Modified couple stress flexural-flexural quasi-static pull-in analysis of large deformable cantilever-based micro-gyroscopes

This paper aims to develop a size-dependent large deformable model to study the flexural-flexural quasi-static motion of cantilever-based micro-gyroscopes based on the modified couple stress theory. The micro-cantilever is assumed to undergo non-planar bending-bending-twisting motion. Based on this assumption, the displacement field associated with a point placed on the cross-section of the micro-beam is introduced. Employing the introduced displacement field, the components of the Green-Lagrange strain and curvature tensors are obtained. Neglecting the terms with the time deriva-tives, the nonlinear, coupled and higher-order equations governing on three-dimensional quasi-static motion of in-extensional cantilever-based micro-gyroscopes and their cor-responding boundary conditions are then obtained. Normalizing the strain and kinetic energies as well as the works done by the external forces, the reduced equations governing the quasi-static motion of micro-gyroscopes are obtained through employing the Ritz method with linear mode-shapes of clamped-free beams as the approximating functions. Vanishing the Jacobian of the reduced equations, the governing stability equations of the system are then obtained. Afterward, employing the achieved reduced equations, both the stable and unstable branches of the micro-gyroscope equilibrium paths are extracted. To validate the accuracy of the present model, several verifications have been conducted. The performed verifications demonstrate that the current model can be considered as a promising tool in capturing the influence of small scales on micro-gyroscopes undergoing extremely large deformations.