Modal properties of fractal trees as recursive analytical solutions

This study analyzes the modal properties of two-dimensional fractal trees under the assumptions of linear-elastic vibrations and axially inextensible branches. The examined fractal trees are of a generalized sympodial architecture with possibly non-zero and non -uniform branching angles. The study first presents the construction of tree geometry via iterated functions and then formulates the trees' dynamic matrices from a renormalization viewpoint. The study proposes a novel recursive analytical solution to capture the trees' modal properties. The proposed solution perceives the modal analysis of the fractal trees as an equivalent static analysis of trees with a modified configuration. These "modified"trees are made from the original trees but with all branches connected to the ground via negative stiffness springs. The analysis shows that identifying the lateral trunk displacements of the modified trees (under a static unit force) is associated with the recursive construction of a new family of functions that are introduced in the study, called the auxiliary P-functions. The results reveal that the modal frequencies of the original fractal trees are given by the roots of all auxiliary P-functions. Crucially, these auxiliary P-functions allow the graphical construction of the trees' mode shapes. Illustrative examples demonstrate how the proposed recursive analytical solutions can be used to determine the trees' modal properties. Lastly, the study concludes that the repetition of branches, even without geometrical symmetry, is a necessary condition for the existence of a unique vibration mitigation mechanism in fractal trees, namely the self-similar modes.