Spectro-spatial analysis of van der Pol-type phononic crystals

Abstract

Abstract The application of phononic chains as metamaterials demonstrates their remarkable capability to manipulate the propagation of waves. These periodic structures yield frequency-dependent behavior of material comprising characteristics with many possible engineering applications. In this paper, we investigate the weak and general nonlinear behaviors of the van der Pol-type damped phononic chains. The analysis of wave propagation is initially conducted for a one-dimensional structure, and subsequently, is extended to consider the wave motion through two-dimensional and three-dimensional lattices. Results are obtained using the method of multiple scales and a Spectro-spatial analysis by employing the numerical method of the 4th-order Runge–Kutta. A new phase-diagram relation within the chain’s unit cell is also introduced aiming to enhance the numerical findings. Our results indicate that in the weakly nonlinear regime, the van der Pol-type damping closely follows the linear dispersion curve, regardless of the initial amplitude. This suggests a symmetry between energy pumping and dissipation modes, where hardening and softening behaviors align with linear characteristics of common damping mechanisms, such as viscous damping. Additionally, the formulation demonstrates the existence of limit-cycle stability in the motion of each mass. For the general damped system, it is observed that a special frequency exists where the system converges, for all wave numbers similar to the synchronization effect. Hence, the motion and the frequency of all masses are synced. Additionally, non-reciprocal wave propagation is observed, resulting in a bandgap structure with a symmetry breaking occurring near the limit cycle. These results are promising in the fields of wave emitters, wave filters, and signal encryption.