On the symplectic superposition method for new analytic free vibration solutions of side-cracked rectangular thin plates

Abstract

In this study, a first attempt is made to develop an up-to-date symplectic superposition method for some new analytic free vibration solutions of side-cracked rectangular thin plates that were not obtained by conventional semi-inverse methods. In contrast with the classical Lagrangian system and Euclidean space, the present method is implemented within the Hamiltonian-system framework and symplectic space. The solution procedure involves expressing the problems in the Hamiltonian system and dividing a side-cracked plate into several sub-plates that are analytically solved by the symplectic superposition method, where the imposed quantities are determined by the plate boundary conditions, free edge conditions along the crack, and interfacial continuity conditions between the sub-plates. In the analytic solution of a sub-plate, specifically, the symplectic eigenvalue problems are formulated, followed by the symplectic eigen expansion. The integration of the solutions of the sub-plates yields the final solution of a side-cracked plate. The rigorous mathematical techniques, without predetermination of solution forms, qualify the present method as an unusual approach for exploring more analytic solutions. Comprehensive natural frequency and mode shape solutions of the side-cracked plates under three representative boundary constraints are provided and well validated by other methods. The new analytic solutions obtained may serve as benchmarks for other potential solution methods.