Physical modeling of Japanese temple bells using thin cylindrical shells involving dynamics of clappers Shumoku

Abstract

In this paper, we present a physical model of Japanese temple bells based on the elasticity theory of thin cylindrical shells. The proposed model involves the dynamics of the clapper, called shumoku, thereby constructing a total physical system of a temple bell. The governing equation of a temple bell is solved semi-analytically using the Fourier series expansion in the circumferential direction. The finite difference method (FDM) is applied to the governing equation in the axial direction. These modeling and numerical treatment are an attempt to reduce computational costs while retaining the physical essence of temple bells. Shumoku is physically modeled as a 3D beam, and numerically analyzed by the finite element method (FEM). Simulation results show that the fundamental frequencies calculated by the proposed method are close to those of existing FDM calculations and measured data, while differences in higher-order partial frequencies are observed, indicating that the differences are attributed to the details of the structural features of the bell omitted in the proposed model. Changing the material of shumoku affects the spectrogram of the vibration of the bell, suggesting the importance of physical modeling of clappers to precisely simulate bell systems.