Free vibration characteristics of piezoelectric cylindrical shells with stepped thickness using an analytical symplectic approach

Abstract

An analytical study on free vibration of stepped-thickness piezoelectric cylindrical shells is performed under the framework of symplectic mechanics. In the proposed symplectic approach, the step-wise thickness in the axial direction is considered by regarding the overall shell as a combination of multiple segments with different uniform thicknesses. The Hamiltonian governing equations for free vibration of each segment are established by defining a new total unknown vector and the Reissner's thin shell theory. In this manner, analytical solutions can be represented by a series of symplectic eigenfunctions with undetermined coefficients. By using boundary conditions at the ends and interfaces, the coefficients of symplectic series are easily obtained by solving a set of algebraic equations. Consequently, the natural frequencies and analytical mode shapes are obtained simultaneously. The methodology is rational and rigorous, and the solution procedure is systematic with a step-by-step definition. Numerical results are compared with those reported in the open literature and good agreements are observed. A comprehensive parametric study of stepped thickness on vibration characteristics of piezoelectric cylindrical shells are carried out also. The results demonstrate that the local thickness thinning will decrease natural frequencies and lead to local deformation modes. Some of them can be served as benchmarks for other numerical or approximate methods.