Sympodial tree-like structures: from small to large-amplitude vibrations

This study deals with tree-like structures that mimic a trunk with either first-order branches or both first and second-order branches of a sympodial tree. The corresponding mechanical model comprises physical pendula coupled with torsional springs and viscous dampers. Natural frequencies and modal shapes are obtained analytically and the effects of a branching angle and a stiffness ratio on their change is analysed. Then, the trunk is harmonically excited and the corresponding structural response is investigated, both for small undamped and damped vibrations, focusing on the concept of dynamic absorbers and the attenuation of the amplitude of the trunk and first-order branches as this attenuation is beneficial for practical engineering applications. The corresponding frequencies at which these amplitudes are equal to zero or in resonance are determined. Finally, free large-amplitude vibrations are examined. Conditions for internal resonances in the structure with first-order branching are determined analytically by using the method of multiple scales. Frequency spectra of large-amplitude vibrations are obtained numerically focusing on the influence of the branching angle on their content. Biomimetic potentials and advantages of designing these coupled branched structures are discussed through the whole study via the influence of the stiffness ratio of the springs, the branching angle and the hierarchy of branches on their response.