A Dynamic Model for Laminated Piezoelectric Microbeam

Piezoelectric beams are widely used in micro-electromechanical systems. At the microscale, the influence of the size effect on a piezoelectric beam cannot be ignored. In this paper, higher-order elasticity theories are considered to predict the behaviors of piezoelectric micro-structures and a size dependent dynamic model of a laminated piezoelectric microbeam is established. The governing equations for the laminated piezoelectric microbeam are derived using the variational principle. The natural frequencies of piezoelectric microbeams are obtained by size-dependent dynamic models. The results reveal that the size effect can enhance the structural stiffness at the microscale. The natural frequency obtained by using the classical model is smaller than that obtained using the size-dependent model. Compared with the modified couple stress model, the modified couple stress model underestimates the size-dependent response. Thus, the modified couple stress model is a simplification of the modified strain gradient model. The influence of beam thickness on the natural frequency is also discussed. With increasing the thickness, the natural frequency of the size-dependent models gradually approaches the result of the classical model. If the value of h/l is greater than 15, the influence of the size effect can be neglected. Additionally, the relative thickness can influence the natural frequency, and if the relative thickness is greater than 5 or less than −5, the bilayer beam can be simplified to a single layer beam.