Free vibration of arbitrarily shaped plates with complex cutouts

Abstract

In this paper, an exact model for free vibration analysis of arbitrarily shaped plates with various cutouts under general elastic boundary conditions is established. It combines the artificial connecting spring technique with the domain segmentation integral method (DSIM) proposed by the author. The arbitrarily shaped plate with cutouts is decomposed into several arbitrarily shaped subdomains based on the cutouts. The continuity conditions at the interconnecting interfaces of the subdomains are realized by artificial coupling springs. The energy expressions of the free vibration of the subdomain with arbitrary shape is calculated analytically or semi-analytically by using the DSIM. That is, in order to simplify the energy expressions into definite integrals, the arbitrarily shaped integration domain of the energy integrals is divided into several trapezoid domains with curved sides, and then the penalty method and Gram–Schmidt orthogonalization process are combined to generate the admissible functions for calculation. When the contour curve equations of the plate enable the energy integrals to be analytically calculated, the model can obtain the exact solution of free vibration of an arbitrarily shaped plate with various cutouts under general elastic boundary conditions; otherwise, the semi-analytic results can be obtained by combining Gauss–Legendre method. The convergence and accuracy of the proposed model are verified by comparing the numerical examples with results available in the literatures. The vibration characteristics of several complex shaped plates with various cutouts are studied. Taking elliptical plates with a rhombic cutout as examples, the effects of cutout size and position on the vibration characteristics of the plate are discussed. These new results can serve as benchmarks for further research.