(Re-)exploring the normal modes of axisymmetric structures: An English church bell case study

The normal modes of axisymmetric structures are of interest to structural engineers, physicists and musical acousticians. Previously, some of the present authors have made studies of church bells, hand bells, elephant bells, various gongs and rings. Group theoretical arguments have been used, with considerable success, in classifying the normal modes of these structures and in understanding how “beats” arise from split degenerate doublets. It is now pointed out that further information can be obtained from group theory using a variety of additional arguments. In particular, it infers a basic distinction between “in-extensional” and “extensional” types of modes. In the present work, we concentrate on the case of an English church bell, as an example axisymmetric structure, whose normal modes were previously measured in a frequency range of up to about 10 kHz. In that earlier work, the results were analyzed with what was then considered a state-of-the-art finite-element package. We now repeat this exercise with a modern finite-element package to explore the differences between the types of modes and validate the Group theory observations. The agreement with experiment is much improved and some new level of understanding of the spectrum of the bell is achieved.