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.

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The model of a clamped-clamped Euler-Bernoulli beam is presented in order to study nonlinear vibration control of electrostatically actuated nanobeam with nanocapacitive sensor, considering primary and superharmonic resonances. The capacitance of nanobeam capacitor changes with the nanobeam deformation. The nanocapacitive sensor is applied to extract vibration signals and to transform enlarged signals into controller to control nanobeam vibrations. The method of multiple scales is used to obtain the first-order approximate solutions and derive the amplitude-frequency equation. The nonlinear vibration characteristics and amplitude-frequency response of nanobeam vibration system are studied under different excitation voltage, feedback gains, and damping. The relationships between amplitude and system parameters are discussed in detail. The presented analytical and numerical simulations show that dynamic response of nanobeam is stable when the appropriate parameters are chosen. This investigation provides a better understanding of the nonlinear vibration of nanoelectromechanical systems devices based on nanobeam.

The dynamic equations of the strongly nonlinear vibration of vehicle suspension with linear and nonlinear feedback controllers are developed. The strongly nonlinear vibration is transformed into the weakly nonlinear vibration by the nonlinear controller. The forced vibration of vehicle suspension is studied by the method of multiple scales. The regions of feedback gains obtained from the stability conditions of eigenvalue equation quantitatively are presented. Taking attenuation ratio and energy function as the objective functions, the control parameters of velocity and displacement are calculated by the minimum optimal method. Illustrative examples are given to show the effectiveness of vibration control.

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.