A free vibration analysis of piezo-electric beams via hierarchical one-dimensional finite elements

In this article, several one-dimensional finite elements for the free vibration analysis of composite beams with piezoelectric layers or patches are presented. The displacement components and the electric potential are approximated above the cross section through Lagrange's polynomials in a layer-wise sense. Thanks to a Unified Formulation, elements' stiffness and mass matrices are derived in a general, nuclear form that does not depend upon the approximation order of the displacements and the electric potential over the cross section as well as the number of element nodes along the axial direction. Higher-order, layer-wise theories accounting for non-classical effects can be, therefore, straightforwardly formulated. Beam with full piezo-electric layers or piezo-patches are investigated. Results are given in terms of natural frequencies and electro-mechanical coupling coefficients of the bending and torsional modes. Several mechanical and electrical boundary conditions as well as stacking sequences are investigated. Comparison with three-dimensional finite element models is provided showing that the proposed class of finite elements is able to yield accurate results.