ВІЛЬНІ КОЛИВАННЯ ОРТОТРОПНИХ ПЛАСТИН

The solution of the problem of free vibrations of a rectangular orthotropic plate by the methods of boundary and finite elements under any boundary conditions. Transformation of the two-dimensional differential equation of free vibrations of an orthotropic rectangular plate to one-dimensional. Determination of the complete system of its fundamental solutions using the numerical-analytical method of boundary elements. Implementation of the algorithm on the example of a specific plate and comparison with the results of finite element analysis in ANSYS. The solution to the problem of natural vibrations of a rectangular orthotropic plate is obtained without any restrictions on the nature of the fixing of its sides. A transcendental frequency equation is obtained whose roots give the full spectrum of natural frequencies. The modeling and calculations of the orthotropic plate by the finite element method are performed. An analysis of the numerical results obtained by the author's method shows a very good convergence with the results of finite element analysis. For a plate with rigid fastening of three sides with a free fourth side, the discrepancy is slightly higher than for a plate with a hinged support along the contour. Under both variants of the boundary conditions, the frequency spectrum calculated by the boundary element method is lower than in the finite element calculations. Analytical expressions of fundamental functions are obtained that correspond to all possible solutions to the differential equation of free oscillations. For the first time, a solution to the problem of free vibrations of a rectangular orthotropic plate is presented by the numerical-analytical method of boundary elements. The results allow us to solve the problem of free vibrations of a rectangular orthotropic plate by two methods under any boundary conditions, including inhomogeneous ones.