Free vibrations and dynamic behavior of the three layered flexoelectric functionally graded microbeam under moving load

In this study, free vibrations and the dynamic responses of the three layered flexoelectric functionally graded microbeam under the effect of moving load are investigated. Using the first order shear deformation and the size-dependent piezoelectricity theories in the framework of Hamilton's principle, the governing equations of motion and the corresponding classical and non-classical boundary conditions are obtained. Next, the equations of motion were discretized and solved using the finite element method. In order to validate the governing equations and the solution method, the obtained results have been compared with the results of previous studies, which shows a good convergence between the current and previous results. The obtained results demonstrate the dynamic behavior of the microbeam is strongly influenced by the geometrical ratios, the flexoelectric layers thickness, the functionally graded power index, the flexoelectric coefficient and the load velocity. So that the escalation in the strain gradient from the augmentation of the flexoelectric layers' thickness leads to a corresponding rise in the electric potential function. Moreover, as the flexoelectric coefficient increases, the microbeam becomes more responsive to polarization, thereby generating a greater amount of electrical energy. In addition, changes in the velocity of the moving load significantly affect the dynamic behavior of the structure in such a way that with the increase in the load velocity, the dynamic deflection of the structure increases.