Computing the dynamic response of a periodic structure coupled with a nonlinear junction using the harmonic balance method and Floquet-Bloch modelling

Structures constituted of assembled sub-components are often subject to nonlinear effects due to the presence of joints or contacts, which can induce higher-order harmonics of the source excitation. In the context of periodic structures, these nonlinearities cannot be tackled using the Floquet-Bloch theory in its usual harmonic formulation. This study hence presents an adaptation of the classical Floquet-Bloch theory to address multi-harmonic systems arising from a finite number of localized nonlinearities in periodic waveguides. We adopt the Harmonic Balance Method to recast the governing equations into a nonlinear algebraic system, which can then be solved using numerical continuation algorithms. To demonstrate the validity and benefit of this approach, the predictions derived from the proposed methodology are compared to results obtained through conventional Finite Element analysis. Excellent agreement and a notable reduction in computational cost are observed. This efficient procedure for analysing the nonlinear dynamic response of periodic waveguides opens up new possibilities in the study of damaged composite structures.