Sound Radiation from a Plate with Embedded Active Acoustic Black Holes

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) can be realised as tapered structural features that focus and absorb vibrational energy. In a plate, the damping e ff ect of ABHs can be used to reduce the radiation e ffi ciency, particularly above the critical frequency of the plate. However, at lower frequencies, there is still significant radiation due to the low order modes of the plate. It has previously been shown that the combination of active control and ABHs can result in an e ff ective broadband control solution with lower computational and power requirements compared to a standalone active configuration. This paper presents an investigation into how two di ff erent active control strategies a ff ect structural response and radiated sound power of a plate with active ABHs (AABHs). It is shown, through an active vibration control strategy and an active noise control strategy, that the AABHs can be used to successfully control either the global structural response of the plate or the radiated sound power. However, it is also shown that controlling either response can lead to an enhancement in the other.

1. INTRODUCTION

Acoustic black holes (ABHs) can be used to provide significant, lightweight, damping in both beams and plates due to the wave speed decrease that occurs along the length of the taper. A small amount of damping is usually required to achieve e ff ective absorption and the performance of an ABH increases with frequency, as the local modal density of the taper increases [1–3]. In plates, ABHs have been investigated through a variety of embedded designs [4–15], a selection of surface-attached vibration absorbers [16–21] and as part of compound or meta-structures [22–24]. In addition to the passive designs referenced above, Active Acoustic Black Holes (AABHs) have been investigated as a means to improve the broadband performance, which is typically limited at lower frequencies by the first local mode of the ABH [25, 26]. AABHs have been shown to provide e ff ective broadband vibration control in both beams and plates [27, 28]. It has also been shown that

1 j.cheer@soton.ac.uk

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

the power and computational requirements of an AABH are also lower compared to a purely active solution. However, sometimes it is desirable to control the sound radiated from a structure rather than the structural vibration. Therefore, this paper presents an investigation into the implementation of both an active vibration control strategy and an active noise control strategy and demonstrates the e ff ect that each control strategy has on both the global structural response and radiated sound power from a plate. The paper is organised in the following way. In Section 2, the experimental setup used in this investigation is described. The multichannel feedforward control strategies are then presented in Section 3 and the results from implementing these control strategies are presented in Section 4. Finally, the conclusions of this study are presented in Section 5.

2. EXPERIMENTAL SETUP

In this investigation, a plate with five ABHs has been used. The dimensions are listed in Table 1 and

Variable Value

l plate 475 mm ± 1 mm

w plate 375 mm ± 1 mm

h plate 6 mm ± 0.1 mm

r abh 50 mm ± 0.1 mm

h min 0.5 mm ± 0.1 mm

δ a 117.5 mm ± 1 mm

δ b 187.5 mm ± 1 mm

δ c 109.5 mm ± 1 mm

δ d 134.5 mm ± 1 mm

100 mm ± 1 mm

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

a diagram is presented in Figure 1. The layout of the ABHs has been selected to remain consistent with a previous study [28]. The profile of each ABH has been defined by rotating the power law taper

! µ + h min , (1)

h ( r ) =  h plate − h min  1 − r r abh

by 360 degrees and embedding this into the plate. Five circular piezoelectric patches with the same operating properties as the PI876-A12 patch described in [29] have been attached to the flat side of the plate, co-located with the ABHs. An inertial actuator has been attached on the opposite side of the plate, as shown in Figure 1. 12 accelerometers have been attached to the flat side of the plate using wax in a 4 × 3 grid with a separation of 100 mm, which corresponds to a flexural wavelength of 5.5 kHz in the plate. The plate has then been mounted on a perspex box using a metal edge clamp, which is shown in Figure 2. A Microflown pu-probe [30] has been set up 10 cm above the mounted plate and used to measure the acoustic pressure and particle velocity above each of the accelerometers, thus giving a 4 × 3 grid of acoustic measurements. The acoustic spacing corresponds to a wavelength of 3.43 kHz in air. The primary structural responses of the plate have been measured by driving the inertial actuator with white noise using a sampling frequency of 20 kHz and measuring the acceleration using the grid of 12 accelerometers. The primary pressure and particle velocity responses of the plate have been measured using the 12 pu-probe locations. The frequency responses between the input signal

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# -

# ,

PZT PZT

ABH ABH

) !"#$%

PZT

ABH

PZT

PZT

∆

ABH ABH Primary Actuator

# #

∆

* !"#$%

# &

Figure 1: A diagram of the plate design showing the ABH locations, piezoelectric patch (PZT) locations and primary actuator location. The ABH geometry is also presented.

to the actuator and each of the signals measured at an accelerometer or at a pu-probe location have been calculated using the H1-estimator. The secondary structural responses have been measured by individually driving each of the piezoelectric patches with white noise using a sampling frequency of 20 kHz and measuring the acceleration at each of the accelerometers. The secondary pressure and particle velocity responses have been measured by individually driving each piezoelectric patch and measuring the responses at each pu location using the pu-probe. The secondary frequency responses between the input signal used to drive each piezoelectric patch and each of the signals measured at an accelerometer or at a pu-probe location have also been calculated using the H1 estimator. Low pass anti-aliasing and reconstruction filters have been used in this investigation with a cut-o ff frequency of 10kHz, which is also the cut-o ff frequency of the pu-probe. The piezoelectric patches have a cut-on frequency of 250 Hz. Therefore, these two limits have been used to set the frequency range of interest for this investigation.

3. CONTROLLER FORMULATION

The multichannel control system described in Section 2 can be represented by the block diagram shown in Figure 3. A tonal controller has been used in this investigation, acting at the frequency ω 0 . A subscript s has been used to refer to the structural signals, a subscript p has been used to refer to the pressure signals and a subscript v has been used to refer to the particle velocity signals. In this system the vector of M complex control signals, u ( ω 0 ), is multiplied by the matrix of L × M complex frequency responses, which is G s for the structural sensors and G pv for the pressure and particle velocity sensors. The resulting signals are summed with their respective vectors of disturbance signals to obtain the respective vectors of error signals, which can thus be expressed as

e s ( ω 0 ) = d s ( ω 0 ) + G s ( ω 0 ) u ( ω 0 ) (2)

and e p + v ( ω 0 ) = d p + v ( ω 0 ) + G p + v ( ω 0 ) u ( ω 0 ) , (3)

where e p + v = e p + e v , d p + v = d p + d v and G p + v = G p + G v .

pu probe

Accelerometers

Stinger mount

Piezoelectric Patches

Perspex Box

Shaker

(a)

(b)

Figure 2: (a) A photo of the experimental setup and (b) a diagram of the experimental setup showing all components.

In this investigation, two di ff erent cost functions are considered, and therefore two di ff erent tonal feedforward controllers are required. The first works to minimise the summation of the mean-squared structural error signals, given by the cost function

L X

l = 1 | e sl ( ω 0 ) | 2 = e s ( ω 0 ) H e s ( ω 0 ) , (4)

J s ( ω 0 ) =

where H denotes the hermitian transpose. The second works to minimise the summation of the mean- squared pressures and particle velocities, given by the cost function

L X

e pl ( ω 0 ) + e vl ( ω 0 )  2 = e p + v ( ω 0 ) H e p + v ( ω 0 ) , (5)

J p + v ( ω 0 ) =

l = 1

Since the system is overdetermined, it will not be possible to completely minimise either of these cost functions. However if it is assumed that G H G is positive definite, di ff erentiating each cost function with respect to the real and imaginary parts of each component of u and setting the derivative to 0 gives the optimum vector of control signals that can be used to minimise each cost function [31]. These are given as u opt s ( ω 0 ) = − [ G s ( ω 0 ) H G s ( ω 0 )] − 1 G s ( ω 0 ) H d s ( ω 0 ) , (6)

which can be used to control the structural response and

u opt p + v ( ω 0 ) = − [ G p + v ( ω 0 ) H G p + v ( ω 0 )] − 1 G p + v ( ω 0 ) H d p + v ( ω 0 ) , (7)

which can be used to control the sum of the pressures and particle velocities. Although the optimal control signal could be reached using an adaptive algorithm, such as the FxLMS algorithm, u opt

has been implemented directly so that any limitations imposed by convergence of an adaptive implementation are negated.

4. RESULTS

The optimum control signals presented in Section 3 have been implemented o ffl ine over a frequency range of 0 – 10 kHz and the results are presented in Figure 4 in terms of the sum of the squared accelerations and an estimate of the radiated sound power. ‘AVC’ (Active Vibration Control) has

𝒅 ! (𝜔 " )

𝒖(𝜔 " )

𝒆 ! (𝜔 " )

∑

𝑮 ! (𝜔 " )

𝒅 # (𝜔 " )

𝒆 # (𝜔 " )

∑ 𝑮 # (𝜔 " )

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Figure 3: A block diagram showing the multichannel control system operating at a single frequency ω 0 .

been used to refer to the case when the structural vibration is controlled and ‘p + v’ has been used to refer to the case when the sum of the pressures and particle velocities is controlled. From the results presented in Figure 4(a), it can be seen there are 3 strongly defined resonances in the uncontrolled structural response at 342 Hz, 624 Hz and 980 Hz, followed by a cluster of resonances at 1510 Hz, 1620 Hz, 1884 Hz and 2230 Hz. When AVC is implemented, the structural response is controlled at most frequencies. Particularly, the 624 Hz resonance is damped by approximately 20 dB. Control is slightly less e ff ective at some of the resonances which could be due to the placement of the control sources. At higher frequencies above 7600 Hz, control is fairly limited which is likely due to the number and spacing of both the error sensors and the control sources. Overall, the mean broadband structural response is reduced by 3 dB when AVC is implemented. From the results presented in Figure 4(b), it can be seen that there are strong resonances in the uncontrolled sound power at 342 Hz, 624 Hz and 980 Hz. Above these, there are less well defined resonances at approximately 1274 Hz, 1440 Hz, 1884 Hz and 2230 Hz. Although most of these resonances are also present in the structural response of the plate, when AVC is implemented it can be seen that the radiated sound power at the first 5 resonance frequencies is not controlled. There is a reduction at both 1884 Hz and 2330 Hz of up to 8 dB, however above these frequencies there is enhancement of up to 15 dB. Overall, the mean broadband sound power is increased by 4 dB when AVC is implemented. If the sum of the pressures and particle velocities is controlled rather than the structural response, the results presented in Figure 4(b) show that significant control is achievable at lower frequencies, particularly at the 342 Hz, 624 Hz and 980 Hz resonances where up to 36 dB of attenuation occurs. At higher frequencies, there is a roll o ff in performance which could be attributed to the number and spacing of the pu-probe measurements. The mean broadband sound power is reduced by approximately 3 dB. From the results presented in Figure 4(a), it can be seen that controlling the sum of the pressures and particle velocities produces an enhancement in the structural response, particularly below the first resonance at 342 Hz. Between 342 Hz and 1300 Hz there is significant enhancement, especially o ff resonance and the first frequency where a reduction in the structural response is achieved is at 2230 Hz. The mean broadband structural response is enhanced by approximately 3 dB. Overall, these results have shown that controlling the structural vibration results in an enhancement in the radiated sound power and controlling the pressure and particle velocity results in an

60

100

Control off AVC Control on p+v Control on

Control off AVC Control on p+v Control on

Structural Response (dB wrt. 1)

Estimated SWL (dB wrt. 10 -12 )

55

90

50

80

45

70

40

60

35

50

30

40

25

20

30

0 2000 4000 6000 8000 10000 Freqeuncy (Hz)

0 2000 4000 6000 8000 10000 Freqeuncy (Hz)

(a)

(b)

Figure 4: (a) The structural responses and (b) The estimated sound power radiated from the plate without control (thick solid grey line), with AVC (thin solid blue line) and with pressure and particle velocity control (thin dashed red line).

enhancement in the structural vibration. the only exceptions are resonances around 1884 Hz and 2230 Hz, where some degree of control can be achieved across both error signals. Further work is required to determine why the structural response and radiated sound power do not appear to be strongly coupled at low frequency resonances since controlling low order structural modes typically results in a reduction in the radiated sound power at these frequencies.

5. CONCLUSIONS

This paper has shown the di ff erence in performance when a plate with 5 embedded AABHs is subjected to an active vibration control strategy or an active noise control strategy. The results have demonstrated that there is a performance trade-o ff between the two control strategies. When the active vibration control strategy was implemented, the overall structural vibration level was reduced by 3 dB and the most significant reductions were achieved at the lower resonance frequencies. At higher frequencies, where fewer resonances were identified, the level of control is reduced, which could be due to the placement and spacing of the structural sensors and piezoelectric actuators. However, the implementation of this active vibration control strategy resulted in an enhancement of the radiated sound power by approximately 4 dB. Over frequency, this enhancement is particularly prevalent above 3 kHz, where the structural and acoustic resonances are less defined. The results have also demonstrated that if the pressure and particle velocity are controlled, large reductions in the radiated sound power can be achieved, particularly below 3 kHz. At higher frequencies, control is more limited which could also be due to the placement and spacing of the acoustic sensing locations and piezoelectric actuators. Overall, a 3 dB reduction can be achieved in the radiated sound power, however, this results in an enhancement of 3 dB in the structural response. Whilst there are a small number of frequency bands where either control strategy results in a reduction of both the structural response and radiated sound power, in general a particular control strategy is required to achieve a reduction in the respective response and so in practice the choice of control strategy will depend on the application.

ACKNOWLEDGEMENTS

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

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