Size-dependent nonlinear dynamics of a microbeam based on the modified couple stress theory

In this paper, the nonlinear dynamics of an electostatically actuated microbeam with von karman geometric nonlinearity and squeeze-film damping are studied. The governing equations and the boundary conditions are developed using the modified couple stress theory and Hamilton's principle. The size-dependent responses are investigated for the primary, super-harmonic, sub-harmonic resonances, and static pull-in voltage. The effects of the thickness, width of the beam, and gap between electrodes on the frequency response of the resonance, the peak amplitude, nonlinearity of the system, and the pull-in voltage are investigated. Special attention is paid to the "softening" and "stiffening" effects of the linear stiffness. These results show that static behavior and forced vibration of the microbeam are highly size dependent. In addition, a delayed state feedback is introduced into the system. It shows appropriate control gain, and time delay can control the vibrational behavior of the microbeam.