Fundamental frequency of an orthotropic rectangular plate with uniformly distributed point supports: Application to the design procedure

In the paper an analytical solution to the problem of determining the fundamental frequency of vibrations of a rectangular orthotropic plate with uniformly distributed point supports is obtained. The deflection and the two rotation angles of the tangents are equal to zero at each point of support of the plate. The edges of the plate are fully clamped. The solution to the required dynamic problem is reduced to the determination of the fundamental frequency of vibrations of a rectangular fragment of a plate supported at four corner points. At each of the edges of such a plate fragment, the angle of rotation of the tangent and the generalised shear force are both zero. The solution to the dynamic problem for the plate fragment was obtained by employing the Ritz method. An approximation of the deflection of a plate fragment supported at four corners was performed using a three-term combination of clamped–clamped beam functions. The implementation of the Ritz method enabled the derivation of a cubic equation, from which the desired frequency of vibration of the corned supported plate fragment was subsequently ascertained by the Cardano method. The analytical solution was utilised to calculate the fundamental frequency of vibrations of the orthotropic plate fragment with given sizes. The found frequency was then compared with the fundamental frequency of vibrations of whole plates with uniformly distributed point supports. The calculation of the latter was performed by means of the finite element method. The comparison demonstrated that the fundamental frequency of vibrations of whole plates slightly exceeds the frequency of vibrations of their fragments. It is evident that the discrepancy between these frequencies diminishes as the number of fragments along the edges of the plates increases. The paper demonstrates the use of the value of the fundamental frequency of fragment vibrations in the design of point-supported plates.