Abstract: The system of coupling structure with controlling is a kind of the dynamic feedback system in which the mechanical load and the actuator/controller interact. The mechanical system and the control system can be considered as a closed system, and the forces actuating the mechanical loads can be taken as inner forces in the system of coupling structure with controlling. Therefore, this method for dealing with the conservative system to simplify greatly modeling and analyses can be used. The relative equilibrium state of motion of the flexible beam under the condition of zero gravity is studied by using energy method. And further, the stability conditions for the relative equilibrium state of the system have been found. If the stability conditions are used as controlling parameters of motion of the system, the kinetic energy of the flexible beam can be converted advantageously into the electromagnetic energy of the actuator/controller. It can not only speed up the braking process of the actuator/controller system but also reduce rapidly the vibration of the flexible structure. The results in the research provide a kind of very simple and effective real-time control method for reducing the vibration of the flexible beam. The method has been verified on many kinds of experimental systems of the flexible beam.

Key words: Flexible beam, Relative equilibrium, Stability condition, Structure and controlling coupling system