This paper presents a mathematical analysis of a modified tubular bell. Tubular bells are often modeled as a vibrating beam with free boundary conditions. However, most orchestral tubular bells contain a single cap attached to one end of the tube. This cap affects both the strike note as well as the relative frequencies of the overtones. However, the literature contains little information regarding the relationship between the mass of the cap and the modal frequencies. This analysis may be used to determine how the presence of the one or two caps affects both the strike note and the ratio of the modular frequencies of a tubular bell.

中文翻译：

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加权管状钟形端部对模块化频率的影响

本文介绍了一种改进的管状钟罩的数学分析。管状钟通常被建模为具有自由边界条件的振动梁。但是，大多数管弦乐队的钟形罩都包含一个连接到管子一端的帽。此上限会影响敲击音符以及泛音的相对频率。但是，文献中几乎没有有关瓶盖质量与模态频率之间关系的信息。可以使用这种分析来确定一个或两个盖的存在如何影响敲击音和管状钟形模数的比率。