On the Dynamics Behavior and a Control Design to a Nonlinear 2-DOF Vibrating Gyroscopic-MEMS Model

In this paper, we deal with the nonlinear dynamics, the transfer of energy and control of the vibrations of a Micro Electro-mechanical System (MEMS) gyroscope. The MEMS are micro-transducers whose operation is based on elastic and electrostatic forces that convert electrical energy into mechanical energy and vice-versa. These systems can be modeled by 2-DOF spring-mass-damper system and the coupling of the system equations is performed by Coriolis force. This coupling is responsible for the energy transfers of the two vibration modes (drive-mode and sense-mode) and for the resonance in MEMS gyroscope. The governing equations of motion have periodic coefficients and as the dimensions of the quantities involved in the system may be inconsistent it is not advisable the use of perturbation methods for the solution of the MEMS gyroscope. For this reason, in the dynamic analysis and control of the vibrations of the MEMS gyroscope, we used a technique based on Chebyshev polynomial expansion, the iterative Picard and transformation of Lyapunov-Floquet (L–F). For the analysis of the dynamic of the micro electro-mechanical system gyroscope, we did the diagram of stability, phase planes and time history of transfer of energy. Finally, we did the control of the unstable orbit to a desired periodic one and compared the phase planes of orbits desired and controlled and time histories of energy transfer of the controlled and non-controlled system.