On a variant of the improved theory of ortotropic plates: nonclassical forms of free fluctuations

This paper offers an improved variant of the equations of free fluctuations of ortotropic plates constructed as a first approximation by the reduction of the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates by using trigonometric basic functions and by satisfying static boundary conditions on boundary surfaces. The solutions to these equations are found for a plate with jointedly supported edges. The equations are divided into two isolated systems of equations. The first system describes the nonclassical shiftless longitudinal-transverse forms of free fluctuations accompanied by a distortion of the flat form of the transverse sections. It is shown that the fluctuation frequencies corresponding to these forms at some geometrical parameters of a plate strongly depend on Poisson's ratio and elasticity module in a transverse direction and, for plates of a medium thickness with the same frequency parameter (tone), can be considerably lower than the frequencies corresponding to the classical longitudinal forms of free fluctuations proceeding with preservation of the flat form of the transverse sections. The second system of equations describes the transverse bend-shift forms of free fluctuations, the frequencies of which decrease at the reduction of the transverse shift module. According to the quality and pithiness, these equations are almost equivalent to the similar equations in the known variants of improved theories, but, unlike them, under the increase in the tone number and the reduction of thickness ratio, lead to the solutions obtained within the classical theory of rods.