Recursive modal properties of fractal monopodial trees, from finite to infinite order

This study examines the modal properties of fractal-idealized monopodial trees, which comprise a main trunk with branches grown laterally and axially from it. Analysis via a renormalization technique shows that monopodial trees possess emerging modes, self-similar modes with lateral branching, and self-similar modes with axial branching. The study introduces a recursive analytical approach, which involves the construction of auxiliary $P$-functions to characterize the modal properties. Results reveal that monopodial trees’ modal frequencies are closely spaced because their emerging modes increase exponentially in number. Under the self-similar modes with lateral branching, the trees inherit all modes from their fractal ancestors, while under the self-similar modes with axial branching, they inherit only the self-similar modes from their ancestors. Hence, the main trunk stands still at self-similar modes, and the trees localize vibration at higher-order lateral branches. Therefore, the monopodial branching architecture is advantageous for reducing the vibration of the main trunk. This study also derives analytical formulas for the modal frequencies of trees with infinite order via a group tree modeling approach. Monopodial trees acquire the highest natural frequency when they have a one-to-one lateral-to-axial-branching ratio. The proposed formulas are verified with independent literature results and are shown to be accurate.