Keywords: beam, viscoelastic, piezoelectric, active constrained layer damping, finite element.

In the last decade active constrained layer damping (ACLD) treatments have been applied to structures. Those are hybrid treatments where constraining layers made of active piezoelectric materials are used to cover the viscoelastic passive layers. One of the unique features of piezoelectric materials is that they can serve both as sensors and actuators. If utilized as actuators, and according to an appropriate control law, the active constraining layer can increase the shear deformation of the viscoelastic layer and overcome some of the passive constrained layer damping (PCLD) limitations. The ACLD treatments combine the high capacity of passive viscoelastic materials to dissipate vibrational energy at high frequencies with the active capacity of piezoelectric materials at low frequencies. Therefore, in the same damping treatment, a broader band control is achieved benefiting from the advantages of both passive (simplicity, stability, fail-safe, low-cost) and active (adaptability, high-performance) systems.

Various configurations of active and passive layers have been proposed in an attempt to improve performance. In general so-called hybrid active-passive (or arbitrary ACLD) treatments involving arbitrary arrangements of constraining and passive layers, integrating piezoelectric sensors and actuators, might be utilized. When designing hybrid active-passive treatments it is important to know the configuration of the structure and treatment that gives optimal damping. For simulation the designer needs a model of the system in order to define the optimal locations, thicknesses, configurations, control law, etc.

The aim of this work is the development of a generic analytical model that can account for the hybrid couplings in an accurate and consistent way. It can therefore be seen as an initial step from which different analytical and discretization methods can be used for the solution of arbitrary hybrid active-passive treatments on beams. We start by presenting the structural analytical model of a composite beam with an arbitrary number of layers of elastic, piezoelectric and viscoelastic materials, attached to both surfaces of the beam. The kinematic assumptions, based in a partial layerwise theory, are first presented. Then, the electric model assumptions for the piezoelectric materials which account for a fully coupled electro-mechanical theory are described. The damping behavior of the viscoelastic layers, modeled by the complex modulus approach (CMA), is presented. Hamilton's principle is utilized to derive the equations of motion and electric charge equilibrium. The strong forms of the general analytical model of the composite beam with an arbitrary number of layers are then presented by a set of partial differential equations governing the motion and electric charge equilibrium. A finite element (FE) model solution is presented and a composite beam FE is derived from the weak forms of the analytical model. The FE is one-dimensional, with the nodal degrees of freedom being the axial and transverse displacements and the rotation of the centreline of the host beam, the rotations of the individual layers and the electric potentials of each piezoelectric layer.

In order to validate the FE model measurements were taken on an aluminium beam with a partial ACLD treatment (viscoelastic layer sandwiched between the piezoelectric patch and the base beam). The analysis concerned free-free boundary conditions. The FE model was implemented in MATLABRand three frequency response functions were measured experimentally and evaluated numerically: acceleration per unit force (accelerance), acceleration per unit voltage into the piezoelectric patch and induced voltage per unit force.

The ability of the method to accurately model the hybrid behavior, elastic, viscoelastic and piezoelectric materials, is shown and validated against the experimental results. The analytical formulation can be used for other solution methods and the FE model can be used in the simulation of active control systems in beams with arbitrary ACLD treatments.

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