The influence of thermo-hygrometric conditions on metamaterials’ acoustic performance: an investigation on a 3-D printed coiled-up resonator

Matteo Cingolani 1 Department of Industrial Engineering, Università di Bologna Viale Risorgimento 2, Bologna, 40136, Italy

Gioia Fusaro 2 Department of Industrial Engineering, Università di Bologna Viale Risorgimento 2, Bologna, 40136, Italy

Massimo Garai 3 Department of Industrial Engineering, Università di Bologna Viale Risorgimento 2, Bologna, 40136, Italy

ABSTRACT In the last decades, coiled-up resonators have become popular within the metamaterial research community for narrow band, low frequency resonances combined to subwavelength thickness. Such structures are particularly suited to one of the most widespread manufacturing processes, i.e. PET-based 3D printing. Acoustic performance of coiled-up resonators depends on the geometrical parameters’ variation, which is influenced by thermo-hygrometric conditions; however, the deformation itself needs to be further investigated. For this reason, the present paper evaluates the correlation between temperature, relative humidity, and the geometrical parameters’ (spiral length and hole diameter) deformations and, consequently, the acoustic performance of a 3D printed coiled-up resonator. A combined approach through analytical, numerical, and experimental measurements quantified the coefficient increasing the temperature (T = 10 – 50 °C) , and the relative humidity (RH = 20 – 50 – 80 %) of the samples. Relative humidity variations turned neglectable discrepancies on sound absorption’s peaks. On the other hand, the increase in temperature caused a frequency peaks’ shift following an exponential trend. This study can be a starting point for practical applications when the thermos-hygrometric variations are of concern.

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1. INTRODUCTION

Metamaterials have become more and more attractive in a wide range of scientific fields. Also in acoustics, the interest in metamaterials is quickly growing due to the variety of possible applications,

1 matteo.cingolani6@unibo.it

2 gioia.fusaro@unibo.it

3 massimo.garai@unibo.it

«te a at inter.noise ausust scornsh eve er ca casPus

from room acoustics to environmental noise control. AMMs are basically structures made by holes, channels, or resonant cavities [1-4], with specific absorption properties that go beyond the λ/ 4 mechanism and overcome the limitations of classic fibrous and porous materials. Coiled-up resonator is a specific kind of AMM based on the Helmholtz’s resonance phenomenon and its narrow band acoustic behavior is quite sensitive to variation of the geometrical dimensions. The geometry can be affected by deformations due to thermo-hygrometric variations [5] and the unwanted deformations can alter, consequently, the acoustic properties which the metamaterial had been designed for. Previous studies already investigated the relationship between tempera ture fluctuations and AMMs’ sound absorption [6,7] but experimental test focusing on thermal and hygrometric conditions could highlight a suitability and customizability of such systems in engineering applications.

The present paper investigates how thermo-hygrometric variations affect the coiled up resonator geometry [8]. First, the analytical model has been used to design a specimen which has been manufactured through PETG 3-D printing. Then, a climate chamber has been employed to alter the specime n’s temperature (T = 10 – 50 °C) and relative humidity (RH = 20 – 50 – 80 %) The alterations in terms of frequency and amplitude of the resonance peaks were reported to quantify the effects of deformations on the sound absorption properties of the metamaterial under study. The outcomes can be a preliminary study for the tuning of AMMs in practical applications when thermo-hygrometric variations are of concern.

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2. METHOD

2.1. Analytical model The coiled-up resonator is made up of a spiral tube embedded in a rigid circular disk and covered by a front panel and a back terminal layer, as shown in Figure 1. The section of the perforation and the spiral is square. The geometrical parameters the acoustical performances depend on are L t the length of the spiral, 2 a the side of the square perforation, and R 0 the radius of the circular sample. The sound

Figure 1: Exploded view of the coiled up resonator: the spiral length L t , the side of the square hole 2 a , and diameter of the sample 2 R 0 are highlighted.

absorption coefficient of the coiled-up resonator is defined as:

1 Z in − Z 0 .

2

(1)

– Z in + Z 0 .

α n =

where Z 0 the characteristic impedance of air. Z in is the input specific acoustic impedance, defined as:

Z in = Z t

Z t = (2 a ) 2

(2)

ϕ

π R 2

0

where ϕ is the porosity of the front panel and Z t is the acoustic impedance at the opening of a tube with rigid termination (8) which depends on the spiral length L t , the effective density ρ ( ω ) and the effective compressibility C ( ω ).

2.2. 3-D printing

Additive manufacturing is a widespread method used in acoustics to design metamaterial’s geometries because of affordable costs and wide variety of polymers that can be used. The coiled-up resonator has been manufactured through Fuse Deposition Modelling (FDM) using PETG. Different printing speeds and different printing qualities have been considered during the manufacturing process. Some printing parameters were kept constant: the extruder temperature (225 ◦ ), the platform temperature (80 ◦ ), the infill percentage (30%) and the infill density (2 mm/s). For all the other printing parameters, the finest setups of the printer have been used to obtain as much accuracy as possible. The variable printing parameters are summarized in Table 1 To obtain the most accuracy

Table 1: Values of the variable printing parameters: the slow-fine combination of parameters has been used to print the specimen.

Printing speed Slow | Fast

Base speed (mm/s) 50 | 80

Travel speed (mm/s) 70 | 100

Top and bottom layer (mm/s) 4 | 2

Printing quality Fine | Coarse

Layer height (mm) 0.12 | 30

as possible, the slow printing settings and the fine quality setting were chosen to print the coiled-up resonator specimen. Figure 2 shows the manufactured specimen with the 3-D printing best quality (slow-fine), along with its cross and longitudinal sections.

Figure 2: View of the 3-D printed coiled up resonator obtained with the slow-fine manufacturing setting. The overall thickness of the specimen is 11 . 1 mm. the diameter of the sample is 2 R 0 = 39 mm, the side of the hole square is 2 a = 5 . 59 mm, the porosity of the front panel is ϕ = 2 . 6%, and the total spiral length is L t = 133 mm

2.3. Measurements

The experimental campaign has been carried out in the impedance tube according to ISO 10534- 2 [9] to quantify the acoustical performance of the specimen in terms of normal incidence sound absorption coefficient. Considering the geometrical dimensions of the tube and the measurement chain used, the frequency range of the tube is 300 5000 Hz. The digital signal processing was developed with MATLAB [10] and the ITA-Toolbox [11] calculation scripts, according to the Transfer- Function Method.

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A climate chamber was employed to impose specific relative humidity (RH = 20 – 50 – 80 %) and temperatures (from T = 10 ◦ C to T = 50 ◦ C with a ∆T = 10 ◦ C) to the specimen. The specimen has been kept for 24 hours in the chamber to grant an uniform distribution of temperature and humidity, keeping constant the environmental condition inside the impedance tube.

2.4. Numerical models In the present work, finite-element simulations were carried out to support the multiphysics analysis based on the experimental results. The numerical model helped to quantify the inner deformations of the specimen in function of the thermo-hygrometric conditions. The simulations, carried out on the commercial software COMSOL Multiphysics [12], provided the determination of the actual thermal deformations by imposing the three relative humidity and the five temperature scenarios and then by determining the corresponding sound absorbing performance for each deformed mesh.

3. RESULTS

First, a calibration between the analytical, numerical and experimental measurements has been done in standard condition (RH= 50% and T = 20 ◦ C): the outcome shown in Figure 3 has been taken as a reference for the successive steps of the study. The resonance peaks’ frequency f R , the highest

Figure 3: Comparison between analytical, numerical and measurements for the coiled up resonator under study.

amplitude of sound absorption coefficient α MAX , and the relative bandwidth of each resonance peak 1 / Q have been taken as parameters to evaluate the sound absorption properties of the specimen. The 1 / Q parameter is defined as:

1

Δ𝑓 𝑅

𝑄 =

𝑓 𝑅 ( 3)

(3)

yualnyja05 uondiosay punos: ‘1000 1800 20002500 3000 3600 000 4600 5000 Frequency (Hz)

where ∆ f R is the absorption bandwidth where the condition α > α max is satisfied. It is the inverse of the quality factor Q , connected to the expression of the half-power points of the power radiation from open-ended pipes [13]. After the calibration, all the thermo-hygrometric settings have been investigated. Three measurements were performed with three different values of relative humidity (RH = 20% - 50% 80%), keeping constant the temperature (T= 20 ◦ C): the acoustic properties of the specimen remained quite constant in terms of resonance peaks frequency and amplitude. This result confirmed that the employed material (PETG) is not hygroscopic and relative humidity does not affect the geometry of the metamaterial and consequently, the sound absorption. The measurements in Figure 4 show the results varying the temperature between the range T = 10 50 ◦ C with a ∆T = 10 ◦ C, keeping constant the relative humidity (RH = 20%) . Heating the specimen leads the PETG to expand

2

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Figure 4: Experimental curves for each resonance peak obtained by increasing the temperature with a ∆T = 10 ◦ C and keeping constant the relative humidity at RH= 50%. The resonance frequency f R , the amplitude α MAX , and the 1 / Q parameter are highlighted for each peak.

causing a decrease of the inner volume of the spiral. Conversely, cooling the specimen cause an increasing of the inner volume: in particular, referring to the analytical model described in Section 2.1, the variation of the spiral length L t is preponderant concerning variation of the inner volume. For this reason, the frequency resonances increase with increasing the temperature and decrease decreasing it. The 1 / Q parameter remains the same for each peak: it means that the bandwidth of each resonance peaks does not change varying the temperature. Numerical analysis, validated by the experimental measurements, provides the actual deformations of the inner spiral length.

= peak ‘Sound Absorption Coaticiont Vx %o soo e800 650 700780800850 io 17501800 1850 1900 1950 2000 20602100 Frequency (H) Frequency (H2) 5° poak peak sa sin 9150 set0Ga50_2900 TED H4OD 50 Froquoncy (2) “ESD 4DD ASO 45108504600 W850 A700 Frequency (Hz)

4. CONCLUSIONS

The present study highlights the acoustic effect of thermo-hygrometric variations on a PETG 3-D printed coiled up resonator. Relative humidity variations returned to be neglectable as expected, whilst temperature variations significantly shifted the sound absorption peaks of the specimen. These outcomes may be a starting point for developing tunable resonant absorbers in specific practical application where the temperature variation has a key role.

ACKNOWLEDGEMENTS

This research was funded by Ministero dell’Istruzione dell’Università della Ricerca (Italy) in the frame of the project PRIN 2017. The authors would like to thank Edoardo Idà, who kindly printed PETG metamaterial specimens, Luca Barbaresi and Dario D’Orazio, who helped to carry out the experimental measurements.

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