Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

We study the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with frequency nominally within the Propagation Zone (i.e., propagation occurs in the absence of the damping). The period of the structure consists of a linear stiffness and a weak linear/nonlinear damping. Adapting the transfer matrix method and using harmonic balance for the nonlinear terms, a four-dimensional linear/nonlinear map governing the dynamics is obtained. We analyze this map by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The approximations are validated by comparing with other analytical solutions for the linear case and full numerics for the nonlinear case. Good agreement is obtained. The method of analysis presented here can be extended to more complex structures.