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摘要:
近年来，利用周期结构中弹性波禁带特性减振的研究思路受到了广泛关注。针对以往周期结构难以实现宽频且可调禁带的问题，设计了一种含电路网络的压电周期结构。该结构中的弯曲波和电路网络中的电波通过压电效应可以产生一个较宽的耦合禁带，且通过调整电路参数就可以达到调节禁带位置的目的。首先，为了高效计算该结构的波动特性，开发了适用于压电周期结构的减缩波有限元算法，该算法可以在保证结果准确性的基础上节约90%以上的计算时间。利用该算法研究了压电材料尺寸和形状对耦合禁带性能的影响。结果表明：相同电学参数下，随着压电片尺寸的增大，耦合禁带向低频移动，且禁带带宽增加；相同面积下含圆形和方形压电片的机电系统内耦合禁带差异较小，即形状对耦合禁带的分布影响不大。其次，针对不同尺寸和形状的机电系统，设计了电学参数使得在同一频率附近产生耦合禁带，并分析了其性能差异。最后，为了证明耦合禁带的减振效果，设计了一种有限压电周期板模型，其强迫响应的结果证明了耦合禁带对结构内弹性波可以进行有效调控。
Abstract:In recent years, researches on the elastic band gaps in periodic structures to reduce vibration have attracted widespread attention. However, it is difficult to design a band gap with wide bandwidth and good tunability. Aiming at this problem, we designed a periodic structure with piezoelectric network. By bonding piezoelectric patches periodically into structure, a coupled band gap can be created between the elastic waves and electric waves thanks to the piezoelectric effect. This band gap can be tailored with the help of external circuits. In order to calculate the propagation characteristics of the structure efficiently, a reduced wave finite element method suitable for piezoelectric periodic structures was developed to improve the calculation efficiency. It was found that more than 90% of the calculation time can be saved with great accuracy. Using this method, the influence of the size and shape of the piezoelectric material on the performance of the coupled band gap was studied. The results show that when fixing the electrical parameters, as the size of the piezoelectric patches increases, the coupled band gap moves to lower frequency range and its bandwidth increases. Moreover, the bandwidth in the system with square patches is slightly wider than that with circular patches. However, these two shapes have little impact on the directional distribution of coupled band gap. Then, the guideline is proposed for designing electrical parameters to make sure that coupled band gaps are generated around the desired frequency for electromechanical systems with different sizes and shapes. Finally, in order to prove the vibration reduction effect of the coupled band gap, a finite periodic piezoelectric plate was employed. The results show that the coupled band gap can effectively control the elastic wave in structure.
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图 1 压电网络板及元胞示意图
Figure 1. Illustration of plate with piezoelectric network and its unit cell
图 2 含正方形压电片元胞有限元模型
Figure 2. Finite element model of unit cell with square lead zirconate titanate patch
图 3 未加电路时系统传播常数分布
Figure 3. Distribution of system propagation constants without circuits
图 4 αf=5时减缩前后频散曲线对比
Figure 4. Comparison of frequency dispersion curves with/without reduction when αf=5
图 6 各频率下模型1内波1减缩前后误差
Figure 6. Errors of wave 1 with/without reduction in model 1 at different frequency
图 7 αf=3时减缩前后传播常数对比
Figure 7. Comparison of propagation constants with/without reduction when αf=3
图 8 αf=5时只考虑弯曲模态减缩前后传播常数对比
Figure 8. Comparison of propagation constants with/without reduction when αf=5 with only bending modes considered
图 16 含圆形压电片元胞有限元模型
Figure 16. Finite element model of unit cell with circular lead zirconate titanate patch
图 18 DP=22.57 mm时耦合禁带方向性对比
Figure 18. Comparison of directional distribution for coupled band gaps with DP=22.57 mm
图 19 DP=39.49 mm时禁带方向性分布
Figure 19. Directional distribution of band gaps with DP=39.49 mm
图 20 DP=45.14 mm时耦合禁带方向性对比
Figure 20. Comparison of directional distribution for coupled band gaps with DP=45.14 mm
图 21 DP=50.78 mm时耦合禁带方向性对比
Figure 21. Comparison of directional distribution for coupled band gaps with DP=50.78 mm
图 22 情况1时耦合禁带方向性对比
Figure 22. Comparison of directional distribution for coupled band gaps with configuration 1
图 23 情况4时耦合禁带方向性对比
Figure 23. Comparison of directional distribution for coupled band gaps with configuration 4
图 27 不同情况下强迫响应能量结果对比
Figure 27. Comparison of forced response energy results in different situations
表 1 压电片材料参数
Table 1. Material parameters of piezoelectric patches
参数 数值 ρp/(kg·m-3) 7 500 s11E/(m2·N-1) 13×10-12 s12E/(m2·N-1) -4.29×10-12 s55E/(m2·N-1) 22×10-12 d31/(C·N-1) -1.86×10-10 ε33T/(F·m-1) 3.009×10-8 
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表 2 减缩模型对比
Table 2. Comparison between full model and reduced model
模型对比 计算时间/s 矩阵规模 自由度数 完整模型 2 843.17 8 365×8 365 1 175 减缩模型 227.98 313×313 297 增益/% 91.98 96.25 74.72
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表 3 压电片结构参数
Table 3. Structural parameters of piezoelectric patches
情况编号 长度/mm 直径/mm 电感/H 覆盖率/% 1 20 22.567 6 0.319 6.25 2 35 39.493 3 0.1 19.14 3 40 45.135 2 0.078 25.00 4 45 50.777 1 0.060 6 31.64 5 50 56.419 0 0.051 2 39.06 注：表中覆盖率为压电片面积与基板总面积之比。
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