Sound isolation via temporal and spatial configuration of piezoelectric elements Xiang Liu 1 Department of Mechanical Engineering, The University of Hong Kong Pokfulam, Hong Kong, China Chunqi Wang Lab of Aerodynamics and Acoustics, HKU-Zhejiang Institute of Research and Innovation Dayuan Road, Hangzhou, Zhejiang, China Yumin Zhang School of Mechatronic Engineering and Automation, Foshan University Nanhai District, Foshan, Guang Dong, China Keming Wu Department of Mechanical Engineering, The University of Hong Kong Pokfulam, Hong Kong, China Lixi Huang Department of Mechanical Engineering, The University of Hong Kong Pokfulam, Hong Kong, China

ABSTRACT To adapt to different incident wave spectra, a time-domain design approach is proposed using the met- al-oxide-semiconductor field-effect-transistor (MOSFET) in the shunt circuit to form a temporal modu- lated material. This new composite material consists of a lightweight structure attached with shunted PZT patches, and the time sequence of the working circuit is controlled with the MOSFET. The materi- al is designed to improve the sound isolation band by optimizing the time sequence of multiple resonant shunts working at different frequencies. The results show that the time-varying shunted PZT can make the electrical resonances effective at multiple frequencies, and the total absorbed energy is distributed to multiple frequencies to broaden the effective bandwidth. Hence, broadband sound isolation is achieved with the proposed temporal modulated material, which is lightweight and without a sensor.

1. INTRODUCTION

The compact and lightweight structures are increasingly used in the instrument, architecture, and au- tomobiles, so intensive work is being devoted to the sound and vibration control of lightweight devices. The lightweight structure with good sound isolation performance is the new requirement for composite material. However, this requirement is very hard because normally a heavier material has a better

1 xliu123@connect.hku.hk

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sound isolation performance. The nature mode of the structure gives rise to a dip in the transmission loss (TL) spectrum at the natural frequency because the structure vibrates intensively and radiates most of the sound energy to the other side. The level of the dip in the TL spectrum is influenced by the damping of the system.

In an effort to reduce the sound transmission through the lightweight structures and suppress the vi- bration of the structures, different kinds of passive resonators are conceived and attached to the struc- tures [1,2]. The acoustics metamaterials (AMM) are also designed as local resonators to achieve dis- tinctive performance [3]. The sound absorption of the structures is improved with AMM [4,5], and they are more and more widely used. The specially designed resonators of AMM are also studied to improve the sound isolation [6,7].

Another promising manner to construct the local resonators is using the shunted piezoelectric (PZT) elements. By attaching PZT patches to the structure, and shunting a circuit to control the PZT, the vi- bration response of the structure can hence be controlled. This method was first proposed by Forward et al. [8] to improve the system damping, and its mechanism of vibration-energy absorption has been well studied [9–11]. More complicated PZT arrays have also been proposed to improve the vibration control [12]. It has also been applied to energy harvesting [13,14] and wave control in waveguides [15,16].

In this study, the PZT arrays are attached to the lightweight structure to improve the sound isolation performance. Unlike the previous vibration control with PZT arrays, the PZT arrays in this chapter are designed with different temporal and spatial configurations. The temporal configuration of the temporal modulation material is controlled with the metal-oxide-semiconductor field-effect-transistor (MOSFET) in the circuit of each shunted PZT patch. The spatial configuration is achieved by designing the pattern of the PZT resonators with different resonant frequencies. In the simulation, only one unit of the material element is simulated, and the periodic boundary condition is applied to compute the acous- tic response of the entire composite material. The incident and transmitted sound waves are also in- cluded in the model. The purpose of this work is to study the sound isolation performance of the smart materials attached with different kinds of shunted PZT arrays. The semi-passive shunt circuits are con- nected to the two poles of PZT patches to function as some local electrical resonators to improve the sound isolation. The circuit is composed of operational amplifiers and passive analog circuit elements, so the system is always stable, and the power consumption is low. The MOSFET devices are used in the shunt circuits connected to the PZT patches, so the time sequence of the working circuit is con- trolled with the MOSFET devices in each resonant shunt. The resonance of PZT patches is controlled with electrical circuits, so the acoustic performance in the low-to-medium frequency range can be con- trolled easily by adjusting the electrical elements. No extra volume or weight is required. The temporal modulation material and spatial configuration method proposed in this study have many potential ap- plications in environmental acoustics, precision instrumentation, automotive sector, and aerospace technologies.

2. MODELLING

As shown in Figure 1 (a), the smart composite material consists of a lightweight structure attached with shunted piezoelectric ceramic (PZT) patches. The PZT patches are marked with blue. The single- layer material is studied to improve the sound isolation band by optimizing the time sequence of multi-

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ple resonant shunts working at different frequencies. The model is constructed in the time domain to simulate the time-domain modulation results. Because of the periodic configuration of the PZT patches, only one unit in the PZT array needs to be simulated. As shown in Figure 1 (b), the size of the single unit of the material is marked, the dashed lines represent the acoustic domains on both sides of the composite structure.

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Figure 1: The configuration of the smart composite material. (a) the smart composite material with PZT arrays. (b) the single unit of the material to be simulated.

The acoustic domains on both sides of the smart composite material are governed by the acoustic wave equation

𝜕 2 𝜙

1 𝑐 0

𝜕 2 𝑡 −∇ 2 𝜙 = 0, (1)

2

where 𝜙 is the velocity potential, and 𝑐 0 is the speed of sound in the air. The sound pressure 𝑝 and acoustic particle velocity 𝒗 is calculated as

𝜕𝜙

൝ 𝑝= −𝜌 0

, (2)

𝜕𝑡 𝒗= ∇𝜙

where 𝜌 0 is the density of air. A pulse signal is prescribed as 𝑝 in to eject to the smart composite materi- al. The local equilibrium equations of the solid domain are given by

𝜎 𝑖𝑗,𝑗 = 𝜌 𝑠 𝑢 𝑖 ሷ (3) where 𝜎 𝑖𝑗 is the stress tensor, 𝜌 𝑠 is the mass density of the structure, and 𝑢 𝑖 is the displacement field. The continuum kinematic condition between solid and acoustic domains on both sides of the composite structure is:

𝜎 𝑖𝑗 𝑛ሬԦ = 𝑣 𝑧 ȁ 𝑧=0+ = 𝑣 𝑧 ȁ 𝑧=0− , (4) where 𝑣 𝑧 is the z -direction component of acoustic particle velocity 𝒗 . The governing equations of pie- zoelectric material and the coupling with shunt circuit are introduced in references [17]. To use the single unit in Figure 1 (b) to simulate the large composite structure in Figure 1 (a), the periodic bounda- ry conditions are applied on the acoustic domains in Figure 1 (b). Hence the sound pressure and acous- tic particle velocity at the front and back sides of the acoustic domains follow

𝜕𝜙 𝐵

𝜕𝑥 = 𝜕𝜙 𝐹

𝜕𝑥 𝑒 −𝑗𝑘 𝑥 𝑊 s , (5a)

𝜙 𝐵 = 𝜙 𝐹 𝑒 −𝑗𝑘 𝑥 𝑊 s , (5b) and at the left and right sides of the acoustics domains follow

𝜕𝜙 𝐿

𝜕𝑦 = 𝜕𝜙 𝑅

𝜕𝑦 𝑒 −𝑗𝑘 𝑦 𝐿 s , (6a)

𝜙 𝐿 = 𝜙 𝑅 𝑒 −𝑗𝑘 𝑦 𝐿 s , (6b) where the subscripts ‘B’, ‘F’, ‘L’, and ‘R’ refer to the back, front, left, and right sides respectively, 𝑘 𝑥 and 𝑘 𝑦 are x and y components of wavenumber in air, and 𝑗= ξ−1 is the imaginary unit. The total sound pressure 𝑝 at the inlet domain above the structure contains the incident wave 𝑝 in and reflected wave 𝑝 r . The incident wave is already fully known, so the reflected wave can be determined by

𝑝 r = 𝑝−𝑝 in . (7) The governing equations and boundary equations defined above are solved with the commercial software of COMSOL Multiphysics.

The energy absorption, reflection, and transmission coefficients are the same as reference [18]. They are calculated with the incident wave 𝑝 in and reflected wave 𝑝 r in the acoustic domain above the struc- ture, and the transmitted wave 𝑝 out in the acoustic domain beneath the structure. The transmission loss TL is determined by the amplitudes of the transmitted wave 𝑝 out and the incident wave 𝑝 in as follows

TL = 20log [ȁ𝑝 in ȁ/ȁ𝑝 out ȁ] . (8)

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3. NUMERICAL RESULTS

3.1 Basic results of PZT resonator

With the above FEM model, the transmission response of the smart composite material attached with PZT resonators can be calculated. The PZT patch is shunted with a resistance-inductance (RL) circuit to work as a resonator and attached to a plate. The parameters of the host plate are listed in Ta- ble 1. The parameters of the PZT patch are listed in references [17].

Table 1: The parameters of the host plate.

Parameters Value

Thickness 𝑡 p (mm) 0.5

Length 𝐿 p (mm) 100

Width 𝑊 p (mm) 40

Density (kg/m 3 ) 2700

Young’s modulus (GPa) 69

Poisson’s ratio 0.34

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The simulation results are shown in Figure 2. Changing the inductance value in the shunt circuit will influence the tuning frequency of ER. This is because the resonance of the PZT resonator is from the tuning between the inductance in the shunt circuit and the capacitance of the PZT patch. When the in- ductive reactance in the shunt circuit cancels with the capacitive reactance of the PZT patch, the current of the circuit reaches the maximum value, and a large amount of energy is transferred to joule heat with the electrical damping. Such ER will negotiate the TL dip caused by the natural mode of the panel. But the anti-resonances of the PZT resonator give rise to two new dips at both sides of the ER frequency. This is because the PZT patches work like local dynamic absorbers on the composite plate, and the anti-resonance will happen when a new resonator is added.

TL(AB) 100150200250 300350 400450500 Frequency

Figure 2: Turing the electrical resonance to different frequencies.

3.2 Temporal configuration

The MOSFET devices are connected in each shunt circuit to control the time-domain modulation and hence to construct the proposed temporal modulation material. The modulation of the time se- quence is shown in Figure 3. In a time-modulation period T, multiple shunts (for example, shunts A and B in Figure 3) work in different time segments in a period. The duty ratio 𝛽 is the percentage of total modulation time for the corresponding shunt to be switched on and can be optimized to improve the performance. More time-switching branches can be added when needed.

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Figure 3: The modulation of the time sequence.

The working frequency of the temporal modulation material is controlled by the inductance values in each shunt, the dissipation is controlled by the electrical resistors, and the percentage of working time of each shunt is controlled by the duty ratios.

When there are two time-varying shunts working in the shunt circuits connected to each PZT reso- nator, the energy absorption happens in turns between these two branches. The inductance values, elec- trical resistance values, and duty ratio are optimized to improve the sound isolation performance. The optimization is conducted with the sequential quadratic programming (SQP) algorithm in MATLAB. The total noise reduction is defined as the difference between inlet the outlet sound pressure levels (SPL) and is used as the objection function in the optimization. The optimization result is shown in Figure 4 and compared with the shunt-off result. The corresponding optimized electrical parameters are: 𝐿 1 = 9.3H, 𝐿 2 = 15.1H, 𝑅 1 = 1182 Ω , 𝑅 2 = 1225 Ω . The duty ratios of each shunt are: DtRt1=0.58, DtRt2=0.42. The modulation frequency fm is 200 Hz, and hence the total modulation time T = 1/fm = 1/200 s. It can be observed that the dip in the TL spectrum is improved in a wide frequency range. The anti-resonance between the PZT resonator and the composite structure is relieved. As a compromise, the peak of ER is also lowered. The noise reduction improvement compared with the shunt-off state is 3.2dB.

Work oe Closed Work feos Closed! 0 ar or

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Figure 4: The optimization result of the temporal modulation material with two switching branches. It is compared with the shunt-off result.

The optimization result for three time-varying branches is shown in Figure 5 and compared with the shunt-off result. It is optimized in the same manner introduced above. The optimized electrical parame- ters are: 𝐿 1 = 9.4H, 𝐿 2 = 15.3H, 𝐿 3 = 6.2H, 𝑅 1 = 1232 Ω , 𝑅 2 = 1192 Ω , 𝑅 3 = 1153 Ω . The duty ratios of each shunt are: DtRt1=0.31, DtRt2=0.28, DtRt2=0.41. The modulation frequency fm is also 200 Hz, so the total modulation time T = 1/fm = 1/200 s. It can be observed that the sound isolation band reflected by TL is further widened but lowered. This is because multiple ERs are adjusted to a wider frequency range, and the sound absorption happens in turn at these frequency components. So, the absorption is split into more frequencies, and the absorption at each frequency is decreased. Consequently, the sound isolation improvement is wider and lower. The improvement of noise reduction is 3.1dB.

TL(AB) 1e switch 150-200-250 «300-350-400 450500 Frequency

Figure 5: The optimization result of the temporal modulation material with three switching branches. It is compared with the shunt-off result.

TL(AB) 1e switch 150-200-250 «300-350-400 450500 Frequency

The power fraction of the two-branches optimization result is shown in Figure 6. The absorption mainly happens at two designed ER peaks, and the level of absorption between the two peaks is also relatively high. Such wide absorption gives rise to the improvement of the sound isolation at the dip in the TL spectrum.

Figure 6: Power fraction of absorption, transmission, and reflection of the optimization result.

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3.3 Spatial configuration

The proposed temporal modulation material is compared with the shunted PZT without controlling of MOSFET devices and arranged in a spatial pattern. By placing the PZT resonators with different tuning frequencies in a periodic pattern on the composite materials, the sound isolation can also be made wider. Such improvement is from the optimized spatial configuration. In the simulation of the spatial configuration, two units of the smart structure demonstrated in Figure 1 (b) are used. These two shunted PZT are tuned at different frequencies, and the shunt circuit of each PZT only contains one a resistance-inductance (RL) branch without the MOSFET devices. The periodic boundary conditions are also applied on the four sides of the spatial configuration model, so it is equivalent to the smart struc- ture with the PZT arrays tuning at the two frequencies. Such spatial configuration is compared with the temporal configuration switching between two tuning frequencies. The parameters of the host plate and PZT patch are the same as the above temporal configuration simulation.

With the same optimization scheme of electrical parameters, the optimization result of the spatial configuration is found to be similar with the temporal configuration. The optimized two inductance values are 9.1 H and 15.6 H, and the resistance values are 1293 Ω and 1167 Ω . The results of optimized spatial configuration and temporal configuration are compared in Figure 7 together with the shunt-off result. The temporal and spatial configuration curves are similar, and the ER peaks of spatial configura- tion are slightly more distinguished. The spatial configuration also has a sound isolation improvement of 3.2 dB, which is the same as the temporal configuration.

es Power fraction 2 150 200 250 300 Frequency 350 sorption = = Transmission fl

Figure 7: The comparison between temporal configuration and spatial configuration together with the shunt-off result.

The reason why these two configurations can have similar performance can be seen from the flexur- al kinetic energy. The instantaneous total flexural kinetic energy 𝐾ሺ𝑡ሻ of both configurations can be expressed as follows

𝑁

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𝐾ሺ𝑡ሻ= 1

+ 1

2 d𝐴 p

2 d𝐴 pzt

(9)

2 න𝜌 p 𝑡 p 𝑢 𝑧

2 ෍න𝜌 pzt 𝑡 pzt 𝑢 𝑧

𝑗=1

𝐴 p

𝐴 pzt

where 𝜌 p , 𝑡 p and 𝜌 pzt , 𝑡 pzt are the density and thickness of the host plate and PZT patches respectively, 𝐴 p and 𝐴 pzt are the surface area of the host plate and PZT patches, 𝑢 𝑧 is the z -component of the vibra- tion velocity of the structure, and 𝑁 is the number of PZT patches. Hence the Power Spectral Density (PSD) 𝑆 𝐾 ሺ𝜔ሻ of the total flexural kinetic energy can be obtained by Fourier transform:

𝑁

𝑆 𝐾 ሺ𝜔ሻ= 1

+ 1

(10)

2 න𝜌 p 𝑡 p ℱ[ℛ 𝑢𝑢 ሺ𝜏ሻ]d𝐴 p

2 ෍න𝜌 pzt 𝑡 pzt ℱ[ℛ 𝑢𝑢 ሺ𝜏ሻ]d𝐴 pzt

𝑗=1

𝐴 p

𝐴 pzt

where ℱ[ ] is the Fourier transform, and ℛ 𝑢𝑢 ሺ𝜏ሻ is the autocorrelation function of structural velocity. Therefore, the kinetic energy spectrum can be obtained as shown in Figure 8. It can be observed that the temporal and spatial configurations reduce the kinetic energy of the structure compared with the shunt-off state. The integration of the kinetic energy of both configurations over the studied frequency range is the same, so the noise reduction performance is the same, and the noise radiated to another side of the structure is also the same.

TLB) ——$§ eemporal configuration spatial conf shunt-off 150 200 250 300 Frequency 350 400-450 500

Figure 8: The kinetic energy spectrum of the temporal configuration and spatial configuration are com- pared together with the shunt-off result.

The above analysis proves that the acoustic performance of the temporal and spatial are similar, but it is argued that the temporal configuration still proposes a new way to achieve the acoustic absorption at multiple frequencies to broaden the noise reduction band. In the practical use of the smart composite materials attached with PZT arrays, the temporal configuration method is easier to be adjusted. And in the simulation to predict and optimize the performance, the simulation of temporal configuration only needs to calculate one unit of the material, which saves more computation resources.

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3.4 Double layer configuration

The double-layer configuration of the proposed temporal modulation material is applied to further improve the sound isolation band. The parameters of the simulation are the same as the above temporal modulation. The two layers are separated by 40 mm. The effective frequency ranges of the first and the second layer are staggered. The result is shown in Figure 9. In the frequency range of 100~500Hz, double-layer temporal modulation material makes a 7.9 dB improvement compared with the shunt-off state. The structure of the two layers is the same, and hence their natural frequencies are the same. When the ER frequencies of the two layers are adjusted to the TL dip range, the sound isolation is fur- ther improved.

Kinetic energy 150 200 250 300 Frequency temporal configuration spatial configuration SS shunt-of? 350 400-450 500

Figure 9: The result of double layer configuration.

4. CONCLUSION

In summary, the time-varying method can make the PZT resonator effect at multiple frequencies, and the total isolated energy is distributed to multiple frequencies to broaden the effective bandwidth. The increase of time-varying resonant shunts makes the isolated energy distributed in each MOSFET- controlled resonant shunt decrease but works in a wider frequency range. It is found that the two- branches shunt circuit can work with a good sound isolation band when it is controlled with MOSFET devices in the time domain. Such time-domain modulation method is like the optimization of the spatial configuration of the PZT resonators. But the time-domain modulation is easier to adjust than the spatial configuration, and only the single unit of the material needs to be simulated in the optimization. The effective frequency ranges of the double-layer temporal modulation material are staggered, so the sound isolation band is further improved, and better performance is observed. These results show that the time-varying method split the over-killed single high resonant peak to broadband. Hence, broad- band sound isolation is achieved with the proposed temporal modulated material. The smart sound iso- lation material in this study is lightweight, and no sensor is used. So, it has many potential applications in environmental acoustics, precise instrument, automotive sector, and aerospace technologies.

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5. ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China projects 51775467 and 52105090, the general research fund project 17210720 from the Research Grants Council of Hong Kong SAR.

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