Musical Instruments - The steeldrums

THE DOME GEOMETRY
SABINA SCHÄRER and FELIX ROHNER - PANArt AG

Fig.1 - The Pang Geometry

1. Introduction
THIS paper presents a new approach on a complex vibrating system. PANArt has worked together with spring producers and engineers, and is so doing, have introduced new terms into the terminology of the vibrating shallow shell.
Frequency ratios of flat plates of all sizes and of different edge conditions, have been studied by physicists. They measured, calculated and simulated with computer programs; the frequency [characteristics] of curved shells. The fact that there are no two pan notes with a similar shape, and the unsolved question about the residual stresses, make us understand the difficult situation [into which] the scientists are placed. The theory of vibrating plates and shells can approach the complex geometry of the pan note; but no more.
The Pang with its dome geometry, which is not a mass loaded vibrating system, probably offers a new approach. It is anyway a chance for physicists, metallurgists, engineers and pan constructors to enter into interdisciplinary collaboration. Science and art can meet; the Pang is the proof.

2. Analyses of the Pang Geometry
In the centre of a Pang note is an elliptical dome. Around the dome is a slightly curved area, which graduates softly over into a [spherical shape or] sphere. This transition [boundary] is the soft edge condition of the shell. The boundary of the shell is almost on a plane, which means that the note is almost round. The area around the dome we call the trunk area; the transition from trunk to sphere, the roots area. The dome is the heart; and where its [base is] connected with the trunk, is the crown. (fig.1)

Fig.2 - Coupled stiffness in a Dome structure

3. The Dome
The engineers teach us that the dome is an architecture which was developed by the Babylonians and later by the Romans. It is a very stiff form when a force is applied [over its] centre. At its base, the dome is much more flexible. The dome structure is suited for stable constructions with little material.
The high stiffness of a dome is the consequence of the coupling of bending stiffness and membrane stiffness (1) (fig.2).
An impact on a dome generates a splitting of energy in different waveforms due to the coupling of bending and membrane stiffness. When a clamped plate is struck, only the bending forces are excited; while exciting a plate from its lateral side, it is the membrane forces that are provoked. In the first case bending waves occur; in the second case longitudinal, or transversal, waves. A stroke on a dome provokes both.

Fig.3 - Trunk stiffness and Industrial Membrane Springs

4. The Trunk
In an ideal Pang note, the trunk has the form of a [wide] ring, which can be considered as a spring. Its form is characterised by a curvature with a change from concave to convex. (2) The spring industry produces so called membrane springs. They are clamped and have a stiff centre. They are used to fix construction elements. Their form can be conical, spherical or they are profiled concentrically. The trunk of a Pang note can show conical and spherical curvature when the elliptical dome is not of the right dimensions. In which case the Pangmaker has to adapt the trunk to the dome. (fig.3)

5. Influence of the Dome on the (0,0) mode
The dome stiffens the whole shell, and requires soft-edge conditions to get good reflection of the bending waves for this fundamental mode. We can consider the centre of a shell as one pole; and the boundary conditions as the other pole. They have to come in a good relationship, with the basic requirement that the fundamental mode gets a lot of energy. There must be a good ratio of impedance between the sphere and shell. This is the case in the dome structure of the Pang, where the sphere is stiff and the note boundaries are much less so. In other words: The dome is a strong centre that determines the edges.
The shell which has equal edge conditions, produces a strong standing wave.

6. Influence of the Dome on the (1,0) and (0,1) mode
On a ideal Pang note the dome is of the correct size to spread the two modes of greatest interest: The octave and the fifth.
As the elliptical edge condition determines the partials 2 and 3 in a steelpan, the elliptical dome determines the ratios of these modes in an almost round shell.

Fig.4 - Vibration modes of a Pang and a Steelpan

Small corrections can be made with the shaping of the trunk. These corrections lead to slightly anticlastic edges. The spread is a consequence of the position of the elliptical dome on the nodal-line. Where the dome intervenes more into the mode, it lowers the frequency, like a longer spring.
Thanks to the dome, energy flows into both modes, which have approximately the same amplitude in the [sound] spectra.

7. Influence of the Dome on the (1,1) and (2,0) mode
The dome has a strong influence on the (1,1) and (2,0) modes. They are both of higher stiffness and therefore, higher in frequency. The (2,0) mode is changed into a twist mode. The round edge condition brings these frequencies closer to each other. The elliptical form of the dome has little influence here. Their [frequencies] come to lie between the fifth above the second and third octaves. In the Pang tuning/note, there is no second octave; these two (1,1) and (2,0) modes are always higher [than the second octave] and contribute [greatly] to the timbre of the Pang sound.
The higher modes convert [similarly] to bell-modes, and are higher in frequency.

8. The Dome as the Point of Energy Impact
Müller (3) explains in his work the accumulator theory of shells, which means that the energy of impact flows into two accumulators: the bending wave and the longitudinal wave accumulators. He describes in his paper a model of the exchange between the two accumulators. Through the coupling of these accumulators, an energy [exchange] is possible between them. The bending waves radiate, and energy from the longitudinal and transversal accumulator flows into the bending wave accumulator.
For a musician it is important to control the sound. The steelpan has its sweet point, the Pang has its dome (fig.4). All nodal-lines of the overtones run over the dome, so they are not excited in their amplitude. Most of the energy flows into the fundamental. The touch on a shell with high stiffness means a touch on a shell with more resistance. The mallet has to be softer, to stay longer on the shell, to transfer more energy. The stroke on a dome excites eigen-modes of the dome; and the hard dome can excite eigen-modes of the mallet. Both can be an attractive part of the timbre.

Fig.5 - Experiment 1

9. Experiments
Two experiments illustrate the relationship between material, shell geometry and boundary condition.

Exp - 1
A clamped elliptical plate (A:B = 1.5) gets a round hole and a successively enlarged elliptical hole. (fig.5)

We observe that a small hole in the centre lowers the (0,0) mode, a larger one has not much influence. The fundamental rises slowly. By contrast, the (1,0) and the (0,1) modes reach the same frequency. This experiment shows that a clamped plate has an inner area which can be used to tune the partials (1,0) and (0,1) while not affecting the fundamental as significantly.

Exp - 2
The following experiment shows that a strong architecture in the centre of the shell offers more tuning possibilities on the edges.

Fig.6 - Experiment 2

Holes of 4mm diameter are bored along the boundary of a Pang note, first along the boundary at the x and y axes, and then successively more holes are bored. (fig.6)

We see that the frequencies rise slowly at first, and more rapidly when the degree of perforation is increased to where the condition of a free edge is approached. When the foundation is weakened, the compressive residual stress is reduced, the shell gets more curvature and the frequencies rise.

10. Conclusion
We have seen that the dome in the Pang structure is a part which determines not only the sound, through its power to spread the partials, but also by forcing a significant change in stiffness around its edge. This boundary condition stores energy. The fundamental sounds [for a] longer [time]. There is damping of the higher overtones through the stiffness of the shell.
The experiments have shown a large degree of freedom to form the boundaries. Residual stresses are only introduced through the foundation.
The dome geometry, the equal thickness and the symmetry of the foundation; avoids distorsion of the sound.
The new shape gives the musician the security [of durability and stability]; but demands on the other hand, high control of the impact.

11. Acknowledgements
The authors wish to thank Dr. A. Varsanyi, Musikethnologe München D; Dipl. Ing. K. Wanke, Scherdel GmbH Marktredwitz D; Dipl. P. U. Müller, Plauen D; Prof. Dr. M. Farshad, EMPA Dübendorf CH.

12. References
1. Dr. M. Farshad, Design and Analysis of Shell Structures, Kluwer Academic Publishers 1992
2. M. Meissner and K. Wanke, Handbuch Federn, Verlag Technik GmbH Berlin München 1993
3. U. Müller, Das Musikinstrument, Frankfurt a. M. Jg. 31, 1982, S. 1424-1432

Compiled & Processed by Islands Research for:
The Steelbands (Pan) of Trinidad & Tobago
http://www.seetobago.org/trinidad/pan/archive/r&d/panart/ir_archivecopy_the_dome_geometry.htm

[2nd Ed] © 2003: tobagojo@gmail.com - 20031026 - 1m20071228 - 2m20140615 Historic Update: 16 November 2003; Last Update: 29 June 2014 02:12:00 TT Processed by: JdeB - Islands Research

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