Multi-parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies

In this paper, we consider periodic waveguides in the shape of a inhomogeneous string or beam partially supported by a uniform elastic Winkler foundation. A multi-parametric analysis is developed to take into account the presence of internal cutoff frequencies and strong contrast of the problem parameters. This leads to asymptotic conditions supporting non-typical quasi-static uniform or, possibly, linear microscale displacement variations over the high-frequency domain. Macroscale governing equations are derived within the framework of the Floquet-Bloch theory as well as using a high-frequency-type homogenization procedure adjusted to a string with variable parameters. It is found that, for the string problem, the associated macroscale equation is the same as that applying to a string resting on a Winkler foundation. Remarkably, for the beam problem, the macroscale behavior is governed by the same equation as for a beam supported by a two-parameter Pasternak foundation.