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I. Kovacic et al.
Using Theorem for the case N = 3, one can calculate that 11 localised modes will
appear. Four modes are localised in the highest-order of hierarchy. There are two
groups of two localised in the penultimate and ultimate order of hierarchy and, consequently, two doubled frequencies. In addition, there are also three modes appearing
at a different frequency. It should be pointed out that this chain with the third-order
hierarchy is characterised by one quadrilateral and two double modal frequencies.
9.5 Conclusions
A chain of bio-inspired hierarchically organised masses on linear springs has been
considered, where the trunk is modelled as a main mass, while branches are symmetrically attached to the masses in pairs via linear springs. The mass and stiffness properties are reduced in each order of hierarchy. This model is characterised by localised
modes, when only certain parts of the chain oscillate while the main mass stays at
rest. So, in the case of a trunk and the first-order branches, only one localised mode
exists: the main mass does not oscillate, while the branches do. In the model with
second-order branches, there are two modes localised in the second-order branches
and two in all branches. As the order of hierarchy increases, the number of localised
modes increases, too. One potential applications of these findings is the case when
the trunk is excited at the frequency tuned to the one that corresponds to a localised
modes Although excited, the trunk will not oscillate, as this case corresponds to the
concept of dynamic absorbers or tuned mass-dampers [3, 7].
Acknowledgements The authors acknowledge support of the Ministry of Education and Science
of Serbia (the project no. 451-03-68/2020-14/200156).
References
1. G. Galilei, Dialogues Concerning Two New Sciences, 1638 (trans. H. Crew and A. de Salvio)
(Dover Publications, Mineola, New York, USA, 1954)
2. E. De Langre, Plant vibrations at all scales: a review. J. Exp. Bot. 70, 3521–3531 (2019)
3. I. Kovacic, M. Zukovic, D. Radomirovic, Sympodial tree-like structures: from small to largeamplitude vibrations. Bioinspir. Biomim. 13, Paper No 026002 (20 pp.) (2018)
4. I. Kovacic, M. Zukovic, Coupled purely nonlinear oscillators: normal modes and exact solutions
for free and forced responses. Nonlinear Dyn. 87, 713–726 (2017)
5. K. James, Dynamic loading of trees. J. Arboric. 29, 165–171 (2003)
6. K.R. James, A Dynamic Structural Analysis of Trees Subject to Wind Loading. PhD thesis,
University of Melbourne, 2010
7. I. Kovacic, D. Radomirovic, Mechanical Vibrations: Fundamentals with Solved Examples
(Willey, 2017)
8. I. Kovacic, M. Zukovic, D. Radomirovic, On a localization phenomenon in two types of bioinspired hierarchically organised oscillatory systems. Nonlinear Dyn. 99, 679–706 (2020)
I. Kovacic et al.
Using Theorem for the case N = 3, one can calculate that 11 localised modes will
appear. Four modes are localised in the highest-order of hierarchy. There are two
groups of two localised in the penultimate and ultimate order of hierarchy and, consequently, two doubled frequencies. In addition, there are also three modes appearing
at a different frequency. It should be pointed out that this chain with the third-order
hierarchy is characterised by one quadrilateral and two double modal frequencies.
9.5 Conclusions
A chain of bio-inspired hierarchically organised masses on linear springs has been
considered, where the trunk is modelled as a main mass, while branches are symmetrically attached to the masses in pairs via linear springs. The mass and stiffness properties are reduced in each order of hierarchy. This model is characterised by localised
modes, when only certain parts of the chain oscillate while the main mass stays at
rest. So, in the case of a trunk and the first-order branches, only one localised mode
exists: the main mass does not oscillate, while the branches do. In the model with
second-order branches, there are two modes localised in the second-order branches
and two in all branches. As the order of hierarchy increases, the number of localised
modes increases, too. One potential applications of these findings is the case when
the trunk is excited at the frequency tuned to the one that corresponds to a localised
modes Although excited, the trunk will not oscillate, as this case corresponds to the
concept of dynamic absorbers or tuned mass-dampers [3, 7].
Acknowledgements The authors acknowledge support of the Ministry of Education and Science
of Serbia (the project no. 451-03-68/2020-14/200156).
References
1. G. Galilei, Dialogues Concerning Two New Sciences, 1638 (trans. H. Crew and A. de Salvio)
(Dover Publications, Mineola, New York, USA, 1954)
2. E. De Langre, Plant vibrations at all scales: a review. J. Exp. Bot. 70, 3521–3531 (2019)
3. I. Kovacic, M. Zukovic, D. Radomirovic, Sympodial tree-like structures: from small to largeamplitude vibrations. Bioinspir. Biomim. 13, Paper No 026002 (20 pp.) (2018)
4. I. Kovacic, M. Zukovic, Coupled purely nonlinear oscillators: normal modes and exact solutions
for free and forced responses. Nonlinear Dyn. 87, 713–726 (2017)
5. K. James, Dynamic loading of trees. J. Arboric. 29, 165–171 (2003)
6. K.R. James, A Dynamic Structural Analysis of Trees Subject to Wind Loading. PhD thesis,
University of Melbourne, 2010
7. I. Kovacic, D. Radomirovic, Mechanical Vibrations: Fundamentals with Solved Examples
(Willey, 2017)
8. I. Kovacic, M. Zukovic, D. Radomirovic, On a localization phenomenon in two types of bioinspired hierarchically organised oscillatory systems. Nonlinear Dyn. 99, 679–706 (2020)