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Effects of weighting the ends of a tubular bell on modular frequencies

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.