Chapter 9
On Localised Modes in Bio-inspired
Hierarchically Organised Oscillatory
Chains
Ivana Kovacic, Dragi Radomirovic, and Miodrag Zukovic
Abstract This study is concerned with bio-inspired hierarchically organised oscillatory chains from the viewpoint of a localisation phenomenon, when only certain
parts of the chain oscillate. The chain consists of block masses attached mutually via
tension-extension Hookean springs. The first- and second-order hierarchy are considered to determine their modes and modal frequencies, with the focus on localised
modes. Then, a chain with an arbitrary number of masses is dealt with. A theorem
is presented that defines the number of its localised modes in terms of the order of
hierarchy.
Keywords Spring-mass system · Coupling · Localisation phenomenon
9.1 Introduction
Mechanical aspects of trees’ static and dynamic behaviour and the role of branches
have been intriguing researchers not only for decades, but for centuries. Even Galileo
[1] noted ‘that an oak two hundred cubits high would not be able to sustain its own
branches if they were distributed as in a tree of ordinary size.’ Branches of trees
belong to slender structures, but cope reasonably well both with small and largeamplitude oscillations caused by a variety of excitations, unlike man-made slender
structures, which are not so robust. Understanding the underlying physical principle
of their behaviour is of interest both for plant science [2] and also for biomimetics
when they are modelled as coupled oscillators [3], as they can be beneficially utilised
in many engineering applications [3, 4].
Of interest for this work are mechanical models of branched trees as biological
oscillators. These mechanical models created and examined so far have been either
I. Kovacic (B) · M. Zukovic
Faculty of Technical Sciences, Centre of Excellence for Vibro-Acoustic Systems and Signal
Processing, University of Novi Sad, Novi Sad, Serbia
e-mail: ivanakov@uns.ac.rs
D. Radomirovic
Faculty of Agriculture, University of Novi Sad, Novi Sad, Serbia
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_9
153
On Localised Modes in Bio-inspired
Hierarchically Organised Oscillatory
Chains
Ivana Kovacic, Dragi Radomirovic, and Miodrag Zukovic
Abstract This study is concerned with bio-inspired hierarchically organised oscillatory chains from the viewpoint of a localisation phenomenon, when only certain
parts of the chain oscillate. The chain consists of block masses attached mutually via
tension-extension Hookean springs. The first- and second-order hierarchy are considered to determine their modes and modal frequencies, with the focus on localised
modes. Then, a chain with an arbitrary number of masses is dealt with. A theorem
is presented that defines the number of its localised modes in terms of the order of
hierarchy.
Keywords Spring-mass system · Coupling · Localisation phenomenon
9.1 Introduction
Mechanical aspects of trees’ static and dynamic behaviour and the role of branches
have been intriguing researchers not only for decades, but for centuries. Even Galileo
[1] noted ‘that an oak two hundred cubits high would not be able to sustain its own
branches if they were distributed as in a tree of ordinary size.’ Branches of trees
belong to slender structures, but cope reasonably well both with small and largeamplitude oscillations caused by a variety of excitations, unlike man-made slender
structures, which are not so robust. Understanding the underlying physical principle
of their behaviour is of interest both for plant science [2] and also for biomimetics
when they are modelled as coupled oscillators [3], as they can be beneficially utilised
in many engineering applications [3, 4].
Of interest for this work are mechanical models of branched trees as biological
oscillators. These mechanical models created and examined so far have been either
I. Kovacic (B) · M. Zukovic
Faculty of Technical Sciences, Centre of Excellence for Vibro-Acoustic Systems and Signal
Processing, University of Novi Sad, Novi Sad, Serbia
e-mail: ivanakov@uns.ac.rs
D. Radomirovic
Faculty of Agriculture, University of Novi Sad, Novi Sad, Serbia
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_9
153