Exact and Approximate Analytical Solutions to a Problem on Plane Modes of Free Oscillations of a Rectangular Orthotropic Plate with Free Edges,with the Use of Triginometric Basis Functions

The linear problem on plane modes of free oscillations of a rectangular orthotropic plate with free unloaded edges is considered. A procedure for constructing displacement functions exactly satisfying the boundary conditions, with the use of double-trigonometric basis functions, is offered. Exact and approximated analytical solutions to the problem formulated are found, which presumably describe all plane modes of free oscillations of the plate in the class of the functions indicated. It is established that, in the use of variational principles, the variations of required functions must be considered not only arbitrary, but also mutually independent. Therefore, the solutions constructed give physically reliable results for the frequencies and modes of free oscillations only if the problem is stated in the form of Bubnov variation equations, which depend on the structure of displacement functions. It is found that the exact analytical solutions of the problem correspond to oscillation modes without shear strains. It is shown that it is possible to select such solutions from them which correspond to trigonometric functions with a zero harmonic in one direction. These solutions describe only flexural oscillation modes of the plate, and the results obtained are equivalent to those given by the classical Kirchhoff model known in the theory of rods, plates, and shells.__________Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 461–488, July–August, 2005.