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Exact solutions of problems describing the plane forms of free oscillations of a rectangular plate with fixed and free edges of the exact solution describing the plane forms of free oscillations of a rectangular plate with fixed and free edges

Samerhanov RZ, VA Ivanov, VN Paimushin

#### Abstract:

For isotropic rectangular plates at all possible variants of boundary conditions formulated by the same on opposite edges, as well known in mathematical physics method constructed exact solutions of problems on the plane forms of free oscillations. They are based on the use of two of the Helmholtz equation with respect to the inner vortex and potential functions, which reduced the initial equations of motion of the plane problem in displacements. Based on the latter, for a plate with all the loose edges, as well as having only two opposing loose and the other two hinged edge, built and other precise analytical solutions of these problems, based on the use of trigonometric functions as basis. It is shown that these solutions have been found by various methods which are exact and do not coincide and may not be unique.

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