Exploring the acoustic potential of 3D printed micro-perforated panels: A comparative analysis

Abstract

In the present study, the sound absorption performance of inhomogeneous Micro-Perforated Panels (MPPs) with multiple cavities is investigated. Two models, a three-cavity system and a four-cavity system, are proposed and a numerical study is performed using MATLAB. The models are validated through experimental analysis in an impedance tube. The study meticulously varies the geometrical parameters, including pore diameter, thickness of the MPP, perforation ratio, and back-cavity length. It is found that MPPs with a greater number of sub-cavities have a better sound absorption coefficient than two-cavity systems. The results suggest that the back air cavity is predominantly responsible for multiple peaks, ensuring wideband sound absorption. It is also found that smaller perforation ratios for sub-cavities with larger pore diameters improve sound absorption performance in the lower frequency region. The study indicates that a pore diameter of less than 0.5 mm should be used for better sound absorption above the range of 800–850 Hz, and back cavity length has greater control than pore diameter between 850 Hz and 2000 Hz to make the curve smooth with less fluctuation. The findings have significant implications for the design of MPPs for real-world applications.

1. Introduction

The landscape of building technologies is continually evolving, with recent advancements focusing on sustainability and efficiency. The integration of innovative technologies in the construction sector, underscoring the pivotal role of circular economy principles in modern building practices [1,2]. Moreover, the exploration of metamaterials as a novel material technologies with tailored properties are gaining momentum predominantly for their attractive acoustic properties [3,4].

Noise control holds paramount importance in the design of a wide array of structural components across industries such as automotive, aerospace, marine, as well as in the production of home appliances, and various electrical/electronic goods. The significant influence of noise on human mental health and children's psychological well-being necessitates a diligent approach to noise management [[5], [6], [7]]. A key tool that researchers and engineers have utilized to counter these noise control challenges is the micro-perforated panels (MPPs). Offering multiple advantages over conventional sound-absorbing materials, such as non-combustibility, appealing aesthetics, ease of manufacturing, and tunability for sound absorption across wide frequency bands, MPPs have proven indispensable in noise control applications.

The MPPs can be effortlessly manufactured using a variety of materials including, but not limited to, plastic, plywood, sheet metal, and acrylic glass [[8], [9], [10]]. The versatility of these panels is further bolstered by the advent of additive manufacturing, which allows for the creation of panels with complex geometrical attributes, thereby expanding the horizons of acoustic energy absorption research [11,12]. The evolutions in 3D printing technology have given rise to an era of highly intricate sound absorption systems with unprecedented complexity.

Over the years, researchers have extensively explored various methodologies to enhance the sound absorption capabilities of MPPs. Fundamentally, an MPP comprises a perforated sheet, an air cavity situated behind this sheet, and a rigid backing at the end. The effects of varying parameters such as pore diameter, perforation ratio, air cavity length, and specimen thickness on the performance of MPPs have been well-studied [13]. The pore diameter in MPPs plays a crucial role, and researchers have conducted comprehensive investigations into the performance of MPPs with varying pore diameters, including large-sized pores, ultra-micro pores, hybrid-arranged perforated panels, micro-lattices, and micro-perforated panels [[14], [15], [16], [17], [18]].

Moreover, researchers have exploited various methodologies to enhance MPP performance, such as incorporating inhomogeneous cavities, multilayer perforation panels perforated panel resonators, porous sound-absorbing materials, and implementing honeycomb structures with different numbers of sub-cavities and geometrical variations [13,[19], [20], [21], [22], [23]]. Each of these methodologies has contributed significantly to the understanding of the MPPs' sound absorption mechanisms, which bear similarity to the Helmholtz resonator (HR) [24].

The advent of 3D printing technology has fostered the development of MPPs with intricate geometrical variations using environmentally friendly, biodegradable materials, albeit with several technical challenges. The exponential potential in the development of MPP through additive manufacturing is a key driver of this research, especially as additive manufacturing processes have evolved to facilitate the creation of complex models not only with polymers but also with various metals and composites.

Previous research has provided noteworthy insights. For example, Sailesh et al. employed the fused deposition modeling (FDM) 3D printing process to generate a series of spherical bubble perforations and found that their sample FG532, featuring decreasing order spherical bubble size, enhanced sound absorption in the lower frequency (<1000 Hz) region [25]. Mosa et al. conducted a study on a two-cavity system and demonstrated that an inhomogeneous cavity system with varying air cavity depth behind the MPP sheet improved the sound absorption coefficient (SAC) of MPP compared to the typical homogeneous cavity system [13]. In another noteworthy study, Yan et al. examined an MPP with a honeycomb structure as the basic cavity and analyzed the effect of four and seven sub-cavities [26]. In the study conducted by Deepak et al., a honeycomb structure was utilized as a part of the research methodology [27]. The team analyzed an MPP backed by a periodic honeycomb structure, fabricated through the Fused Filament Fabrication (FFF) technique. It was discerned from their findings that triangular perforation shapes exhibited superior Sound Absorption Coefficient (SAC) when compared to square and circular perforation shapes. Further, a research investigation undertaken by Carbajo et al. examined a heterogeneous perforated panel [28]. Their research findings underscored that a macro-perforation with geometric variation significantly enhanced the sound absorption capacity of the panel.

Further advancements in 3D printing have opened up opportunities to explore non-traditional sound-absorbing designs. Gao et al. leveraged this technology to develop an MPP with a helix-shaped cavity using micro-helix metamaterial (MHM), finding that increasing the cavity depth with a decrease in the pitch of the helix improved sound absorption [29]. Similarly, Edith et al. analyzed a complex micro-lattice periodic structure, preparing rigid micro-rods through additive manufacturing and placing them in different orientations [30]. They found that alternating the orientation of the filament in each layer could improve the broadband acoustic sound absorption capability.

Adaptable porous sound absorbers were utilized in a study by Kamil et al., designed to dissipate energy through a small steel ball situated within a periodic structure [31]. This method effectively blocks micro-holes in response to varying conditions. On a different note, a study by Almeida et al. underscored the efficacy of Micro-Perforated Panels (MPPs) in absorbing low-frequency sounds [32]. These researchers constructed specialized coiled-up spaces, which led to increased sound energy absorption due to improved viscous friction.

Nevertheless, the pursuit of broadband sound absorption with the aid of MPPs still poses a considerable challenge. This issue is primarily due to the intricate balance that must be struck among various parameters. An interesting approach to this problem was explored by Qian et al., where a series-parallel coupled MPP absorber was used. By organizing MPPs in both series and parallel combinations, their experimental findings revealed a potential solution for achieving wideband sound absorption through enhanced acoustic energy dissipation in parallel MPPs [33]. Findings from a study conducted by Shanlin et al. affirmed that the efficiency of sound absorption is primarily influenced by a series of parameters in a descending order of sensitivity [34]. The order includes cavity depth, aperture, perforation ratio, and panel thickness, highlighting the fundamental role these variables play in modulating the sound absorption capacity.

This extensive body of research strongly indicates that to enhance the sound absorption performance of sound-absorbing panels, geometrical parameters play a pivotal role. Different geometrical parameters affect the dissipation of acoustic energy in varying ways, and it has become apparent that no single parameter dictates the acoustic performance of the MPP. Rather, achieving optimal SAC often requires a trade-off among different parameters.

This study seeks to expand upon the existing body of research by analyzing the acoustic characteristics inherent in inhomogeneous micro-perforated panels (MPPs) featuring multiple back-cavities. Of particular interest in this investigation are the sound absorption traits of inhomogeneous MPPs composed of three and four back-cavities. It is crucial to acknowledge that an inhomogeneous MPP comprises numerous sub-cavities, each one segregated from the others by a slender wall. In contrast to this, traditional homogeneous MPPs incorporate pores with a constant perforation ratio, resulting in a solitary impedance value and consequently, a considerably narrow band of sound absorption.

Contrarily, inhomogeneous MPPs encapsulate a multitude of sub-cavities, each varying in terms of back cavity lengths, diameters of pores, and perforation ratios. This culminates in the creation of multiple impedances within one MPP, thus ensuring wideband sound absorption with superior sound absorption coefficient (SAC). A comprehensive analysis of parameters is undertaken within the framework of Maa's classical theory of sound absorption and the equivalent electro-acoustic model [35,36]. The range of this parameter study encompasses investigations into the effects of perforation diameter, perforation ratio, back-cavity length, and thickness of MPP on sound absorption traits. Finally, to validate the theoretical models, comparisons are drawn between theoretical and experimental outcomes for two pre-selected scenarios.

2. Methodology

2.1. Theoretical analysis

This study is designed to augment the sound absorption potential of an inhomogeneous cavity system by integrating three or four cavities. These cavities are distinctly partitioned by thin walls, which results in varying cavity lengths. The theoretical framework for this study expands upon the electro-acoustic model employed by Mosa et al. for a two-cavity homogeneous system, adapting it to accommodate three or four cavity homogeneous systems [13]. A wireframe representation of a three-cavity system is provided in Fig. 1(a). The equivalent circuit diagram for an inhomogeneous MPP with an array of parallel cavities is presented in Fig. 1(b). Each individual circuit comprises an initial resistance component, followed by a reactance component, and concluding with the resistance engendered by the air cavity located behind the MPP.

Fig. 1.

(a) Wireframe model of inhomogeneous MPP with three sub-cavities in which d is pore diameter, L is back cavity length, t thickness of panel and D is diameter of MPP, (b) Equivalent electro-acoustic model of an IMPP.

2.1.1. Acoustic impedance of inhomogeneous cavity system

The total impedance of MPP is derived by summing the acoustic impedance produced by MPP ($Z_{mpp}$) with that of the air cavity behind the MPP $\left(Z_L\right)\text{,}$ as specified by equations (1), (2), (3), (4), (5), (6) [36]:

$$Z_{total}=Z_{mpp}+Z_L$$ (1)

where the acoustic impedance produced by MPP is given by

$$Z_{mpp}=R_{mpp}+j\omega m$$ (2)

where, $R_{mpp}$ is normalized specific acoustic resistance, $\omega$ is angular frequency measured in rad/sec and $m$ is normalized acoustic reactance. The impedance by air when the air cavity depth behind the MPP is L, is given by

$$Z_L=-j\cot \left(\frac{\omega L}{C}\right)$$ (3)

where, $c$ is the speed of sound in air measured in m/sec. The normalized specific acoustic resistance is given by

$$R_{mpp}=\frac{32\mu t}{p\rho c{d}^2}\left(\sqrt{1+\frac{x^2}{32}}+\frac{\sqrt{2}}{8}x\frac{d}{t}\right)$$ (4)

where, $\mu$ is air viscosity of air in Pa.s, and t is thickness of MPP in m, $p$ is perforation ratio, $\rho$ is the density of air kg/m³, $d$ is the diameter of pore in m.

The normalized specific acoustic reactance is given by

$$m=\frac{t}{\rho c}\left[1+{\left(9+\frac{x^2}{2}\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+0.85\frac{d}{t}\right]$$ (5)

Similarly, the perforation constant which is given by

$$x=\frac{d}{t}\left(\frac{\omega \rho }{\mu }\right)$$ (6)

2.1.2. Acoustic impedance of an inhomogeneous three-cavity and four cavity system

The sub-cavities of the inhomogeneous MPP work as a separate MPP and individual sub-MPP possesses all the characteristics of a standard MPP. Therefore, the electrical model proposed by MAA (1987, 1988) can be easily implemented on these sub-cavities. Moreover, each sub-system has its own acoustic impedance which can be given by following expressions,

$$Z_i=R_i+j\omega m_i+Z_{D_i}$$ (7)

In Equation (7), $i$ indicates the $i^{th}$ number of sub-cavity in the inhomogeneous MPP. In Fig. 1, it is depicted that all sub-cavities of MPP are in parallel connection. For three cavity system, the equivalent acoustic resistance is given by equation (8):

$$Z_{equ}=\frac{1}{\sum _{i=1}^n\left(\frac{{\phi }_i}{Z_i}\right)}$$ (8)

where, φ is the ratio of area covered by the particular sub-MPP to the total area of MPP. The sound absorption coefficient of inhomogeneous cavity system is given by equation (9):

$$\mathrm{\alpha }=\frac{4Re\left\{Z_{equ}\right\}}{{\left[1+Re\left\{Z_{equ}\right\}\right]}^2+{\left[Imag\left\{Z_{equ}\right\}\right]}^2}$$ (9)

2.2. Experimental studies

2.2.1. Material

This study utilizes Polylactic Acid (PLA), a biodegradable material derived from vegetable waste, for the fabrication of Inhomogeneous Micro-Perforated Panels (IMPPs). The PLA filament, with a diameter of 1.7 mm, was procured from IPRO3D. This filament was then used to construct the IMPPs using 3D printing technology. For the experimental investigation, two samples with distinct parameters, as outlined in Table 1, were selected.

Table 1.

Specifications of samples.

Sample	Sample diameter (mm)	Panel thickness t (mm)	Pore diameter d1 = d2 (mm)	Pore diameter d3 (mm)	Perforation ratio p1 = p2	Perforation ratio p3	Back cavity length L1 = L2 (mm)	Back cavity length L3 (mm)
IMPP-1	99	1.0	1.0	1.5	2.5%	1.0%	20	60
IMPP-2	99	1.0	1.0	1.5	1.5%	2.5%	20	60

2.2.2. 3D printing

The 3D printing process and the resultant samples, highlighting the intricate details and the measured back cavity lengths, are vividly illustrated in Fig. 2(a–h). The 3D printing of the samples was carried out using the Prusa i3 MK3S+ 3D printer. This printer is renowned for its reliability, power, and silent operation. It features a build volume of 250 × 210 × 210 mm³ and a nozzle diameter of 0.4 mm. The printing process was conducted at a nozzle temperature of 210 °C, a bed temperature of 60 °C, and an ambient temperature maintained between 25 °C – 28 °C.

Fig. 2.

(a) Ultimaker cura 4.0, (b) 3D printer, (c) and (d) printed samples, (e) back cavity of the inhomogeneous MPP, (f) back cavity length with measuring scale, (g) and (h) MPP on the back cavity.

The Prusa i3 MK3S+ is equipped with a SuperPINDA probe for fully-automatic Mesh Bed Leveling, ensuring precise first layer adhesion. It also features high-quality bearings and a reliable belt tensioning mechanism on the X-axis for improved stability and performance. The printer's ability to handle a wide range of materials, including PLA, makes it an ideal choice for this study. The CAD models for the IMPPs were prepared using Creo 7.0. These CAD files were then converted to.stl formats to generate the G-codes necessary for 3D printing. For this conversion, PrusaSlicer 2.3 was employed. PrusaSlicer is a powerful slicing software developed by Prusa Research.

2.2.3. SAC prediction

The experimental setup for measurement of SAC of MPPs is shown in Fig. 3(a–c) in which the impedance tube (manufactured by PLACID Instruments BV, Netherlands) and two microphones (diameter ¼″ Class 20 Hz to 20 kHz with SMB to BNC connector) are used as per the ASTM E1050-08 and ISO10543-2 standards in the frequency range of 50 Hz–1600 Hz. The white noise is produced by a 20W and 4 Ω speaker which connected to 50 W power amplifier with the help of 2 m Banana cable. The incident and reflected sound pressure are measured by two microphones which are connected to DAQ (with 4 Channels) system and the microphones are calibrated with the help of a sound calibrator at 94 dB at 1000 Hz. Two samples (99 mm diameter) with back cavity length are tested in this experiment.

Fig. 3.

(a) Impedance tube for experimental analysis, (b) sample fitted in the impedance tube, (c) back cavity length determination on a moveable piston.

3. Validation study

The current study employs an equivalent electro-acoustic model to derive theoretical results. This model has been previously utilized by Mosa et al. in their investigation of two-cavity inhomogeneous Micro-Perforated Panels (MPPs) [13]. To validate the efficacy of the MATLAB code developed in this study, the results obtained by Mosa et al. are used as a benchmark for comparison.

The specific parameters of the two-cavity inhomogeneous MPP analyzed by Mosa et al. are as follows: L1 = 40 mm, L2 = 75 mm, d1 = 0.6 mm, d2 = 0.3 mm, p1 = 0.6%, and p2 = 4.0%. These parameters are replicated in the present study for the purpose of validation. The comparison of the results is depicted in Fig. 4.

Fig. 4.

Comparison of the SAC predicted using present approach with Mosa et al. [9].

Upon examination of Fig. 4, it is clear that the results obtained from the current approach align well with those of Mosa et al. [13]. This agreement validates the effectiveness of the MATLAB code developed in this study for modeling two-cavity inhomogeneous MPPs using the equivalent electro-acoustic model.

It is worth noting that the equivalent electro-acoustic model has been widely used in the field of acoustics, particularly in the design of wideband acoustic absorbers composed of parallel-arranged MPPs. These absorbers are known for their high absorptive performance without the need for any additional fibrous or porous material. The model has been theoretically validated and implemented on various prototypes, showing good agreement between the prediction model and the experimental data.

The validation study confirms the accuracy and reliability of the equivalent electro-acoustic model used in this study for predicting the acoustic performance of two-cavity inhomogeneous MPPs. Future work will focus on further refining the model and expanding its application to other types of MPPs.

4. Numerical results and discussions

The impact of the number of inhomogeneous cavities on sound absorption performance is a critical area of study due to its direct influence on the Sound Absorption Coefficient (SAC) across a broad frequency range. This investigation is conducted in two parts: the analysis of a three-cavity system (Section 4.1) followed by a four-cavity system (Section 4.2). These analyses are executed using the analytical approach detailed in the methodology section.

A comprehensive parameter study is performed, examining the effects of key parameters such as pore diameter (d), Micro-Perforated Panel (MPP) thickness (t), perforation ratio (p), and back-cavity length (L). Subsequently, two samples are fabricated using 3D printing, and experimental results are obtained using the impedance tube method. The final part of this section presents a comparison of the analytical and experimental results.

4.1. Studies on three cavity system

The circular disc sample, typically employed in the impedance tube method for sound absorption studies, is partitioned evenly into three distinct sections. Each section, characterized by its unique pore diameter, thickness, back-cavity length, and perforation ratio, is delineated in Fig. 5. It is noteworthy to mention that these cavities are physically isolated from one another by thin, 1 mm walls. Each parameter associated with a specific cavity is symbolized by an appropriate subscript number, facilitating a clear understanding of the variations across the sample.

Fig. 5.

IMPP with three sub-cavity system.

4.1.1. Effect of perforation ratio

To scrutinize the influence of the perforation ratio, the selection of other parameters is meticulously undertaken. The air cavity depths for the three cavities are standardized as L₁ = L₂ = 20 mm, L₃ = 60 mm. Drawing from insights gleaned from two-cavity system studies [13], a wider sound absorption bandwidth emerges when the differences among the air cavity depths are pronounced. This phenomenon is attributed to the creation of multiple Helmholtz resonators by these varied depths, each tuned to different sound wavelengths. The thickness of the MPP is fixed at 1 mm, as variations in thickness exhibit the least impact on sound absorption improvement. It is acknowledged that for the absorption of medium and low-frequency waves, the diameter of pores should be submillimeter [20]. Therefore, the diameters of the pores associated with the first and second cavities are set as d₁ = d₂ = 0.4 mm, while for the third cavity, it is fixed at d₃ = 0.8 mm.

To evaluate the effect of perforation ratio on the SAC of three-cavity systems, the perforation ratio of the third cavity (p₃) is kept constant, while those of the first two cavities are equated (p₁ = p₂) and varied from 0.5% to 2.5%. The corresponding results are presented in Fig. 6(a–c) which reveal the perforation ratios of the first two cavities (p₁ and p₂) increase, the second peak shifts toward the higher frequency range, and the bandwidth broadens. However, the first peak exhibits minimal variation in terms of peak value and bandwidth. This is because the specific acoustic resistance at the first peak (480 Hz–510 Hz) remains equal for all the samples, as depicted in Fig. 6(d) and e. Furthermore, with an increase in p₁ and p₂, the anti-peak's amplitude significantly diminishes, widening the bandwidth. A similar variation in SAC is noted when p₁ = p₂, and p₃ is varied, as observed in Fig. 6(f).

Fig. 6.

Effect of perforation: L₁ = L₂ = 20 mm, L₃ = 60 mm, d₁ = d₂ = 0.4 mm, d₃ = 0.8 mm, t = 1 mm. (a, b, c): sound absorption curves, (d, e, f): and corresponding specific acoustic resistance and specific acoustic reactance curves.

The resonance frequency of an acoustic cavity ascends with the perforation ratio as illustrated by Eqn. (10), corroborated by the results in Fig. 6. Concurrently, the specific acoustic reactance remains below zero, and a further decrement in specific acoustic reactance is noticed as the perforation ratios p₁ = p₂ increase, as seen in Fig. 6(d, e, f), facilitating wideband sound absorption.

The resonance frequency and perforation ratio relationship is expressed by:

$$f_r=\sqrt{\frac{pc}{8\pi Dt}}$$ (10)

where, $f_r$ denotes the resonance frequency in kHz, p represents perforation ratio, c signifies the speed of sound in air (344 m/s), D is the air cavity depth in mm, and t symbolizes the thickness of MPP in mm.

The application of pore diameters exceeding 1 mm permits researchers and engineers to leverage 3D printing technology to produce unconventional sound absorption systems characterized by complex geometrical variations. The influence of the perforation ratio for pore diameter values beyond the sub-millimeter range is demonstrated in Fig. 7. For this examination, a sample with d₁ = d₂ = 1 mm, d₃ = 1.5 mm, and L₁ = L₂ = 20 mm; L₃ = 60 mm, is employed. The perforation ratio of the first two cavities (p₁ = p₂) is varied from 0.5% to 2.5%, while p₃ is kept constant.

Fig. 7.

Effect of perforation when pore diameter is greater than 1 mm. L₁ = L₂ = 20 mm, L₃ = 60 mm, (a) (b) (c) d₁ = d₂ = 1 mm, d₃ = 1.5 mm, t = 1 mm. (d) d₁ = d₂ = 0.4 mm, d₃ = 0.8 mm.

The SAC variation mimics that observed in the samples with sub-millimeter pore diameters (as depicted in Fig. 6). The first peak remains consistent, while the second peak shifts towards higher frequency ranges with increased perforation ratios of the first two cavities, as previously discussed. However, anti-peaks are observed with very low SAC values (less than 0.4), indicating a significant downside of MPPs with pore diameters exceeding 1 mm.

The effect of unequal perforation ratios on SAC (i.e., p1 ≠ p2 ≠ p3) is showcased in Fig. 7(d). In this scenario, the sample with pore diameters d1 = d2 = 0.4 mm, d3 = 0.8 mm, back cavity lengths L1 = L2 = 20 mm, L3 = 60 mm, and MPP thickness t = 1 mm is considered. Perforation ratio p1 is varied from 0.4 to 0.8, with p2 and p3 being the second and third multiples of p1, respectively. The results portrayed in Fig. 7(d) align with those of the studies conducted on samples with sub-millimeter pore diameter and samples with pore diameters greater than 1 mm, as delineated in Fig. 6, Fig. 7, respectively.

The depiction presented in Fig. 7(d) further substantiates that as the perforation ratios ascend, the peaks shift towards higher frequency regions with wider bandwidth and SAC values above 0.8. Nonetheless, a minor decrement in peak SAC value is observed with increased perforation ratios. Significantly, different perforation ratios (p1 = 0.8%, p2 = 1.6%, p3 = 2.4%) have substantially improved the SAC at the anti-peaks (SAC ≥ 0.8) when compared to the outcomes presented in Fig. 6, Fig. 7a, b, c). This study unambiguously illustrates that the perforation ratio of a three-cavity system can be tailored based on the frequency region of interest.

4.1.2. Effect of pore diameter

Our investigation of the effect of pore diameter on SAC employs the sample where p₁ = p₂ = p₃ = 2%, as analyzed in Section 4.1.1. This sample exhibited superior SAC compared to other cases under study. The other associated parameters are set at L₁ = L₂ = 20 mm, L₃ = 60 mm, and t = 1 mm. The pore diameter for the first two cavities is held constant while varying the diameter of the third cavity. The ensuing results are displayed in Fig. 8.

Fig. 8.

Effect of pore diameter: p₁ = p₂ = p₃ = 2%, L₁ = L₂ = 20 mm, L₃ = 60 mm; t = 1 mm (a) when d₁ = d₂ = 0.5 mm, (b) when d₁ = d₂ = 0.7 mm, (c) when d₁ = d₂ = 0.9 mm.

The depiction in Fig. 8(a) elucidates that the second peak and its associated bandwidth (SAC exceeding 0.75) remain largely unaffected by the change in the third cavity's pore diameter. This outcome can be traced back to the negligible variations in the specific acoustic resistance related to the second peak when the pore diameter varies. Conversely, the first peak demonstrates significant variations in peak values and bandwidth with changes in the pore diameter of the third cavity. This is attributable to the substantial alterations in specific acoustic resistance and specific acoustic reactance accompanying pore diameter changes. Further, as d3 escalates, a consistent increment in specific acoustic resistance is noted, leading to the formation of deeper anti-peaks between the two peaks. Moreover, as the equal diameters d₁ = d₂ are incremented from 0.5 mm to 0.9 mm, there is a noticeable reduction in the amplitude at the second peak, as indicated in Fig. 8(a, b, c).

The study on the impact of pore diameter variations on SAC, employing different configurations of pore diameters across the sub-cavities, is detailed with results shown in Fig. 9(a) for homogeneous pore diameters and in Fig. 9(b) for the effects of inhomogeneous cavity depths on SAC. To conduct this analysis, a sample with p₁ = p₂ = p₃ = 2%, L₁ = L₂ = 20 mm, L₃ = 60 mm, and t = 1 mm is considered. Three distinct pore diameters—0.4 mm, 0.6 mm, and 0.8 mm—are examined in various combinations, and the consequential effects on SAC are shown in Fig. 9(a–b).

Fig. 9.

Effect of pore diameter when all three diameters are unequal and p₁ = p₂ = p₃ = 2%, t = 1 mm (a) L₁ = L₂ = 20 mm, L₃ = 60 mm (b) L₁ = 20 mm, L₂ = 60 mm, L₃ = 40 mm.

The SAC of the heterogeneous MPP is observed to be influenced by the variation in pore diameter, as shown in Fig. 9. The dip in SAC, with values less than 0.8, is most significant when the pore diameters of the first two cavities are similar, given the identical back-cavity lengths, as observed in the d₁ = 0.4 mm, d₂ = 0.6 mm, and d₃ = 0.8 mm case. However, this situation results in a second peak with a wider bandwidth. Moreover, SAC is poor when the pore diameter is large and the associated back-cavity length is small, as seen in the d₁ = 0.8 mm, d₂ = 0.6 mm, and d₃ = 0.4 mm case with a back cavity length of 60 mm. Contrastingly, superior SAC is recorded when both the pore diameter and corresponding back-cavity length are small, that is, for the d₁ = 0.4 mm, d₂ = 0.8 mm, and d₃ = 0.4 mm case with a back cavity length of 20 mm. In this instance, there is a distinct divergence in the pore diameters of the first two cavities.

The impact of pore diameter on the SAC performance of a heterogeneous MPP with variable back-cavity lengths is depicted in Fig. 9(b). Under these conditions, greater fluctuations in SAC are registered when the pore diameter is small, and the corresponding back-cavity length is large for any one of the cavities. Similarly, an almost flat SAC curve with a wide bandwidth is observed when the pore diameter is large and the corresponding back-cavity length is small for any one of the cavities. These findings underscore the intricate interactions between pore diameter and back-cavity length in determining the SAC of an MPP system.

4.1.3. Effect of air cavity depth

The SAC associated with lower frequency waves can be enhanced significantly by providing appropriate air cavity depth behind the MPP. In order to accomplish this study, the perforation ratios and diameters of pores are set as p₁ = p₂ = p₃ = 2% and d₁ = d₂ = d₃ = 0.5 mm respectively. The air cavity depth of first two cavities (L₁ and L₂) is varied from 20 to 40 mm while the third air cavity depth is kept constant. When L₁ is equal to L₂ and L₃ is varied, two clear peaks in the SAC curve is observed, only when there is a significant difference in the back cavity lengths, i.e., L₁ = L₂ << L₃ as shown in Fig. 10(a, b, c). This may be attributed to specific acoustic impedance variation with respect to the back-cavity length. This also clearly indicate that in a multi-cavity system, the SAC can be improved by keeping the air-cavity depths with large variation. Furthermore, the increase in L₁ and L₂ also improves the SAC associated with frequencies higher than the first peak frequency in general. Based on this, for the study on L₁ ≠ L₂ ≠ L₃, case, three cavity lengths 15 mm, 30 mm, and 60 mm are varied in different combinations and the results are given in Fig. 10(d) and the SAC is greater than 0.9 due to huge variation in air cavity depths in sample (L₁ = 60 mm, L₂ = 30 mm, L₃ = 15 mm), as well as, the broadband sound absorption is achieved.

Fig. 10.

Effect of air cavity depth: p₁ = p₂ = p₃ = 2%, t = 1 mm when (a), (b) (c): d₁ = d₂ = d₃ = 0.5 mm. (d) d₁ = 0.4 mm, d₂ = 0.6 mm, d₃ = 0.8 mm with different air cavity depths.

4.1.4. Effect of thickness of MPP

It is well established fact that the variation in thickness has the least effect on SAC [9] and thicker panels are required to absorb the low frequency waves. For this analysis, a sample with p₁ = p₂ = p₂ = 2%, d₁ = d₂ = 0.4 mm, d₃ = 0.8 mm is considered. The results shown in Fig. 11(a and b) are correspond to the parameters L₁ = L₂ = 20 mm, L₃ = 55 mm, with variation of t₁ = t₂ and t₃. Similarly, the results shown in Fig. 11(c) are corresponds to L₁ = 60 mm, L₂ = 30 mm L₃ = 15 mm. Furthermore, the effect of thickness on SAC is very minimal as shown in Fig. 11(b) in which as the thickness is increasing form 0.5 mm–0.9 mm the second peak is shifting towards the lower frequency region. However, the bandwidth of absorption decreases and similar trend can be seen in Fig. 11(a). In Fig. 11(c) when the thickness is very close to 1 mm then only a smooth curve is achieved and a big variation in thickness is creating undesirable anti-peaks as well as the variation in thickness has reduced the sound absorption band. Hence, to manufacture the MPPs, it is an appropriate step to not vary the thickness which eventually make the MPP industry friendly.

Fig. 11.

Effect of thickness: p₁ = p₂ = p₂ = 2%, (a) (b) d₁ = d₂ = d₃ = 0.5 mm, and L₁ = L₂ = 20 mm, L₃ = 55 mm, (c) d₁ = d₂ = 0.4 mm, d₃ = 0.8 mm, and L₁ = 60 mm, L₂ = 30 mm, L₃ = 15 mm.

4.2. Four cavity system

A typical four cavity system having different parameters and perforation ratios is shown in Fig. 12.

Fig. 12.

A four cavity MPP panel containing four sub cavities.

4.2.1. Effect of perforation ratio

In order to analyze the effect of the perforation ratio, the perforation ratio of each sub-cavity is varied in different configurations. For example, in the first configuration, all the perforations ratios are kept equal, the second configuration has two pairs of perforation ratios, the first pair (p₁, p₂) and the second pair (p₃, p₄) are varied in a specific way to maximize the sound absorption capacity in the third configuration different perforation ratios are assumed for different cavity. To accomplish this study, the diameters d₁, d₂, d₃ and d₄ are kept constants as 0.4 mm, 0.4 mm, 0.8 mm and 0.8 mm respectively and the back cavity lengths L₁, L₂, L₃ and L₄ are set as 15 mm, 30 mm, 45 mm and 60 mm respectively and the thickness of MPP is kept as 1 mm. The reason for keeping such values of back cavity lengths is the observation from the three cavity systems. From three-cavity system study, it is observed that for the inhomogeneous MPP system, back-cavities with significant difference in lengths enhances the sound absorption. From the study on three cavity system it is also observed that a sub-cavity having higher perforation ratio than the other sub-cavities consists of more number of small sized pores and a sub-cavity with lower perforation ratio consists of fewer number of pores with larger diameter enhances the SAC. This factor has been considered throughout the present study while selecting the parameters to ensure better sound absorption performance by the inhomogeneous MPPs.

The effect of the perforation ratio when all the perforation ratios are equal and varied from 1.5% to 3% is shown in Fig. 13(a). All sub-cavities with same perforation ratio makes the MPP as least inhomogeneous. Because of less inhomogeneity better sound absorption is observed in a very narrow band which is 700 Hz–1300 Hz. Furthermore, as the perforation ratio increases the peak value of SAC is shifting towards higher frequency regions with the increment in the bandwidth can be seen in Fig. 13(a). However, the first peak value of SAC of each case decreases with the perforation ratio and SAC at the second peak is increasing and the bigger anti-peaks are formed.

Fig. 13.

Effect of perforation ratio: (a) for p₁ = p₂ = p₃ = p₄, (b) p₁ = p₂ and p₃ = p₄, (c) p₁≠ p₂≠ p₃≠ p₄.

In order to improve inhomogeneity of the MPP two pairs of sub-cavities having same perforation ratio (i.e., first pair p₁ = p₂ and second pair p₃ = p₄) is considered. In this study, the perforation ratio of the first pair is varied from 1.5% to 2.5% and for the second pair it is varied from 1% to 2%. It should be noted here that p₁ and p₂ correspond to the sub-cavities having 0.4 mm pore diameter and p₃ and p₄ correspond to the sub-cavities having 0.8 mm pore diameter. Moreover, it is evident from results in Fig. 13(b) that sound absorption curves shift towards higher frequency region as well as overall sound absorption frequency band is reduced when the perforation ratio of first pair (p₁ = p₂) is kept constant and perforation ratio of pair second (p₃ = p₄) is varied between 1% and 2%. Conversely, if the perforation ratio of the second pair is reduced and p₁ and p₂ are equal and kept constant then the first peak further shift towards the lower frequency region and eventually demonstrate wideband sound absorption behaviour, the dip next however, in SAC to first peak should be taken care as it is significant. These phenomena clearly suggest that if better sound absorption is intended over a broader frequency range with less concern about lower frequency regions then the perforation ratios should be smaller for sub-cavities which consists of relatively larger pore diameter than the other sub-cavities. This study also reveals that increased inhomogeneity achieved by varying the perforation ratio of pair of cavities enhances the SAC over wider bandwidth as seen in Fig. 13(b).

In order to increase the inhomogeneity of MPP further, all the four perforation ratios are kept unequal and also to ensure the non-repetitiveness of data increments of 20% and 40% in the perforation ratio are made in the base value and the results are shown in Fig. 13(c). Due to increased inhomogeneity of MPP, the sound absorption performance is improved drastically not only with higher SAC but also with wider sound absorption bandwidth as seen in Fig. 13(c). Moreover, as the perforation ratios increase the peaks are shifting towards the higher frequency side and this trend can be comprehended with the help of Eq. (10). On the other hand, the first sample with p₁ = 2.5%, p₂ = 2%, p₃ = 1.5% and p₄ = 1% consists of bigger diameter pores (for p₃ = 1.5% and p₄ = 1%) exhibited first peak in the lower frequency side compared to the other samples. Finally, the sample which contains perforation ratios as p₁ = 3%, p₂ = 2.4%, p₃ = 1.8% and p₄ = 1.2% is considered for the further parametric studies of MPP.

4.2.2. Effect of pore diameter

The use of sub-millimeter size pores gives better sound absorption which is observed in the study of three cavities system as well. Three different configurations of pore diameters are studied similar to the three cavity system, are considered to analyze effect of the perforation ratio. In the first configuration, it is assumed that all the four cavities have same pore diameters. In the second configuration, it is assumed that d₁ = d₂ and d₃ = d₄ and in the third configuration each sub-cavity has different pore diameter. For this study the back cavity lengths are kept constant as L₁ = 15 mm, L₂ = 30 mm, L₃ = 45 mm, and L₄ = 60 mm and thickness of MPP is taken as 1 mm. For the first configuration, the results are shown in Fig. 14(a) when the diameter is varied from 0.25 mm to 1 mm very flat response is observed for the MPPs with smallest pore diameter (0.25 mm). However, when the diameter is further increased, multiple peaks and troughs are observed and the flatness of curves above the value of SAC of 0.75 has been reduced significantly. Despite less flatness, the sample with pore a diameter of 0.5 mm shows better SAC.

Fig. 14.

Effect of pore diameter: (a) for d₁ = d₂ = d₃ = d₄, (b) d₁ = d₂ and d₃ = d₄, (c) d₁≠ d₂≠ d₃≠ d₄. (d) comparison of results.

For the study on samples with second configuration, the pore diameters of first pair (d₁ = d₂) is varied between 0.4 mm and 0.6 mm while for the second pair (d₃ = d₄) it is varied between 0.7 mm and 0.9 mm and the results are shown in Fig. 14(b). It can be observed that the third and fourth peak values or regions above 850 Hz are more susceptible to variation in pore diameter of the first pair. It is also evident from the results that for a constant pore diameter of 0.7 mm of the second pair with varying pore diameter of first pair there is a distinct variation in the third and fourth peaks while the first and second peaks remain majorly unchanged. In a similar manner, the first and second peaks are more susceptible to change in pore diameter of second pair (d₃ and d₄) for constant d₁ = d₂ as shown in Fig. 14(b). Finally, it is observed that the effect of pore diameter on SAC in the region approximately above 1000 Hz is not very effective as significant fluctuation in SAC curves with the deeper trough are not sensitive to the variation in pore diameter. This study clearly suggests that in the regions above the frequency range 800–850 Hz when the diameters of pores are sub-millimeter size, the variation in diameter of pores is not a prominent factor to improve the sound absorption performance. From the results the sample (d₁ = d₂ = 0.4 mm and d₃ = d₄ = 0.7 mm) shows better results in terms of wide band sound absorption with moderate trough.

Influence of unequal pore diameter in different sub-cavities on SAC of inhomogeneous MPP is shown in Fig. 14(c). To accomplish this study, the base value of the four sub-cavity pore diameters is kept as d₁ = 0.2 mm, d₂ = 0.3 mm, d₃ = 0.4 mm and d₄ = 0.5 mm then 30%, 60% and 90% increment in the base value of pore diameters is analyzed. From the results, it is clear that the fluctuations in SAC curve increases with the size of the pore diameter and even an increase of 0.5 mm of any one of the cavities manifolds the fluctuation. This analysis indicates that for the multi-cavity inhomogeneous system with sub-millimeter pore and in order to obtain the better SAC, the pore diameter should be less than 0.5 mm and the other parameters should be varied accordingly. Best performing samples in terms of higher SAC with wider bandwidth and lesser fluctuation from each configuration is selected and compared in Fig. 14(d) and found that sample with d₁ = d₂ = d₃ = d₄ = 0.5 mm performs better in terms of wide-band sound absorption and lesser fluctuation in the curve.

4.2.3. Effect of back cavity length

The back cavity length is a powerful parameter which influences the SAC of MPPs significantly compared to the other parameters. In order to accomplish this study, the pore diameter of all the cavities are kept as 0.5 mm with the perforation ratios are p₁ = 3%, p₂ = 2.4%, p₃ = 1.8%, p₄ = 1.2% and the thickness is 1 mm. In this analysis three configurations are analyzed. In first configuration, all the four back cavity lengths are same, in the second configuration contains two pairs of back cavity lengths and the third configuration contains all the four back cavities with different length and the results are shown in Fig. 15.

Fig. 15.

Effect of back cavity length: (a) for L₁ = L₂ = L₃ = L₄, (b) L₁ = L₂ and L₃ = L₄, (c) L₁≠ L₂≠ L₃≠ L₄. (d) comparison of results.

The results of first configuration in which all the four back cavity lengths are equal and varied from 15 mm to 60 mm are shown in Fig. 15(a). From the results, it is evident that increase in back cavity length shifts the SAC curve towards the lower frequency range. This also indicate that sound absorption in a smaller frequency range needs a longer back cavity and sound absorption in a higher frequency region requires a shorter back cavity length the results of the second configuration are shown in Fig. 15(b). In this configuration the length of first pair of cavities, (L₁ and L₂) is varied from 15 mm to 25 mm while for the second pair (L₃ and L₄) it is varied from 50 mm to 55 mm. Because of the significant difference in two of the sub-cavities the inhomogeneous of the MPP increases drastically. Due to this, better sound absorption is achieved when compared to the first configuration in which all the back-cavity have same length. From results, it can be observed that peak value of SAC, as well as the broader frequency range, remains unchanged and only anti-peaks increase when first pair (L₁ = L₂) is kept constant and second pair (L₃ = L₄) is varied from 50 mm to 55 mm. The difference between back cavity lengths of first and second pair influences the peak values and bandwidth significantly less difference leads to poor performance which larger differences leads to better performance, hence L₁ = L₂ = 25 mm and L₃ = L₄ = 50 mm shows poor SAC with narrow frequency band sound absorption performance to a small frequency region. Hence, again from the analysis on the four cavity system, it is evident that significant difference in back cavity lengths can improve SAC of the inhomogeneous MPP drastically.

Effect of different back cavity length for different sub-cavity is studied in the third configuration. For this purpose, sample with following back cavity lengths L₁ = 15 mm, L₂ = 25 mm, L₃ = 35 mm, L₄ = 45 mm is chosen. Then the subsequent samples having back-cavity length with 5%, 10%, 15% and 20% increment from base value are analyzed. Due to the highest degree of inhomogeneity achieved as a result of different back-cavity lengths SAC has been improved tremendously and the results are shown in Fig. 15(c).

Best performing sample in each configuration in terms of better SAC value with wider bandwidth is chosen and compared in Fig. 15(d). From Fig. 15(d), it can be noted that the amplitude of SAC beyond the frequency 800–850 Hz can be efficiently controlled by back cavity length and better sound absorption performance can be achieved by keeping the pre diameter as less than 0.5 mm. In particular, the sample with back cavity lengths as L₁ = 18 mm, L₂ = 30 mm, L₃ = 42 mm and L₄ = 54 mm exhibits better SAC peaks values and even anti-peak values are higher more than 0.8 and it is best suited for this particular frequency range in real-world applications.

5. Comparison of experimental and theoretical results

To validate the theoretical assumptions and provide a robust foundation for the conclusions drawn, two samples, IMPP-1 and IMPP-2, were fabricated using 3D printing technology, as per the specifications presented in Table 1. The pore diameters were carefully chosen to minimize typical printing errors associated with the Fused Deposition Modeling (FDM) process, thereby facilitating a more accurate comparison of results.

The comparison of the SAC results for IMPP-1 and IMPP-2 samples is illustrated in Fig. 17. A cursory glance at Fig. 17 reveals that the trend of experimental results follows the trajectory of the numerical results. In Fig. 17(a), the experimental results exhibit two major peaks at 434 Hz and 1101 Hz, with the former peak aligning well with the numerical results. The increased SAC observed in the experimental results can be attributed to surface irregularities such as fiber inside the cavity, surface roughness, and the presence of micro-porosity and air gaps around the specimen [37].

Fig. 17.

Experimental and numerical comparison of (a) IMPP-1 and (b) IMPP-2.

Similarly, in Fig. 17(b), there are two peaks in the experimental results at 656 Hz and 996 Hz. The trend of the numerical results is in good agreement with the experimental results, with a slight variation at the mid-region of the two peaks due to non-perfectly circular pores and some surface irregularities, as shown in Fig. 16.

Fig. 16.

Sample IMPP-2 with surface irregularities.

The use of 3D printing technology was instrumental in this phase of the research. The ability to fabricate samples with precise parameters allowed for a detailed comparison of experimental and theoretical results. Furthermore, the rapid prototyping capability of 3D printing technology enabled quick iterations on the designs and the gathering of a wealth of empirical data. However, it is important to note that 3D printing technology, while offering numerous advantages, also introduces certain challenges. As observed in the study, surface irregularities and non-perfectly circular pores can occur during the printing process. These irregularities can affect the sound absorption performance of the MPPs and introduce discrepancies between the experimental and theoretical results.

Despite these challenges, the study demonstrates the immense potential of 3D printing technology in the field of acoustics research. By enabling the precise and rapid fabrication of MPPs with varying parameters, 3D printing technology has opened up new avenues for the exploration and optimization of sound absorption performance. It is anticipated that 3D printing will continue to play a pivotal role in advancing the understanding of sound absorption and in the development of more effective sound absorption materials and devices.

The comparative analysis of experimental and theoretical results provided valuable insights into the performance of MPPs with varying parameters. The findings underscored the importance of careful parameter selection and precise fabrication in achieving optimal sound absorption performance. As the field moves forward, it is expected that this research will contribute significantly to the field of acoustics, paving the way for the development of more effective sound absorption materials and devices.

6. Conclusion

The research undertaken in this study led to the proposal of two models, specifically a three-cavity system and a four-cavity system, based on the validation study. MATLAB was utilized for the numerical examination of these models, followed by an empirical assessment conducted in an impedance tube.

A careful adjustment of the geometric parameters led to several notable conclusions:

• •Micro-perforated panels (MPPs) encompassing a larger count of sub-cavities exhibited a superior sound absorption coefficient in comparison to a two-cavity system.
• •The back air cavity primarily instigates multiple peaks, thereby assuring wideband sound absorption. The study suggests that greater variations among air cavity depths significantly augment the sound absorption coefficient. Notably, in this research, sound absorption coefficients with three peaks (α > 0.8) and wideband sound absorption were accomplished.
• •For an inhomogeneous MPP (IMPP), when an enhancement in sound absorption performance is desired in the lower frequency region along with wide-band sound absorption, the perforation ratios should be smaller for sub-cavities that bear relatively larger pore diameters than other sub-cavities.
• •The research indicates that for better sound absorption above the 800–850 Hz range, a pore diameter of less than 0.5 mm should be employed. Furthermore, the back cavity length has a more significant influence than pore diameter between 850 Hz and 2000 Hz in smoothing the curve with less fluctuation.

One of the crucial novelties of this study resides in the application of 3D printing technologies, enabling a complex geometry for MPPs. The employment of 3D printing introduces a new frontier in sound absorption research, as it allows for precise manipulation and fine-tuning of geometric parameters, thus enhancing the efficacy and performance of the resulting MPPs. This innovative approach opens up new avenues for more complex and efficacious noise control solutions.

Funding

There are no funding sources associated with this work.

Data availability statement

All the data generated/analyzed during this study has been included in the article in the form of graphs and tables. However, any further data can be obtained from the corresponding author on request.

CRediT authorship contribution statement

Deepak: Data Curation, Formal Analysis. Jeyaraj Pitchaimani: Validation. Raghukiran Nadimpalli: Resources. Lenin Babu Mailan Chinnapandi: Methodology.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.