Modal interaction and higher harmonic generation in a weakly nonlinear, periodic mass-spring chain

Wave propagation in a nonlinear periodic material is investigated, by considering an infinite chain of two-mass unit cells with cubic stiffness nonlinearity. The chain is analysed using the method of multiple scales, predicting the dispersion shift in the band structure due to nonlinear self-interaction. The solution further reveals modest higher harmonic generation within the limits of the solution approach, proportional to the strength of nonlinearity and energy level in the chain. The possibility for controlling the higher harmonic generation by changing the distribution of the cubic nonlinearity is investigated. The predictions based on the analytical model are verified by numerical simulations, which also explores the limits of the infinite, analytical model.