Hamiltonian system-based analytical solutions for the free vibration of edge-cracked thick rectangular plates

Abstract

Analytical modeling for cracked plates’ free vibration is of great significance as a crack alters the properties of plate structures and thus greatly affects the mechanical behaviors of such key engineering components. This paper presents novel analytical solutions for the free vibration of thick rectangular plates with a crack normal to an edge, a kind of typical edge-cracked plates. To resolve the jump discontinuity caused by the edge crack, an original plate is decomposed into four elementary plates, whose free vibration problems are solved analytically via an unusual symplectic superposition method (SSM). The SSM is implemented in the Hamiltonian system-based symplectic space, where several mathematical treatments such as the variable separation and the symplectic eigenvector expansion are valid, rendering rigorous step-by-step derivations without pre-defining solution forms. By incorporating the solutions of the elementary plates, the eventual solutions of the edge-cracked thick plates are obtained. Such a symplectic framework provides a new route to solving various complicated plate problems that cannot be analytically handled previously. Comprehensive free vibration results such as natural frequencies and vibration modes of edge-cracked thick plates are presented. High accuracy and fast convergence of the framework are confirmed. The present analytical solutions are also utilized to study the effects of both the crack length and crack location on free vibration behaviors of edge-cracked thick plates.