Free and forced dynamics of a class of periodic elastic systems

The free and forced motions of ordered and disordered layered systems are analyzed. The structure of propagation and attenuation zones (PZs and AZs) of the finite system depends on two non-dimensional parameters, v and τ. The parameter v is the ratio of the times of wave propagation at phase velocity through each layer, whereas parameter τ denotes the ratio of mechanical impedances of the materials forming the two layers. Systems with finite values of v and large or small values of τ are weakly coupled and possess narrow PZs and wide AZs. Depending on the value of v, a distinction is made between 'degenerate' and 'non-degenerate' PZs. The 'degenerate' PZs are wider than 'non-degenerate' ones. For small or large values of τ, the finite periodic system includes dense 'clusters' of natural frequencies; when forced by a trapezoidal pulse, the maximum compressional force of the first arrival of the stress wave is localized close to the point of application of the excitation. For values of τ of order unity, this localization is eliminated. The effect on the free and forced response of disorder is then investigated. For a sufficiently strong disorder, a natural frequency shifts from a PZ to an AZ of the ordered system. This results in a spatial localization of the corresponding eigenfunction at the disorder. Finally, the transient responses of disordered and ordered systems are computed and compared.