Some New Analytic Solutions For Free Vibration Of Rectangular Plates

Elastic rectangular plate structures play an important role in engineering applications such as aeronautics,bridge decks,structure engineering and foundation beds.Solution of their free vibration has long been one of the key issues in mechanics research,and there is a large mathematical challenge on solving the problems.However,except the plates with two opposite edges simply supported,few of the others have been analytically solved by the classical approaches.In this dissertation,the free vibration problems of rectangular plates with typical boundary conditions,including those based on the Kirchhoff theory(thin plate theory),the orthotropic thin plate theory and Mindlin theory(moderately thick plate theory),are respectively introduced into the Hamiltonian system.Accordingly,a symplectic superposition method,based on the combination of the symplectic geometry method and superposition method,is applied to solve the problems of rectangular plates with complex boundary conditions.For the free vibration problems of thin plates,orthotropic thin plates and moderately thick plates,respectively,the Hamiltonian system is constructed from the governing equations,with the basic mechanical quantities as the symplectic variables,where the mathematical techniques in the symplectic space are applied,including the separation of variables,which is sometimes invalid in the Euclid space,and symplectic eigen expansion.Based on the obtained analytic solutions,skillful superposition is imposed to yield the analytic solutions of the original problems.There are some boundary conditions to be satisfied in equating the superposition of the fundamental solutions to the final solution of a cantilever plate,which leads to the frequency equation.The mode shapes are then readily obtained.Many numerical results are presented to show the convergence and accuracy of the new analytic solutions by excellent comparison with those from the literature,if any,and the finite element method(FEM).They are expected to serve as the benchmarks for validation of the other methods.The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation,without assuming any trial solutions;thus,it is regarded as rational,and its applicability to more bending,buckling and vibration problems of plates with more complex shapes and boundary conditions may be expected.