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Acoustic properties of periodic micro-structures obtained by additive manufacturing
Applied Acoustics 148 (2019) 322–331
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Acoustic properties of a periodic micro-structures obtained by additive manufacturing Edith Roland Fotsing a,⇑, Arnaud Dubourg a, Annie Ross a, Jacky Mardjono b a b
LAVA, Department of Mechanical Engineering, ÉcolePolytechnique de Montréal, P.O. Box 6079 Station Centre-ville, Montréal, Québec H3C 3A7, Canada Safran Aircraft Engines, Villaroche, Rond Point René Ravaud – Réau, 77550 Moisy-Cramayel Cedex, France
a r t i c l e
i n f o
Article history: Received 13 August 2018 Received in revised form 11 December 2018 Accepted 19 December 2018
Keywords: Acoustic properties Acoustic impedance Absorption coefficient Additive manufacturing
a b s t r a c t Periodic acoustic structures of rigid micro-rods produced via additive manufacturing were analyzed using an impedance tube in the 500 Hz–6000 Hz frequency range. The acoustical properties such as the porosity, the airflow resistivity, the tortuosity, the viscous characteristic length and the thermal characteristic length were determined by comparing an analytical model and the experimental results. The model which takes into account the interaction between the acoustic wave and a rigid periodic structure was proven to be an effective tool to correlate the final acoustic performance of the structure and the printing parameters. It was shown that broadband acoustic capabilities can be improved over the studied frequency range with the smallest filament possible or by reducing the lattice parameter and/or by changing the orientation of the filament from one layer to the next. This work was motivated by the need to control the manufacturing parameters of acoustic materials with broadband absorption capabilities. Furthermore the present investigation paves the way to explore new avenues to replace stochastic porous materials by new structures with combined mechanical and acoustic properties. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Due to their broadband absorption properties, polymer and metallic foams are studied and considered in the aeronautic sector as a viable replacement for Helmhotz resonators [1,2]. When flow is not taken into account, efficient foams providing high absorption coefficient and low surface impedance do generally display open and interconnected porosities and a high level of pores per inch (PPI). This implies that high performance foams are essentially made of voids within a very thin skeleton of the bulk material, which in turn, limit the mechanical properties. Metal foams offer high thermal and chemical stability along with superior mechanical properties, but they add significant mass to the initial structure [3]. Added mass is especially problematic in application fields such as aeronautics. Acoustic polymer foams, on the other hand, are light but have inferior mechanical strength combined with low thermal and dimensional stability [4]. Arrangement of cylindrical rods or platelets-like beams to form a three dimensional structure can offer an intermediate solution where the structural and acoustic properties are combined if produced with the appropriate material. Such a periodic three dimensional arrangement of cylindrical
⇑ Corresponding author. E-mail address: [email protected] (E.R. Fotsing). https://doi.org/10.1016/j.apacoust.2018.12.030 0003-682X/Ó 2018 Elsevier Ltd. All rights reserved.
rods bearing some resemblance to sonic crystals [5] can be used to attenuate or filter sound waves. Another issue with foams is the complexity of the foaming processes [6,7] which, in some cases, can be expensive and time consuming. These processes do not always provide foams with the high and repeatable quality required for high performance acoustic materials. Moreover foaming processes use blowing agents such as chlorofluoro carbon (CFC) or generate greenhouse gases such CO2 which are known to cause ozone depletion in the upper atmosphere. Additive manufacturing [8,9] offers the possibility of creating 3D materials which characteristics are close to those of standard stochastic foams. 3D printing is a rapidly emerging manufacturing technology defined as the process of joining materials layer by layer to form objects from 3D model data. Its enables the creation of printed parts of almost any shape that could be very difficult, even impossible to produce otherwise, leading to time and cost reduction. Multiple layers of perforated membranes supported by a network of micro-lattice were recently proposed by Cai [10]. Narrow tube periodic structures used as quarter resonators to attenuate acoustic energy of discrete frequencies were investigated in [11,12]. Structures made of periodic arrangement of rods obtained by 3D printing with controlled filling fraction (open porosity) can serve as sound absorbers over a wide frequency range, similar to regular foams. This periodic structures with lattice
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greater than the acoustic wavelength but capable of causing enough thermal and viscous losses can be used to replace stochastic foams since they provide acoustic energy absorption while conserving an acceptable level of mechanical strength. The amount of loss depends on the open porosity, the static airflow resistivity and the tortuosity which are the most important parameters defining a
porous material. It was shown that the maximal absorption of micro-lattice periodic structure can be obtained when the pore size is twice the viscous boundary layer thickness [10,13]. Researches combining experimental and numerical approaches have been conducted to correlate the acoustic parameters with the microstructure. The dynamic viscous permeability was modeled
Fig. 1. Schematic representation of the three configurations along with the corresponding actual samples obtained by additive manufacturing (a) orthogonal direct (OD), (b) orthogonal alternate (OA) and (c) configuration (Sh); a is the lattice parameter, 2ro is the diameter of the rods and h corresponds to the change of orientation of the rods.
Table 1 Printing parameters and actual properties of the periodic structures. Sample #
1 2 3 4 5 6 7 8
Config
OD OD OD OD OD OA S30 S30
Thickness (mm)
0.45esp 0.40esp 0.41esp 0.43esp 0.40esp 0.37esp 0.39esp 0.40esp
Lattice parameter (microns)
1.5asp 1.75asp 2 asp 2.25asp 2.5asp 2asp 2asp 2.5asp
Filament Diameter (microns)
Porosity (%)
Printed
Analytical model
Experimental
Analytical model
0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp
0.7Dsp 0.75Dsp 0.675Dsp 0.65Dsp 0.74Dsp 0.73Dsp 0.73Dsp 0.72Dsp
51.7 58.4 61.4 65.4 69.4 51.8 59.4 65.1
55 63 70 72 75 55 64 70
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in 2D for hexagonal lattice of solid fibers with circular crosssection by Perrot et al [14]. Here, X-ray microtomography was used to characterize the macroscopic parameters. Recently, it was show that the orientation of the porosity can be used as controlling parameter to lower the frequency of quarter wavelength resonator [15]. Tarnow [16] proposed a model for the resistivity of cylindrical rods perpendicular and parallel to the to the propagation sound wave. For rigid fibrous media consisting of parallel cylinders, Allard [17] derived an analytical relationship between the filling fraction, the resistivity and the tortuosity. In both models, the interaction
Frequency analysis system
Signal generator
between sound wave and fibrous media were exploited. In this work, acoustic absorbing materials consisting of arrangement of rigid micro-rods are printed by additive manufacturing. Three different configurations with rods perpendicular to the normal incident sound wave were investigated. Acoustic properties were evaluated using an impedance tube to demonstrate its applicability as broadband sound absorber. An analytical model describing the interaction of acoustic wave with a fibrous network was compared to the experimental results in order to extract the physical parameters of the periodic structure. Additionally, the inverse method based on the Johnson-Champoux-Allard model (JCA) for stochastic foams, was also applied to the experimental results. The two modelling approaches were compared and discussed with respect to manufacturing parameters.
Signal conditioner
Power amplifier
Sample Mic1
Mic2
Speaker Hard backing Fig. 2. Measurement set-up with the 30 mm-diameter impedance tube.
2. Experimental procedures Cylindrical samples with a diameter of 30 mm were produced using 3D printer from FISNAR consisting of a computerized 4axis robot (I&J 2200N-4) to which a dispensing nozzle (diameter between 50 mm and 3 mm) was connected. The printer also has a microscope equipped with image processing for live observation during processing. The polymer used was a bi-component
Fig. 3. Absorption coefficient (a) normalized acoustic resistance (b) and reactance (c) of samples printed in configuration OD.
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thermoset. The formulation of the resin (ratio between the hardener and the resin) was designed to prevent premature hardening during the deposition. The viscosity of the thermoset resin was adjusted to ensure that a continuous material filament was extruded from the nozzle. The 3D printer was equipped with pressure multiplier that enables adjusting extrusion pressure and eases the printing of samples with thickness ranged between 10 mm and 40 mm. Printing parameters include the diameter of the filament (d), the lattice parameter which is the distance between two filaments (a) and the in-plane orientation of the filament (h). Three configurations were produced. A schematic representation of the each configuration is shown on Fig. 1. The first configuration named orthogonal direct (OD) in which series of parallel cylindrical rods were printed alternately in orthogonal x and y directions; the filaments of any given layer in one direction are directly above the filaments of the preceding layer in the same direction. The second configuration follows a similar scheme, with orthogonal layers, except that the positions of the filaments were staggered from one layer to the next one in the same direction, through the thickness of the sample. This configuration is named orthogonal alternate (OA). In the third configuration, the orientation of the filaments was increased by a fixed angle h from one layer to next until reaching the desired thickness of the sample. This configuration is called (Sh). Different filling fractions were obtained by varying the lattice parameter from 300 mm to 1500 mm. For the remaining sections, microstructures will be defined by a specific filament diameter Dsp , a specific thickness esp and a specific lattice parameter asp . The specific thickness corresponds to a nominal acoustic treatment thickness whereas Dsp and asp are characteristic values leading to acoustically efficient microstructures. The nominal diameter of the filament (nozzle diameter) was kept constant at 0.625Dsp microns for all samples. All samples produced in this work with the corresponding printing parameters are gathered in Table 1. The density of the bulk material was measured using a gas pycnometer (AccyPYc 1340) from Micromeritics. The porosity of a the periodic structure was calculated by
/exp ¼ 1
qm qb
materials in which fluid is flowing. Based on this theory, porous media can be modelled as an equivalent fluid having an equivalent density and equivalent bulk modulus. The equation describing the wave propagation can then be written as
Dp þ x 2
q~eq
K~eq
p¼0
ð3Þ
~ eq are the equivalent density and the equivalent where q~eq and K bulk compressibility modulus and p corresponds to the acoustic pressure. The equivalent density represents the inertial viscous effects whereas the equivalent bulk modulus represents the thermal viscous effects. Both are complex numbers and account for the energy dissipation in the porous media. The model proposed by Allard [17] to describe the interaction between the acoustic wave and a network of rigid fibrous media consisting of parallel cylinders give the general expressions of the equivalent density and the equivalent bulk modulus as
q~ eq ðxÞ ¼ q0
a1 /
F~ 1j x~s
ðc 1Þ P0 K~eq ðxÞ ¼ /
1þ
!
ð4Þ ðc 1Þd ð 1 j Þ K0
ð5Þ
where q0 ¼ 1:225kg=m3 is the density of air, P0 = 105 Pa the ambient static pressure, c ¼ 1:4 the ratio of specific heats and / is the
ð1Þ
where qb is the density of the bulk material measured using the pycnometer and consistent with the value given by the manufacturer. qm is the density of structure. Knowing the weight (m) as well as the diameter (2r0) and the thickness (h) of a cylindrical sample, the density of the printed periodic structure was calculated using the following expression
qm¼
m
pr20 h
ð2Þ
The acoustic parameters including the absorption coefficient and the acoustic impedance were measured using an impedance tube with a 30 mm diameter. Following the ASTM E1050 standard, all tests were performed using the two microphones with hard backing configuration as shown in Fig. 2. The measurements were made between 500 Hz and 6000 Hz. 3. Modelling 3.1. Acoustic model for periodic structures Biot [18–20] developed the theory describing the wave propagation in fluid saturated porous materials having a rigid skeleton and interconnected pores. The theory was used originally to predict the time evolution of stresses and deformations of porous
Fig. 4. Absorption coefficient: (a) Comparison between (OD) and (S30) configurations for lattice parameter of 2asp microns and 2.5asp microns; (b) comparison between (OD), (S30) and (OA) configurations for lattice parameter of 2asp microns.
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porosity. The characteristic thermal length ~ s are defined by and x
/ K ¼ r0 1/ 0
F~ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~s jMx 1þ 2
x~s ¼
xq0 a1 /r
V0
and the functions F~
ð6Þ
ð7Þ ð8Þ
gC p j
qffiffiffiffiffiffiffiffiffiffi is the Prandtl number and kt ¼ ð1 jÞ xq2g0 Pr
is defined as the thermal wave number. Cp = 1004 J/K.kg and
j ¼ 0:062 W=m:K represent the specific heat and the thermal conductivity of the air, respectively. For parallel rigid cylinders perpendicular to airflow Tarnow [16] developed an expression of the air flow resistivity for a periodic structure as follows.
4gð1 /Þ
r¼ h
Þ r 20 12 ln ð1 /Þ þ 14 / ð1/ 4
8a1 g V2 /r
ð9Þ
where x is the frequency, r0 is the radius of the filament, g ¼ 1:8 105 Pa:s is the dynamic viscosity of the air. The viscous boundary layer around each fiber d, the characteristic viscous length V and the high frequency limit of the tortuosity a1 are defined by
pffiffiffiffiffi Pr d ¼ ð1 jÞ kt ^
ð12Þ
In Eq. (10), Pr ¼
with
M¼
a1 ¼ 2 /
¼ r0
/ð2 /Þ 2ð1 /Þ
ð10Þ ð11Þ
2
i
ð13Þ
Given the expressions of acoustic parameters, it can be seen that the equivalent density and bulk modulus do not contain any empirically derived coefficients and depend only on the porosity of the periodic structure and the diameter of printed filament. The sound absorption coefficient (A) can be calculated as
A¼1
Z~ Z 0 Z~ þ Z 0
ð14Þ
~ is the surface acoustic impedance of the sample In Eq. (12), Z and Z 0 ¼ q0 c0 is the characteristic impedance of air. c0 is the sound velocity in air. For hard backing behind the sample in the impe~ can be calculated with the following expression dance tube, Z
Fig. 5. Comparison between measured and modeled acoustic absorption for samples in OD configuration (a), OA configuration (b) and S30 configuration (c).
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~ Z~ ¼ jZ~c cot ke
ð15Þ
where e is the thickness of the sample, Z~c is the characteristic impe~ the wave number of the media given dance of the media and k
absorption coefficient and the predicted values. The prediction model also based on the Biot theory and is designed for stochastic porous materials. The equivalent density and bulk modulus are given by the following expressions
0
respectively by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z~c ¼ q~eq K~eq
ð16Þ
sffiffiffiffiffiffiffiffiffi ~ ~ ¼ x qeq k K~eq
ð17Þ
An optimization routine was implemented to determine the values of r0 and the filling fraction (wf = 1 /) providing the best fit of the experimental impedance and absorption coefficient.
q~ eq ðxÞ ¼ B @ K~eq ðxÞ ¼
11 2 / aðxÞ 1 C þ A q0 aðxÞ /q þ /2 q 1 1 0 0 aðxÞ
cP0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 c ðc 1Þ 1 j 2Hx 1 þ j Hx0
With
aðxÞ ¼ a1 1 j
3.2. Inverse method
/r
xq0 a1
ð18Þ
ð19Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1þj
x
ð20Þ
H
V
This approach consists of adjusting a set of independent parameters, namely the porosity (/), the resistivity (r), the tortuosity V (a1 ), the viscous characteristic length ( ) and thermal characteris0
Normalized acoustic impedance
tic length (K ) in order to reach the minimum of the merit function that measures the agreement between the experimental sound
H¼ 0
H ¼
r2 2 /2 4a21 gq0
ð21Þ
16g V0 2
ð22Þ
P2r
q0
5
0
Model R Model X Exp R (sample 2) Exp X (sample 2)
-5
(a) -10 0
1000
2000
3000
4000
5000
6000
Normalized acoustic impedance
Frequency (Hz) 5
0
Model R Model X Exp R (sample 6) Exp X (sample 6)
-5
(b) -10 0
1000
2000
3000
4000
5000
6000
Normalized acoustic impedance
Frequency (Hz) 5
0
Model R Model X Exp R (sample 7) Exp X (sample 7)
-5
(c) -10 0
1000
2000
3000
4000
5000
6000
Frequency (Hz) Fig. 6. Comparison between measured and modeled acoustic impedance for samples in OD configuration (a), OA configuration (b) and S30 configuration (c).
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As in the model for acoustic periodic structure, it can be seen that the equivalent density and the bulk modulus take into account the inertial interaction between the fluid and the solid phase as well as the viscous and thermal losses materialized by the quantiV V ties and 0 . The absorption coefficient and the surface impedance are given by Eqs. (14) and (15), respectively. The inverse method is implemented in the commercial software FOAM-X.
the propagation of the acoustic wave. The same observations were made for experimental acoustic impedance for S30 and OA configurations. The low surface impedance also indicates that the acoustic wave easily penetrates the periodic structure before being dissipated through the viscous friction and thermal exchange with rod walls.
4.2. Modelling
4. Results and discussions 4.1. Experimental results Fig. 3a shows the absorption coefficients for periodic microstructure with cylindrical rods in orthogonal direct (OD) configuration. Due to artefacts at lower frequencies, all experimental measurements are taken between 500 Hz and 6000 Hz. Generally, the periodic acoustic structures exhibit the same frequencydependant behaviour as stochastic porous media. The absorption coefficients increase from low frequencies, then reach a maximum at medium frequencies before decreasing slightly at higher frequencies. However, a slight increase at high frequencies is observed for sample 1. The maximum absorption coefficient (0.97 at 2720 Hz) is observed for the sample with the lowest porosity (sample 1) and decreases as the lattice parameter and the porosity increase. The lowest absorption coefficient (0.65 at 4000 Hz) is obtained for structure with lattice parameter of 2.5asp microns. It is also observed that the maximum absorption peak tends to shift towards higher frequencies (from 2700 to 4000 Hz) as the porosity increases. Fig. 4a compares periodic structures with the configurations OD and S30 with lattice parameters of 2asp microns and 2.5asp microns. It comes out that regardless of the lattice parameter, changing the filament orientation by 30 degree from one layer to next raises of the acoustic energy dissipation. It can be seen that the absorption coefficient rises from 0.65 (sample 5) to 0.85 (sample 8) and from 0.84 (sample 3) to 0.94 (sample 7). This improvement can be explained by an enhancement of the airflow resistivity and the tortuosity, which tends to increase the viscous and thermal losses. On Fig. 4b, samples with OD, S30 and OA configurations are compared. For the same lattice parameter (2asp microns) the orthogonal alternate (OA) configuration provides the highest absorption (1.00 at 3500 Hz). As in the S30 configuration, alternating the positions of the cylindrical rods increases the source of viscous and thermal losses within the porous material. The real and the imaginary parts of the normalized acoustic impedance are shown on Fig. 3b and c for (OD) configuration. Low values of the impedance were measured with the normalized acoustic resistance ranging between 1 and 5 in the 500 Hz– 1000 Hz frequency range. Above 1000 Hz, the resistance was close to zero. The acoustic reactance is found between 10 and 1 in the 500 Hz-1000 Hz range and close to 0 above 1000 Hz. This suggests that the printed structure offers little resistance and less inertia to
The acoustic analytical model for periodic structures was use to evaluate the physical properties of the printed samples. A curve fitting was achieved by minimizing the error between the measured acoustic absorption and the modeled values using the least squares method. The variable parameters used in the optimization procedure were the filling fraction (wf) defined as (1 /) and the diameter (2r0) of the printed filament. Figs. 5 and 6 show the measured absorption coefficient and the normalized impedance compared with the modeled values. For the sake of clarity only one example for each configuration is shown. The values of the porosity and the filament diameter corresponding to the best fit are reported in Table 1. It can be seen that for each configuration, the model and the experiments are in good agreement for both the absorption and the impedance, in the frequency range between 500 Hz and 6000 Hz. Only small differences can be seen between 500 Hz and 3000 Hz for sample 2 and sample 6 and between 2500 Hz and 3500 Hz for sample 7. The best fit
Sound absorption average between 2000-10000Hz
Fig. 7. Sound absorption average at high frequencies as function of volume fraction and filament diameter; positions of the 3D-printed samples 1, 6 and 8 are represented at the intersections of the dashed lines.
Table 2 Acoustic properties as obtained from the analytical model and the inverse method. Sample #
1 2 3 4 5 6 7 8
Viscous characteristic length (lm)
Thermal characteristic length (lm)
Airflow resistivity (N.s.m4) 104
Tortuosity
Analytical model
Inverse method
Analytical model
Inverse method
Analytical model
Inverse method
Analytical model
Inverse method
1.4 1.1 0.9 0.9 0.5 3.1 1.4 0.8
– 1 0.9 0.8 0.6 2.5 1 0.7
1.4 1.4 1.3 1.3 1.2 1.4 1.3 1.3
– 1.3 1.1 1.2 1.1 1.1 1.3 1.2
130.5 173.2 207.7 203.6 277.5 129.8 177.1 220
– 193.6 144.8 249.5 249.3 48.7 104.4 120
182 252.4 315 317.5 444 179 260.4 338.3
– 210.1 227.3 249.5 299.5 241.5 244.4 273.5
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describing the experimental results was obtained with values of the diameter that are always higher (between 4% and 20%) than the set values for printing which is 0.625Dsp microns (the nozzle diameter). Moreover the model tends to overestimate the porosity of the periodic structure by 6% to 14%. The measured values of the porosity (using Eq. (1)) are consistently lower than the value of porosity corresponding to the best curve fitting. The larger diameters and higher densities obtained with the model can be explained by the fact that during manufacturing, each layer is not yet cure when new layers are printed on top of it. Due to the weight on the lower layers, the micro-rods are being deformed. After complete curing, instead of being cylindrical, the final structure ends up with ovoid rods.
With optimal variable couple (2r0, wf) and using Eqs. (6), (11), (12) and (13) the corresponding values of the airflow resistivity, the tortuosity, the viscous and thermal characteristic lengths were calculated. These values are gathered in Table 2. For orthogonal direct (OD) configuration, a decrease of the porosity, the tortuosity and the resistivity with increasing lattice parameter was observed. This result was foreseeable, since the distance between micro-rods is increasing with lattice parameter, thus creating fewer obstacles to sound propagation. Meanwhile, the viscous and thermal characteristic lengths are also increasing. This observation implies that as the lattice parameter increases, the visco-inertial effects at low frequencies (resistivity) and at high frequencies (tortuosity) become less dominant and that the acoustic losses are controlled by vis-
1
(a) Absorption coefficient (α)
0.8
0.6
0.4
0.2
0 500
900
1300
1700
2100
2500
2900
3300
3700
4100
4500
4900
5300
5700
Frequency (Hz) 1
Absorption coefficient (α)
(b) 0.8
0.6
0.4
0.2
0 500
900
1300
1700
2100
2500
2900
3300
3700
4100
4500
4900
5300
5700
4900
5300
5700
Frequency (Hz) 1
Absorption coefficient (α)
(c) 0.8
0.6
0.4
0.2
0 500
900
1300
1700
2100
2500
2900
3300
3700
4100
4500
Frequency (Hz) Fig. 8. Absorption coefficient: Curve fitting with the inverse method for sample in OD configuration (a), OA configuration (b) and S30 configuration (c).
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cous and thermal effects appearing in the medium to high frequencies range. Alternating the position of the cylindrical rods (OA configuration) seems to significantly increase the tortuosity and the airflow resistivity but reduces the characteristic lengths (sample 6). This observation is consistent with the increase of the acoustic absorption from low to medium frequencies to reach 1 at 3500 Hz. The main printing parameters are the set diameter (2r0) of the nozzle and the lattice parameter (a) which have an influence on the filling fraction and consequently on the acoustic performance of the final structure. Since there is a direct correlation between the printing parameters and the acoustic properties, the analytical model can be used to find the optimal printing parameters needed to achieve the best acoustic performance possible. For this investigation, an average value of the sound absorption coefficient was evaluated in the frequency range from 2000 to 10000 Hz. For each pair of values (2r0, wf) corresponding to a frequency dependant absorption coefficient, the average absorption over the entire measurement frequency range can be expressed as
I¼
N 1 X aðf i Þ N i¼1
ð23Þ
where N is the number of discrete frequencies in the domain of interest. This quantity bears some resemblance with the sound absorption average (SAA) as defined in the ASTM C423 [21] standard and its values range from 0 (no absorption) to 1 (full absorption). Fig. 7 represents the parametric plots of the average sound absorption for each pair of values (2r0, wf) for frequencies ranged between 2000 Hz and 10000 Hz, based on the model for periodic structure. It should be noted that this analysis was made only for orthogonal direct (OD) configuration. However, there is no restriction to apply this analysis to the two other configurations. In the investigated frequency domain, it is observed that the average absorption reaches a maximum of 0.95. The zone providing the best performance is uniform with the filling fraction between 5% and 60% and the corresponding the filament diameter varying between 0.05Dsp and 1.25Dsp. This zone is relatively large in terms of filling fraction and the diameter, which provides more flexibility for acoustic microstructures. For samples produced with a filament diameter of 0.625Dsp, the experimental values are in agreement with the performance index predicted by the model. As example, the positions of samples 1, 6 and 8 are shown on Fig. 7. It should be mentioned that the model can be easily applied to lower and mid frequencies domains to determine the filling fraction and the corresponding filament diameter necessary to reach the optimal acoustic absorption. This model is an effective tool for the selection of the printing parameters leading to the best acoustic properties possible. It can be also used to assess the reliability of the design of micro-lattice by predicting, for example, the decline in acoustic performance due to manufacturing imperfections. The inverse method for stochastic porous media was used to fit the experimental values of the absorption coefficient for the periodic structure. The best fit was sought by varying the tortuosity, the airflow resistivity as well as the characteristic lengths while keeping the porosity constant. The experimental value of the porosity calculated from Eq. (1) was used. Keeping the porosity constant suggests that flattening of the printed rods may not significantly change the periodic structure to influence the outcoming acoustical parameters. Fig. 8 shows examples of curve fitting for OD, OA and S30 configurations, from which the acoustic properties were extracted. It also reveals that the model behind the inverse method is in good agreement with the experimental results. Although the general trend and the order of magnitude are the same for all parameters, differences are observed for the values of the viscous and thermal characteristic lengths (sample 6, sample
7 and sample 8 in Table 2). Despite the differences between the analytic model and the inverse method, this analysis suggests that acoustic periodic structure can be fully characterized by models designed for materials with randomly distributed and interconnected porosities. 5. Conclusions The acoustic properties of a periodic structure produced by additive manufacturing was analyzed using an impedance tube and a model taking into account the interaction of an acoustic wave with periodic medium. This work was motivated by the need to control the manufacturing parameters of acoustic materials with broadband absorption capabilities. The following main results were obtained: Periodic acoustic structures produced via additive manufacturing exhibits broadband acoustic properties comparable to those of porous materials. The printing parameters such as the lattice parameter and the diameter of the printed filament can be adjusted to produce a structure with high absorption capabilities. High absorption can be achieved by using smaller filament diameter, by reducing the lattice parameter or/and by changing the orientation of the printed filament. The proposed analytical model was used as effective tool to define the best printing parameters necessary to produce structure with optimal acoustic performance. Analysis with analytical model for periodic structure reached the same conclusions as the model for stochastic porous materials. The tortuosity and the airflow resistivity are less dominant as the lattice parameter increases. Alternating the position of the rod from one layer to the other increase energy losses in the low to medium frequencies. Changing the orientation of the rods from one layer to the next appears to improve the absorption capability in the whole frequency range.
Acknowledgments The authors kindly acknowledged Safran Aircraft Engines and the Natural Sciences and Engineering Research Council of Canada (NSERC) for supporting and funding this research. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.apacoust.2018.12.030. References [1] Mao Q, Pietrzko S. Experimental study for control of sound transmission through double glazed window using optimally tuned Helmholtz resonators. Appl Acoust 2010;71:32–8. [2] Zhao D, Morgans AS. Tuned passive control of combustion instabilities using multiple Helmholtz resonators. J Sound Vib 2009;320:744–57. [3] Lefebvre L-P, Banhart J, Dunand D. Porous metals and metallic foams: current status and recent developments. Adv Eng Mater 2008;10:775–87. [4] Lee LJ, Zeng C, Cao X, Han X, Shen J, Xu G. Polymer nanocomposite foams. Compos Sci Technol 2005;65:2344–63. [5] Guild MD, García-Chocano VM, Kan W, Sánchez-Dehesa J. Acoustic metamaterial absorbers based on multilayered sonic crystals. J Appl Phys 2015;117:114902. [6] Sauceau M, Fages J, Common A, Nikitine C, Rodier E. New challenges in polymer foaming: a review of extrusion processes assisted by supercritical carbon dioxide. Prog Polym Sci 2011;36:749–66. [7] Jacobs LJ, Kemmere MF, Keurentjes JT. Sustainable polymer foaming using high pressure carbon dioxide: a review on fundamentals, processes and applications. Green Chem 2008;10:731–8.
E.R. Fotsing et al. / Applied Acoustics 148 (2019) 322–331 [8] Wong KV, Hernandez A. A review of additive manufacturing. ISRN Mech Eng. 2012;2012. [9] Vaezi M, Seitz H, Yang S. A review on 3D micro-additive manufacturing technologies. Int J Adv Manuf Technol 2013;67:1721–54. [10] Cai X, Yang J, Hu G, Lu T. Sound absorption by acoustic microlattice with optimized pore configuration. J Acoustical Soc Am 2018;144:EL138–43. [11] Akiwate DC, Date MD, Venkatesham B, Suryakumar S. Acoustic properties of additive manufactured narrow tube periodic structures. Appl Acoust 2018;136:123–31. [12] Cai X, Guo Q, Hu G, Yang J. Ultrathin low-frequency sound absorbing panels based on coplanar spiral tubes or coplanar Helmholtz resonators. Appl Phys Lett 2014;105:121901. [13] Cai X, Yang J, Hu G. Optimization on microlattice materials for sound absorption by an integrated transfer matrix method. J Acoustical Soc Am 2015;137:EL334–9. [14] Perrot C, Chevillotte F, Panneton R. Dynamic viscous permeability of an opencell aluminum foam: computations versus experiments. J Appl Phys 2008;103:024909.
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[15] Attenborough K. Microstructures for lowering the quarter wavelength resonance frequency of a hard-backed rigid-porous layer. Appl Acoust 2018;130:188–94. [16] Tarnow V. Airflow resistivity of models of fibrous acoustic materials. J Acoustical Soc Am 1996;100:3706–13. [17] Allard JF, Champoux Y. New empirical equations for sound propagation in rigid frame fibrous materials. J Acoustical Soc Am 1992;91:3346–53. [18] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J Acoust Soc Am 1956;28:168–78. [19] Biot MA. ‘‘Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J Acoust Soc Am 1956;28:179–91. [20] Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 1962;33:1482–98. [21] ASTM S, Standard test method for sound absorption and sound absorption coefficients by the reverberation room method, C423-90a, 1990.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Acoustic properties of a periodic micro-structures obtained by additive manufacturing Edith Roland Fotsing a,⇑, Arnaud Dubourg a, Annie Ross a, Jacky Mardjono b a b
LAVA, Department of Mechanical Engineering, ÉcolePolytechnique de Montréal, P.O. Box 6079 Station Centre-ville, Montréal, Québec H3C 3A7, Canada Safran Aircraft Engines, Villaroche, Rond Point René Ravaud – Réau, 77550 Moisy-Cramayel Cedex, France
a r t i c l e
i n f o
Article history: Received 13 August 2018 Received in revised form 11 December 2018 Accepted 19 December 2018
Keywords: Acoustic properties Acoustic impedance Absorption coefficient Additive manufacturing
a b s t r a c t Periodic acoustic structures of rigid micro-rods produced via additive manufacturing were analyzed using an impedance tube in the 500 Hz–6000 Hz frequency range. The acoustical properties such as the porosity, the airflow resistivity, the tortuosity, the viscous characteristic length and the thermal characteristic length were determined by comparing an analytical model and the experimental results. The model which takes into account the interaction between the acoustic wave and a rigid periodic structure was proven to be an effective tool to correlate the final acoustic performance of the structure and the printing parameters. It was shown that broadband acoustic capabilities can be improved over the studied frequency range with the smallest filament possible or by reducing the lattice parameter and/or by changing the orientation of the filament from one layer to the next. This work was motivated by the need to control the manufacturing parameters of acoustic materials with broadband absorption capabilities. Furthermore the present investigation paves the way to explore new avenues to replace stochastic porous materials by new structures with combined mechanical and acoustic properties. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Due to their broadband absorption properties, polymer and metallic foams are studied and considered in the aeronautic sector as a viable replacement for Helmhotz resonators [1,2]. When flow is not taken into account, efficient foams providing high absorption coefficient and low surface impedance do generally display open and interconnected porosities and a high level of pores per inch (PPI). This implies that high performance foams are essentially made of voids within a very thin skeleton of the bulk material, which in turn, limit the mechanical properties. Metal foams offer high thermal and chemical stability along with superior mechanical properties, but they add significant mass to the initial structure [3]. Added mass is especially problematic in application fields such as aeronautics. Acoustic polymer foams, on the other hand, are light but have inferior mechanical strength combined with low thermal and dimensional stability [4]. Arrangement of cylindrical rods or platelets-like beams to form a three dimensional structure can offer an intermediate solution where the structural and acoustic properties are combined if produced with the appropriate material. Such a periodic three dimensional arrangement of cylindrical
⇑ Corresponding author. E-mail address: [email protected] (E.R. Fotsing). https://doi.org/10.1016/j.apacoust.2018.12.030 0003-682X/Ó 2018 Elsevier Ltd. All rights reserved.
rods bearing some resemblance to sonic crystals [5] can be used to attenuate or filter sound waves. Another issue with foams is the complexity of the foaming processes [6,7] which, in some cases, can be expensive and time consuming. These processes do not always provide foams with the high and repeatable quality required for high performance acoustic materials. Moreover foaming processes use blowing agents such as chlorofluoro carbon (CFC) or generate greenhouse gases such CO2 which are known to cause ozone depletion in the upper atmosphere. Additive manufacturing [8,9] offers the possibility of creating 3D materials which characteristics are close to those of standard stochastic foams. 3D printing is a rapidly emerging manufacturing technology defined as the process of joining materials layer by layer to form objects from 3D model data. Its enables the creation of printed parts of almost any shape that could be very difficult, even impossible to produce otherwise, leading to time and cost reduction. Multiple layers of perforated membranes supported by a network of micro-lattice were recently proposed by Cai [10]. Narrow tube periodic structures used as quarter resonators to attenuate acoustic energy of discrete frequencies were investigated in [11,12]. Structures made of periodic arrangement of rods obtained by 3D printing with controlled filling fraction (open porosity) can serve as sound absorbers over a wide frequency range, similar to regular foams. This periodic structures with lattice
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greater than the acoustic wavelength but capable of causing enough thermal and viscous losses can be used to replace stochastic foams since they provide acoustic energy absorption while conserving an acceptable level of mechanical strength. The amount of loss depends on the open porosity, the static airflow resistivity and the tortuosity which are the most important parameters defining a
porous material. It was shown that the maximal absorption of micro-lattice periodic structure can be obtained when the pore size is twice the viscous boundary layer thickness [10,13]. Researches combining experimental and numerical approaches have been conducted to correlate the acoustic parameters with the microstructure. The dynamic viscous permeability was modeled
Fig. 1. Schematic representation of the three configurations along with the corresponding actual samples obtained by additive manufacturing (a) orthogonal direct (OD), (b) orthogonal alternate (OA) and (c) configuration (Sh); a is the lattice parameter, 2ro is the diameter of the rods and h corresponds to the change of orientation of the rods.
Table 1 Printing parameters and actual properties of the periodic structures. Sample #
1 2 3 4 5 6 7 8
Config
OD OD OD OD OD OA S30 S30
Thickness (mm)
0.45esp 0.40esp 0.41esp 0.43esp 0.40esp 0.37esp 0.39esp 0.40esp
Lattice parameter (microns)
1.5asp 1.75asp 2 asp 2.25asp 2.5asp 2asp 2asp 2.5asp
Filament Diameter (microns)
Porosity (%)
Printed
Analytical model
Experimental
Analytical model
0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp 0.625Dsp
0.7Dsp 0.75Dsp 0.675Dsp 0.65Dsp 0.74Dsp 0.73Dsp 0.73Dsp 0.72Dsp
51.7 58.4 61.4 65.4 69.4 51.8 59.4 65.1
55 63 70 72 75 55 64 70
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in 2D for hexagonal lattice of solid fibers with circular crosssection by Perrot et al [14]. Here, X-ray microtomography was used to characterize the macroscopic parameters. Recently, it was show that the orientation of the porosity can be used as controlling parameter to lower the frequency of quarter wavelength resonator [15]. Tarnow [16] proposed a model for the resistivity of cylindrical rods perpendicular and parallel to the to the propagation sound wave. For rigid fibrous media consisting of parallel cylinders, Allard [17] derived an analytical relationship between the filling fraction, the resistivity and the tortuosity. In both models, the interaction
Frequency analysis system
Signal generator
between sound wave and fibrous media were exploited. In this work, acoustic absorbing materials consisting of arrangement of rigid micro-rods are printed by additive manufacturing. Three different configurations with rods perpendicular to the normal incident sound wave were investigated. Acoustic properties were evaluated using an impedance tube to demonstrate its applicability as broadband sound absorber. An analytical model describing the interaction of acoustic wave with a fibrous network was compared to the experimental results in order to extract the physical parameters of the periodic structure. Additionally, the inverse method based on the Johnson-Champoux-Allard model (JCA) for stochastic foams, was also applied to the experimental results. The two modelling approaches were compared and discussed with respect to manufacturing parameters.
Signal conditioner
Power amplifier
Sample Mic1
Mic2
Speaker Hard backing Fig. 2. Measurement set-up with the 30 mm-diameter impedance tube.
2. Experimental procedures Cylindrical samples with a diameter of 30 mm were produced using 3D printer from FISNAR consisting of a computerized 4axis robot (I&J 2200N-4) to which a dispensing nozzle (diameter between 50 mm and 3 mm) was connected. The printer also has a microscope equipped with image processing for live observation during processing. The polymer used was a bi-component
Fig. 3. Absorption coefficient (a) normalized acoustic resistance (b) and reactance (c) of samples printed in configuration OD.
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thermoset. The formulation of the resin (ratio between the hardener and the resin) was designed to prevent premature hardening during the deposition. The viscosity of the thermoset resin was adjusted to ensure that a continuous material filament was extruded from the nozzle. The 3D printer was equipped with pressure multiplier that enables adjusting extrusion pressure and eases the printing of samples with thickness ranged between 10 mm and 40 mm. Printing parameters include the diameter of the filament (d), the lattice parameter which is the distance between two filaments (a) and the in-plane orientation of the filament (h). Three configurations were produced. A schematic representation of the each configuration is shown on Fig. 1. The first configuration named orthogonal direct (OD) in which series of parallel cylindrical rods were printed alternately in orthogonal x and y directions; the filaments of any given layer in one direction are directly above the filaments of the preceding layer in the same direction. The second configuration follows a similar scheme, with orthogonal layers, except that the positions of the filaments were staggered from one layer to the next one in the same direction, through the thickness of the sample. This configuration is named orthogonal alternate (OA). In the third configuration, the orientation of the filaments was increased by a fixed angle h from one layer to next until reaching the desired thickness of the sample. This configuration is called (Sh). Different filling fractions were obtained by varying the lattice parameter from 300 mm to 1500 mm. For the remaining sections, microstructures will be defined by a specific filament diameter Dsp , a specific thickness esp and a specific lattice parameter asp . The specific thickness corresponds to a nominal acoustic treatment thickness whereas Dsp and asp are characteristic values leading to acoustically efficient microstructures. The nominal diameter of the filament (nozzle diameter) was kept constant at 0.625Dsp microns for all samples. All samples produced in this work with the corresponding printing parameters are gathered in Table 1. The density of the bulk material was measured using a gas pycnometer (AccyPYc 1340) from Micromeritics. The porosity of a the periodic structure was calculated by
/exp ¼ 1
qm qb
materials in which fluid is flowing. Based on this theory, porous media can be modelled as an equivalent fluid having an equivalent density and equivalent bulk modulus. The equation describing the wave propagation can then be written as
Dp þ x 2
q~eq
K~eq
p¼0
ð3Þ
~ eq are the equivalent density and the equivalent where q~eq and K bulk compressibility modulus and p corresponds to the acoustic pressure. The equivalent density represents the inertial viscous effects whereas the equivalent bulk modulus represents the thermal viscous effects. Both are complex numbers and account for the energy dissipation in the porous media. The model proposed by Allard [17] to describe the interaction between the acoustic wave and a network of rigid fibrous media consisting of parallel cylinders give the general expressions of the equivalent density and the equivalent bulk modulus as
q~ eq ðxÞ ¼ q0
a1 /
F~ 1j x~s
ðc 1Þ P0 K~eq ðxÞ ¼ /
1þ
!
ð4Þ ðc 1Þd ð 1 j Þ K0
ð5Þ
where q0 ¼ 1:225kg=m3 is the density of air, P0 = 105 Pa the ambient static pressure, c ¼ 1:4 the ratio of specific heats and / is the
ð1Þ
where qb is the density of the bulk material measured using the pycnometer and consistent with the value given by the manufacturer. qm is the density of structure. Knowing the weight (m) as well as the diameter (2r0) and the thickness (h) of a cylindrical sample, the density of the printed periodic structure was calculated using the following expression
qm¼
m
pr20 h
ð2Þ
The acoustic parameters including the absorption coefficient and the acoustic impedance were measured using an impedance tube with a 30 mm diameter. Following the ASTM E1050 standard, all tests were performed using the two microphones with hard backing configuration as shown in Fig. 2. The measurements were made between 500 Hz and 6000 Hz. 3. Modelling 3.1. Acoustic model for periodic structures Biot [18–20] developed the theory describing the wave propagation in fluid saturated porous materials having a rigid skeleton and interconnected pores. The theory was used originally to predict the time evolution of stresses and deformations of porous
Fig. 4. Absorption coefficient: (a) Comparison between (OD) and (S30) configurations for lattice parameter of 2asp microns and 2.5asp microns; (b) comparison between (OD), (S30) and (OA) configurations for lattice parameter of 2asp microns.
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porosity. The characteristic thermal length ~ s are defined by and x
/ K ¼ r0 1/ 0
F~ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~s jMx 1þ 2
x~s ¼
xq0 a1 /r
V0
and the functions F~
ð6Þ
ð7Þ ð8Þ
gC p j
qffiffiffiffiffiffiffiffiffiffi is the Prandtl number and kt ¼ ð1 jÞ xq2g0 Pr
is defined as the thermal wave number. Cp = 1004 J/K.kg and
j ¼ 0:062 W=m:K represent the specific heat and the thermal conductivity of the air, respectively. For parallel rigid cylinders perpendicular to airflow Tarnow [16] developed an expression of the air flow resistivity for a periodic structure as follows.
4gð1 /Þ
r¼ h
Þ r 20 12 ln ð1 /Þ þ 14 / ð1/ 4
8a1 g V2 /r
ð9Þ
where x is the frequency, r0 is the radius of the filament, g ¼ 1:8 105 Pa:s is the dynamic viscosity of the air. The viscous boundary layer around each fiber d, the characteristic viscous length V and the high frequency limit of the tortuosity a1 are defined by
pffiffiffiffiffi Pr d ¼ ð1 jÞ kt ^
ð12Þ
In Eq. (10), Pr ¼
with
M¼
a1 ¼ 2 /
¼ r0
/ð2 /Þ 2ð1 /Þ
ð10Þ ð11Þ
2
i
ð13Þ
Given the expressions of acoustic parameters, it can be seen that the equivalent density and bulk modulus do not contain any empirically derived coefficients and depend only on the porosity of the periodic structure and the diameter of printed filament. The sound absorption coefficient (A) can be calculated as
A¼1
Z~ Z 0 Z~ þ Z 0
ð14Þ
~ is the surface acoustic impedance of the sample In Eq. (12), Z and Z 0 ¼ q0 c0 is the characteristic impedance of air. c0 is the sound velocity in air. For hard backing behind the sample in the impe~ can be calculated with the following expression dance tube, Z
Fig. 5. Comparison between measured and modeled acoustic absorption for samples in OD configuration (a), OA configuration (b) and S30 configuration (c).
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~ Z~ ¼ jZ~c cot ke
ð15Þ
where e is the thickness of the sample, Z~c is the characteristic impe~ the wave number of the media given dance of the media and k
absorption coefficient and the predicted values. The prediction model also based on the Biot theory and is designed for stochastic porous materials. The equivalent density and bulk modulus are given by the following expressions
0
respectively by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z~c ¼ q~eq K~eq
ð16Þ
sffiffiffiffiffiffiffiffiffi ~ ~ ¼ x qeq k K~eq
ð17Þ
An optimization routine was implemented to determine the values of r0 and the filling fraction (wf = 1 /) providing the best fit of the experimental impedance and absorption coefficient.
q~ eq ðxÞ ¼ B @ K~eq ðxÞ ¼
11 2 / aðxÞ 1 C þ A q0 aðxÞ /q þ /2 q 1 1 0 0 aðxÞ
cP0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 c ðc 1Þ 1 j 2Hx 1 þ j Hx0
With
aðxÞ ¼ a1 1 j
3.2. Inverse method
/r
xq0 a1
ð18Þ
ð19Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1þj
x
ð20Þ
H
V
This approach consists of adjusting a set of independent parameters, namely the porosity (/), the resistivity (r), the tortuosity V (a1 ), the viscous characteristic length ( ) and thermal characteris0
Normalized acoustic impedance
tic length (K ) in order to reach the minimum of the merit function that measures the agreement between the experimental sound
H¼ 0
H ¼
r2 2 /2 4a21 gq0
ð21Þ
16g V0 2
ð22Þ
P2r
q0
5
0
Model R Model X Exp R (sample 2) Exp X (sample 2)
-5
(a) -10 0
1000
2000
3000
4000
5000
6000
Normalized acoustic impedance
Frequency (Hz) 5
0
Model R Model X Exp R (sample 6) Exp X (sample 6)
-5
(b) -10 0
1000
2000
3000
4000
5000
6000
Normalized acoustic impedance
Frequency (Hz) 5
0
Model R Model X Exp R (sample 7) Exp X (sample 7)
-5
(c) -10 0
1000
2000
3000
4000
5000
6000
Frequency (Hz) Fig. 6. Comparison between measured and modeled acoustic impedance for samples in OD configuration (a), OA configuration (b) and S30 configuration (c).
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As in the model for acoustic periodic structure, it can be seen that the equivalent density and the bulk modulus take into account the inertial interaction between the fluid and the solid phase as well as the viscous and thermal losses materialized by the quantiV V ties and 0 . The absorption coefficient and the surface impedance are given by Eqs. (14) and (15), respectively. The inverse method is implemented in the commercial software FOAM-X.
the propagation of the acoustic wave. The same observations were made for experimental acoustic impedance for S30 and OA configurations. The low surface impedance also indicates that the acoustic wave easily penetrates the periodic structure before being dissipated through the viscous friction and thermal exchange with rod walls.
4.2. Modelling
4. Results and discussions 4.1. Experimental results Fig. 3a shows the absorption coefficients for periodic microstructure with cylindrical rods in orthogonal direct (OD) configuration. Due to artefacts at lower frequencies, all experimental measurements are taken between 500 Hz and 6000 Hz. Generally, the periodic acoustic structures exhibit the same frequencydependant behaviour as stochastic porous media. The absorption coefficients increase from low frequencies, then reach a maximum at medium frequencies before decreasing slightly at higher frequencies. However, a slight increase at high frequencies is observed for sample 1. The maximum absorption coefficient (0.97 at 2720 Hz) is observed for the sample with the lowest porosity (sample 1) and decreases as the lattice parameter and the porosity increase. The lowest absorption coefficient (0.65 at 4000 Hz) is obtained for structure with lattice parameter of 2.5asp microns. It is also observed that the maximum absorption peak tends to shift towards higher frequencies (from 2700 to 4000 Hz) as the porosity increases. Fig. 4a compares periodic structures with the configurations OD and S30 with lattice parameters of 2asp microns and 2.5asp microns. It comes out that regardless of the lattice parameter, changing the filament orientation by 30 degree from one layer to next raises of the acoustic energy dissipation. It can be seen that the absorption coefficient rises from 0.65 (sample 5) to 0.85 (sample 8) and from 0.84 (sample 3) to 0.94 (sample 7). This improvement can be explained by an enhancement of the airflow resistivity and the tortuosity, which tends to increase the viscous and thermal losses. On Fig. 4b, samples with OD, S30 and OA configurations are compared. For the same lattice parameter (2asp microns) the orthogonal alternate (OA) configuration provides the highest absorption (1.00 at 3500 Hz). As in the S30 configuration, alternating the positions of the cylindrical rods increases the source of viscous and thermal losses within the porous material. The real and the imaginary parts of the normalized acoustic impedance are shown on Fig. 3b and c for (OD) configuration. Low values of the impedance were measured with the normalized acoustic resistance ranging between 1 and 5 in the 500 Hz– 1000 Hz frequency range. Above 1000 Hz, the resistance was close to zero. The acoustic reactance is found between 10 and 1 in the 500 Hz-1000 Hz range and close to 0 above 1000 Hz. This suggests that the printed structure offers little resistance and less inertia to
The acoustic analytical model for periodic structures was use to evaluate the physical properties of the printed samples. A curve fitting was achieved by minimizing the error between the measured acoustic absorption and the modeled values using the least squares method. The variable parameters used in the optimization procedure were the filling fraction (wf) defined as (1 /) and the diameter (2r0) of the printed filament. Figs. 5 and 6 show the measured absorption coefficient and the normalized impedance compared with the modeled values. For the sake of clarity only one example for each configuration is shown. The values of the porosity and the filament diameter corresponding to the best fit are reported in Table 1. It can be seen that for each configuration, the model and the experiments are in good agreement for both the absorption and the impedance, in the frequency range between 500 Hz and 6000 Hz. Only small differences can be seen between 500 Hz and 3000 Hz for sample 2 and sample 6 and between 2500 Hz and 3500 Hz for sample 7. The best fit
Sound absorption average between 2000-10000Hz
Fig. 7. Sound absorption average at high frequencies as function of volume fraction and filament diameter; positions of the 3D-printed samples 1, 6 and 8 are represented at the intersections of the dashed lines.
Table 2 Acoustic properties as obtained from the analytical model and the inverse method. Sample #
1 2 3 4 5 6 7 8
Viscous characteristic length (lm)
Thermal characteristic length (lm)
Airflow resistivity (N.s.m4) 104
Tortuosity
Analytical model
Inverse method
Analytical model
Inverse method
Analytical model
Inverse method
Analytical model
Inverse method
1.4 1.1 0.9 0.9 0.5 3.1 1.4 0.8
– 1 0.9 0.8 0.6 2.5 1 0.7
1.4 1.4 1.3 1.3 1.2 1.4 1.3 1.3
– 1.3 1.1 1.2 1.1 1.1 1.3 1.2
130.5 173.2 207.7 203.6 277.5 129.8 177.1 220
– 193.6 144.8 249.5 249.3 48.7 104.4 120
182 252.4 315 317.5 444 179 260.4 338.3
– 210.1 227.3 249.5 299.5 241.5 244.4 273.5
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describing the experimental results was obtained with values of the diameter that are always higher (between 4% and 20%) than the set values for printing which is 0.625Dsp microns (the nozzle diameter). Moreover the model tends to overestimate the porosity of the periodic structure by 6% to 14%. The measured values of the porosity (using Eq. (1)) are consistently lower than the value of porosity corresponding to the best curve fitting. The larger diameters and higher densities obtained with the model can be explained by the fact that during manufacturing, each layer is not yet cure when new layers are printed on top of it. Due to the weight on the lower layers, the micro-rods are being deformed. After complete curing, instead of being cylindrical, the final structure ends up with ovoid rods.
With optimal variable couple (2r0, wf) and using Eqs. (6), (11), (12) and (13) the corresponding values of the airflow resistivity, the tortuosity, the viscous and thermal characteristic lengths were calculated. These values are gathered in Table 2. For orthogonal direct (OD) configuration, a decrease of the porosity, the tortuosity and the resistivity with increasing lattice parameter was observed. This result was foreseeable, since the distance between micro-rods is increasing with lattice parameter, thus creating fewer obstacles to sound propagation. Meanwhile, the viscous and thermal characteristic lengths are also increasing. This observation implies that as the lattice parameter increases, the visco-inertial effects at low frequencies (resistivity) and at high frequencies (tortuosity) become less dominant and that the acoustic losses are controlled by vis-
1
(a) Absorption coefficient (α)
0.8
0.6
0.4
0.2
0 500
900
1300
1700
2100
2500
2900
3300
3700
4100
4500
4900
5300
5700
Frequency (Hz) 1
Absorption coefficient (α)
(b) 0.8
0.6
0.4
0.2
0 500
900
1300
1700
2100
2500
2900
3300
3700
4100
4500
4900
5300
5700
4900
5300
5700
Frequency (Hz) 1
Absorption coefficient (α)
(c) 0.8
0.6
0.4
0.2
0 500
900
1300
1700
2100
2500
2900
3300
3700
4100
4500
Frequency (Hz) Fig. 8. Absorption coefficient: Curve fitting with the inverse method for sample in OD configuration (a), OA configuration (b) and S30 configuration (c).
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cous and thermal effects appearing in the medium to high frequencies range. Alternating the position of the cylindrical rods (OA configuration) seems to significantly increase the tortuosity and the airflow resistivity but reduces the characteristic lengths (sample 6). This observation is consistent with the increase of the acoustic absorption from low to medium frequencies to reach 1 at 3500 Hz. The main printing parameters are the set diameter (2r0) of the nozzle and the lattice parameter (a) which have an influence on the filling fraction and consequently on the acoustic performance of the final structure. Since there is a direct correlation between the printing parameters and the acoustic properties, the analytical model can be used to find the optimal printing parameters needed to achieve the best acoustic performance possible. For this investigation, an average value of the sound absorption coefficient was evaluated in the frequency range from 2000 to 10000 Hz. For each pair of values (2r0, wf) corresponding to a frequency dependant absorption coefficient, the average absorption over the entire measurement frequency range can be expressed as
I¼
N 1 X aðf i Þ N i¼1
ð23Þ
where N is the number of discrete frequencies in the domain of interest. This quantity bears some resemblance with the sound absorption average (SAA) as defined in the ASTM C423 [21] standard and its values range from 0 (no absorption) to 1 (full absorption). Fig. 7 represents the parametric plots of the average sound absorption for each pair of values (2r0, wf) for frequencies ranged between 2000 Hz and 10000 Hz, based on the model for periodic structure. It should be noted that this analysis was made only for orthogonal direct (OD) configuration. However, there is no restriction to apply this analysis to the two other configurations. In the investigated frequency domain, it is observed that the average absorption reaches a maximum of 0.95. The zone providing the best performance is uniform with the filling fraction between 5% and 60% and the corresponding the filament diameter varying between 0.05Dsp and 1.25Dsp. This zone is relatively large in terms of filling fraction and the diameter, which provides more flexibility for acoustic microstructures. For samples produced with a filament diameter of 0.625Dsp, the experimental values are in agreement with the performance index predicted by the model. As example, the positions of samples 1, 6 and 8 are shown on Fig. 7. It should be mentioned that the model can be easily applied to lower and mid frequencies domains to determine the filling fraction and the corresponding filament diameter necessary to reach the optimal acoustic absorption. This model is an effective tool for the selection of the printing parameters leading to the best acoustic properties possible. It can be also used to assess the reliability of the design of micro-lattice by predicting, for example, the decline in acoustic performance due to manufacturing imperfections. The inverse method for stochastic porous media was used to fit the experimental values of the absorption coefficient for the periodic structure. The best fit was sought by varying the tortuosity, the airflow resistivity as well as the characteristic lengths while keeping the porosity constant. The experimental value of the porosity calculated from Eq. (1) was used. Keeping the porosity constant suggests that flattening of the printed rods may not significantly change the periodic structure to influence the outcoming acoustical parameters. Fig. 8 shows examples of curve fitting for OD, OA and S30 configurations, from which the acoustic properties were extracted. It also reveals that the model behind the inverse method is in good agreement with the experimental results. Although the general trend and the order of magnitude are the same for all parameters, differences are observed for the values of the viscous and thermal characteristic lengths (sample 6, sample
7 and sample 8 in Table 2). Despite the differences between the analytic model and the inverse method, this analysis suggests that acoustic periodic structure can be fully characterized by models designed for materials with randomly distributed and interconnected porosities. 5. Conclusions The acoustic properties of a periodic structure produced by additive manufacturing was analyzed using an impedance tube and a model taking into account the interaction of an acoustic wave with periodic medium. This work was motivated by the need to control the manufacturing parameters of acoustic materials with broadband absorption capabilities. The following main results were obtained: Periodic acoustic structures produced via additive manufacturing exhibits broadband acoustic properties comparable to those of porous materials. The printing parameters such as the lattice parameter and the diameter of the printed filament can be adjusted to produce a structure with high absorption capabilities. High absorption can be achieved by using smaller filament diameter, by reducing the lattice parameter or/and by changing the orientation of the printed filament. The proposed analytical model was used as effective tool to define the best printing parameters necessary to produce structure with optimal acoustic performance. Analysis with analytical model for periodic structure reached the same conclusions as the model for stochastic porous materials. The tortuosity and the airflow resistivity are less dominant as the lattice parameter increases. Alternating the position of the rod from one layer to the other increase energy losses in the low to medium frequencies. Changing the orientation of the rods from one layer to the next appears to improve the absorption capability in the whole frequency range.
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