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 1st and 2nd order expansions are reported in closed form, while results for the 3rd 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 2nd order initial conditions as compared to 1st order ones. Furthermore, chains given 3rd order initial conditions exhibit smaller variance when compared to those given 1st and 2nd 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.