New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

Backward Error Analysis for Eigenproblems Involving Conjugate Symplectic Matrices

Conjugate symplectic eigenvalue problems arise in solving discrete linear-quadratic optimal control problems and discrete algebraic Riccati equations. In this article, backward errors of approximate pairs of conjugate symplectic matrices are obtained from their properties. Several numerical examples are given to illustrate the results.

A SYMPLECTIC HAMILTONIAN APPROACH FOR THERMAL BUCKLING OF CYLINDRICAL SHELLS

The paper deals with the thermal buckling of cylindrical shells in a uniform temperature field based on the Hamiltonian principle in a symplectic space. In the system, the buckling problem is reduced to an eigenvalue problem which corresponds to the critical temperatures and buckling modes. Unlike the classical approach where a predetermined trial shape function satisfying the geometric boundary conditions is required at the outset, the symplectic eigenvalue approach is completely rational where solutions satisfying both geometric and natural boundary conditions are solved with complete reasoning. The results reveal distinct axisymmetric buckling and nonaxisymmetric buckling modes under thermal loads. Besides, the influence for different boundary conditions is discussed.