Acoustic Black holes in Curved Plates

K. Hook, J. Cheer 1 and S. Daley Institute of Sound and Vibration Research, University of Southampton, University Rd, Highfield, Southampton, UK, SO17 1BJ.

ABSTRACT Acoustic Black Holes (ABHs) have been shown to produce significant vibration damping in both beams and flat plates. However, it is interesting to determine how more complex structures with singularly curved surfaces are damped by the addition of ABHs. Therefore, this paper presents an investigation into three di ff erent ABH designs that can be implemented into a curved plate. The performance of each ABH design is compared in terms of the global structural response of the curved plate.

1. INTRODUCTION

Acoustic black holes can be used to e ff ectively damp structures such as beams and plates. Currently, investigations into ABHs integrated into flat plates have included designs such as embedded circular ABHs [1–13] and surface attached ABH dampers [14–19]. In curved structures, ABHs have been investigated as part of cylindrical shells [20, 21], circular beams [22] and aircraft panels [23, 24]. However, further research is needed to determine how di ff erent ABH designs compare in terms of performance and thus which design is the most suitable for implementation into singly curved plates. Therefore, this paper presents an investigation into three di ff erent designs of ABH that can be embedded into a singly curved panel. Circular ABHs have been compared to annular ABHs and longitudinal ABHs. To carry out this investigation, a Finite Element (FE) model of a curved plate has been created to test each ABH design. The total kinetic energy of the plate has been calculated and compared to the total kinetic energy of the plate with each of the embedded ABH designs. The paper is structured as follows. Section 2 contains a description of the FE model used in this investigation and includes a mesh convergence study. Section 3 contains the results that have been calculated using the FE model and is accompanied with a discussion of the findings. Finally, the conclusions of this investigation are presented in Section 4.

1 j.cheer@soton.ac.uk

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

2. MODEL DESCRIPTION

In this investigation, an FE model has been used to investigate the di ff erent ABH designs that can be applied to a curved plate. The base plate design consists of a constant thickness plate with a curvature in one dimension with a radius of 0.5 m. The dimensions of the model are listed in Table 1 and a picture of the completed model is shown in Figure 1. The plate has been designed to match the size

Table 1: Dimensions of the pipes used in the experimental setup.

Variable Value

l plate 475 mm

w plate 375 mm

h plate 6 mm

r plate 500 mm

r abh 50 mm

l abh 275 mm

h min 0.5 mm

50 mm

µ 3

∆

∆

+

* !"#$%

& !"#$%

' !"#$%

ℎ !"#$%

Figure 1: A diagram of the FE model, showing the curved plate structure.

of the plate used in [27], however in this investigation curvature has been introduced. A total force of 1 N has been applied near one corner of the plate, directed towards the origin of the curvature radius. The boundary conditions of the plate have been set to free on all sides. Three di ff erent ABH designs have been integrated into the plate, resulting in a total of four models including the constant thickness plate. To implement these ABHs, two di ff erent power law profiles

are required. The first is the standard ABH profile, defined as

! µ + h min , (1)

h flat ( r ) = ( h plate − h min ) r

r abh

which runs along the flat cross-section of the plate. The second ABH profile is slightly more complex, as the curvature of the plate must be taken into account in order to preserve the power law profile along the curved cross-section. The modified profile is defined as

! µ + h min

h curve ( r ) = ( h plate − h min ) r plate r

! cos( r ) . (2)

r abh

The radius of each ABH has been set to 5 cm, equating to an ABH cut-on frequency of approximately 830 Hz, which has been calculated using the method outlined in [25, 26]. The first ABH design is a traditional circular embedded ABH, which has been defined by lofting the power law profile given in Equation 1 90 degrees to the modified power law profile given in Equation 2, via an arc along the surface of the curved plate. A mirroring function has then been used to create the other 270 degrees of the ABH. The finalised model is shown in Figure 2(a) where the ABH is located in the centre of the plate. The second ABH design is a longitudinal ABH, which has been implemented by extending the power law profile given in Equation 2 along the length of the plate, perpendicular to the curvature. The ABH is located in the centre of the plate and has a total length of 275 mm, which has been chosen so that the ends of the ABH are at least a distance of r abh away from the edges of the plate. The finalised model is shown in Figure 2(b). The third ABH design is an annular ABH, which has been implemented by rotating the power law profile given in Equation 1 around the curvature of the plate to give a total ABH length of 275 mm. The length of the annular ABH has been kept the same as the length of the longitudinal ABH so that the two are comparable. The finalised model is shown in Figure 2(c). For each ABH plate configuration, a thin damping layer with a thickness of 0.5 mm has been attached to the surface of the ABH. This has been highlighted in orange in Figure 2. The damping material has been modelled based on Henley’s yellow compound [28], which has been used experimentally in [27,29].

* #&'

& #&'

* #&'

& #&'

* #&'

(a)

(b)

(c)

Figure 2: The curved plate with an embedded (a) circular ABH, (b) longitudinal ABH and (c) annular ABH. A damping layer, highlighted in orange, has been attached to the surface of each ABH.

Each of the four plate configurations have been meshed using quadratic mesh elements and a convergence study has been carried out where the total kinetic energy of the plate has been calculated for a range of maximum element sizes. The study has been carried out at the highest frequency of interest, 10 kHz, and the maximum element sizes have been defined with respect to the flexural

-110

Plain Plate Longitudinal ABH Annulus ABH Circular ABH

Total Kinetic Energy (dB wrt. 1)

-115

-120

-125

-130

-135

-140

-145

-150

0 1 2 3 4 5 Minimum number of elements per wavelength

Figure 3: The mesh convergence study, showing how the minimum number of elements per wavelength a ff ects the total kinetic energy of the system when excited at 10 kHz.

wavelength in the plate and ABH at 10 kHz. The results are presented in Figure 3. From these results, it can be seen that the total kinetic energy for all models has converged when the minimum number of elements per wavelength is 3. Therefore, the mesh for each plate configuration has been set to contain at least 3 elements per wavelength. The following section contains the results from an investigation into the total kinetic energy of each plate over frequency.

3. RESULTS

The total kinetic energy for each of the ABH plate configurations in Figure 4 has been calculated and presented over frequency with respect to the constant thickness plate for clarity. From the results shown in Figure 4(a), it can be seen that there are many resonances in the kinetic energy response of the undamped curved plate and these resonances span the full frequency bandwidth examined. This is expected due to the lack of damping treatment. The results presented in Figure 4(a) also show that the addition a circular ABH provides significant damping at many of the resonances above the cut-on frequency of the ABH (830 Hz), however there is also enhancement at approximately 257 Hz, 507 Hz, 3.369 kHz and 4.501 kHz. Overall, there is a slight shift down in the resonance frequencies, indicating that the addition of the circular ABH slightly reduces the sti ff ness of the structure. From the results shown in Figure 4(b), it can be seen that the addition of the longitudinal ABH considerably dampens all of the resonance frequencies above the cut-on frequency of the ABH. In addition, there is also some damping achieved below the cut-on frequency of the ABH, which may be partially due to the larger area of damping material which covers the ABH. It can be seen that the ABH is less e ff ective at particular frequencies, such as 2.385 kHz and 4.540 kHz. Examining the structural modes at these frequencies shows that the nodal lines are oriented perpendicularly to the length of the longitudinal ABH, which limits the energy entering the ABH. The longitudinal ABH also produces the largest shift down in resonance frequencies, indicating that it reduces the sti ff ness of the structure more than the other ABH designs. This could be due to the ABH spanning most of the plate width, compared to the other designs considered. From the results presented in Figure 4(c), it can be seen that the annular ABH also produces considerable damping above the cut-on frequency of the ABH. Compared to the longitudinal ABH, the annular ABH has a slightly poorer performance at lower frequencies but is more e ff ective at controlling the resonances at 2.385 kHz and 4.540 kHz. At these frequencies, the nodal lines of the modes are parallel to the length of the annular ABH, causing flexural displacement in the tapering section. However, the annular ABH is less e ff ective when the modal nodes are oriented perpendicularly to its length as seen, for example, at 4754 Hz. Similarly to the other ABH designs, the annular ABH produces a slight downward shift in the resonance frequencies, which is more than the circular ABH and less than the longitudinal ABH. Overall, the longitudinal and annular ABHs are

-20

-20

Plain Plate Circular ABH

Plain Plate Longitudinal ABH

Total Kinetic Energy (dB wrt. 1)

Total Kinetic Energy (dB wrt. 1)

-40

-40

-60

-60

-80

-80

-100

-100

-120

-120

-140

-140

-160

-160

0 2000 4000 6000 8000 10000 Frequency (Hz)

0 2000 4000 6000 8000 10000 Frequency (Hz)

(a)

(b)

-20

Plain Plate Annulus ABH

Total Kinetic Energy (dB wrt. 1)

-40

-60

-80

-100

-120

-140

-160

0 2000 4000 6000 8000 10000 Frequency (Hz)

(c)

Figure 4: The total kinetic energy of the constant thickness curved plate compared to the sample plate with an embedded (a) circular ABH, (b) longitudinal ABH and (c) annular ABH.

more e ff ective than the circular ABH, although this may be in part due to their larger size. The choice between the longitudinal and annular styles will depend on the orientation of the flexural modes excited by the disturbance.

4. CONCLUSIONS

An investigation into a singularly curved plate with three di ff erent embedded ABH designs has been presented in this paper. It has been shown that, although a standard circular ABH is e ff ective, longitudinal or annular ABHs produce better damping over the frequency range examined. Longitudinal ABHs are slightly limited in damping flexural vibrations when the nodal lines are oriented perpendicularly to the length of the longitudinal ABH and annular ABHs are slightly limited in damping vibrations when the nodal lines are oriented perpendicularly to the length of the annular ABH. This investigation has shown that the orientation of the flexural modes in a plate can determine whether a longitudinal or annular ABH is more suited to tackle a particular vibration problem. It may also be beneficial to combine the two approaches.

ACKNOWLEDGEMENTS

This work was supported by the Intelligent Structures for Low Noise Environments (ISLNE) EPSRC Prosperity Partnership (EP / S03661X / 1).

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