Reduced-order modeling for dispersion analysis of elastic waveguides coupled to arrays of nonlinear systems

This work introduces a reduced-order modeling (ROM) approach for deriving the dispersion relation of elastic waveguides hosting an array of nonlinear resonating systems with multiple degrees of freedom. The ROM is constructed using the invariant manifold method, which captures the effects of nonlinear restoring forces from secondary “slave” modes as functions of the displacement and velocity coordinates of a selected “master” mode. This enables the efficient computation of the transfer function of the resonating system and its coupling with the waveguide to compute the wave dispersion relation. The results demonstrate that incorporating mode interactions through the ROM provides accurate estimates of the behavior of the full nonlinear model, ensuring accurate prediction of the dispersion relation while maintaining computational efficiency. Conversely, neglecting mode interactions would lead to significant errors in both resonator response and waveguide dispersion estimation, potentially resulting in suboptimal metamaterial design. The proposed approach provides a robust framework for designing nonlinear metamaterials with complex resonator configurations.