The Bagley-Torvik（B-T） equation is a differential equation of motion with fractional （3/2）-order derivative terms that is applied to describe the motion of a rigid plate in Newtonian，viscous fluid. In this paper，we develop non-stationary analytic solutions of the B-T equation whose inhomogeneous term is a stochastic process. The B-T equation is transformed into a half-order state-space equation in matrix form and eigen-analysis is performed to obtain complex eigenvalues and eigenvectors. Subsequently，the generalized coordinate transformation is introduced to decouple the equation into a system of independent 1/2-order differential equations which are solved by Laplace transform to obtain the solution in generalized coordinates； The generalized coordinate solution is converted into a natural coordinate solution to obtain the impulse or step response function. When the inhomogeneous term of the equation is a stochastic process，the Laplace transform can be used to derive the time-varying frequency response function from which the analytical solution of the non-stationary stochastic response can be obtained by relying on the relationship between the excitation and the response power spectral density. The correctness of the method is verified by numerical cases using the Spanos-Solomos fully non-statoionary stochastic excitation as an example.

Environmental vibration energy is a kind of renewable and clean energy with abundant reserves and wide distribution. Through energy harvesting technology，the mechanical energy in the environment is converted into electrical energy to power low-power electronic devices and wireless sensor networks，which is an effective solution to break the limitations of traditional power supply methods. In this paper，the bursting oscillation and energy capture efficiency of a mechanical nonlinear multistable piezoelectric cantilever beam device are studied under low frequency excitation. By analyzing the potential energy of the system，it can be seen that the system has multi-stable characteristics with the change of system parameters. According to the fast and slow dynamic analysis method，the external excitation term is regarded as a slow variable and control parameter to adjust the dynamic behavior of the fast subsystem，and the time history diagram，phase diagram and transition phase diagram of the system are obtained. The motion state and energy capture performance of the system under low frequency excitation are analyzed by numerical method. The results show that the system exists bursting oscillation under low frequency excitation，and the system has good energy capture characteristics when the system is bistable. In addition，the time-delay feedback control can control the clustering phenomenon and ensure the stable operation of the system.

The disturbance induced by the rotation of dual axis flex solar wing is the important factor which impacts satellite attitude and pointing accuracy and stability of satellite precise payload. The flexible multibody system method based on recursive formulation is proposed to solve the dynamics problem induced by satellite dual axis flexible solar wing rotation. The satellite dynamics equations are established by considering orbit mechanics，satellite configuration，solar wing flexibility and solar array drive assembly and momentum wheels. As an example of a satellite with dual axis solar wing，the research on multibody dynamics simulation of satellite electromechanical coupling is developed，and the simulated attitude angle and angular velocity of satellite are compared with the corresponding telemetry data. The research shows that the simulated attitude angle and angular velocity of satellite agree well with the corresponding telemetry data，which proves the correctness of the model，The rotation of flexible dual axis solar wing will produce big disturbance torque and attitude angle，the control torque needs to be distributed to momentum wheel assembly for the accurate simulation results，and the disturbance torque of floating satellite caused by the rotation of solar wing was obtained by simulation，which can give the reference condition of ground test verification of solar wing rotation.

In order to solve the engineering problems of strict convergence condition of traditional decentralized algorithm and huge computation amount of centralized algorithm in vibration active control of complex systems，this paper combines network topology cooperation strategy and FxLMS algorithm to design a novel active vibration control algorithm based on network topology cooperation，and selects a simplified airframe model of a helicopter as the controlled object. The simulation study of active vibration control with a scale of 20×20 （20 actuators and 20 error sensors） is carried out. The results show that the algorithm based on network topology can achieve the same vibration reduction effect as the centralized algorithm while significantly reducing the computation amount，which is an advantage that the decentralized algorithm and the centralized algorithm do not have. The mean vibration decreases about 34.3 dB under single-frequency control，and about 12.6 dB under multi-frequency control. At the same time，the characteristics of secondary path coupling are properly simplified，which is conducive to the value of convergence coefficient，and the effectiveness and superiority of this algorithm for active vibration control of helicopter complex system are fully verified.

Statistical energy analysis （SEA） is a widely used method for analyzing the high-frequency dynamic response of mechanical structures. The reasonable division of subsystems is one of the critical basises for SEA. In this paper，an automatic identification method of SEA subsystem based on order-reduced modal energy density and hierarchical cluster analysis is developed. First，the structural modal energy densities in the high-frequency band are obtained through the discrete finite element model. Then，the main features of the modal energies are extracted through the proper orthogonal decomposition. The similarity of the modal energy density between different elements is analyzed by hierarchical cluster analysis. Finally，the number of statistical energy analysis subsystems and the corresponding structural elements are identified. The T-shaped plates，I-shaped plates，and engine combustion chamber are taken as simulation models to verify the effectiveness of the proposed method. Simulation results show that the coupling relationship between components，the number of subsystems，and corresponding elements can be automatically identified by the proposed method. Then，the SEA model can be established efficiently and accurately.

Active magnetic bearings （AMBs） are ideal bearings for high speed and high power rotating machinery for its adjustable stiffness and damp. In this paper，a dynamic model of AMBs-flexible rotor system is established. Aiming at suppressing vibration displacement of the rotor system in passing through the first bending critical speed region，a control which combines a decentralized PID controller and input second filter in series is designed and the controller performances are simulated. The experiments in simulated rotation and real acceleration operations are carried out in a platform of AMBs-flexible rotor system. The rotor system can smoothly pass through its first bending critical speed region and the maximum rotor vibration displacement in acceleration operation is less than half of backup bearing gap. The rotor vibration displacement and current responses of the rotor in different unbalances are measured in order to analyses the influence of the rotor unbalance on vibration characteristics of AMBs-flexible rotor system. It is shown that the proposed controller can make the rotor system smoothly pass through its first bending critical speed region. The rotor imbalance has a significantly influence on the control performance and stability of AMBs-flexible rotor system. The experiment results give a support on the high-performance control strategy of AMBs-flexible rotor system.

A milling unbalance correction method based on a discrete vector model is proposed to address the issue of poor performance of traditional dynamic balancing methods when the initial unbalance of the micro motor rotor is large. A discrete vector model is established based on the parameters of the milling cutter and rotor，and the corresponding relationship between the equivalent cutting mass and cutting depth under second time cutting is obtained by integrating the discrete points of the edge curve. By comparing with the 3D model simulation data，it is verified that the deviation rate of the model is low. Experimental verification shows that when the initial unbalance on one side of the rotor exceeds 100 mg，a total weight removal rate of over 90% can be achieved，and the equivalent mass of the remaining unbalance on one side can be controlled below 10 mg. All rotors meet the G1 accuracy level. This indicates that this proposed method can improve the dynamic balance accuracy of the micro motor rotor when the initial unbalance of the micro motor rotor is large.

Research on magnetically levitated rotors has been heavily influenced by studies on slender shaft magnetic levitated rotors. In the study on a certain magnetically levitated flat rotor for a centrifugal pump structure，both experiments and finite element analysis revealed that the support characteristics of the radial permanent magnetic bearings，with the same dual-ring structure，exhibited the significant differences from the known experience when applied to flat rotors. The translational stiffness and torsional stiffness showed substantial variations. This paper analyzes the variations in translational and torsional stiffness of permanent magnetic radial bearings for flat rotors based on changes in the bearing’s structural dimensions. Based on the analysis，a flat rotor magnetic levitation structure is proposed，which can increase and adjust the torsional stiffness of the permanent magnetic bearings，while also allowing for a rational ratio between translational and torsional stiffness. A finite element analysis is used to identify the structural conditions that yield maximum translational and rotational stiffness. The effectiveness of the proposed methodology is subsequently validated.

To investigate the effect of waviness on the slippage and vibration characteristics of the full ceramic bearing，displacement excitation and thermal deformation are coupled to propose the dynamic waviness model. The Hertz contact theory and time-varying displacement excitation are combined to obtain the calculation method of time-varying contact stiffness coefficient，and the stiffness coefficient is analyzed in detail. The effects of time-varying contact stiffness coefficient and time-varying displacement excitation are also taken into account to model the slipping dynamic of the full ceramic bearing. The effects of rotational speed and waviness on the slippage and nonlinear vibration characteristics of the full ceramic bearing are analyzed. The results show that an increase in rotational speed，waviness amplitude and wave number all lead to an enlarged contact stiffness coefficient between the ball and the raceway. The contact stiffness coefficient is more sensitive to changes in wave number. The increase in rotational speed can exacerbate slippage. Both the increase in waviness amplitude and wave number can have the effect of inhibiting slippage. However，the waviness amplitude and wave number can be too large resulting in abnormal vibration of the inner ring. The maximum fundamental frequency deviation between simulation and test is 2.75 Hz，the maximum error is 0.37%. This research can be used for the optimal design of the full ceramic bearing structures as well as for health monitoring.

During real-time hybrid simulation（RTHS） of nonlinear specimens，the interaction between the specimen and the loading system can lead to variations in the specimen’s behavior，consequently affecting the time delay in the servo system. Online estimation of the system’s time delay enables the application of an adaptive time-delay compensation method for controlling time-varying systems. Nevertheless，during the initial stages of parameter identification，the estimated values frequently exhibit notable fluctuations，which can have a detrimental impact on the effectiveness of control. To this end，a two-stage adaptive time-delay compensation method driven by the inverse model for RTHS is proposed. Firstly，the inverse model controller of the system is used to perform coarse compensation to eliminate the test error caused by the main time delay. Then，the adaptive delay compensation method based on recursive least squares is used to compensate the remaining delay to further control the accuracy. By using the two-story shear frame as the prototype and the self-centering viscous dampers as the specimens，a time-delay compensation RTHS is carried out simultaneously on the two experimental substructures. Numerical simulations and experimental results show that the control accuracy of the proposed method is higher than that of the single-stage time-delay compensation method，and it can be applied to RTHS involving multiple experimental substructures.