Dynamics regularization with tree-like structures

The dynamics of a fully nonlinear, tree-structured resonator and its response to a broadband forcing of the branches is examined. It is shown that the broadband forcing yields a transfer of energy between the parts of the spectrum so that the spectrum becomes progressively more narrow-band for each level of the tree-like structure in the direction of the stem. We show that this behavior is in contrast to the response of a linear oscillator, which simply filters out the harmonics away from the resonance. We term such behavior “regularization” and examine its significance for two- and three-dimensional motion using a Lagrangian framework. Key to our analysis is to investigate the dependence of the spectrum of motion, and its narrowing, on the parameters of the tree-like structure, for instance the lengths of different branches. Model predictions are obtained for idealized wind forcing characterized by an airflow that is interrupted at random time intervals. Our numerically-derived results are then compared against the data collected from select analogue laboratory experiments, which confirm the robust nature of the vibration regularization.