Keywords: sandwich, shell, electroactive, finite element, vibrations, piezoelectricity, viscoelasticity, active control, modal damping.

Sandwich panels and beams with a viscoelastic core layer are used in many applications for vibration and noise control thanks to their superior capability in energy absorption. Recent developments in adaptive structures and their potential industrial applications have resulted in significant improvements in vibration modeling and vibration control problems.

In this paper, a shell finite element is proposed for piezoelectric and viscoelastic sandwich structures, in which a thin viscoelastic layer is sandwiched between identical elastic layers with thin piezoelectric layers bonded on their top and bottom surfaces to act as distributed sensor and actuator. The sandwich finite element is obtained by assembling five elements throughout the thickness of the sandwich structure, more exactly piezoelectric/elastic/viscoelastic/elastic/piezoelectric layers are considered. The theoritical dynamic formulation is first derived from D'Alembert variational principle which includes the total structural potential energy and the electrical potential energy of the piezoelectric material, involving both mechanical and electrical variables. The upper and the lower layers play roles of sensor and actuator respectively and are connected via some feedback control laws. The shear deformation results from the difference between the in-plane displacements of the elastic layers. Using some usual assumptions [1], [2] and [3] for sandwich, and considering that for displacement there is an exact continuity between layers; the displacement field in any layers will be deduced from the displacement field in the elastic layers.

Assuming a linear strain field through each layer and that the transverse displacement and the rotations are the same in the elastic layers and piezoelectric layers; the number of degrees of freedom will be reduced to eight mechanical ones. In modeling of the electrical quantities, we assumed a linear electric potential field in the two piezoelectric layer through the thickness direction, also two electrical degrees of freedom per node will be necessary to describe the electrical condition on a node. The electric and mechanical quantities have been coupled through the constitutive equations then electrical terms will be condensed in respect to the control algorithm so that one reduces to eight the number of degrees of freedom (DOF) per node. Direct proportional feedback and velocity feedback algorithms are used to actively control the dynamic response of the smart structure. Based on the formulation, a finite-element code has been generated and the code has been validated by comparison with a commercial code (ABAQUS) and then with analytical results [4] in some specific cases.

We considered first a proportional control and we used the modal strain energy method (MSEM) coupled with a 2D finite element (FE) performed on the ABAQUS standard code to obtain the first modal properties (flexural frequency and associated damping ratios) from the sandwich beam with different values of the viscoelastic shear modulus. The results show that the new shell element gives a good agreement with the FE model on ABAQUS code.

Considering both proportional and velocity feedback control we used an analytical model [4] and compared the numerical results, for different values of viscoelastic loss factor and different values of the control law gain, with our numerical shell results. In all the different studied cases we observe good agreement between analytical and numerical results.

This study presented a shell finite element for the analysis of modal damping values of hybrid controlled sandwiches. The obtained solutions for proportionnal and velocity feedback control law are in adequation with existing analytical models and prove its validity.

1

P. Cupial and J. Niziol, "Vibration and damping analysis of three-layered composite plate with viscoelastic mid-layer", Journal of Sound and Vibration, 183(1):99-114, 1995. doi:10.1006/jsvi.1995.0241

2

B.A. Ma and J.F. He, "Finite element analysis of viscoelastically damped sandwich plates", Journal of Sound and Vibration, 52:107-123, 1992. doi:10.1016/0022-460X(92)90068-9

3

D.K. Rao, "Frequency and loss factor of sandwich beams under various boundary conditions", Journal of Mechanical Engineering Science, 20(5):271-282, 1978. doi:10.1243/JMES_JOUR_1978_020_047_02

4

L. Duigou, H. Boudaoud, E.M. Daya, S. Belouettar and M. Potier-Ferry, "Equivalent stiffness and damping of sandwich viscoelastic and piezoelectric beams submitted to active control", To be submitted.

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