Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

Abstract

The nonlinear dynamics of a weakly coupled pendulums chain is investigated under primary resonance. The coupled equations governing the nonlinear vibrations are normalized and transformed into a set of coupled complex algebraic equations using the multiple scales method coupled with standing wave decomposition. A model reduction method is proposed to calculate the dominant dynamics without significant loss of accuracy compared to the full model. The validity of the proposed semi-analytical method is verified, and its role in identifying the type of the solution branches is highlighted. The modal interactions and their effects on the nonlinear dynamics are studied in the frequency domain in order to emphasize the large number of multimode solution branches and the bifurcation topology transfer between the modal intensities. Basins of attraction analysis have been performed, showing that the distribution of the multimodal solution branches generated by all modes collectively increases by increasing the number of coupled pendulums.

Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain

“…The periodicity of the structure is broken by presence of pendulums containing parametric uncertainties which can, for instance, be the pendulum's length as illustrated in Figure 1. In perfect periodicity case, one can refer to works performed in [12,15,28,29] to investigate the collective dynamics of the periodic nonlinear coupled-pendulums chain. Nevertheless, such analyzes are no longer suitable if periodicity is disturbed.…”

“…and The displacement φ n can be expressed as a sum of standing wave modes with slowly varying amplitudes [12,15,28,29]. Taking into account the boundary conditions φ 0 = φ N+1 = 0, the standing wave modes are:…”

“…Subsequently, two possible configurations occur regarding including or not the stability analysis. To solve similar problem, Bitar et al [15,29] applied the Asymptotic Numerical Method (ANM) [53][54][55][56] in graphical interactive software named MANLAB [57] and included stability analysis. The complexity of the study was underlined in presence of multiplicity of stable and unstable solutions.…”

“…The authors show that adding an external excitation increases the existence region of solitons. Bitar et al [28,29] investigated the collective nonlinear dynamics of perfectly periodic coupled pendulum structure under primary resonance using multiple scales and standing-wave decomposition. The authors studied the effects of modal interactions on the nonlinear dynamics.…”

“…Under the hypothesis of small imperfections, the collective dynamics and the localization phenomenon due to the weak coupling of components is preserved. To investigate the collective dynamics of perfectly periodic nonlinear N degrees of freedom systems and control modal interactions between coupled components, previous works [12,15,28,29] proposed discrete analytical models combining the multiple scales method and standing-wave decomposition. The main objective of the present work is to extend these methodologies to the presence of imperfections by proposing a more generic discrete model.…”