Flexural wave rainbow trapping effect in the periodic non-uniform Euler-Bernoulli beams and its application in energy harvesting

The rainbow trapping of elastic wave enables spatial frequency shunting and energy concentration phenomenon, which implies that the broadband vibration will forbid propagating forward and occur energy concentration at different positions of metamaterial structure. This paper proposes a novel metamaterial consisting of the periodic non-uniform Euler-Bernoulli beams with an arbitrary profile section to achieve the rainbow trapping effect of flexural wave and application in piezoelectric energy harvester. Firstly, the differential quadrature method is introduced to solve a partial differential equation with variable coefficients. The convergence of this method is systematically demonstrated, and the correctness of band structures is validated by comparing theoretical results calculated by the differential quadrature method with those from the finite element method for different profiles. Based on band structures analysis, the working mechanism of the flexural wave rainbow trapping effect is attributed to the group velocity modulation; thereby, the constructed periodic array of non-uniform beams achieves integration of specific-band vibration reduction and energy enhancement at specific positions. Secondly, simulations indicate that the resonance rainbow trapping frequencies demonstrate more intense concentration of flexural wave energy compared to the counterpart with initial frequencies. Finally, the resonance rainbow trapping phenomenon demonstrates superiority in vibration energy harvesting. Specifically, when piezoelectric films are pasted on specific-position with enhanced energy density, the maximum output voltage in the independent circuit connection reaches up to 2.30V for the resonance rainbow trapping frequency of 8752Hz and the PVDF film position of x = 670 mm. Furthermore, simulations illustrate that the maximum output voltage and power are up to 4.52V and 1894.12 nW for the Series-A circuit connection, respectively, which is approximately the sum of the individual maximum output electrical energy generated by independent circuit connection at positions A and B. The proposed metamaterial beams can offer new selection guidelines for highperformance vibration energy harvesters.