Optimal control of nonlinear vibration resonances of single-walled nanotube beams

An optimal time-delay feedback control method is provided to mitigate the primary resonance of a single-walled carbon nanotube (SWCNT) subjected to a Lorentz force excited by a longitudinal magnetic field. The nonlinear governing equations of motion for the SWCNT under longitudinal magnetic field are derived and the modulation equations are obtained by using the method of multiple scales. The regions of the stable feedback gain are worked out by using the stability conditions of eigenvalue equation. Taking the attenuation ratio as the objective function and the stable vibration regions as constrained conditions, the optimal control parameters are worked out by using minimum optimal method. The optimal controllers are designed to control the dynamic behaviors of the nonlinear vibration systems. It is found that the optimal feedback gain obtained by the optimal method can enhance the control performance of the primary resonance of SWCNT devices.