Waveform Invariance in Nonlinear Periodic Systems Using Higher-Order Multiple Scales

This paper carries-out a higher-order, multiple scales perturbation analysis on nonlinear monoatomic and diatomic chains with the intent of predicting invariant waveforms. The chains incorporate linear, quadratic, and cubic force-displacement relationships, and linear dampers. Multi-harmonic results for 1^st and 2^nd order expansions are reported in closed form, while results for the 3^rd order are computed numerically on a case-by-case basis, thus avoiding difficulties associated with large symbolic expressions. Dimensionless parameters are introduced which characterize the amplitude-dependent nonlinear nature of a given chain. Interpretation of the perturbation solutions suggests that the nonlinear chains support certain waveforms which propagate invariantly; i.e., the spectral content does not change significantly over time and space. Numerical simulations confirm this finding using initial conditions corresponding to a specific order of the perturbation solution, and subsequent FFT’s of the response track the growth (or decay) of spatial harmonic content. A variance parameter computes mean fluctuation of the harmonics about their initial values. For a variety of parameter sets, the numerical studies confirm that spectral variance reduces when waves receive 2^nd order initial conditions as compared to 1^st order ones. Furthermore, chains given 3^rd order initial conditions exhibit smaller variance when compared to those given 1^st and 2^nd order ones. The studies’ results suggest that introducing higher-order multiple scales perturbation analysis captures long-term, non-localized invariant waves (or cnoidal waves), which have the potential for propagating coherent information over long distances.