【作者】 杨帆；

【导师】 谭柱华；

【作者基本信息】 河北工业大学 ， 力学， 2022， 硕士

【摘要】 声子晶体是由不同弹性介质按周期性排列组成的人工声学结构。在周期性势场的作用下,声子晶体会出现类似固体中的电子能带结构。通过设计声子晶体的散射体几何形状、材料和结构等参数,可以调控声子晶体的能带,从而实现聚焦、成像、隐身、减振降噪等功能。随着功能检测需求的增长,声子晶体透镜的研究受到越来越多的关注,在医学、探伤和能量回收领域有着广泛的应用前景。本文基于声子晶体的能带理论,设计了三瓣形旋转散射体,通过旋转角度实现了对折射率的调控,在此基础上分别设计了梯度正折射率和梯度负折射率的透镜,分析了透镜的聚焦效应,并利用压电材料的正压电效应实现了能量回收。主要工作和结论具体如下:（1）设计了三瓣形旋转散射体,研究了声子晶体与散射体旋转角度的变化关系,提出了梯度折射率的薄板声子晶体透镜。结果表明,随着旋转角度的增大,A₀模式能带第一条和第二条分支逐渐升高。因声子晶体等效折射率是根据能带计算而来,故可通过调控散射体的旋转角来实现梯度折射率。相比传统的梯度透镜,其不变的材料参数和晶格填充率为透镜的制造和组装带来便捷。（2）基于三瓣形旋转散射体设计了梯度正折射率的薄板声子晶体透镜,并利用压电材料对聚焦的能量进行了回收。分析了A₀能带第一条分支与散射体旋转角的变化关系,提出了其对应等效折射率的双曲正割分布,设计了梯度声子晶体透镜。开展了数值计算分析,分析了透镜的聚焦效应和能量回收效应,探究了不同工作频率下的透镜工作效果。结果表明,透镜焦距同理论解析解吻合;相比传统的透镜,工作频率范围（Δf=1.7 k Hz）较大且焦距随频率的变化较小;基于该透镜的能量回收在整个工作频率范围内都能获得较强的能量捕获效应。（3）采用三瓣形旋转散射体设计了梯度负折射率声子晶体透镜,并对聚焦的能量进行了压电能量回收。之后,通过调节透镜透射端均匀板的厚度对透镜的焦距进行了优化。研究了A₀能带第二条分支对应的等效负折射率随散射体旋转角的变化规律,根据等效路径原理对不同旋转角的散射体进行排布。开展数值计算分析,分析了不同工作频率和不同透镜尺寸下的透镜工作效果,探究了透射端板厚和焦距的变化关系,研究了透镜的能量回收效应。结果表明:透镜焦距和理论预测吻合且工作波长（λ<4 a）较小;焦点的输出电功率相比均质板,可以有数量级（10.8倍）的提升;通过增大透射端板厚可以在增大焦距调控范围的同时保持良好的压电能量回收效应。更多还原

【Abstract】 Phononic crystals are artificial acoustic structures composed of different elastic media arranged periodically.Under the action of periodic potential field,phononic crystals will have band structures similar to that of electrons in solids.By designing the geometry,material and structure of the phononic crystal scatterer,the band of the phononic crystal can be controlled to achieve the functions of focusing,imaging,stealth and vibration suppression.With the increasing demand for functional testing,phononic crystal lens has attracted more and more attention,and has a wide application prospect in the fields of medicine,flaw detection and energy harvesting.Based on the theory of band gap of phononic crystal,this paper designed three disc rotating scatterer,the effective refraction-index of phononic crystals is manipulated by rotating the angle of scatterers.Based on this,this paper respectively designed graded positive refraction-index lens and the graded negative refraction-index lens.The focusing efficiency of the proposed lenses is analyzed,and the positive piezoelectric efficiency of piezoelectric materials are used for the energy harvesting of the lens.The main work and conclusions are as follows:（1）A three-lobe rotational scatterer is designed.The relation between the rotation angle of phononic crystals and the scatterer is studied,and a thin plate phononic crystal lens with graded refraction-index is proposed.The results show that the first and second branches of A₀ mode band gradually increase with the increase of the rotation angle.Since the effective refraction-index of phononic crystals is calculated according to the band,the graded refraction-index can be realized by adjusting the rotation angle of scatterers.Compared with the traditional graded lens,its invariable material parameters and lattice filling fiction bring convenience to the fabrication and assembly of the lens.（2）A thin plate phononic crystal lens with graded positive refraction-index was designed based on rotating a three-lobe scatterer,and the focused energy was harvested by piezoelectric materials.The relationship between the first branch of A₀ band and the rotation angle of the scatterer is analyzed.The hyperbolic secant distribution of the effective refraction-index is proposed,and the graded phononic crystal lens is designed.Numerical analysis was carried out to analyze the focusing efficiency and energy harvesting efficiency of the lens,and to explore the working effect of the lens at different working frequencies.The results show that the focal length of the lens is consistent with the theoretical analytical solution.Compared with the traditional lens,the operating frequency range（Δf=1.7 k Hz）is larger and the focal length varies less with the increase of the frequency.The energy harvesting based on the lens can obtain strong energy harvesting efficiency in the entire operating frequency range.（3）A graded negative refraction-index phononic cystal lens was designed by rotating a three-lobe scatterer,and the focused energy was harvested by piezoelectric materials.Then,the focal length of the lens is optimized by adjusting the thickness of the uniform plate at the transmission end of the lens.The rule of the effective negative refraction-index,corresponding to the second branch of A₀ band with rotation angle of scatterers,is studied,and scatterers with different rotation angle are arranged according to the principle of equivalent path.Numerical analysis is carried out to analyze the working effects with different working frequencies and different lens sizes,explore the relationship between the thickness of the transmission end plate and the focal length,and study the energy harvesting efficiency of the lens.The results show that the focal length of the lens is consistent with the theoretical prediction and the working wavelength（λ<4 a）is small.Compared with the homogenous plate,the output power of the focal point can be increased by order of magnitude（10.8 times）.By increasing the thickness of the transmission end plate,the piezoelectric energy harvesting efficiency can be maintained while increasing the focal length range.更多还原