Axial-Delayed Control of Nonlinear Resonance of Nanobeams with Graphene Sensor

Nonlinear resonance response of an electrostatic-actuated nanobeam is controlled by using a delayed axial electrostatic force with near-half the natural frequency. A graphene sensor pasted on the surface of the nanobeam is used to extract the vibration voltage signal. An axial-delayed capacitive controller is designed to produce delayed axial force to control the nonlinear vibration of the nanobeam. The vibration voltage signal from the graphene sensor is input to the axial-delayed capacitive controller to attenuate the nonlinear vibration of the nanobeam. The dynamic response of the resonator is investigated by using the method of multiple scales directly. The sufficient conditions of guaranteeing the system stability and the saddle-node bifurcation are studied. The attenuation ratio is defined as the ratio of the peak amplitude of the nonlinear vibration system with control to that without control. A critical feedback gain is given, which can shift the frequency–amplitude curves from the nonlinear vibration to a linear vibration. An optimal method in which the attenuation ratio is taken as objective function and the aforementioned sufficient conditions as the constraint conditions is given to calculate the optimal feedback gains. Numerical simulations are conducted for uniform nanobeams.