RAINBOW TRAPPING OF FLEXURAL WAVES AND ITS APPLICATION IN ENERGY HARVESTING
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摘要:弹性波在色散关系经过设计的梯度结构中传播时会产生空间分频现象和波场能量增强现象, 即不同频率的弹性波会在结构的不同位置停止向前传播并发生能量聚集, 这就是弹性波彩虹捕获效应. 其相关研究成果可以促进结构健康监测、振动控制以及能量俘获等领域的发展. 本文通过所设计的梯度结构梁, 系统地研究了弯曲波彩虹捕获效应及其在压电能量俘获中的应用. 首先, 利用传递矩阵法获得了梯度结构梁元胞能带结构的解析解, 进而分析了弯曲波彩虹捕获效应的产生机理: 不同频率的弯曲波会在不同元胞附近群速度减小到零, 从而停止向前传播并发生反射; 入射波和反射波的叠加, 以及群速度减小带来的能量聚集, 会显著增强反射处的波场能量. 其次, 通过有限元仿真和实验验证了弯曲波彩虹捕获效应的空间分频现象和波场能量增强现象. 最后, 通过有限元多物理场耦合仿真和实验, 研究了粘贴PVDF压电薄膜的梯度结构梁相对于均匀梁的弯曲波能量俘获效果及其随入射波频率的变化规律. 结果表明, 在弯曲波彩虹捕获效应发生频带内, 粘贴在梯度结构梁上的PVDF压电薄膜的输出电压约为粘贴在均匀梁相应位置处的PVDF压电薄膜的输出电压的2倍.Abstract:When broadband elastic waves with different frequencies propagate in a graded structure with a special dispersion relation, the spatial frequency separation and the wave field energy enhancement will occur, that is, the waves will stop propagating forward and have energy accumulation at different positions of the structure, which is called the rainbow trapping of elastic waves. Researches on the rainbow trapping will promote the development of structural health monitoring, vibration control and energy harvesting. In this paper, the rainbow trapping of flexural waves and its application in piezoelectric energy harvesting are studied by using the designed beam with graded pillars. Firstly, band structures of unit cells of the beam are analytically solved by the transfer matrix method. According to the band structures, the mechanism of the rainbow trapping of flexural waves is analyzed: the group velocities of flexural waves with different frequencies will reduce to zero near different cells, thereby the flexural waves will stop forward propagation and be reflected; the superposition of incident and reflected waves, and the energy accumulation resulted by the reduction of group velocity, will significantly enhance the wave field at the reflection position. Secondly, the spatial frequency separation and the wave field energy enhancement of the rainbow trapping of flexural waves are verified by finite element simulations and experiments. Finally, we quantitatively evaluate the energy harvesting performance of the beam with graded pillars and its variation with the frequency of the incident wave comparing to the corresponding bare uniform beam, where PVDF piezoelectric films are pasted on their surfaces, by using finite element multiphysics coupling simulations and corresponding experiments. The results illustrate that the output voltage of the PVDF piezoelectric film of the beam with graded pillars is about 2 times that of the corresponding bare uniform beam in the bandwidth of the rainbow trapping of flexural waves.
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图 3频域有限元仿真模型(单位: mm)
Figure 3.Finite element simulation model in frequency domain (unit: mm)
图 5f0为2876 Hz的弯曲波在梯度结构梁中的传播过程
Figure 5.Propagation of flexural waves with center frequencyf0of 2876 Hz in beam with graded pillars
图 6波场能量增强现象的产生机理验证(续)
Figure 6.Generation mechanism of wave field energy enhancement (continued)
图 10各单元处电压比与弯曲波频率的关系(续)
Figure 10.Relationship between voltage ratio at each unit and flexural wave frequency (continued)
图 12f0为4681 Hz的弯曲波在第1个单元处的能量俘获实验
Figure 12.Energy harvesting from flexural wave with center frequency of 4681 Hz at the 1st unit
图 13f0为2253 Hz的弯曲波在第8个单元处的能量俘获实验
Figure 13.Energy harvesting from flexural wave with center frequency of 2253 Hz at the 8th unit
图 14f0为1906 Hz的弯曲波在第15个单元处的能量俘获实验
Figure 14.Energy harvesting from flexural wave with center frequency of 1906 Hz at the 15th unit
表 1元胞的零群速度点频率
${\boldsymbol{f}}_{\boldsymbol{n}}$ 和两相邻元胞零群速度点频率的中间频率${\boldsymbol{f}}_{\boldsymbol{n}}^{{\boldsymbol{n}}{\bf{+1}}}$ Table 1.Zero group velocity point frequency
${f}_{n}$ of unit cells and intermediate frequency${f}_{n}^{n+1}$ of zero group velocity point frequencies of two adjacent unit cellsn Hn/mm fn/Hz fnn+ 1/Hz 1 6.0 4851 4681 2 6.4 4510 4352 3 6.8 4194 4049 4 7.2 3904 3771 5 7.6 3637 3516 6 8.0 3394 3283 7 8.4 3171 3070 8 8.8 2969 2876 9 9.2 2783 2699 10 9.6 2614 2537 11 10.0 2459 2389 12 10.4 2318 2253 13 10.8 2187 2127 14 11.2 2067 2012 15 11.6 1957 1906 16 12.0 1855 − 
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表 2PVDF的参数
Table 2.Parameters of PVDF
Parameter Value size/mm3 20 × 15 × 0.11 d31/(C·N−1) 2.3 × 10−11 relative permittivity 12 elastic modulus/GPa 2 density/(kg·m−3) 1780 下载:
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表 3梯度结构梁与均匀梁俘能效果
Table 3.Energy harvesting performance of beam with graded pillars and uniform beam
n f0/Hz Type Normalized voltage
output rate /
(mV·mm−1)Voltage ratio
$\left(\dfrac{ { {\text{graded} } } }{ { {\text{uniform} } } }\right)$1 4681 graded 22.36 1.95 uniform 11.47 8 2876 graded 22.96 1.91 uniform 12.01 15 1906 graded 30.72 2.45 uniform 12.52 下载:
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