Investigating the stability of oscillations of rectangular plates in an infinitely long rectangular parallelepiped with an ideal fluid

The object of this study is two thin elastic isotropic rectangular plates in an infinitely long rectangular parallelepiped with an ideal fluid. The first plate is the upper base of the rectangular parallelepiped, and the second one horizontally separates ideal fluids that have different densities. The subject of the study is the normal joint plane vibrations of elastic rectangular plates and an incompressible fluid and the conditions that enable the stability of these vibrations. In the linear statement, the frequency spectrum of normal plane vibrations of two elastic isotropic plates in an infinitely long rectangular parallelepiped with an ideal incompressible fluid has been investigated. The frequency equation of joint vibrations of the plates and the ideal fluid was reduced to the form of an eighth-order determinant for arbitrary cases of fixing the contours of the plates. The case of clamped contours of the plates and the case of rebirth of the plates into membranes is analyzed. Based on analytical studies of infinite series in the transcendental frequency equation, exact stability conditions for the combined oscillations of plates and liquid were established. It has been shown that instability of oscillations of plates and liquid occurs when a heavier liquid is above a less heavy liquid. The derived stability conditions for symmetric and asymmetric oscillations of plates and liquid do not depend on the elastic parameters of the upper plate, the mass characteristics of the plates and the depths of filling liquids. The analytically obtained exact stability conditions for the combined oscillations of the plate and liquid generalize the previously obtained approximate stability conditions for this problem. The numerical calculations of the frequency equation confirmed the analytical studies of the stability conditions. The results could be used in the calculation and design of mechanical objects related to the storage and transportation of liquid cargo