OSCILLATIONS OF A RECTANGULAR PLATE SEPARATING IDEAL LIQUID OF DIFFERENT DENSITY IN A RECTANGULAR CHANNEL AND AN ELASTIC BASE - Kononov, Yu. M.; Lymar, A. A. (Computer Science and Applied Mathematics, 2017)
Was investigated the frequency equation of the natural oscillations of elastic plate, horizontally dividing the perfect liquid of varying density in a rectangular channel with an elastic bases in the form of rectangular plates. We consider an arbitrary contours the case of fixed plates and various cases of degeneration of the plates in the membrane in an absolutely rigid, the lack of one of the plates, no top or bottom of the liquid. For example, the lack of a lower fluid and the two membranes shows that the frequency equation splits into two equations describing the odd and even frequency. In this case, the frequency spectrum of frequencies for asymmetrical consists of two sets of frequencies corresponding to oscillations of the upper and lower bases, and the frequency spectrum of frequencies for symmetrical consists of three sets of frequencies corresponding to oscillations of the upper and lower bases as well as fluctuations of the liquid column as a whole. On the basis of analytical studies to draw general conclusions about the oscillations of a rectangular plate that separates the ideal liquid of different density in a rectangular channel with an elastic foundation. Consider a flat oscillations of an elastic rectangular plate horizontally separating ideal incompressible heavy fluid of different density in a hard rectangular channel with elastic top and bottom of the base. Elastic bases are presented in the form of rectangular plates. All three plates are considered to be thin and isotropic. The plates are under an action of tensile forces acting on a middle surface. The contours of the plates can be clamped, simply supported or free. The movement of the liquid is potential. The problem is considered in the linear formulation. Plates bending problem is represented as a sum of static and dynamic bending. Joint oscillations of plates and liquid are considered unseparated. Spectral problem is represented as a system of three inhomogeneous equations. These equations govern coupled vibrations of plates and liquid. The general solution to each equation is the sum of the general solution of the related homogeneous equation and the particular solution of inhomogeneous . Mechanical parameters of the plates and the density of the liquid define the general solution of the homogeneous equation which contains an unknown frequency. Particular solution of the inhomogeneous equation takes the form of a series expansion in the eigenfunctions of ideal liquid oscilation in the rectangular channel. From the condition of fixing of contours of the plates, compatibility condition and condition of incompressibility of the liquid follows the frequency equation of natural oscillations of plate, elastic bases and the ideal liquid. In general this transcendental equation takes the form of a determinant of the fifteenth order. On the basis of analytical investigations it may be concluded that: 1. Clamped plates or membranes frequency equation is divided into odd and even frequency. 2. The equation for the even more complex frequency. It includes a main part of the equation for odd frequencies recorded for the even indices, as well as the oscillations of the liquid column as one whole. 4. The frequency spectrum of asymmetric oscillations consists of three sets of frequencies corresponding to oscillations of three plates. 5. The frequency spectrum of the symmetric oscillations is comprised of four sets of frequencies corresponding to oscillations of the three plates of the liquid column as one whole.