Local autoparametric resonances in one-dimensional periodic waveguides with inertia amplification

In the nonlinear dynamics of periodic microstructured systems, the propagation of elastic waves in mechanical metamaterials is attracting increasing interest. The paper investigates the free and forced harmonic response of a one-dimensional waveguide with pantographic mass amplification. These systems are characterized by quadratic and cubic nonlinearities and have been demonstrated to develop a rich scenario of amplitude-dependent phenomena, including softening/hardening backbones, essentially nonlinear waveforms, superharmonic depolarization. According to a novel geometric perspective, the periodic pantographic microstructures can be regarded as longitudinal-to-transversal motion transformers. This distinctive kinematic behavior enables the integration of local resonators activated by autoparametric resonances. The mathematical problem is governed by a pair of coupled differential (in time) difference (in space) nonlinear equations, which can be analytically approached trough asymptotic perturbation techniques. By properly tuning internal resonance conditions, the propagation of longitudinal waves can be effectively mitigated by virtue of a saturation mechanism, following the onset of a dynamic bifurcation induced by quadratic intracellular coupling.