ON REFINING THE CONDITIONS OF VIBRATION STABILITY A RECTANGULAR PLATE DIVIDING A TWO-LAYER IDEAL LIQUID WITH A FREE SURFACE - Lymar, О. O. (Computer Science and Applied Mathematics, 2021)
In a linear formulation, the frequency equation of natural vibrations of a plate horizontally separating a two-layer ideal fluid with a free surface in a rectangular channel is derived and investigated. The contours of the plate can have arbitrary fixings. Joint vibrations of an elastic plate and a two-layer fluid with a free surface are reduced to a system of integro-differential equations. For the clamped contours of the plate, a unified form of the frequency equation is obtained for both symmetric and asymmetric joint vibrations of the plate and fluid. It is shown that if the depth of filling the upper liquid is greater than the channel width, then the effect of the free surface on the frequency spectrum can be neglected. The limiting cases of degeneration of a plate into a membrane and its absence are considered. The conditions for the stability of joint vibrations of a plate and a fluid with a free surface are specified for three cases: the absence of tensile forces in the plate, degeneration of the plate into a membrane, and the case of fluids with the same densities under compressive forces acting on the plate. It is shown that, in the first case, the approximate values of the stability conditions are underestimated for asymmetric frequencies by a factor of 1.050, and for symmetric frequencies – by a factor of 1.075. In the case of a membrane, the approximate values are underestimated for unbalanced frequencies by 1.002 times, and for symmetrical frequencies by 1.010 times. In the third case, the approximate values are underestimated for unbalanced frequencies by 1.002 times, and for symmetric ones – by 1.010 times. Thus, for a plate, the approximate values of the stability conditions coincide with the exact ones with sufficient accuracy for practice, and in the case of a membrane, for approximate calculations it is necessary to take into account more than two terms of the series. Based on the studies carried out and the results of previous works, it can be argued that the approximate and exact conditions for the stability of joint vibrations of a plate and a two-layer liquid do not depend on the presence of a free surface, its absence, and even in the presence of an elastic plate on the free surface of the liquid.