Theory of Vibratory Bending for Elastic and Viscoelastic Layered Finite-Length Beams

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.