Far less is known about the control of random vibration, especially nonlinear random vibration. Through the survey of these literatures, it can be found that most of the studies on vibration control using piezoelectric stack inertial actuator mentioned above are limited to the study of the dynamic characteristics of the actuator itself or the vibration control of linear structure under the action of deterministic load. With this criterion, the piezoelectric smart SFM system has a better single modal controllability and observability and has a good result on the vibration suppression. proposed an optimal placement criterion for piezoelectric actuators. Introducing the modal H 2 norm and change rate of natural frequencies, Lu et al.
studied and designed an actuator that can bear the bending stress, which greatly improved the control effect with the condition of large force and low frequency vibration. The traditional piezoelectric inertia actuator can only bear the force in the axial direction. obtained an actuator with stable linear motion performance using integrated piezoelectric vibrator and MRF control structure. proposed a new type of inertial piezoelectric actuator which has a miniaturization structure and dynamic performance of high precision and high load capacity. used a piezoelectric rotary inertia actuator to control the vibration of the rotating structure, which effectively reduced the noise propagation of the structure. applied a piezoelectric stack actuator to an active shaft transverse vibration control system with large reduction of housing vibrations. applied a piezoelectric stack inertial actuator to the vibration control of simply supported beam at both ends and achieved good control effectiveness. This way is commonly used and has been applied by many scholars in some different areas. The inertial mass can effectively isolate unnecessary interference and also can protect the pressure sensor from being damaged by excessive force. The other way is to use as an inertial actuator, where one side is combined with an inertial mass and the other side is bonded to a structure. Although this kind of actuator has large output force and an easily determined control law, it could bring new excitation sources to the structure. One is the direct actuator, where one side of the piezoelectric stack is fixed and the other is bonded to the structure. Generally, there are two basic approaches when a piezoelectric stack actuator is used as an actuator. Piezoelectric stack actuators have been widely used in vibration control of mechanical structures due to their fast response and high precision, such as aerospace, precision machining, biomedical engineering, and semiconductor manufacturing. Numerical results show that our proposed control strategy is effective for random vibration reduction of the nonlinear structures using piezoelectric stack inertial actuator, and the theoretical method is verified by comparing with the simulation results. The responses of optimally controlled and uncontrolled systems are obtained by solving the Fokker–Planck–Kolmogorov (FPK) equation to evaluate the control effectiveness of the proposed strategy. The proposed control law is analytical and can be fully executed by a piezoelectric stack inertial actuator. The optimal control law is determined by establishing and solving the dynamic programming equation. Then, using the stochastic averaging method, this quasi-non-integrable-Hamiltonian system is reduced to a one-dimensional averaged system for total energy. First, the dynamic model of the nonlinear structure considering the dynamics of a piezoelectric stack inertial actuator is established, and the motion equation of the coupled system is described by a quasi-non-integrable-Hamiltonian system. An optimal control strategy for the random vibration reduction of nonlinear structures using piezoelectric stack inertial actuator is proposed.