Size-dependent geometric nonlinearity and modal coupling in rotating silicon timoshenko microbeam

This study investigates the nonlinear dynamic behavior of a rotating silicon microbeam, incorporating shear deformation, rotary inertia, and size-dependent effects within the framework of the Nonlocal Strain Gradient Theory (NSGT) and Timoshenko Beam Theory (TBT). By utilizing Hamilton's principle and the von Kármán strain-displacement relations, the coupled nonlinear partial differential equations of motion are derived and subsequently discretized into a reduced-order model via the Galerkin projection method. Numerical simulations reveal that the system exhibits significant geometric nonlinearity, characterized by a distinct “hardening” phenomenon where the fundamental frequency shifts considerably (e.g., increasing from approximately 0.7 MHz in the linear regime to 8.5 MHz a large dimensionless amplitude of W m a x h =15). Furthermore, the analysis demonstrates a strong interplay between rotation speed and stiffness (centrifugal stiffening) and unveils a unique modal coupling mechanism, where energy transfer between the first and third vibration modes results in toroidal trajectories in the three-dimensional phase space. These findings highlight the critical necessity of employing higher-order continuum theories and nonlinear analysis for the accurate design and reliability of high-precision MEMS gyroscopes and rotating microsensors operating in nonlinear regimes.