Theoretical analysis of longitudinal vibrations of piezoelectric/elastic composite rods

The dynamic behaviors of piezoelectric/elastic composite rods are mainly analyzed by the commercial multiphysics softwares based on finite element method (FEM) in current times. This paper extends the method of reverberation-ray matrix (MRRM) to the analysis of free vibration and proposes the corresponding mode superposition method (MSM) for the analysis of forced vibration of piezoelectric/elastic composite rods. First, with considering four kinds of electric boundary conditions, the governing equations and their solutions of the piezoelectric rod members are derived. The formulation of MRRM for the analysis of natural frequencies and modes can then be established. Second, the obtained natural modes corresponding to different frequencies are proved to be orthogonal with respect to the structural mass and stiffness properties. Third, the MSM is proposed for the analysis of transient responses due to external excitations. Finally, numerical examples are provided to validate the proposed MRRM and MSM for the analysis of vibrations and to study the influences of electrical boundary conditions on the dynamic behaviors of the piezoelectric/elastic composite rods.