A Nonlinear Normal Mode Approach for Studying Waves in Nonlinear Monocoupled Periodic Systems

The free dynamics of a mono-coupled layered nonlinear periodic system of infinite extent is analyzed. It is shown that, in analogy to linear theory, the system possesses nonlinear attenuation and propagation zones (AZs and PZs) in the frequency domain. Responses in AZs correspond to standing waves with spatially attenuating, or expanding envelopes, and are synchronous motions of all points of the periodic system. These motions are analytically examined by employing the notion of “nonlinear normal mode,” thereby reducing the response problem to the solution of an infinite set of singular nonlinear partial differential equations. An asymptotic methodology is developed to solve this set. Numerical computations are carried out to complement the analytical findings. The methodology developed in this work can be extended to investigate synchronous attenuating motions of multi-coupled nonlinear periodic systems.