Tunable and reconfigurable mechanical transmission-line metamaterials via direct active feedback control

We consider the problem of using active feedback control to create tunable and reconfigurable mechanical metamaterials capable of supporting unconventional wave propagation mechanisms. The nominal system chosen for this analysis is a one-dimensional homogeneous bar having periodically distributed force actuators and subject to longitudinal waves. We note that this system does not make use of local inclusions, instead the actuators are attached directly to the beam and achieve metamaterial properties by means of periodically applied forces. We design control algorithms in order to achieve desired constitutive parameters for the metamaterial in closed loop. In particular, the control system is designed to generate either zero or negative values of the effective properties (i.e. stiffness and mass) to obtain a metamaterial with different dynamic behaviors. Four different regimes of effective properties are considered in this study: single negative or zero, double negative, double zero, double positive. The constitutive parameters achievable via the direct control approach are therefore defined only by the feedback algorithms and are not confined to any particular properties that would otherwise be imposed by local resonators. We model and analyze the system by fractional order transfer functions, which explicitly exhibit the special characteristics of the metamaterial, including the constitutive parameters, dispersion, and transmission and reflection properties. We illustrate the stability and performance of the control system through numerical simulations. Finally we note that, despite the choice of the mechanical system used in this study, the proposed results have general applicability to all those systems described by the second order wave equation in their nominal uncontrolled state.