Symplectic Superposition Approach For The Problems Of Rectangular Thin Plates

As the important engineering structures,elastic rectangular thin plates are widely used in many fields such as aerospace engineering,civil engineering,ocean engineering and mechanical engineering.Static bending and free vibration problems of elastic rectangular thin plates are two types of fundamental issues in engineering applications.However,their analytic solutions are hard to obtain due to the difficulty in solving the governing partial differential equations in mathematics.In this dissertation,the symplectic superposition approach is developed to obtain the analytic solutions of static bending and free vibration problems of elastic rectangular thin plates with complex boundary conditions.For the static bending problems of elastic rectangular plates,the Hamiltonian dual equation is derived from the Hellinger-Reissner variational principle for static bending problems of a thin plate.The symplectic geometry approach is used to obtain the symplectic analytic solutions of plates with two opposite edges simply supported and plates with an edge simply supported and its opposite edge slidingly clamped.Then we obtain accurate analytic static bending solutions of rectangular thin plates with a corner point-supported and rectangular thin plates with mixed boundary conditions by the superposition method.For the free vibration problems of elastic rectangular plates,the Hamiltonian system is constructed from the basic equations for free vibration problems of thin plates.The symplectic geometry approach is used to obtain the symplectic analytic solutions of plates with two opposite edges slidingly clamped and plates with an edge simply supported and its opposite edge slidingly clamped.Then we obtain accurate analytic free vibration solutions of rectangular thin plates resting on multiple point supports and rectangular thin plates with mixed boundary conditions by the superposition method.In this dissertation,the symplectic superposition approach starts from the governing equations of elastic rectangular thin plates.In the Hamiltonian system,using the symplectic geometry approach,the approach of separation of variables as well as symplectic eigen expansion is used to obtain the symplectic analytic solutions of static bending and free vibration problems of elastic rectangular thin plates with simple boundary conditions.Then we obtain accurate analytic solutions of static bending and free vibration problems of elastic rectangular thin plates with complex boundary conditions by superposition method.The symplectic superposition approach combines the Hamiltonian system-based symplectic method and the superposition method;it has the advantages of rationality and regularity.This enables more analytic solutions of some difficult problems to be explored,which cannot be obtained by the classical methods.