This paper is concerned with testing and validating tessellated continuum mechanics (TCM) through holey-structure vibration case studies. There presently exists no means of performing vibration analysis of holey structures in the framework of classical continuum mechanics. TCM is as a new approach that introduces an equivalent continuous model on a tessellated continuum that exactly replicates the physics of a corresponding discontinuous pre-fractal holey structure. In this study pre-fractal beams from the Cantor-dust set (E3) are considered as illustrative examples of holey pre-fractal structures. The results of the study confirm the validity of the approach for classical-beam structures subject to forced vibration under clamped-free boundary conditions.