The vibrational characteristics, natural frequencies, and associated composite loss factor of a finite-length laminated beam having alternate elastic and viscoelastic layers, are investigated. An auxiliary equation which accounts for the effect of the viscoelastic layers is derived. The use of this equation in conjunction with the ordinary bending equations encountered for homogeneous beams, allows one to solve static and dynamic bending problems for laminated beams in the same manner as for homogeneous beams. The resulting equations are complex expressions since the shear modulus of the viscoelastic material is a complex quantity. The use of the auxiliary equation in conjunction with the loading equation for a freely vibrating beam yields a sixth-order, complex, homogeneous differential equation. The solution of this equation, subject to satisfying the boundary conditions, yields the desired natural frequencies and associated composite loss-factors.

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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.

The governing differential equation and natural and rigid boundary conditions have been obtained for a three-layer beam having a middle layer which is primarily shear carrying, using a variational method. Results have been obtained for a fixed-fixed and symply supported beam having a uniform load. For beams having these boundary and loading conditions, curves are presented showing the effect of varying the physical and geometrical properties of the laminated beam on the maximum displacement and stress.