An Analytical Model for the Vibration of Laminated Beams Including the Effects of Both Shear and Thickness Deformation in the Adhesive Layer

The equations of motion governing the vibration of a beam consisting of two metal layers bonded together with a soft viscoelastic damping adhesive are derived and solved. The adhesive is assumed to undergo both shear and thickness deformations during the vibration of the beam. In previous investigations the thickness deformation has been assumed to have negligible effect on the total damping. However, if the adhesive is very soft, and if at least one of the metal layers is stiff in bending, the thickness deformation in the adhesive can become the dominant damping mechanism. The analysis presented here comprises an extension of the well-known sixth order theory of DiTaranto, Mead, and Markus to include thickness deformation. The equations of motion are derived using Hamilton’s Principle and solutions are obtained by the Ritz method. It is shown that the use of a lightweight constraining layer which is stiff in bending will result in a design which is considerably more damped than a conventional configuration in which the adhesive is undergoing predominant shear deformation.