12] is applied for the analysis of free and forced vibrations of shell structures.
For solving the forced oscillation problem we consider the Bernoulli equation (2.1) on a free surface. We represent the velocity potential in form (3.1), and write the function describing the position of the free surface as follows: The functions φ k are modes of free fluid oscillation in a rigid tank. They are obtained by solving boundary-value problem (2.2). We substitute expressions (3.1) and (3.2) into the Bernoulli integral (2.1), taking into account that The values⎟ k are the frequencies of free oscillation of the fluid in the rigid tank. It should be noted that according to [7,10], the frequencies and modes of oscillations of the fluid are considered to correspond to certain harmonics, it is also assumed that the velocity potential and the function describing the level of lifting of the free surface. The formulas are presented below In equality (3.5) the wave number characterizes the harmonic number, or the number of nodal diameters [7,10].
Of especial importance is a value = 0, which determines the axisymmetric fluctuations of the fluid in the rigid reservoir, as well as = 1 which corresponds to antisymmetric oscillations.

Numerical results
The liquid vibrations in the cylindrical-conical shell are considered. Suppose that R 1 = H 1 = H 2 = 1m and R 2 = 1m. Let horizontal and vertical accelerations be    The results depicted in Fig. 4.3, 4.4 demonstrate that the parameters of sloshing process correspond to the region of instability.

Conclusions
The method of estimation of forced vibrations of liquid in the shells of rotation based on the use of the method of boundary integral equations and the Fourier series decomposition has been developed. The integral equations have been solved using the method of discrete singularities. The analysis of the results has shown that the phenomenon of parametric resonance takes place. The joint effect of vertical and horizontal excitations has been considered.