This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp(A) ∪σp1(-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.
The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.
The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved.The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators.A modified method of separation of variables is proposed for a separable Hamiltonian system.As an application of the theorem,the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained.Finally,a numerical test is analysed to show the correctness of the results.
A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differential form and integral form of the separable Hamiltonian system, the biorthogonal relationships of the eigenfunctions are presented. Based on the biorthogonal relationships, a novel complete biorthogonal expansion of the Mindlin plate bending problems with two opposite sides simply supported is proposed through the products of operator matrices. The exact solutions to deflections and bending moments for the Mindlin plate with fully simply supported sides are obtained. A numerical example is illustrated to verify the accuracy and validity of the expansion method.
In this paper, we consider the eigenvalue problem of a class of fourth-order operator matrices appearing in mechanics, including the geometric multiplicity, algebraic index, and algebraic multiplicity of the eigenvalue, the symplectic orthogonality, and completeness of eigen and root vector systems. The obtained results are applied to the plate bending problem.
