Dynamic equilibrium equations of linear piezoelectric Euler–Bernoulli beams

A one-dimensional model for the dynamics of linear piezoelectric straight prismatic beam based on the Euler–Bernoulli’s theory and appropriate hypotheses on the electric displacement field is developed. The equations of motion of longitudinal and flexural vibrations are formulated in terms of one-dimensional mechanical and electrical displacement fields. The formulation uses the beam equilibrium equations including inertia forces and one-dimensional versions of Gauss–Maxwell equations. Applied approach yields a technical theory of linear piezoelectric beams, it rests on ad hoc assumptions in order to neglect certain terms, the mechanical displacements and axial component of the electric displacement are assumed to be linear functions of the cross-sectional coordinates.