Abstract Acoustic black hole (ABH) indentations have proven to be an efficient way for reducing vibrations on straight beams and plates. However, many built-up structures in the naval and aerospace sectors involve curved beams and shells, so there is a need to check whether the ABH effect could be also beneficial for them. To date, such issue has not been yet addressed and in this work some initial steps are proposed towards it. In particular, an analysis is made of the vibrational behavior of a circular beam with embedded ABHs. A semi-analytical model is suggested in the framework of the Rayleigh-Ritz method, making use of Gaussian trial functions. The key to the approach is building a basis of Gaussian functions that fulfill the periodic boundary conditions on the beam, to approximate its radial and tangential motions. It is shown how this can be done quite directly and efficiently. The validity of the proposed approach is then tested by comparison with finite element simulations, showing very good matching. The Gaussian expansion is then applied to study the dependence of the bending vibration of an ABH circular beam with frequency. After that, the influence of curvature and boundary conditions are established by comparing the performance of close and open circular ABH beams and straight periodic beams. Circular beams containing different number of ABHs are also studied and the appearance of frequency stopbands is reported.

中文翻译：

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一种表征嵌入声学黑洞的圆形梁振动的半解析方法

摘要 声学黑洞 (ABH) 压痕已被证明是减少直梁和板振动的有效方法。然而，海军和航空航天领域的许多建筑结构都涉及弯曲的梁和壳，因此需要检查 ABH 效应是否也对它们有益。迄今为止，此类问题尚未得到解决，在这项工作中提出了一些初步步骤。特别是，对嵌入 ABH 的圆形梁的振动行为进行了分析。在 Rayleigh-Ritz 方法的框架内建议了一个半解析模型，利用了高斯试验函数。该方法的关键是建立满足梁上周期性边界条件的高斯函数的基础，以近似其径向和切向运动。它显示了如何可以非常直接和有效地完成此操作。然后通过与有限元模拟的比较来测试所提出方法的有效性，显示出非常好的匹配。然后应用高斯展开来研究 ABH 圆形梁的弯曲振动与频率的相关性。之后，通过比较闭合和开放圆形ABH梁和直线周期梁的性能，建立曲率和边界条件的影响。还研究了包含不同数量 ABH 的圆形光束，并报告了频率阻带的出现。然后应用高斯展开来研究 ABH 圆形梁的弯曲振动与频率的相关性。之后，通过比较闭合和开放圆形ABH梁和直线周期梁的性能，建立曲率和边界条件的影响。还研究了包含不同数量 ABH 的圆形光束，并报告了频率阻带的出现。然后应用高斯展开来研究 ABH 圆形梁的弯曲振动与频率的相关性。之后，通过比较闭合和开放圆形ABH梁和直线周期梁的性能，建立曲率和边界条件的影响。还研究了包含不同数量 ABH 的圆形光束，并报告了频率阻带的出现。