Suppression of motion-induced residual vibration of a cantilever beam by input shaping

Input shaping is an effective means for suppressing motion-induced residual vibration of lightly damped structures. Here, to demonstrate the ideas of various input shaping schemes for continuous structures, the model system of a cantilever beam, whose base is to be displaced by a prescribed distance, is considered. The cantilever-beam motion is modeled by the damped Bernoulli-Euler beam equation, and is then decomposed into normal vibration modes. For the particular system set up here, the modal equations of motion are linear and uncoupled, and consequently are integrated analytically. It is then shown that, by completing the cantilever base movement in a series of properly calculated steps (i.e., by shaping the input command of the dynamical system), so as to annihilate the dominant vibration modes through destructive interference, the overall induced vibration of the cantilever can be significantly suppressed. In particular, the 'zero-vibration' (ZV) and 'zero-vibration-and-derivative' (ZVD) input shaping schemes previously proposed for discrete systems are adapted and applied to the continuous beam here. The theoretical results are also supported by experiments.