Forced liquid vibrations in prismatic tanks under vertical and horizontal loads

Abstract

The method of studying forced vibrations of a liquid in rigid prismatic tanks partially filed by a liquid is offered. It is supposed that the liquid is an ideal and incompressible one, and its motion, caused by the action of external influences, is irrotational. In these assumptions, there exists a velocity potential that satisfies the Laplace equation. The boundary value problem for this potential is formulated. On the wetted surfaces of the tank the non-penetration conditions are chosen. On the free surface of the liquid, the kinematic and static conditions are specified. The static condition consists in the equality of pressure on the  free surface  to atmospheric one. The liquid pressure is determined from the Cauchy-Lagrange integral. To formulate the kinematic condition, an additional unknown function is introduced, which describes the motion of the free surface. The kinematic condition is the equality of the velocity of the liquid, which is described by the velocity potential, and the velocity of the free surface itself. These modes of free vibrations are used as a system of basic functions in solving problems of forced  fluid vibrations in reservoirs. Unknown functions are presented as series of the basic functions. The coefficients of these series are generalized coordinates. Periodic excitation forces acting in the vertical and horizontal directions are considered. If vertical excitation is studied, this leads to appearance of additional acceleration. Here we obtain a system of unbounded differential equations of the Mathieu type. This allows us to investigate the phenomena of parametric resonance. The effect of parametrical resonance is considered when the vertical excitation frequency is equal to double own frequency of liquid vibrations Dependences of change in the level of free surface via time under both separate and mutual action of horizontal and, vertical forces of are obtained. The phase portraits of a dynamic system with indication of resonances are presented. The method allows us to carry out the adjustment of undesired excitation frequencies at the design stage at reservoir producing in order to prevent the loss of stability.