Modal properties of fractal sympodial trees: insights and analytical solutions using a group tree modeling approach

This study examines the intricate relationship between the hierarchical architecture of fractal trees and their dynamic properties, namely modal frequencies and mode shapes. It focuses on idealized sympodial trees under small, linear elastic lateral vibrations. This study proposes a group tree modeling approach that condenses the branches into simple structures, reducing the complexity of the model as appropriate. With the aid of a renor-malization technique, the analysis proves that the sympodial tree is characterized by two families of vibration modes, i.e., the "emerging" and the "self-similar" modes. It further shows that the sympodial tree inherits all modes from the previous tree order (i.e., frac-tal level). Crucially, these inherited modes are self-similar, and beneficial in mitigating the vibration of the trunk by localizing the vibration of the tree at higher order branches. The study derives recursive closed-form solutions for both self-similar modal frequencies and mode shapes. Further, the analysis of sympodial trees with infinite tree order shows that their modal frequencies converge to stable values as the tree order increases. Accordingly, the study proposes analytical formulas to estimate these frequency limits from the fractal parameters of the sympodial trees. Results reveal that the natural frequency of a sympodial tree with infinite tree order and infinite branching number approaches asymptotically to that of a branchless tree. The proposed approach is then validated with numerical simula-tions and compared with independent studies. These findings are promising towards new bio-mimicking design concepts for civil engineering structures with improved dynamic be-haviour.(c) 2022 Elsevier Inc. All rights reserved.