Complex Valued State Space Model for Weakly Nonlinear Duffing Oscillator with Noncollocated External Disturbance

Abstract

This paper proposes a slow flow model for a weakly nonlinear and parametrically driven Duffing oscillator and a complex valued state space model for the oscillator with noncollocated external disturbances. The combination of the parametrically driven duffing oscillator and noncollocated disturbances can be observed in resonant MEMS mirrors with a reinforcement structure to reduce dynamic mirror deformation. The model is based on a rational function approximation for the angular derivative of the out-of-plane comb drive capacitance, enabling qualitative analysis at large amplitudes while the stability analysis is maintained as the conventional cubic function approximation. The slow flow model is extended including a single tone noncollocated disturbance and is linearized at an equilibrium point for a small disturbance. The linearized disturbance model is reformulated by a complex valued state space model to cope with general wideband disturbances, allowing various analytic methods in traditional system theory. The simulation results demonstrate a good agreement between the proposed models and the ODE simulation, verifying the accuracy and benefits of the proposed models.