The frequency and forms of natural vibrations of a hinged beam-strip fabricated from a composite material with small-scale curved structures have been investigated using the plate theory in the framework of the Timoshenko hypothesis. The mechanical relationships of the composites are described by the Akbarov and Gut continuum theory. Using the corresponding variational principle, a method of solution of the considered problem is presented. Specific numerical investigations have been carried out for the case when the strip material consists of alternating equidistantly located and curved isotropic layers. The effect of existence of the curving in the beam-strip structure on its frequencies and forms of natural vibrations is studied. Cases of periodic and local curvings in the beam-strip material structure are considered separately.