Nonlinear wave transmission in a monocoupled elastic periodic system

Abstract

The free dynamics of a monocoupled nonlinear periodic system of infinite extent is analyzed. The system under consideration consists of an infinite number of elastic layers with material nonlinearities, coupled by means of linear stiffnesses. 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. Motions in PZs correspond to traveling waves, i.e., to nonsynchronous oscillations. These motions are analyzed by applying the method of multiple scales in space and time. Numerical computations are carried out to complement the analytical findings. The analytical and numerical methodologies developed in this work can be applied to the study of the free motions of a general class of monocoupled nonlinear periodic systems, and can be extended to investigate motions of nonlinear periodic systems with more than one coupling coordinate.