VIBRATION CONTROL OF A FLEXIBLE BEAM USING A ROTATIONAL INTERNAL RESONANCE CONTROLLER .1. THEORETICAL DEVELOPMENT AND ANALYSIS

In this paper, an unconventional technique to control the vibrations of a cantilevered flexible beam, based on a non-linear dynamics phenomenon known as internal resonance, is proposed. The controller consists of a DC motor, itself a part of a simple regulated feedback system, with a rigid beam/tip mass configuration attached to the motor shaft. The addition of the controller to the tip of the flexible beam introduces quadratic, dynamic non-linearities into an otherwise linear system. Under the proper circumstances, these non-linearities can be used to generate a coupling effect between the modes of vibration of the system. An internally resonant state exists if the equations of motion are characterized by frequency-amplitude interactions and the first two natural frequencies of the linear portion of the non-linear equations of motion are commensurable or nearly commensurable. Once a resonance condition is established, a transfer of energy transpires between the modes of vibration. Thus, due to modal coupling, energy is, in effect, transferred from the flexible beam to the secondary beam, where it is dissipated through velocity feedback of the motor. Theoretical analysis predicts that the planar oscillations of a cantilevered beam displaced at its tip a distance equal to 18 percent of its length can be reduced to a relatively small amplitude in approximately four cycles. This controller has proven to be most effective in controlling large amplitude, low frequency oscillations which are typical for large flexible structures. © 1993 Academic Press Limited.