声子晶体板复能带计算方法

Calculation method of complex band structure of phononic plates

声子晶体具有弹性波带隙,可以用于结构振动与噪声控制。声子晶体传统能带算法一般给定波矢k在不可约布里渊区边界取值,然后求解特征频率ω,得到ω-k曲线。因而,传统方法中波矢k只取实数,只能求解实能带。为了求解复能带,一般需要给定频率ω,求解特征波矢k,从而得到k-ω曲线。提出了一种参数变换方法,解决了特征波矢求解中复杂的非线性特征值问题,实现了复能带的快速求解。最后,采用两个算例对文中算法进行了验证,包括布拉格声子晶体板和局域共振声子晶体板,研究了带隙内衰减常数随波传播方向的变化和阻尼对带隙的影响。

Phononic crystals possess elastic wave band-gaps, which can be used to control the vibration and noise of structures. To obtain the band structures of the phononic crystals, the conventional procedures are as follows: given the wave vector k, whose value sweep the boundary of Brillouin zone, the eigenfrequency ω can be evaluated, resulting in the k-ω curve. However, this method can only yield the real band structure. In order to get the complex band structure, the frequency ω usually is given and then the eigenvalue of wave vector k is calculated, resulting in the k-ω curve. This work proposes a parameter transforma-tion method, which can resolve the complicated non-linear eigenvalue problem and achieve the rapid solution of complex band structure. Finally, two examples, i. e., a Bragg phononic plate and a locally resonant phononic plate, are adopted to validate the proposed method. The variation of the attenuation constant along with the wave direction in the band gap and the influence of damping on the band gap are investigated in detail.