Keywords: vibration control, Bernoulli-Euler beam, rigid-body motion, piezoelectric sensing and actuation, piezoelectric patches, shape control.

This paper concerns the control of flexural vibrations in smart beams subject to external and inertial forces. For example, a flexible beam with integrated piezoelectric actuators is considered. The equations of motion are derived within the framework of Bernoulli-Euler beam theory considering arbitrary large rigid-body motions with super-imposed small elastic deformations, [1]. The piezoelectric actuation is taken into account in the beam starting from a continuum mechanics interpretation of the Euler elastica theory, using stress-strain relations at the local level and integration over the cross section of the beam. Piezoelectric actuation strains are considered using an eigenstrain formulation such that the results can be easily applied to other kinds of actuation effects.

From the equations of motion the shape control solution is derived, i.e. the necessary distribution of actuation strains is computed in order to completely compensate the flexible vibrations of the beam. This shape control solution represents a feed-forward control strategy.

In the next step, it is assumed that in practical applications masses, geometrical parameters and the excitations are not known exactly, and that the distribution of actuation strains is approximated using a limited number of piezoelectric patches. It is thus necessary to add a feedback control strategy in order to compensate the remaining flexible vibrations. For this sake, strain gauges are used as sensors to measure the curvature in the beam, and a P-control algorithm is introduced.

Finally, experimental results are given. Using the numerical simulation model presented in [2], parameter studies have been performed to find a suitable configuration. An aluminium beam of rectangular cross-section with eighteen piezoelectric patches has been built up, and the control algorithm has been implemented within a dSpace environment. In a first step, the beam is clamped at one end and subject to concentrated transversal forces. The control algorithm is designed such, that the deflection of the beam vanishes in selected points. As a second example, the beam is fixed on a rotating shaft and subject to inertial forces. The results show that with the help of the implemented control strategy it is possible to significantly reduce the flexural vibrations of the considered smart beam.

1

C. Zehetner, H. Irschik, "Displacement compensation of beam vibrations caused by rigid-body motions", Smart Materials and Structures, 14, 862-868, 2005. doi:10.1088/0964-1726/14/4/046

2

C. Zehetner, J. Gerstmayr, "Control of flexible vibrations in a two-link robot by piezoelectric actuation", in I. Troch, F. Breitenecker, (Editors), "Proceedings of the 6th Vienna Converence on Mathematical Modelling", Vienna, Austria, 988-998, 2009.

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