Neil M. McLachlan - neil.mclachlan@rmit.edu.au
School of Aerospace, Manufacturing and Mechanical Engineering,
RMIT, GPO Box 2476V, Melbourne Vic. 3000, Australia
Popular version of paper 4pMU4
Presented Thursday afternoon, May 27, 2004
147th ASA Meeting, New York, NY
The Federation Bell Projects were commissioned by the Melbourne International Festival of the Arts to celebrate the Centenary of Australia's Federation in 2001. They enabled a group of artists, scientists and engineers led by Australia Bell Pty Ltd to rethink the bell, including its vibrational behavior, acoustical perception, sculptural and musical presence and cultural meaning (see www.ausbell.com). This paper will introduce a theoretical insight arising from these studies of bell vibration, and present new harmonically tuned bell designs.
The general increase in computing power over recent decades has dramatically improved computer modelling of vibration. This has enabled the behavior of complex 3 dimensional forms such as bells to be understood in much greater detail, and to be analysed over a broader range of shape variants. The behavior of shapes ranging between simple cylinders, through cones of varying wall profiles to disks was explored during the Federation Bell Project. Arising from such studies, behaviors have been reported that substantially updates the reported literature on the vibration of bells. These data also suggest the presence of a relatively simple physical mechanism that could not be predicted from theories that have formed the basis of this branch of engineering from the late-19th to mid-20th centuries.
The vibrational behavior of cylindrical forms is described by complex sets of differential equations. The formulation of these equations involved sets of assumptions that prior to the advent of modern computing were necessary to enable solutions to be found; but limited their compass. For examples of relatively thick walled cylinders analogous to bell geometry, two types of cylindrical vibration have been shown to exhibit correlated behavior in computational studies. The behavior is observed as the summing of modal frequency variations due to changes in the bending stiffness across the length of the cylinder and around its circumference. These two types of vibrational modes were usually described by applying equations derived from either the behavior of thin shells or of beams, and have generally remained unrelated in the literature. The underlying mechanism for this behavior involves the flexural behavior of solids, which requires more intensive computation to solve. However once elucidated this behavior may provide a simple and useful guide to designers concerned with the vibration of bells and other complex forms.
During the Federation Bell Projects a new type of musical bell was invented. This is a bell with a harmonic timbre that could fit comfortably within symphony orchestras or other musical ensembles. This bell was designed by using a combination of the analyses described above, and shape optimization software which is a form of artificial design intelligence that iterates towards the required design objective through a series of analyses and adaptations of a computer model. However, these first harmonic bell designs have proved to be too expensive to manufacture for most musicians and musical educators to afford, and so a new design algorithm was employed to design harmonic bells that could be more cheaply manufactured. The first prototypes of these new bells will be presented at the conference and contrasted to earlier designs.