The present paper is concerned with displacement tracking of a smart cantilever beam. The beam is subjected to a rigid body motion and to external forces. Additionally, a distributed actuation by piezoelectric layers is considered. The scope of the present work is to design the piezoelectric actuation such that the displacements of the beam relative to the rigid body motion follow a prescribed field of trajectories, despite the presence of the external forces and the inertia forces induced by the rigid body motion. We study slender beams, such that the Bernoulli-Euler theory of beams is applicable. The effect of geometric stiffening is taken into account according to v. Karman. In a first step, the problem of displacement tracking is analytically solved in the framework of the latter beam theory. The problem of tracking zero displacements, i.e. the shape control problem, is included as a special case. The practical relevance of the displacement tracking solution is afterwards validated by means of Finite Element computations for a three-layer beam made of two PZT actuation layers and an aluminum substrate layer. The example beam is subjected to a resonant transverse rigid body translation and is assumed to fall in its own weight initially. A harmonic vibration with limited amplitude about a static deflection is tracked. The structure is modeled by means of plane-stress Finite Elements, taking into account electromechanical coupling, where the electric potential resulting from the beam analysis is applied to the electrodes of the PZT layers. In this numerical study, the goal of displacement tracking is reached with a high accuracy, despite the piezoelectric actuation has been designed in the framework of beam theory, while the numerical computations were performed using a refined FE-model. This gives excellent evidence for the appropriateness of the proposed method.
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- Tracking of Displacements in Smart Elastic Beams Subjected to Rigid Body Motions
- Springer Netherlands
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