Model study and active control of a rotating flexible cantilever beam

Abstract

For a dynamic system of a rotating flexible cantilever beam, the traditional model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the beam are uncoupled. This traditional hybrid coordinate model is referred as the zero-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of a flexible cantilever beam with a high rotational speed. In this paper, a first-order approximation coupling model is presented to analyze the dynamics of rotating flexible beam system, which is based on the Hamilton theory and the finite element discretization method. The proposed model for the system considers the second-order coupling quantity of the axial displacement caused by the transverse displacement of the beam. The dynamic characteristics of the rotating beam system when using the zero-order approximation coupling model are compared with those when using the first-order approximation coupling models through numerical simulations. In addition, the applicability of the two dynamic models for control design are studied by using the classical optimal control method. Simulation and comparison studies show that, for the case without control for the system, there exists big difference between the result using the zero-order approximation coupling model and that using the first-order approximation coupling model even for the case of small angular velocity of the system. The larger is the angular velocity, the bigger is the difference. Vibration frequency of the beam by using the first-order approximation coupling model is higher than that by using the zero-order approximation coupling model. When the angular velocity of the system is close to or is larger than the fundamental frequency of the beam without rotation motion, the zero-order approximation coupling results in a wrong result, while the first-order approximation coupling model is valid. For the case with control for the system, the applicability of the zero-order approximation coupling model can be much broadened. The critical angular velocity of the system for validity of the zero-order approximation coupling model is much larger than that without control for the system. The first-order approximation coupling model is available not only for the case of small angular velocity but also for the case of large angular velocity of the system, and is applicable to the cases with or without control for the system.