Harmonic balance formulation for nonlinear wave-scattering clusters

Nonlinear oscillators can be used as a scattering mechanism to passively manipulate propagating elastic waves in an energy-dependent manner. However, the interaction of incident elastic waves with clusters of discrete nonlinear oscillators introduces considerable modeling challenges that must be overcome to leverage this passive scattering mechanism. This paper proposes a harmonic balance (HB) approach based on the theory of multiple scattering to solve the nonlinear scattering problem. Specifically, we compute periodic solutions for finite arrays of nonlinear oscillators coupled to a linear elastic continuum to resolve the nonlinear scattering wavefields as well as their stability properties. Additionally, we show that the problem can be recast into an equivalent set of nonlinear oscillators that are coupled through the Green's function of the elastic continuum. Moreover, the formulation is general in the sense that it is applicable to a general class of linear elastic continua with a known Green's function. We demonstrate our HB approach with an infinite Euler–Bernoulli beam example to demonstrate the passive tunability of the scattering array's frequency response with respect to energy and other system parameters, and we confirm the HB beam solutions with direct finite element simulations. Furthermore, we highlight the ubiquity of this approach by studying a nonlinear scattering array of oscillators on a infinite plate and show the sensitivity of the HB solutions to configuration symmetry by comparing the nonlinear solutions of perfectly symmetric to perturbed scattering arrays. The efficacy of the discussed methodology for a broad array of nonlinear scattering problems is thus emphasized.