Physical Modeling and Dynamical Simulation of a 13th Century Bell

Dated 1287, the bell from the church of S. Pedro de Coruche is the oldest recognized bell in Portugal. Unfortunately, being fragmented and incomplete, its original sound cannot be heard anymore. This paper deals with the dynamical analysis of this 13th century bell, as well as a reconstruction of the possible original bell response through physical modeling techniques. As a first step, based on the measured bell shape and material properties of bell bronze, numerical modal computations through the Finite Elements Method (FEM) where used to predict the relevant dynamic properties of the bell and to access the tuning frequencies of its partials, for an adequate number of modes. Then, using the modal parameters estimated through the FEM analysis, modal vibration synthesis techniques have been applied in order to simulate the dynamical bell responses to percussive excitations. Although the sound radiation is not addressed in the present analysis, realistic "sounds" can be produced based on the model vibratory responses. Several possible dynamical responses were computed, for different assumptions concerning the unknown bell clapper mass, shape and contact properties, as well as the strike location. From the musicological point of view, the bell of interest to the present study is located in a time period of transition between the use of two different bell shape profiles, which would lead to major musical consequences. In this context, the frequency ratios found for the first modes (partials) of this bell are of historical significance - as is, in the same vein, the greater or lesser musical attractiveness of the simulated response sounds.