Based on the method of separation of variables in Hamiltonian system and superposition method, the series expansion solution of the free vibration problem of orthotropic rectangular thin plates (ORTPs) with four clamped edges (CCCC) on two-parameter elastic foundation is obtained. The original vibration problem is decomposed into two subproblems with two opposite sides simply supported, and the general solution of each subproblem is obtained by using the expansion of symplectic eigenvectors. Then by superposing these two general solutions, the series expansion solution of the original problem is obtained. The advantage of this method is that the solution process is carried out in symplectic space, and the validity of variable separation and symplectic eigenvectors expansion ensures the rationality of the solution process, while avoiding the presetting of the solution form. Finally, the correctness of symplectic superposition solution obtained in this paper is verified by calculating three concrete examples of fully clamped rectangular thin plates.
The eigenfunction system of the Hamiltonian operator appearing in the free vibration of rectangular Kirchhoff plates with two opposite edges simply supported is studied. The governing differential equations for free vibration of rectangular Kirchhoff plates is rewritten as a Hamiltonian system based on the known results, and the associated Hamiltonian operator is obtained. Then, in the sense of Cauchy's principal value, the completeness of the symplectic eigenfunction system is proved. It offers a theoretical guarantee of the feasibility of variable separation methods based on the Hamiltonian system for the problem. The exact general solution for the corresponding Hamiltonian system of the problem is given by the symplectic eigenfunction expansion method. The general solution is more simple and convenient than the existing result. Examples are given to illustrate that, combining the general solution with the corresponding boundary conditions, the frequency equations and the transverse displacement functions for Lévy-type plates can be directly derived. Furthermore, the boundary conditions for the problem, which can be solved by this approach, are indicated.
In this paper, new numerical radius inequalities for 2× 2 operator matrices are proved. These numerical radius inequalities refine the existing upper bounds. Mathematics subject classification (2020): 47A05, 47A12, 47A30.
The free vibration of orthotropic rectangular thin plates with four free edges on two-parameter elastic foundations is studied by the symplectic superposition method. Firstly, by analyzing the boundary conditions, the original vibration problem is converted into two sub-vibration problems of the plates slidingly clamped at two opposite edges. Based on slidingly clamped at two opposite edges, the fundamental solutions of these two sub-vibration problems are respectively derived by the separation variable method of the corresponding Hamiltonian system, and then the symplectic superposition solution of the original vibration problem is obtained by superimposing the fundamental solutions of the two sub-problems. Finally, the symplectic superposition solution obtained in this study is verified by calculating the frequencies and mode functions of several concrete rectangular thin plates with four free edges.
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