Dynamics of elastic-piezoelectric two-layer beams using spectral element method

It is important to predict the dynamic characteristics of a piezoelectrically actuated beam very accurately for successful vibration controls. It has been well recognized that spectral elements provide very accurate solutions for such simple structures as beams. Thus, this paper introduces a spectral element method (SEM) and a spectral- element based modal analysis method (MAM) for elastic-piezoelectric two-layer beams. The axial-bending coupled equations of motion are derived ®rst by using Hamiltonʹs principle and the spectral element matrix is formulated from the spectrally formulated exact eigenfunctions of the coupled governing equations. For MAM, the orthogonality of the eigenfunctions (i.e. natural modes) is proved. Present solution approaches are veri®ed by comparing their results with the conventional FEM results. It is shown that the results by MAM and FEM converge to those by SEM as the number of superposed natural modes and the number of discretized ®nite elements are increased, respectively. It is also shown that, as the thickness of piezoelectric layer vanishes, the axial-bending coupled problems are decoupled to yield the solutions for two independent problems: the pure axial-motion problem and the pure bending-motion problem. 7