Using the Tochen polynomial in modeling the oscillatory response of a drum cymbal under the action of mechanical impulses

Cymbals are thin, slightly curved metal discs that are part of the percussion instrument family. They can be struck against each other or with special sticks, and upon impact, the metal vibrates, generating sound waves that propagate in the form of a clear, bright, and often shrill sound. The intensity and duration of the sound depend on the size, thickness, and material of the cymbals, as well as on the way they are struck. This paper analyzes the behavior of a 400mm diameter and 1mm thick brass cymbal from the point of view of vibrations produced by hitting the cymbal with a drumstick, the cymbal having the support point at its center. The system is considered damped.To evaluate the dynamic behavior of the cymbal, a formulation based on the Tocher polynomial was used, this approach allowing a precise estimation of the displacements in the nodes of the finite elements, also providing a detailed description of the distribution of deformations and vibrations in the structure. An essential aspect of the study was the use of a specialized CAD program, which allowed the determination of the natural vibration modes of the cymbal, essential for understanding the dynamic behavior of the structure. The simulation of real conditions of use (the point of impact between the drumstick and the cymbal) was performed by randomly varying nine different positions on the surface of the analyzed element. This approach was carried out to capture the natural distribution of the hitting position, which significantly influences the dynamics of the vibrations produced. Each impact position generates a different response of the system, which contributes to a more comprehensive understanding of the vibrational behavior of the cymbal. To streamline the analysis process, a specialized program was developed capable of simultaneously performing the dynamic analysis of the discretized element for all nine impact positions. This program automates the process of generating initial conditions, calculating the time response and interpreting the results, allowing a rapid and comparative evaluation of the effects produced by the different impact points. The results of this study can be useful both in the design of cymbals with controlled acoustic responses and provide a valuable framework for future research in the field of acoustics of these musical instruments.