A semi-analytical solution for in-plane free waves analysis of rectangular thin plates with general elastic support boundary conditions

Abstract

A semi-analytical solution for in-plane free waves analysis of rectangular thin plates with general elastic support boundary conditions (BCs) is developed. Based on the classical thin plate theory, the governing differential equations of the rectangular thin plate are introduced. The semi-analytical solution is obtained by superposition of in-plane waves of different patterns and different propagation directions. And it is validated by comparisons of the present results with those given in literatures. Eigensolutions for general BCs and the specific expressions of characteristic equations for various classical BCs are demonstrated. Characteristic curves of the dimensionless wavenumbers of the in-plane longitudinal wave and the in-plane shear wave of the rectangular thin plate with various BCs are presented and discussed. Perfect orthogonality of the in-plane longitudinal wave and the in-plane shear wave of different mode numbers is presented in the characteristic curves of the rectangular thin plate with two opposite sides simply supported. The characteristic curves corresponding to the rest of the classical BCs and all of the non-classical BCs show non-orthogonality of different extents, which reveals the formation mechanism of the variety of the mode shapes of the rectangular thin plate with various BCs. The strengthening and weakening process of the orthogonality and the formation of the intertwinement or independence of the characteristic curves of different mode numbers are elaborated through analysis on elastic support BCs intermediate between two classical BCs, which shows the influence of the BCs on the intrinsic characteristics of in-plane waves of the rectangular thin plate.