Symplectic Superposition Method For The Free Vibration Problems Of Irregular Plates

As basic structural elements,the irregular plates,represented by triangular plates,L-shape plates and rectangular plates with cutouts,are widely used in marine engineering,bridge structures and aerospace engineering.The analytical solutions to the free vibration problems of irregular plates are important for both rapid analyses and preliminary designs of structures.However,due to the difficulty in finding solutions that satisfy both high order partial differential equations and the boundary conditions of the plate,current knowledge about the analytical solutions is limited.In this dissertation,the free vibration problem of irregular plates is solved analytically.The symplectic superposition method is applied to divide an original problem into sub-problems reasonably.For a triangular plate,the triangular domain is divided into two fixed-angle superimposed rectangular domains.For an L-shape plate and a rectangular plate with a rectangular cutout,based on domain decomposition,the irregular domain is constructed by concatenation of multiple rectangular domains.The superposition system is constructed.Then the sub-problems are imported into Hamiltonian system based on the basic governing equations of the plate.Accordingly,with the symplectic geometry methodology via reasonably imposing the variable separation and the eigen expansion,the sub-problems are solved analytically.Finally,the analytical frequency and mode shape solutions to the free vibration problems of irregular plates are then obtained by the requirement of the equivalence between the original problem and the superposition of sub-problems.The analytical results of the natural frequencies and mode shapes are given and show good agreement with the numerical results obtained by the finite element method.It turns out that this method is suitable for dealing with the vibration problems of plates with different shapes and boundary conditions and has high precision.This method is rational and analytical because of its strict mathematical derivation,without pre-determination of the solution forms.The results obtained with this method can be used as benchmarks for validation of the other analytical and numerical methods.The symplectic superposition method is expected to be extended to obtain analytical solutions to the mechanics problems of plates with more complex materials,geometries and boundary conditions.