Delayed Feedback Transverse Vibration Control Of A Flexible Cantilever Beam

A flexible cantilever structure is a common structure in industry.However,it often vibrates when it works which will affect its normal work.Generally,the time delay is inevitable in the vibration active control loop.However,the time delay is usually ignored in the vibration control systems,which can lead to the failure of achieving desired vibration control effect,or even causes huge damage to the original system.Therefore,it is of certain significance to consider the delay effect of the control system in the first place,when designing the vibration control law of a flexible cantilever structure.In chapter one of this dissertation,we introduce the application background of the flexible cantilever structure,and then summarize the research results on the vibration control of flexible cantilever structures.In view of the shortcomings in the current researches,we puts forward the research content of this paper.In chapter two,we first describe the mechanism of the piezoelectric effect,and we take the uniform cross-section flexible cantilever beam as the research object and establish its dynamic differential equation under the action of double piezoelectric actuators.In the third chapter,the vibration control of a flexible cantilever beam is studied.Firstly we use modern control theory to obtain the state space equation of the vibration control system,where we use state and state derivative with time delays to control the system vibrations.Secondly,the control law design of the delayed system is derived through the idea of segmentation and equivalence.And lastly it is realized of two sets of systems with different time delays and actuator positions with the same control effect by pole assignment.In the fourth chapter,we firstly derive the optimal state derivative feedback control law of the delayed flexible cantilever system based on the linear quadratic optimal control theory,and then discuss the delay effect of on the optimal control of the system.The results show that,the existence of the time delays does not change the control input required by the system at the current moment,but only delays the action time of the actuator,which finally affects the control effect of the optimal control.Also,increasing the system damping can reduce this effect.Secondly,we ues the integral value judgment method and the Nyquist graphical method to judge the stability of the delayed flexible cantilever with control.It is found that the former method possesses more advantages in the stability judgment.Finally,we apply the integral value discriminant method to investigate the influence of the time delay,the actuator position and the delayed feedback gain on the stabilities of the controled system.When the stability interval of the parameters is obtained,we can investigate the influence of these parameters on the control effect,which is utilized in the parameter optimization.The results show that,a suitable delay can sometimes stabilize an unstable system and achieve good control effects,while the selected parameter worsens the control effect when it is close to its unstable region.