The dynamic properties of elastic and viscoelastic periodic composites involving inner resonance have beenstudied previously by using multiple scale expansions. For the case of composites whose constituents present a large contrast ofelasticity tensor, it was shown that inner resonance induces cut-off frequencies. Multiple scale expansions show also that thecut-off frequencies correspond to the eigen-frequencies of the domain of inner inclusions constrained by fixed boundaryconditions. A recent development [10] has shown that the case of composite inclusions containing a massive inner inclusionsurrounded by a soft material has cut-off frequencies which can be obtained by studying the static elastic response of the massspringsystem constituted by the composite inclusion. The comparison is made between the case of uniform and compositeinclusions. Then, a full solution is given for the lowest cut-off frequency (or resonance frequency) in the case of inclusionsmade of composite fibers.