本文基于一维超材料梁的理论设计思想，将薄膜超材料简化成具有周期性的谐振单元和一维梁的结构。根据薄板的Kirchhoff理论，建立了整体结构的运动方程，计算了周期性谐振单元作用力及薄膜两侧载荷的声压差。利用牛顿第二定律及虚功原理，得到了薄板不同阶的振动模态幅值An所满足的运动方程。通过引入等效密度概念，得到等效密度共振形式受薄膜的刚度k、厚度h、谐振单元周期常数L及附加质量M等因素影响，并针对每种影响因素进行了等效密度峰值的分析。得出在一定尺寸范围内，得到了薄膜厚度h越薄，谐振单元刚度与质量的比值越小，共振频率越低的结论。而且周期常数L越大，薄膜自身的共振形式与谐振单元的共振形式同时存在，且薄膜本身的振动形式越强烈。最后，通过探讨谐振单元质量与透射峰值的实验验证，采用实验值与数值模拟值进行比较，结果吻合较好，表明质量-弹簧谐振子的共振作用对薄膜超材料的透射吸收起到至关重要作用。
Abstract
In this paper, based on the theory of one dimensional metamaterials beam, we present a metamaterial membrane consisting of resonance unit with periodic and one-dimensional beam structure. With the Kirchhoff plate theory , the equations of motion about the overall structure are established, and the concentrated force by mass-spring resonator and acoustic loading are calculated. By using Newton's second law and the virtual work principle, the vibration modes of different order for plate amplitude An can be computed through the equations of motion. By introducing the concept of equivalent density, the equivalent density of resonance forms is affected by plate stiffness k 、plate thickness h 、resonance unit periodic constant L and added mass M, the arbitrary affected factor is analyzed about the equivalent density peak. Within the scope of a certain size, the thickness of membrane is thinner, or the ratio between resonance unit and stiffness is smaller, the resonance frequency is lower. As the periodic constant L is larger, the resonance forms between membrane and resonance unit are coexistence, and the membrane vibration form is stronger. Finally, by discussing the resonant unit quality and transmission peak, the numerical simulation value is verified by the test results, the experiment shows the local resonance for mass-spring system has an essential role for metamaterial membrane transmission absorption.
关键词
薄膜超材料 /
谐振单元 /
虚功原理 /
等效密度 /
周期常数 /
透射峰值
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Key words
metamaterial membrane /
resonance unit /
virtual work principle /
equivalent density /
periodic constant /
transmission peak
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