Dynamic shape control of flexural beam vibrations: An experimental set-up

The topic of the present contribution is an experimental verification of the active control of flexural vibrations of smart beams. The spatial distribution of the piezoelectric actuator is determined in such a way that deformations induced by assigned forces with a given spacewise distribution and an arbitrary but known time-evolution are exactly eliminated by the piezoelectric actuation. In the present paper, the theoretical solution of this dynamic shape control problem is first derived from an electromechanically coupled theory in a three-dimensional setting, where we make use of the theorem of work expended, and from Graffi's theorem. This more general formulation is specialized to the case of beams, where the kinematic hypothesis of Bernoulli-Euler and a uni-axial stress state are assumed, and the direct piezoelectric effect is neglected. We thus re-derive some results for beams published by our group in earlier contributions. It has been found that if the piezoelectric actuator shape-function is chosen as the spanwise distribution of the quasi-static bending moment due to assigned transverse forces, and if additionally the time-evolution of the applied electrical potential difference is chosen to be identical to the negative time-evolution of the assigned forces, the beam deflections due to these forces are exactly eliminated by the piezoelectric actuation. In the present paper, the validity of this theoretical solution is studied in an experimental set-up. As a result of the performed experiments, the elimination of force-induced vibrations of smart beams by shaped piezoelectric actuators is demonstrated for various time-evolutions of exciting single forces. The obtained experimental results give evidence for the validity of the presented theoretical solution of the dynamic shape control problem.