An optimal delayed feedback control methodology is developed to mitigate the primary and super harmonic resonances of a flexible simply-simply supported beam with piezoelectric sensor and actuator. Stable vibratory regions of the feedback gains are obtained by using the stability conditions of eigenvalue equation. Attenuation ratio is used to evaluate the performance of vibration control by taking the proportion of peak amplitude of primary or super harmonic resonances for the suspension system with and without controllers. Optimal control parameters are obtained using an optimal method, which takes attenuation ratio as the objective function and the stable vibratory regions of the time delay and feedback gains as constraint conditions. The piezoelectric optimal controllers are designed to control the dynamic behaviour of the nonlinear dynamic system. It is found that the optimal feedback gains obtained by the optimal method result in a good control performance.

An optimal control method is provided to mitigate the cubic strongly nonlinear vibration of vehicle suspension with velocity and displacement feedback controllers. The forced vibration of the vehicle suspension is studied utilizing the methods of modified Lindstedt-Poincaré and multiple scales. Ranges of feedback gains that can keep the vibration system stable are worked out by the stability conditions of eigenvalue equation. Taking the decay rate and the energy function as the objective functions and the ranges of stable vibration feedback gains as constrained conditions, the optimal feedback gains of velocity and displacement are calculated by the method of minimum method. The simulation results show that the control method can have optimal control results.

A two-degree-of-freedom nonlinear vibration system of a quarter vehicle suspension system is studied by using the feedback control method considered the fractional-order derivative damping. The nonlinear dynamic model of two-degree-of-freedom vehicle suspension system is built and linear velocity and displacement controllers are used to control the nonlinear vibration of the vehicle suspension system. A case of the 1:1 internal resonance is considered. The amplitude–frequency response is obtained with the multiscale method. The asymptotic stability conditions of the nonlinear system can be gotten by using the Routh–Hurwitz criterion and the ranges of control parameters are gained in the condition of stable solutions to the system. The simulation results show that the feedback control can effectively reduce the amplitude of primary resonance, weaken or even eliminate the nonlinear vibration characteristics of the suspension system. Fractional orders have an impact on control performance, which should be considered in the control problem. The study will provide a theoretical basis and reference for the optimal design of the vehicle suspension system.

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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.