High-fidelity dynamics of piezoelectric covered metamaterial Timoshenko beams using the spectral element method

Piezoelectric metamaterial beams have received enormous research interest for the applications of vibration attenuation and/or energy harvesting in recent years. This paper presents a generic modelling approach for predicting the high-frequency dynamics of piezoelectric metamaterial beams. The spectral element method (SEM) is used to derive the dynamic stiffness matrix of a composite piezoelectric beam segment. Boundary condition implementations are demonstrated. Both band structure and transmittance analyses are realized. Several case studies for piezoelectric metamaterial beams configured in different geometric/electrical forms are carried out. The corresponding finite element (FE) models are built for verification, and a comparison study with the transfer matrix method (TMM) is conducted. For the uniform configurations, an almost indistinguishable difference is noted between the theoretical and FE results. For the stepped configurations, only minor discrepancies are observed in the high-frequency responses. The improved robustness and stability of the SEM method compared to the TMM method are demonstrated. A further discussion has been provided to explain the cause of the high-frequency discrepancies: sudden changes in the cross-section of the beam result in the stress concentration effect and reduce the bending stiffness at the junction connection. Finally, the value of the high-fidelity modelling approach is reflected through a parametric-based optimization study towards merging the Bragg scattering and locally resonant band gaps in an example piezoelectric metamaterial beam to achieve a wide band gap.