Wave analysis of torsional vibration in triangular gyroscopic metamaterial

In the gyroscopic phononic crystal, the gyroscopic structure breaks the time-reversal symmetry by applying a constant torque, analogous to the quantum spin Hall effect, thus endowing the structure with chiral characteristics and enabling it to possess new wave characteristics. The dispersion and wave characteristics of twisted waves in triangular gyroscopic phononic crystals based on a triangular discrete structure is proposed and studied in this paper. The dynamic equations at the nodes are established using the angular momentum theorem, and the dispersion relationship between the angular frequency and wave vector is derived using the Bloch theorem. The transmission coefficient of the finite structure is calculated numerically and the dispersion relationship is verified. The results show that the existence of the gyroscopic torque causes the dispersion surface to gradually separate as the torque parameter increases, and two dispersion surfaces appear with a bandgap. The group velocity, phase velocity, and wave propagation diagram are analyzed for the triangular gyroscopic phononic crystal which shows that combining the group velocity and phase velocity can provide a good prediction of wave propagation in actual structures. And the wave propagation diagram shows that the lower dispersion surface has a more concentrated propagation pattern along the X-type direction, while the upper dispersion surface propagates more evenly in all directions, and the waves near the bandgap frequency decay rapidly. The research results provide theoretical support for precise control of wave characteristics of phononic crystals, and this structure can also be used for dynamic control of the bandgap frequency range.