Aspects of numerical techniques for the design of musical structures.
Katherine Legge, Joseph. Petrolito · 2006
Numerical techniques are used extensively by structural engineers to both analyse and design physical structures. In particular, numerical optimization procedures are used to improve the efficiency of a design. Improved design in musical structures is a somewhat different concept and applying numerical optimisation procedures requires careful consideration of what one is trying to achieve. In this paper we consider a general procedure for applying constrained optimisation techniques to tuned idiophones. The procedure involves choosing a suitable balance between constraints and desirable attributes and is illustrated through application to a range of structures from a percussive bar to a church bell. Introduction Numerical optimisation is a tool employed by engineers in the design of physical structures such as a building or a bridge. The optimal design is generally one that minimises cost or the amount of construction material used. To the design engineer, a musical instrument is a physical structure and as such its design can be performed using numerical optimisation techniques. To the designer of musical instruments however, the application of numerical techniques may be constrained by pre-conceived ideas of what they are attempting to achieve. The designers of musical structures have foremost in their minds that they require optimal tuning and this, coupled with the more traditional use of numerical techniques to analyse vibrational frequencies, seems to lead to the use of an optimising function in terms of the required frequencies. Such a strategy inevitably leaves the user with the task of determining the suitability of the optimal solution, that is, how close to the desired frequency should each mode be and which mode is more important to get correct? Instead, we suggest the application of a constrained optimisation procedure whereby the frequency requirements are written as constraints and the governing equations are solved subject to the constraints being met. We have previously presented a general procedure for applying numerical optimising techniques to oneand twodimensional models of musical structures (Petrolito & Legge, 1997, 2005b). This paper re-visits the general approach and extends its applications to the design of church bells. Mathematical Model The analysis of the motion of any vibrating structure begins with the governing equations. These equations depend on the model adopted, and the rst decision is therefore the choice of model to be used to described the structure. For example, the dominant response of a struck xylophone bar is transverse to its longitudinal axis, and the motion is therefore beam-like in nature. Hence, a one-dimensional model that can account for shear deformation is usually considered suitable (Petrolito & Legge, 1997). A plate on the other hand can generally be satisfactorily modelled as a two-dimensional structure, so long as its thickness is small in comparison to a typical plan dimension (Petrolito & Legge, 2005a). A bell is a three-dimensional structure. However, provided that the thickness is not too large in comparison with the radius, it may be modelled as a two-dimensional surface with appropriate stiffness and mass characteristics. Except in the case of very simple structures, solutions of the governing equations cannot be obtained analytically, and numerical techniques are employed. Finite element analysis is generally chosen as the most appropriate numerical