On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories

Size dependent static and dynamic behavior of a fully clamped micro beam under electrostatic and piezoelectric actuations is investigated. The microbeam is modeled under the assumptions of Euler–Bernoulli beam theory. Viscous damping and nonlinearities due to electrostatic actuation and mid-plane stretching are considered. Residual stress and fringing field effect are taken into account as well. Governing equation of motion is derived using Hamilton’s principle along with the strain gradient theory (SGT), which is a non-classical continuum theory capable of taking size effect of elastic materials into account. Reduced order model of the partial differential equations of the system is obtained using Galerkin method. Static deflection, pull-in voltage and the primary resonance of the microbeam are examined and the effect of piezoelectric voltage and its polarization on the size dependent static and dynamic response is studied. It is found that the piezoelectric voltage can effectively change the flexural rigidity of the system which in turn affects the pull-in instability regime. The effect of material length scale parameter is examined by comparing the results of the SGT with the modified couple stress (MCST) and classical theory (CT), both of which are special cases of the former. Comparison demonstrates that the CT underestimates the stiffness and consequently the pull-in voltage and overestimates the amplitude of periodic solutions. The difference between the results of classical and non-classical theories becomes more and more as the dimensions of the system gets close to the length scale parameter. Non-classical theories predict more realistic behaviors for the micro system. The results of this paper can be used in designing microbeam based MEMS devices.