Dark solitons, modulation instability and breathers in a chain of weakly nonlinear oscillators with cyclic symmetry

Abstract

In the aerospace industry the trend for light-weight structures and the resulting complex dynamic behaviours currently challenge vibration engineers. In many cases, these light-weight structures deviate from linear behaviour, and complex nonlinear phenomena can be expected. We consider a cyclically symmetric system of coupled weakly nonlinear undamped oscillators that could be considered a minimal model for different cyclic and symmetric aerospace structures experiencing large deformations. The focus is on localised vibrations that arise from wave envelope modulation of travelling waves. For the defocussing parameter range of the approximative nonlinear evolution equation, we show the possible existence of dark solitons and discuss their characteristics. For the focussing parameter range, we characterise modulation instability and illustrate corresponding nonlinear breather dynamics. Furthermore, we show that for stronger nonlinearity or randomness in initial conditions, transient breather-type dynamics and decay into bright solitons appear. The findings suggest that significant vibration localisation may arise due to mechanisms of nonlinear modulation dynamics.

Reconstruction of Governing Equations from Vibration Measurements for Geometrically Nonlinear Systems

“…Notice that for all panels the excitation frequency is = 12.3 Hz and the base acceleration amplitude = [1.93,1.52,1.37,1.43] m/s 2 respectively for panels (a,b,c,d). Those measurements clearly show that nonlinear localization and solution multiplicity may appear in nonlinear systems experiencing hardening type stiffness nonlinearities, as it was shown in [32,33], and in nonlinearly damped structures [34,35]. Next, SINDy is employed to identify the governing equations of the experimental system through sparse regression.…”

Multistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators

“…qualitative changes in system dynamics (see e.g. [19][20][21][22][23][24][25][26][27][28]). Nonlinear localization has been shown to appear not only in conservative systems [19] but also in friction-excited chains of weakly coupled oscillators [20,21], where the authors showed that the localization phenomenon is strongly related to the bistable behaviour of the single oscillator in the chain.…”

Predicting ocean rogue waves from point measurements: An experimental study for unidirectional waves

“…Although rare (approximately 3 waves per day in a single point measurement, using linear theory and an average wave period of 10 seconds) these waves can have dramatic effects on ships and other ocean structures [6,7]. Therefore, predicting such extreme events is an important challenging topic in the field of ocean engineering, as well as other fields of wave physics including plasma [8], solids [9] and optics [10,11,12]. In addition, from a mathematical viewpoint the short-term prediction problem of extreme events in nonlinear waves presents particular interest due to the stochastic character of water waves but also the inherent complexity of the governing equations.…”