Modal Analysis of a Toothpick Pattern Structure

Work in the field of vibrations has extensively explored the dynamical behavior of periodic structures, which allow insights into vibration attenuation, noise control, wave guiding, and non-reciprocal wave propagation. In comparison, aperiodic structures have generally received less research attention, despite their potential to realize interesting vibration and wave phenomena, such as mode localization, topological wave pumping, edge states, and corner modes. As such, this research studies novel aperiodic toothpick-pattern structures with varying generational progressions when subjected to vibrations. The toothpick pattern is built following a replication algorithm starting from a single slender beam (i.e., a toothpick), representing the first generation, with each subsequent generation adding new toothpicks to every free end of the structure, creating a unique and complex structure. Previous work investigated the dynamical behavior of aperiodic structures inspired by the Ulam-Warburton cellular automaton. The dynamics of such structures revealed interesting behaviors, such as corner modes, repeated natural frequencies across different generations, and symmetry in the eigenfrequency spectrum. These behaviors have been difficult to categorize for the toothpick pattern even though it follows a similar generational algorithm. Motivated by this, numerical analyses of the toothpick patterns at varying generations have been conducted, emphasizing the evolution of natural frequencies as the structure grows through additional generations. The effective stiffness of such aperiodic structures is also of interest, and correlations between the number of generations and effective stiffness were investigated. Furthermore, the existence of corner modes in toothpick structures through solid continuum and discretized models was explored via numerical models. These corner modes allow the corners to be excited without affecting the rest of the structure. This research offers new insights into the study of aperiodic structures and how algorithm-based aperiodicity affects their vibratory behavior.