Vibratory Bending of Damped Laminated Plates

Abstract

The natural frequencies and associated composite loss factor have been determined for a finite-length laminated plate having alternate elastic and viscoelastic layers. Partial differential equations in terms of the variables of the plate are derived and, with the loading equation for a freely vibrating plate, a set of simultaneous partial differential equations is formed. Of two solutions considered the first is general and the second satisfies the boundary condition for a simply supported plate. In both cases, the resulting algebraic simultaneous equations are complex since the shear modulus of the viscoelastic material is a complex expression. In the first case, the expressions could not be solved directly since the value of the eigenvalues depended upon the boundary conditions, whereas the eigenvalues for the simply supported plate could be easily chosen. The simply supported case is solved and the results plotted for specific dimensionless parameters.

Dynamic effective thickness in laminated-glass beams and plates

“…Ross, Ungar, and Kerwin, the first to study the flexural vibration of a sandwich configuration [18,19], proposed an effective complex flexural stiffness which can be used to determine the modal parameters of a sandwich beam using the equations and the wavenumbers corresponding to an Euler-Bernoulli beam. DiTaranto [20,21] and Mead and Markus [22,23] demonstrated that the flexural motion of a sandwich beam is governed by a sixth-order linear homogeneous differential equation. Rao derived a similar equation of motion using Hamilton's principle [24].…”

“…Thus, the behavior of the interlayer is modeled by means of the shear modulus in the time domain or in the frequency domain [5][6][17][18][19][20][21][22][23][24][25][26][27][28].…”

“…• The interlayer exhibits a linear-viscoelastic behavior [18,19,20,21,22,23,24] described by the shear modulus 9 or 9 . This implies that all the variables considered in the model (stiffness, deflections, stresses, effective thicknesses, etc.)…”

“…Unlike the mass and stiffness characteristics of a structural systems, damping does not relate to a single physical phenomenon and there are as many damping mechanisms as there are modes converting mechanical energy into heat [48]. In laminated-glass elements, the material damping contribution comes mainly from the interlayer and the contribution of glass to damping is usually neglected in the analytical models [18][19][20][21][22][23][24]. However, this contribution also depends on the material used for the interlayer.…”

Natural frequencies and damping ratios of multi-layered laminated glass beams using a dynamic effective thickness

“…They assumed that the wave motion in a constrained layer configuration can be described by a fourth-order differential equation. Ditaranto [23,24] and Mead and…”