The collective dynamics of a periodic structure of coupled Duffing-Van Der Pol oscillators is investigated under simultaneous external and parametric excitations. An analytico-computational model based on a perturbation technique, combined with standing wave decomposition and the asymptotic numerical method is developed for a finite number of coupled oscillators. The frequency responses and the basins of attraction are analyzed for the case of small arrays, demonstrating the importance of the multimode solutions and the robustness of their attractors. This model can be exploited to design periodic structure-based smart systems with high performance, by taking advantage of the multi-modes induced by the collective dynamics.

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.

Perfect structural periodicity is disturbed in presence of imperfections. The present paper is based on a realistic modeling of imperfections, using uncertainties, to investigate the robustness of the collective nonlinear dynamics of a periodic coupled pendulums chain. A generic discrete analytical model combining multiple scales method and standing-wave decomposition is proposed. To propagate uncertainties through the established model, the generalized Polynomial Chaos Expansion is used and compared to the Latin Hypercube Sampling method. Effects of uncertainties are investigated on the stability and nonlinearity of two and three coupled pendulums chains. Results prove the satisfying approximation given by the generalized Polynomial Chaos Expansion for a significantly reduced computational time, with respect to the Latin Hypercube Sampling method. Dispersion analysis of the frequency responses show that the nonlinear aspect of the structure is strengthened, the multistability domain is wider, more stable branches are obtained and thus multimode solutions are enhanced. More fine analysis is allowed by the quantification of the variability of the attractors' contributions in the basins of attraction. Results demonstrate benefits of presence of imperfections in such periodic structure. In practice, imperfections can be functionalized to generate energy localization suitable for several engineering applications such as vibration energy harvesting.

The present research work aims to design a passive vibration control based on nonlinear energy pumping. An extended asymptotic approach is introduced based on the invariant manifold approach for the case of 1:1 resonance. It consists in introducing an extended form of Manevitch's complex variables, taking into consideration higher harmonics, enabling the detection of the invariant manifold of the system at fast time scale. At the slow time scale, equilibrium points and singularities are identified analytically in order to predict periodic regimes and strongly modulated responses. The example of a passive shunt loudspeaker using a nonlinear absorber is studied. Unlike classical investigations, the first and third harmonics are taken into consideration. It is demonstrated that the presence of the third harmonic improves the approximations of the results. Different cases are considered, where the obtained analytical results are in good agreement with those obtained via direct numerical integration of the principal system of equations.