Propuesta metodológica de diseño mecánico y geométrico de membranófonos basada en su comportamiento vibratorio

Abstract

The manufacture of musical instruments is a sector in which the material plays a leading role. The continuous development of luthiers and manufacturers, due to the need to offer different acoustic characteristics, has resulted in a very extensive offer of materials. In the case of chordophones, aerophones and idiophones, the materials traditionally used are still in place, such as certain species of wood, normally of high density, metals, especially copper alloys, and in recent years some composite materials. But in the case of membranophones, where until the beginning of the 20th century the material used was only wood, the current range of materials has become incredibly extensive. A single manufacturer can offer a large catalogue of wood species, almost the entire range of metals, such as titanium alloys, steel, bronze or aluminium, and polymers and composite materials such as carbon fibre, glass fibre or polycarbonates. While in other families of instruments there is a classification criterion for the materials and only some of them are suitable for manufacture, the large offer in the case of membranes denotes a trial-and-error method, and therefore a lack of knowledge of the influence of the mechanical properties of the shell on the final acoustics. Due to simplifications made in previous investigations, no direct relationship has been established between the shell material and the final spectrum and therefore the influence of the shell, compared to the membrane, has been considered minimal. In this controversial framework, it is essential that both manufacturers and users have tools to establish material and geometry selection criteria according to the desired vibration behaviour. Therefore, the objective of this thesis is based on establishing a direct relationship between the design variables and the vibratory behavior of the membranes. The research is focused on aspects such as the understanding of the functioning of a membrane, the influence of geometry, mechanical properties and the production process of the shell, and on the vibratory behaviour of the upper membrane. Design methodologies are established to provide control over the effects of shell material and geometry, and their link to membrane vibrations. Sound qualities (pitch, intensity, timbre and duration) are closely related to physical aspects such as sound pressure levels, frequency spectrum or decay, so it is possible to characterize the final sound of a membrane by intervening on the aspects responsible for its vibration. In order to analyse this effect, a combination of numerical simulations by finite elements and their experimental validation is carried out, based on the modal, mobility (admittance) and transient study of the membrane, from which the dynamic behaviour is evaluated and then the different interactions between structures that play a decisive role in the vibration of the whole instrument are analysed. Both the material and the geometry of the shell are capable of modifying the resonances and therefore the resulting spectrum. The characteristic of the material with the greatest potential for this particular use is based on its wave speed. The study in this field allows us to establish criteria for the selection of materials and to analyse their suitability, improving aspects such as the dynamic characterisation of different types of materials including those with visco-elastic behaviour. As regards geometry, any modification of the dimensional characteristics of the shell has the capacity to widely modify its resonance, which allows the identification of design points where the possible combinations of geometric variables represent advantages of different kinds in the field of manufacturing, without sacrificing its vibratory behaviour. From the point of view of the manufacture of this type of instruments, both the knowledge of the influence of the material and of the geometry represents a great advance that offers great vibratory configurability of the shell. Although the transmission of the membrane vibrations to the shell and their magnitude is known, the results of this research show effects unknown until now. It is necessary to consider the exact geometries of both the shell and the membrane, without resorting to simplifications. This allows the evaluation of the actual stresses to which the shell is subjected, and also allows the definition of the actual boundary conditions of the shell, which in turn allows the detection of several effects that previous investigations did not address. After the initial impact of the upper membrane, the shell vibrates by generating its own resonance frequencies. These have sufficient capacity to excite the upper membrane so that the shell has a direct participation in the total spectrum, which reveals the importance of the shell in the sound nuances of the final spectrum of the membrane. The detection of this effect confirms the direct participation of the resonances of the shell in the membrane, and therefore of its mechanical and geometric characteristics. This thesis proposes a new descriptive model of the functioning of membranophones based on a new modal coupling rule, as well as a new methodology for shell design and evaluation based on its material and geometry. A new tool that allows the analysis and prediction of the influence of a shell on the final vibratory behaviour of a membrane based on the material and geometry used in its manufacture.