Finite Element Modeling of Support Symmetry Effects on the Natural Frequencies of Periodic Beams

Truncation resonances are natural frequencies that arise in finite periodic structures and show strong energy localization near the structure boundaries. These resonances commonly occur inside bandgaps, i.e., frequency ranges where wave propagation is suppressed. In periodically bounded beams, truncation resonances have been linked to unit‑cell asymmetry and to how the structure is truncated at its ends. Two periodic boundary‑condition types were examined: pinned and rotationally locked (i.e., sliders). To study the effect of boundary‑condition symmetry on the natural frequencies of both boundary‑condition types, a finite‑element model (FEM) was implemented in MATLAB. The model first varied the number of connected unit cells to identify bandgaps. Each unit cell was defined as a beam segment with prescribed constraints. Then, the spacing (symmetry) of constraints within the unit cell was varied to observe changes in natural frequencies and the appearance or shifting of truncation resonances. Mode shapes were extracted to confirm localization patterns at truncation resonances. All FEM results were validated against SolidWorks frequency analyses.