# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 4 Conclusions  
A generic discrete analytical model combining multiple scales method and standing-wave decomposition was proposed in this paper. The model accounts for parametric uncertainties and permits to investigate the robustness of the collective dynamics of nonlinear coupled pendulums chain. Uncertainties allow a realistic modeling of imperfections, which can disturb the perfect periodicity of real engineering structures. In probabilistic framework, dispersion analyses of the frequency responses and the basins of attraction of two and three coupled pendulums chains were performed. Statistical evaluations of the frequency responses, in both modal and physical coordinates, quantified the variability of the frequency and amplitude intervals of the multistability domain. The complexity of the frequency responses detected in terms of modal interactions and bifurcation topology transfer, in deterministic case, was amplified in presence of uncertainty. The envelopes of the multimode branches are enlarged, and even overlapped, making difficult the detection of the extreme statistics of adjacent branches. This complexity emphasizes the large number of multimode solutions, even for chains of only two and three coupled pendulums, and the modal localization generated by uncertainty. Moreover, the presence of imperfection in a periodic chain widens its multistability domain and strengthens its nonlinear aspect. Particularly, the stability and the nonlinearity of the responses of the pendulums containing uncertainty are strongly enhanced. Furthermore, the robustness analysis of the basins of attraction was investigated through the dispersion analysis of the attractors' contributions. Finer analysis was thus allowed highlighting the complexity of the problem and the modal localization generated on the response of the pendulum containing uncertainty. These advantages can serve as a hint of the important modal localization, expected, in presence of uncertainties, for large number of pendulums. Imperfections can be functionalized to generate energy localization suitable for several engineering applications such as vibration energy harvesting, mass sensing in micro and nanotechnology, energy scavenging or trapping applications.  
The generalized Polynomial Chaos Expansion was compared to the Latin Hypercube Sampling method, considered as reference, for uncertainty propagation. Results proved the satisfying approximations given by the former method for up to 97% of computational time reduction with respect to the later.  
The robustness analysis becomes, both analytically and numerically, more complex and prohibitive when large-size structures and high dispersion levels are considered. Future works will focus on these challenges in order to improve analytical and numerical analyses.  
The robustness of the collective dynamics against the uncertainty of some chosen structural input parameters was discussed in this paper. Extending the study to parametric analysis, with the aim of investigating other possible influential factors with respect to system robustness, is an interesting perspective that will be addressed in future works.  
This project has been performed in cooperation with the Labex ACTION program (contract ANR-11-LABX-01-01).  
This paper is developed by a continuous and synergic collaboration among all authors. All authors read, revised and approved the final manuscript.  
The authors declare no conflict of interest.