# On the Correlation of Symbols' Vibrational Behavior and Manufacturing Processes
## 1 Introduction  
The vibration of percussion instruments has been the topic of interest for many studies in the field of musical acoustics. Cymbals are among the most popular percussion instruments and have been studied by various methods [1, 2, 3]. The weight, size, shape of the bell, and the material of construction all affect the sound of the cymbal, while an intriguing aspect is their nonlinear vibrational behavior, which has been the focus of numerous studies [1, 2, 3]. A cymbal is a struck idiophone that belongs to the same category of percussion instruments as gongs, chimes, sound plates, and triangles. These instruments are typically played by striking them with mallets or sticks [3]. Cymbals and gongs [4]follow similarly complex manufacturing stages, while chimes [5], sound plates [6], and triangles [7] are relatively simpler to construct. Cymbals are made from alloys [8], which are primarily based on copper combined mainly with tin (bronze), nickel (nickel silver), and zinc (brass). Other materials, such as silver, gold, and phosphorus are also used but in much smaller proportions. Cymbals are typically crafted from cast ingots, which are shaped into discs, which in turn are hot-forged and lead to the formation of roughly shaped symbols. These roughly shaped symbols are rolled and formed into a dome shape by cupping and spinforming, and are generally called blanks. Blanks are hardened by hammering to semi-finished symbols, which are finished by lathing, leading to thickness reduction [8]. Finishing may include additional thickness adjustments and polishing. Hammering is a crucial step since it creates tension and stress within the material. Large hammer marks cause discontinuities in the material that are similar in scale to the wavelength of cymbal vibrations, thereby influencing the sound [3]. In the literature, the presence of mixed modes in cymbals was investigated in [9]. Aiming to synthesize the sound of cymbals, a model was proposed [10] using a modal approach. This study also highlighted the importance of incorporating damping in such models. In another study on sound synthesis, the importance of the geometrical properties of cymbals was discussed [11]. A model focusing on thickness variations in cymbals was also presented in [12]. The vibrational behavior of cymbals was also studied, using the finite element method (FEM) and sound synthesis [13]. This approach was further enhanced by incorporating vibration measurements and accounting for geometry variations in cymbals [14]. The emitted sound from cymbals was also the topic of a study that highlighted the importance of the instrument's geometry [15], while vibroacoustic numerical simulations of motion-driven cymbal-drumstick interactions were shown in [16]. Furthermore, the effect of the hammering process on the emitted sound was studied via experiments and simulations in [17]. Based on the aforementioned studies, the majority of research focuses on commercially available finished cymbals, which offer clear, observable acoustic outcomes that researchers can measure and analyze, making them ideal for studies involving sound characteristics, resonance, and tonal quality. In contrast, less extensive research has explored the manufacturing process to investigate how, and to what extent, each stage of manufacturing affects the resulting sound. To the authors' knowledge, one comprehensive study has examined the correlation between the emitted sound and the distribution of residual stress and strain in cymbals [18]. Since idiophones do not need to be processed to a particular finished state to be played, the study of raw symbols during their manufacturing is demanded to trace how changes in shape, thickness, and material composition evolve and affect sound quality throughout the production process. Well-established brands are nowadays providing commercial unfinished cymbals [19,20] to meet the needs of the music market.