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This work has demonstrated that it is possible to achieve mode localisation or mode confinement through manipulation of the bowl-wall curvature surrounding a relatively flat central region. This controlled entrapment of modes to a selected region has implications for the future manufacturing of steelpans. Knowledge of this mechanism of entrapment may also find application in other areas of structural vibrations particularly where the transfer of vibrational energy from one section of a structure to another needs to be minimised. It could be said that there are two facets of steelpan note tuning: 1) mode entrapment and 2) mode tuning. The quality of the instrument may be influenced on how well tuned modes are confined, hence the importance of this work.

It is first important to draw an analogy to the steelpan in order for the findings of this work to be relevant and applicable. Recall, steelpan notes are flat or slightly concave shells embedded in a dish [10]. The test-notes created in this work were an attempt to replicate this important feature of the steelpan note. This was one of the main deciding factors in the choice of geometry. Another important aspect of this work was that the initial horizontal diameter of each test-bowl was fixed at

260mm. Steelpans of all types are usually crafted using oil drums of the same diameter. This also allowed for some comparison between the test-bowls and actual pans. The investigation revealed that the mode confinement was stronger in the test-notes of the test-bowls with the highest curvature (see Figure 5.6c and Figure 5.7a and c). In full size pans, tuners usually tune at most one or two modes in the notes of the low-pitched shallower drums and up to four in the outer notes of the deeper high-pitched pans. It is not certain whether tuning in this way is done deliberately or whether the degree of curvature surrounding the note acts as a constraint.

Some limitations of the findings in this work must also be stated. For instance, the majority of notes in full size steelpans are typically located on the walls of the dish and not at the base. In this work, the test-notes were constructed at the base of the surrounding test-bowl or dish. The influence of note position on mode confinement is yet another topic for consideration and is recommended for future work. It may be difficult to manufacture this geometry in a controlled manner. However, an initial approach would be analysis through finite element studies. Another limitation was that the notes produced in this work were circular mainly because this geometry could be easily reproduced as a CAD model in addition to being easily transformed into a tool-path code for manufacture of the structures. The majority of steelpan notes resemble elliptical or rectangular plates. Further work should also consider mode confinement in rectangular or elliptically shaped notes.

Springback was identified as being a possible contributor to the percentage difference observed between the measured and predicted frequencies of the first confined modes in the test-notes (see Figure 5.8a-f). However, after accounting for this, the percentage difference between the measured and predicted frequencies of the first confined modes of these modes were still relatively high (see Table 5.5). This may indicate that there are other influences besides springback. It is reasonable to assume that the presence of springback may also be a manifestation of the presence of residual stresses in the test-bowl. It is not certain to what extent residual stresses affect the modal properties of the structure but this is another avenue for further investigation. The FE model did not account for the residual stresses that may have been created during the forming of the test-notes. The geometry of the structures was created as CAD profiles for mode studies in ABAQUS_®. A better approach might have been to simulate the incremental forming of the test-bowls in ABAQUS®

before doing a vibration analysis. This might have helped to provide estimates of material springback.

Since all of the test-notes had the same diameters, it was assumed that as the amount of pillowing increases, the amount of difference between the measured and predicted frequencies would also

this was not realised as the test-notes with the smallest amount of pillowing manifested the largest deviations between the measured and predicted frequencies of their first confined modes (see Figure 5.8f). Although the updated FE results showed improved agreement between measured and predicted natural frequencies for the first confined modes, a similar trend between pillow height and frequency deviation was maintained. The amount of springback may also be influenced by how firmly the structures were clamped within the forming jig. This clamping may have affected the amount of pillowing in each case. There was no means of monitoring or controlling the level of holding force as the forming jig made use of a bolt-and-nut arrangement to secure the plates to be formed.

50°

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Bowl wall

Flange

Test-note

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Actual Point of inflection

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Figure 5.11: 50° test-bowl: (a) Ideal bowl with no springback effects and (b) Actual bowl with springback effects that creates a point of inflection between the test-note region and the bowl wall There was also springback in the bowl walls and the flange sections of the test-bowls. While this was not expected to directly influence the behaviour of the test-notes, the springback resulted in the creation of a point of inflection between the test-note region and the bowl wall. This is illustrated in Figure 5.11. Here it is clear that in the ideal geometry (Figure 5.11a) there is no such point of inflection in which there is a change of the curvature in the test-note vicinity from concave to convex. In the actual geometry, this point of inflection may be the main mechanism that is responsible for mode confinement and not the change of curvature that occurs between the test-note and bowl-wall. The level of confinement may depend on the rate of change of bowl depth with respect to the bowl width on either side of the inflection point. This needs to be investigated further.

The level of resolution obtained from the CMM measurements was not sufficient to examine the profile of the note in this region in finer detail. Future work could incorporate the use of a stereovision camera which would capture the entire 3D shape and offer finer resolution. Calculated control in the way in which mode confinement is produced in steelpan might assist in the development of dishes in which the notes require minor adjustment for tuning. To achieve this would also require a close study in conjunction with other geometrical parameters that affect mode

tuning in steelpans. The benefits of this cannot be overlooked. This would reduce the time it takes to tune a steelpan and would also effectively reduce the amount of exposure to high noise levels and hand-arm vibrations currently experienced by pan makers. There would also be some debate on how many modes should be strongly confined in the pan note. The limiting factor would be on how many modes could be practically tuned and therefore a high level of mode confinement may be not be necessary. However, it would be beneficial to determine the effect of strongly entrapping several modes to a single note region while only two or three are tuned.

The light damping displayed by the confined modes, in accordance with Scott and Woodhouse [157] may be an indication that the damping is solely governed by acoustic and material damping.

There was good agreement between the Q-factor values for some of the corresponding modes in test-bowls supported in free-free and clamped arrangements (see Figure 5.10). However, the damping for some of the confined modes appeared to be affected by the boundary conditions which were not in the immediate vicinity of the test-note. This is probably a result of the coupling to bowl-wall or particularly flange modes close in frequency to the confined modes.

It is also important to indicate that the flange regions, to which clamping was applied, were not perfectly flat. This could be seen in Figure 5.8f. There is between 0.5 and 1.5mm of deviation of the flanges, in the edge region, from its ideal shape (Figure 5.8f). Distortions in the flange may have arisen during the release of the part from the forming jig. During the modal tests, clamping of these uneven flange regions may have resulted in added distortion of the entire structure. These distortions may have altered the boundary conditions in the test-note vicinity and consequently affect the damping properties. It is also seen that in some cases, the mode frequencies of the confined modes were higher than in the clamped condition (see Figure 5.10). The free-free tests were conducted after the tests with the test-bowl in the clamped condition. Any added distortion during the modal test with the bowl clamped may have resulted in local stiffening in the test-note region thereby raising the natural frequencies of the confined modes.