On the vibration of size-dependent rotating flexoelectric microbeams

Flexoelectricity is one of the significant electromechanical properties present in the majority of materials with any centrosymmetric crystal structures. Due to the importance of this effect, the free vibrations of rotating microbeams under the influence of flexoelectricity have been studied based on the assumptions of the Euler–Bernoulli beam theory. Assuming small strains and average rotation, the strain–displacement relations of the von Kármán type have been employed. Initially, by applying Hamilton's principle on the electric enthalpy density and kinetic energy, the equations of motion and boundary conditions were extracted. Subsequently, using the generalized differential quadrature method and its application to the equations of motion, the dimensionless natural frequencies of the first and second transverse modes were obtained. Among the innovations and results of this research is the derivation of the static deformation equation due to rotation about the longitudinal axis under the influence of flexoelectricity. Additionally, various effects were examined, including dimensionless rotation speed, different dimensionless slenderness ratios, and other parameters affecting the natural frequency. The results indicated that increasing structural stiffness parameters, such as higher dimensionless rotation speed and larger size effect parameters, and smaller slenderness ratio, lead to an increase in the natural frequency. Finally, by comparing the parameters of rotation speed, flexoelectric coefficient, and slenderness ratio, it is evident that the flexoelectric coefficient has a more pronounced effect on the natural frequency of the microbeam compared to rotation speed and the slenderness ratio.